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       "filename": "1.1.1.1 (a+b x)^m.m",
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            "lhs": "Int[x_^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[m, x] && NeQ[m, -1]"
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       "filename": "1.1.1.1 (a+b x)^m.m",
            "rhs": "Log[RemoveContent[a + b*x, x]]/b",
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            "lhs": "Int[(a_. + b_.*x_)^m_, x_Symbol]",
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         "givens": "FreeQ[{a, b, m}, x] && NeQ[m, -1]"
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            "rhs": "1/Coefficient[u, x, 1]*Subst[Int[(a + b*x)^m, x], x, u]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*u_)^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m}, x] && LinearQ[u, x] && NeQ[u, x]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "d*x*(a + b*x)^(m + 1)/(b*(m + 2))",
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            "lhs": "Int[(a_ + b_.*x_)^m_.*(c_ + d_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && EqQ[a*d - b*c*(m + 2), 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "Int[1/(a*c + b*d*x^2), x]",
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            "lhs": "Int[1/((a_ + b_.*x_)*(c_ + d_.*x_)), x_Symbol]",
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "b/(b*c - a*d)*Int[1/(a + b*x), x] - d/(b*c - a*d)*Int[1/(c + d*x), x]",
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            "lhs": "Int[1/((a_. + b_.*x_)*(c_. + d_.*x_)), x_Symbol]",
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       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "x*(a + b*x)^m*(c + d*x)^m/(2*m + 1) + 2*a*c*m/(2*m + 1)*Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x]",
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            "lhs": "Int[(a_ + b_.*x_)^m_*(c_ + d_.*x_)^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "x/(a*c*Sqrt[a + b*x]*Sqrt[c + d*x])",
     "rulenumber": 0,
            "lhs": "Int[1/((a_ + b_.*x_)^(3/2)*(c_ + d_.*x_)^(3/2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "-x*(a + b*x)^(m + 1)*(c + d*x)^(m + 1)/(2*a* c*(m + 1)) + (2*m + 3)/(2*a*c*(m + 1))* Int[(a + b*x)^(m + 1)*(c + d*x)^(m + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_)^m_*(c_ + d_.*x_)^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && ILtQ[m + 3/2, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "Int[(a*c + b*d*x^2)^m, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_)^m_.*(c_ + d_.*x_)^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || GtQ[a, 0] && GtQ[c, 0])"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "(a + b*x)^ FracPart[m]*(c + d*x)^FracPart[m]/(a*c + b*d*x^2)^FracPart[m]* Int[(a*c + b*d*x^2)^m, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_)^m_*(c_ + d_.*x_)^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 0] && Not[IntegerQ[2*m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_.*(c_. + d_.*x_)^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && (Not[IntegerQ[n]] || EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0] || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_)^m_.*(c_. + d_.*x_)^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] && Not[IGtQ[n, 0] && LtQ[m + n + 2, 0]]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "(a + b*x)^(m + 1)*(c + d*x)^(n + 1)/((b*c - a*d)*(m + 1)) - d*Simplify[m + n + 2]/((b*c - a*d)*(m + 1))* Int[(a + b*x)^Simplify[m + 1]*(c + d*x)^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && Not[LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n])] && (SumSimplerQ[m, 1] || Not[SumSimplerQ[n, 1]])"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "-4/(5*b*(a + b*x)^(5/4)*(c + d*x)^(1/4)) - d/(5*b)*Int[1/((a + b*x)^(5/4)*(c + d*x)^(5/4)), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_ + b_.*x_)^(9/4)*(c_ + d_.*x_)^(1/4)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && NegQ[a^2*b^2]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "(a + b*x)^(m + 1)*(c + d*x)^n/(b*(m + 1)) - d*n/(b*(m + 1))*Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] && Not[IntegerQ[n] && Not[IntegerQ[m]]] && Not[ILeQ[m + n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])] && IntLinearQ[a, b, c, d, m, n, x]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "-2/(b*(a + b*x)^(1/4)*(c + d*x)^(1/4)) + c*Int[1/((a + b*x)^(5/4)*(c + d*x)^(5/4)), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_ + b_.*x_)^(5/4)*(c_ + d_.*x_)^(1/4)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && NegQ[a^2*b^2]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "(a + b*x)^(m + 1)*(c + d*x)^n/(b*(m + n + 1)) + 2*c*n/(m + n + 1)*Int[(a + b*x)^m*(c + d*x)^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_)^m_*(c_ + d_.*x_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0] && IGtQ[n + 1/2, 0] && LtQ[m, n]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "(a + b*x)^(m + 1)*(c + d*x)^n/(b*(m + n + 1)) + n*(b*c - a*d)/(b*(m + n + 1))* Int[(a + b*x)^m*(c + d*x)^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] && Not[IGtQ[m, 0] && (Not[IntegerQ[n]] || GtQ[m, 0] && LtQ[m - n, 0])] && Not[ILtQ[m + n + 2, 0]] && IntLinearQ[a, b, c, d, m, n, x]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "(a + b*x)^(m + 1)*(c + d*x)^(n + 1)/((b*c - a*d)*(m + 1)) - d*(m + n + 2)/((b*c - a*d)*(m + 1))* Int[(a + b*x)^(m + 1)*(c + d*x)^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && Not[LtQ[ n, -1] && (EqQ[a, 0] || NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n])] && IntLinearQ[a, b, c, d, m, n, x]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "ArcCosh[b*x/a]/b",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_ + b_.*x_]*Sqrt[c_ + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "Int[1/Sqrt[a*c - b*(a - c)*x - b^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_ + b_.*x_]*Sqrt[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[b + d, 0] && GtQ[a + c, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "2/Sqrt[b]* Subst[Int[1/Sqrt[b*c - a*d + d*x^2], x], x, Sqrt[a + b*x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_. + b_.*x_]*Sqrt[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "With[{q = Rt[(b*c - a*d)/b, 3]}, -Log[RemoveContent[a + b*x, x]]/(2*b*q) - 3/(2*b*q)*Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)] + 3/(2*b)* Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)]]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_. + b_.*x_)*(c_. + d_.*x_)^(1/3)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "With[{q = Rt[-(b*c - a*d)/b, 3]}, Log[RemoveContent[a + b*x, x]]/(2*b*q) - 3/(2*b*q)*Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)] + 3/(2*b)* Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1/3)]]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_. + b_.*x_)*(c_. + d_.*x_)^(1/3)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NegQ[(b*c - a*d)/b]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "With[{q = Rt[(b*c - a*d)/b, 3]}, -Log[RemoveContent[a + b*x, x]]/(2*b*q^2) - 3/(2*b*q^2)*Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)] - 3/(2*b*q)* Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)]]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_. + b_.*x_)*(c_. + d_.*x_)^(2/3)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "With[{q = Rt[-(b*c - a*d)/b, 3]}, -Log[RemoveContent[a + b*x, x]]/(2*b*q^2) + 3/(2*b*q^2)*Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)] + 3/(2*b*q)* Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1/3)]]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_. + b_.*x_)*(c_. + d_.*x_)^(2/3)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NegQ[(b*c - a*d)/b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "With[{q = Rt[d/b, 3]}, -Sqrt[3]*q/d* ArcTan[2*q*(a + b*x)^(1/3)/(Sqrt[3]*(c + d*x)^(1/3)) + 1/Sqrt[3]] - q/(2*d)*Log[c + d*x] - 3*q/(2*d)*Log[q*(a + b*x)^(1/3)/(c + d*x)^(1/3) - 1]]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_. + b_.*x_)^(1/3)*(c_. + d_.*x_)^(2/3)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[d/b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "With[{q = Rt[-d/b, 3]}, Sqrt[3]*q/d* ArcTan[1/Sqrt[3] - 2*q*(a + b*x)^(1/3)/(Sqrt[3]*(c + d*x)^(1/3))] + q/(2*d)*Log[c + d*x] + 3*q/(2*d)*Log[q*(a + b*x)^(1/3)/(c + d*x)^(1/3) + 1]]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_. + b_.*x_)^(1/3)*(c_. + d_.*x_)^(2/3)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NegQ[d/b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "(a + b*x)^ m*(c + d*x)^m/(a*c + (b*c + a*d)*x + b*d*x^2)^m* Int[(a*c + (b*c + a*d)*x + b*d*x^2)^m, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_ + d_.*x_)^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[3, Denominator[m], 4] && AtomQ[b*c + a*d]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "(a + b*x)^m*(c + d*x)^m/((a + b*x)*(c + d*x))^m* Int[(a*c + (b*c + a*d)*x + b*d*x^2)^m, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_ + d_.*x_)^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[3, Denominator[m], 4]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "With[{p = Denominator[m]}, p/b*Subst[Int[x^(p*(m + 1) - 1)*(c - a*d/b + d*x^p/b)^n, x], x, (a + b*x)^(1/p)]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a, b, c, d, m, n, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "c^n*(b*x)^(m + 1)/(b*(m + 1))* Hypergeometric2F1[-n, m + 1, m + 2, -d*x/c]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*x_)^m_*(c_ + d_.*x_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, m, n}, x] && Not[IntegerQ[m]] && (IntegerQ[n] || GtQ[c, 0] && Not[EqQ[n, -1/2] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "(c + d*x)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m)* Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*x/c]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*x_)^m_*(c_ + d_.*x_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, m, n}, x] && Not[IntegerQ[n]] && (IntegerQ[m] || GtQ[-d/(b*c), 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "c^IntPart[n]*(c + d*x)^FracPart[n]/(1 + d*x/c)^FracPart[n]* Int[(b*x)^m*(1 + d*x/c)^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*x_)^m_*(c_ + d_.*x_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, m, n}, x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && Not[GtQ[c, 0]] && Not[GtQ[-d/(b*c), 0]] && (RationalQ[m] && Not[EqQ[n, -1/2] && EqQ[c^2 - d^2, 0]] || Not[RationalQ[n]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "(-b*c/d)^ IntPart[m]*(b*x)^FracPart[m]/(-d*x/c)^FracPart[m]* Int[(-d*x/c)^m*(c + d*x)^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*x_)^m_*(c_ + d_.*x_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, m, n}, x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && Not[GtQ[c, 0]] && Not[GtQ[-d/(b*c), 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "(b*c - a*d)^n*(a + b*x)^(m + 1)/(b^(n + 1)*(m + 1))* Hypergeometric2F1[-n, m + 1, m + 2, -d*(a + b*x)/(b*c - a*d)]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_)^m_*(c_ + d_.*x_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && NeQ[b*c - a*d, 0] && Not[IntegerQ[m]] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "(a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n)* Hypergeometric2F1[-n, m + 1, m + 2, -d*(a + b*x)/(b*c - a*d)]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_)^m_*(c_ + d_.*x_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || Not[RationalQ[n] && GtQ[-d/(b*c - a*d), 0]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "(c + d*x)^ FracPart[ n]/((b/(b*c - a*d))^IntPart[n]*(b*(c + d*x)/(b*c - a*d))^ FracPart[n])* Int[(a + b*x)^m*Simp[b*c/(b*c - a*d) + b*d*x/(b*c - a*d), x]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_)^m_*(c_ + d_.*x_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && (RationalQ[m] || Not[SimplerQ[n + 1, m + 1]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.2 (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.2 (a+b x)^m (c+d x)^n.m",
            "rhs": "1/Coefficient[u, x, 1]*Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, u]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*u_)^m_.*(c_. + d_.*u_)^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && LinearQ[u, x] && NeQ[Coefficient[u, x, 0], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "Int[ExpandIntegrand[(e + f*x)^p/((a + b*x)*(c + d*x)), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^p_./((a_. + b_.*x_)*(c_. + d_.*x_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "Int[(a*c + b*d*x^2)^m*(e + f*x)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_)^m_.*(c_ + d_.*x_)^n_.*(e_. + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "b*(c + d*x)^(n + 1)*(e + f*x)^(p + 1)/(d*f*(n + p + 2))",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_)*(d_.*x_)^n_.*(e_ + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] && Not[ILtQ[n + p + 2, 0] && GtQ[n + 2*p, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_)*(d_.*x_)^n_.*(e_ + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && NeQ[b*e + a*f, 0] && (Not[IntegerQ[n]] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || GeQ[n + p + 2, 0] && RationalQ[a, b, d, e, f]) && (NeQ[n + p + 3, 0] || EqQ[p, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)*(c_ + d_.*x_)^n_.*(e_. + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && (ILtQ[n, 0] && ILtQ[p, 0] || EqQ[p, 1] || IGtQ[p, 0] && (Not[IntegerQ[n]] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "-(b*e - a*f)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e)) - (a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e))* Int[(c + d*x)^n*(e + f*x)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && (Not[LtQ[n, -1]] || IntegerQ[p] || Not[IntegerQ[n] || Not[EqQ[e, 0] || Not[EqQ[c, 0] || LtQ[p, n]]]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "-(b*e - a*f)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e)) - (a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e))* Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, p}, x] && Not[RationalQ[p]] && SumSimplerQ[p, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "b*(c + d*x)^(n + 1)*(e + f*x)^(p + 1)/(d*f*(n + p + 2)) + (a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d* f*(n + p + 2))*Int[(c + d*x)^n*(e + f*x)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "b*(c + d*x)^(n + 1)*(e + f*x)^(p + 1)*(2*a*d*f*(n + p + 3) - b*(d*e*(n + 2) + c*f*(p + 2)) + b*d*f*(n + p + 2)*x)/(d^2* f^2*(n + p + 2)*(n + p + 3))",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^2*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && NeQ[n + p + 3, 0] && EqQ[d* f*(n + p + 2)*(a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1)))) - b*(d*e*(n + 1) + c*f*(p + 1))*(a*d*f*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2))), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "a*Int[(a + b*x)^n*(c + d*x)^n*(f*x)^p, x] + b/f*Int[(a + b*x)^n*(c + d*x)^n*(f*x)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_.*(c_. + d_.*x_)^n_.*(f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[m - n - 1, 0] && Not[RationalQ[p]] && Not[IGtQ[m, 0]] && NeQ[m + n + p + 2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "(b*e - a*f)/(b*c - a*d)* Int[(e + f*x)^(p - 1)/(a + b*x), x] - (d*e - c*f)/(b*c - a*d)*Int[(e + f*x)^(p - 1)/(c + d*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^p_./((a_. + b_.*x_)*(c_. + d_.*x_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && LtQ[0, p, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "f*(e + f*x)^(p - 1)/(b*d*(p - 1)) + 1/(b*d)* Int[(b*d*e^2 - a*c*f^2 + f*(2*b*d*e - b*c*f - a*d*f)* x)*(e + f*x)^(p - 2)/((a + b*x)*(c + d*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^p_/((a_. + b_.*x_)*(c_. + d_.*x_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "f*(e + f*x)^(p + 1)/((p + 1)*(b*e - a*f)*(d*e - c*f)) + 1/((b*e - a*f)*(d*e - c*f))* Int[(b*d*e - b*c*f - a*d*f - b*d*f*x)*(e + f*x)^(p + 1)/((a + b*x)*(c + d*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^p_/((a_. + b_.*x_)*(c_. + d_.*x_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "b/(b*c - a*d)*Int[(e + f*x)^p/(a + b*x), x] - d/(b*c - a*d)*Int[(e + f*x)^p/(c + d*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^p_/((a_. + b_.*x_)*(c_. + d_.*x_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p}, x] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "Int[ExpandIntegrand[(e + f*x)^ FractionalPart[p], (c + d*x)^ n*(e + f*x)^IntegerPart[p]/(a + b*x), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^p_/(a_. + b_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 0] && LtQ[p, -1] && FractionQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_.*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] || GtQ[m, 0] && GeQ[n, -1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "(b*c - a*d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1)/(d^2*(d*e - c*f)*(n + 1)) - 1/(d^2*(d*e - c*f)*(n + 1))*Int[(c + d*x)^(n + 1)*(e + f*x)^p* Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^2*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || Not[SumSimplerQ[p, 1]]))"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "b*(a + b*x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1)/(d*f*(n + p + 3)) + 1/(d*f*(n + p + 3))*Int[(c + d*x)^n*(e + f*x)^p* Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^2*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "With[{q = Rt[(d*e - c*f)/(b*e - a*f), 3]}, -Sqrt[3]*q* ArcTan[1/Sqrt[3] + 2*q*(a + b*x)^(1/3)/(Sqrt[3]*(c + d*x)^(1/3))]/(d*e - c*f) + q*Log[e + f*x]/(2*(d*e - c*f)) - 3*q*Log[q*(a + b*x)^(1/3) - (c + d*x)^(1/3)]/(2*(d*e - c*f))]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_. + b_.*x_)^(1/3)*(c_. + d_.*x_)^(2/3)*(e_. + f_.*x_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "b*f*Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_. + b_.*x_]*Sqrt[c_. + d_.*x_]*(e_. + f_.*x_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d*e - f*(b*c + a*d), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "With[{q = Denominator[m]}, q*Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_/(e_. + f_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "(a + b*x)^(m + 1)*(c + d*x)^ n*(e + f*x)^(p + 1)/((m + 1)*(b*e - a*f)) - n*(d*e - c*f)/((m + 1)*(b*e - a*f))* Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && Not[SumSimplerQ[p, 1] && Not[SumSimplerQ[m, 1]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1), 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f)) + (a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f))* Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "(a + b*x)^(m + 1)*(c + d*x)^ n*(e + f*x)^p/(b*(m + 1)) - 1/(b*(m + 1))* Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^(p - 1)* Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1)) + 1/(b*(b*e - a*f)*(m + 1))* Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p* Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "(a + b*x)^(m + 1)*(c + d*x)^ n*(e + f*x)^(p + 1)/((m + 1)*(b*e - a*f)) - 1/((m + 1)*(b*e - a*f))* Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p* Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1)/(d* f*(m + n + p + 1)) + 1/(d*f*(m + n + p + 1))* Int[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p* Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "(a + b*x)^m*(c + d*x)^ n*(e + f*x)^(p + 1)/(f*(m + n + p + 1)) - 1/(f*(m + n + p + 1))* Int[(a + b*x)^(m - 1)*(c + d*x)^(n - 1)*(e + f*x)^p* Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a*f) + b*n*(d*e - c*f))*x, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_.*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (IntegersQ[2*m, 2*n, 2*p] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1)/(d* f*(m + n + p + 1)) + 1/(d*f*(m + n + p + 1))* Int[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p* Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f)) + 1/((m + 1)*(b*c - a*d)*(b*e - a*f))* Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p* Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f)) + 1/((m + 1)*(b*c - a*d)*(b*e - a*f))* Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p* Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
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       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "With[{q = Rt[b*(b*e - a*f)/(b*c - a*d)^2, 3]}, -Log[a + b*x]/(2*q*(b*c - a*d)) - Sqrt[3]* ArcTan[1/Sqrt[3] + 2*q*(c + d*x)^(2/3)/(Sqrt[3]*(e + f*x)^(1/3))]/(2* q*(b*c - a*d)) + 3*Log[q*(c + d*x)^(2/3) - (e + f*x)^(1/3)]/(4*q*(b*c - a*d))]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_. + b_.*x_)*(c_. + d_.*x_)^(1/3)*(e_. + f_.*x_)^(1/3)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d*e - b*c*f - a*d*f, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "b*(a + b*x)^(m + 1)*(c + d*x)^(2/ 3)*(e + f*x)^(2/3)/((m + 1)*(b*c - a*d)*(b*e - a*f)) + f/(6*(m + 1)*(b*c - a*d)*(b*e - a*f))* Int[(a + b*x)^(m + 1)*(a*d*(3*m + 1) - 3*b*c*(3*m + 5) - 2*b*d*(3*m + 7)*x)/((c + d*x)^(1/3)*(e + f*x)^(1/3)), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_/((c_. + d_.*x_)^(1/3)*(e_. + f_.*x_)^(1/3)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d*e - b*c*f - a*d*f, 0] && ILtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "Int[(a*c + b*d*x^2)^m*(f*x)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_.*(c_. + d_.*x_)^n_.*(f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[m - n, 0] && GtQ[a, 0] && GtQ[c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "(a + b*x)^ FracPart[m]*(c + d*x)^FracPart[m]/(a*c + b*d*x^2)^FracPart[m]* Int[(a*c + b*d*x^2)^m*(f*x)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_.*(c_. + d_.*x_)^n_.*(f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[m - n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*x)^n*(c + d*x)^n*(f*x)^ p, (a + b*x)^(m - n), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_.*(c_. + d_.*x_)^n_.*(f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && IGtQ[m - n, 0] && NeQ[m + n + p + 2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_.*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, p}, x] && (IGtQ[m, 0] || ILtQ[m, 0] && ILtQ[n, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f)) + 1/((m + 1)*(b*c - a*d)*(b*e - a*f))* Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p* Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && ILtQ[m + n + p + 2, 0] && NeQ[m, -1] && (SumSimplerQ[m, 1] || Not[NeQ[n, -1] && SumSimplerQ[n, 1]] && Not[NeQ[p, -1] && SumSimplerQ[p, 1]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "With[{k = Denominator[p]}, k/e*Subst[ Int[x^(k*(p + 1) - 1)*(a + b*x^k/e)^m*(c + d*x^k/e)^n, x], x, (e*x)^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^p_*(a_ + b_.*x_)^m_*(c_ + d_.*x_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && FractionQ[p] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "(b*c - a*d)^ n*(a + b*x)^(m + 1)/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1))* Hypergeometric2F1[m + 1, -n, m + 2, -(d*e - c*f)*(a + b*x)/((b*c - a*d)*(e + f*x))]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "(a + b*x)^(m + 1)*(c + d*x)^ n*(e + f*x)^(p + 1)/((b*e - a*f)*(m + 1))*((b*e - a*f)*(c + d*x)/((b*c - a*d)*(e + f*x)))^(-n)* Hypergeometric2F1[m + 1, -n, m + 2, -(d*e - c*f)*(a + b*x)/((b*c - a*d)*(e + f*x))]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_*(e_. + f_.*x_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[m + n + p + 2, 0] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "c^n*e^p*(b*x)^(m + 1)/(b*(m + 1))* AppellF1[m + 1, -n, -p, m + 2, -d*x/c, -f*x/e]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*x_)^m_*(c_ + d_.*x_)^n_*(e_ + f_.*x_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "(c + d*x)^(n + 1)/(d*(n + 1)*(-d/(b*c))^ m*(d/(d*e - c*f))^p)* AppellF1[n + 1, -m, -p, n + 2, 1 + d*x/c, -f*(c + d*x)/(d*e - c*f)]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*x_)^m_*(c_ + d_.*x_)^n_*(e_ + f_.*x_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && GtQ[-d/(b*c), 0] && (IntegerQ[p] || GtQ[d/(d*e - c*f), 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "c^IntPart[n]*(c + d*x)^FracPart[n]/(1 + d*x/c)^FracPart[n]* Int[(b*x)^m*(1 + d*x/c)^n*(e + f*x)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*x_)^m_*(c_ + d_.*x_)^n_*(e_ + f_.*x_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && Not[GtQ[c, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "(b*e - a*f)^ p*(a + b*x)^(m + 1)/(b^(p + 1)*(m + 1)*(b/(b*c - a*d))^n)* AppellF1[m + 1, -n, -p, m + 2, -d*(a + b*x)/(b*c - a*d), -f*(a + b*x)/(b*e - a*f)]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_)^m_*(c_. + d_.*x_)^n_*(e_. + f_.*x_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && IntegerQ[p] && GtQ[b/(b*c - a*d), 0] && Not[GtQ[d/(d*a - c*b), 0] && SimplerQ[c + d*x, a + b*x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "(c + d*x)^ FracPart[ n]/((b/(b*c - a*d))^IntPart[n]*(b*(c + d*x)/(b*c - a*d))^ FracPart[n])* Int[(a + b*x)^m*(b*c/(b*c - a*d) + b*d*x/(b*c - a*d))^ n*(e + f*x)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_)^m_*(c_. + d_.*x_)^n_*(e_. + f_.*x_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && IntegerQ[p] && Not[GtQ[b/(b*c - a*d), 0]] && Not[SimplerQ[c + d*x, a + b*x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "(a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^ n*(b/(b*e - a*f))^p)* AppellF1[m + 1, -n, -p, m + 2, -d*(a + b*x)/(b*c - a*d), -f*(a + b*x)/(b*e - a*f)]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_)^m_*(c_. + d_.*x_)^n_*(e_. + f_.*x_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && Not[IntegerQ[p]] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && Not[GtQ[d/(d*a - c*b), 0] && GtQ[d/(d*e - c*f), 0] && SimplerQ[c + d*x, a + b*x]] && Not[GtQ[f/(f*a - e*b), 0] && GtQ[f/(f*c - e*d), 0] && SimplerQ[e + f*x, a + b*x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "(e + f*x)^ FracPart[ p]/((b/(b*e - a*f))^IntPart[p]*(b*(e + f*x)/(b*e - a*f))^ FracPart[p])* Int[(a + b*x)^m*(c + d*x)^ n*(b*e/(b*e - a*f) + b*f*x/(b*e - a*f))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_)^m_*(c_. + d_.*x_)^n_*(e_. + f_.*x_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && Not[IntegerQ[p]] && GtQ[b/(b*c - a*d), 0] && Not[GtQ[b/(b*e - a*f), 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "(c + d*x)^ FracPart[ n]/((b/(b*c - a*d))^IntPart[n]*(b*(c + d*x)/(b*c - a*d))^ FracPart[n])* Int[(a + b*x)^m*(b*c/(b*c - a*d) + b*d*x/(b*c - a*d))^ n*(e + f*x)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_)^m_*(c_. + d_.*x_)^n_*(e_. + f_.*x_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && Not[IntegerQ[p]] && Not[GtQ[b/(b*c - a*d), 0]] && Not[SimplerQ[c + d*x, a + b*x]] && Not[SimplerQ[e + f*x, a + b*x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.3 (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "1/Coefficient[u, x, 1]* Subst[Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x, u]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*u_)^m_.*(c_. + d_.*u_)^n_.*(e_ + f_.*u_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && LinearQ[u, x] && NeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)*(g + h*x), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_.*(c_. + d_.*x_)^ n_.*(e_ + f_.*x_)*(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h}, x] && (IGtQ[m, 0] || IntegersQ[m, n])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "(b^2*d*e*g - a^2*d*f*h*m - a*b*(d*(f*g + e*h) - c*f*h*(m + 1)) + b*f*h*(b*c - a*d)*(m + 1)*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)/ (b^2*d*(b*c - a*d)*(m + 1)) + (a*d*f*h*m + b*(d*(f*g + e*h) - c*f*h*(m + 2)))/(b^2*d)* Int[(a + b*x)^(m + 1)*(c + d*x)^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_.*(e_ + f_.*x_)*(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1] && Not[SumSimplerQ[n, 1] && Not[SumSimplerQ[m, 1]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "(b^2*c*d*e*g*(n + 1) + a^2*c*d*f*h*(n + 1) + a*b*(d^2*e*g*(m + 1) + c^2*f*h*(m + 1) - c*d*(f*g + e*h)*(m + n + 2)) + (a^2*d^2*f*h*(n + 1) - a*b*d^2*(f*g + e*h)*(n + 1) + b^2*(c^2*f*h*(m + 1) - c*d*(f*g + e*h)*(m + 1) + d^2*e*g*(m + n + 2)))*x)/ (b*d*(b*c - a*d)^2*(m + 1)*(n + 1))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1) - (a^2*d^2*f*h*(2 + 3*n + n^2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(2 + 3*m + m^2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(6 + m^2 + 5*n + n^2 + m*(2*n + 5))))/ (b*d*(b*c - a*d)^2*(m + 1)*(n + 1))* Int[(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_*(e_ + f_.*x_)*(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h}, x] && LtQ[m, -1] && LtQ[n, -1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "(b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g + e*h) + d*e*g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)/ (b^2*(b*c - a*d)^2*(m + 1)*(m + 2))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1) + (f*h/ b^2 - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/ (b^2*(b*c - a*d)^2*(m + 1)*(m + 2)))* Int[(a + b*x)^(m + 2)*(c + d*x)^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_.*(e_ + f_.*x_)*(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || EqQ[m + n + 3, 0] && Not[LtQ[n, -2]])"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "(a^2*d*f*h*(n + 2) + b^2*d*e*g*(m + n + 3) + a*b*(c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b*f*h*(b*c - a*d)*(m + 1)*x)/ (b^2*d*(b*c - a*d)*(m + 1)*(m + n + 3))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1) - (a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/ (b^2*d*(b*c - a*d)*(m + 1)*(m + n + 3))* Int[(a + b*x)^(m + 1)*(c + d*x)^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_.*(e_ + f_.*x_)*(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (GeQ[m, -2] && LtQ[m, -1] || SumSimplerQ[m, 1]) && NeQ[m, -1] && NeQ[m + n + 3, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)/ (b^2*d^2*(m + n + 2)*(m + n + 3)) + (a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/ (b^2*d^2*(m + n + 2)*(m + n + 3))* Int[(a + b*x)^m*(c + d*x)^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_.*(c_. + d_.*x_)^ n_.*(e_ + f_.*x_)*(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_*(e_. + f_.*x_)^ p_*(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m}, x] && (IntegersQ[m, n, p] || IGtQ[n, 0] && IGtQ[p, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^ n*(e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1)) - 1/(b*(b*e - a*f)*(m + 1))* Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p* Simp[b*c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_*(e_. + f_.*x_)^ p_*(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^ n*(e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1)) - 1/(b*(b*e - a*f)*(m + 1))* Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p* Simp[b*c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_*(e_. + f_.*x_)^ p_*(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f)) + 1/((m + 1)*(b*c - a*d)*(b*e - a*f))* Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p* Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x]",
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            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_*(e_. + f_.*x_)^ p_*(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f)) + 1/((m + 1)*(b*c - a*d)*(b*e - a*f))* Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p* Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_*(e_. + f_.*x_)^ p_*(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "h*(a + b*x)^ m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1)/(d*f*(m + n + p + 2)) + 1/(d*f*(m + n + p + 2))* Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p* Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_*(e_. + f_.*x_)^ p_*(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "h*(a + b*x)^ m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1)/(d*f*(m + n + p + 2)) + 1/(d*f*(m + n + p + 2))* Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p* Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_*(e_. + f_.*x_)^ p_*(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2*n, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f)) + 1/((m + 1)*(b*c - a*d)*(b*e - a*f))* Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p* Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_*(e_. + f_.*x_)^ p_*(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m + n + p + 2, 0] && NeQ[m, -1] && (SumSimplerQ[m, 1] || Not[NeQ[n, -1] && SumSimplerQ[n, 1]] && Not[NeQ[p, -1] && SumSimplerQ[p, 1]])"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "(b*g - a*h)/(b*c - a*d)* Int[(e + f*x)^p/(a + b*x), x] - (d*g - c*h)/(b*c - a*d)*Int[(e + f*x)^p/(c + d*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^p_*(g_. + h_.*x_)/((a_. + b_.*x_)*(c_. + d_.*x_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "h/b*Int[(c + d*x)^n*(e + f*x)^p, x] + (b*g - a*h)/b* Int[(c + d*x)^n*(e + f*x)^p/(a + b*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^n_*(e_. + f_.*x_)^p_*(g_. + h_.*x_)/(a_. + b_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "h/f*Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]), x] + (f*g - e*h)/f* Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)/(Sqrt[a_. + b_.*x_]*Sqrt[c_ + d_.*x_]* Sqrt[e_ + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h}, x] && SimplerQ[a + b*x, e + f*x] && SimplerQ[c + d*x, e + f*x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "h/b*Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p, x] + (b*g - a*h)/ b*Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_*(e_. + f_.*x_)^ p_*(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n, p}, x] && (SumSimplerQ[m, 1] || Not[SumSimplerQ[n, 1]] && Not[SumSimplerQ[p, 1]])"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "(a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[e + f*x]* Sqrt[g + h*x]/(b*(m + 1)) - 1/(2*b*(m + 1))* Int[(a + b*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]* Sqrt[g + h*x])* Simp[d*e*g + c*f*g + c*e*h + 2*(d*f*g + d*e*h + c*f*h)*x + 3*d*f*h*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*Sqrt[c_. + d_.*x_]*Sqrt[e_. + f_.*x_]* Sqrt[g_. + h_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IntegerQ[2*m] && LtQ[m, -1]"
  },
  {
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        "comment": false,
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         "givens": "FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c*f)/d, 0]"
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        "comment": false,
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "b*(a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[e + f*x]* Sqrt[g + h*x]/((m + 1)*(b*e - a*f)*(b*g - a*h)) + 1/(2*(m + 1)*(b*e - a*f)*(b*g - a*h))* Int[((a + b*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]* Sqrt[g + h*x]))* Simp[2*a*c*f*h*(m + 1) - b*(d*e*g + c*(2*m + 3)*(f*g + e*h)) + 2*(a*d*f*h*(m + 1) - b*(m + 2)*(d*f*g + d*e*h + c*f*h))*x - b*d*f*h*(2*m + 5)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_* Sqrt[c_. + d_.*x_]/(Sqrt[e_. + f_.*x_]*Sqrt[g_. + h_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IntegerQ[2*m] && LeQ[m, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "(b*e - a*f)/(b*c - a*d)* Int[(e + f*x)^(p - 1)*(g + h*x)^q/(a + b*x), x] - (d*e - c*f)/(b*c - a*d)* Int[(e + f*x)^(p - 1)*(g + h*x)^q/(c + d*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^ p_*(g_. + h_.*x_)^q_/((a_. + b_.*x_)*(c_. + d_.*x_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, q}, x] && LtQ[0, p, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "Int[ExpandIntegrand[ 1/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), (a + b*x)^ m*(c + d*x)^(n + 1/2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^ m_.*(c_. + d_.*x_)^n_/(Sqrt[e_. + f_.*x_]*Sqrt[g_. + h_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h}, x] && IntegerQ[m] && IntegerQ[n + 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_*(e_. + f_.*x_)^ p_*(g_. + h_.*x_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && IntegersQ[p, q]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "h/b*Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*(g + h*x)^(q - 1), x] + (b*g - a*h)/b* Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^(q - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_*(e_. + f_.*x_)^ p_*(g_. + h_.*x_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n, p}, x] && IGtQ[q, 0] && (SumSimplerQ[m, 1] || Not[SumSimplerQ[n, 1]] && Not[SumSimplerQ[p, 1]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "CannotIntegrate[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_.*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^ p_.*(g_. + h_.*x_)^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n, p, q}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "1/Coefficient[u, x, 1]* Subst[Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x], x, u]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*u_)^m_.*(c_. + d_.*u_)^n_.*(e_. + f_.*u_)^ p_.*(g_. + h_.*u_)^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n, p, q}, x] && LinearQ[u, x] && NeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "(i*(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q)^ r/((a + b*x)^(m*r)*(c + d*x)^(n*r)*(e + f*x)^(p*r)*(g + h*x)^(q* r))* Int[(a + b*x)^(m*r)*(c + d*x)^(n*r)*(e + f*x)^(p*r)*(g + h*x)^(q*r), x]",
     "rulenumber": 0,
            "lhs": "Int[(i_.*(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_*(e_. + f_.*x_)^ p_*(g_. + h_.*x_)^q_)^r_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, i, m, n, p, q, r}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "Int[ExpandToSum[u, x]^m, x]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[m, x] && LinearQ[u, x] && Not[LinearMatchQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "Int[ExpandToSum[u, x]^m*ExpandToSum[v, x]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*v_^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{m, n}, x] && LinearQ[{u, v}, x] && Not[LinearMatchQ[{u, v}, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "Int[ExpandToSum[u, x]^m*ExpandToSum[v, x]^n*ExpandToSum[w, x]^p, x]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*v_^n_.*w_^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{m, n, p}, x] && LinearQ[{u, v, w}, x] && Not[LinearMatchQ[{u, v, w}, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.4 (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "Int[ExpandToSum[u, x]^m*ExpandToSum[v, x]^n*ExpandToSum[w, x]^p* ExpandToSum[z, x]^q, x]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*v_^n_.*w_^p_.*z_^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{m, n, p, q}, x] && LinearQ[{u, v, w, z}, x] && Not[LinearMatchQ[{u, v, w, z}, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.5 P(x) (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.5 P(x) (a+b x)^m (c+d x)^n.m",
            "rhs": "Int[Px*(a*c + b*d*x^2)^m, x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_. + b_.*x_)^m_.*(c_. + d_.*x_)^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] && EqQ[m, n] && (IntegerQ[m] || GtQ[a, 0] && GtQ[c, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.5 P(x) (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.5 P(x) (a+b x)^m (c+d x)^n.m",
            "rhs": "(a + b*x)^ FracPart[m]*(c + d*x)^FracPart[m]/(a*c + b*d*x^2)^FracPart[m]* Int[Px*(a*c + b*d*x^2)^m, x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] && EqQ[m, n] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.5 P(x) (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.5 P(x) (a+b x)^m (c+d x)^n.m",
            "rhs": "Int[PolynomialQuotient[Px, a + b*x, x]*(a + b*x)^(m + 1)*(c + d*x)^n, x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_. + b_.*x_)^m_.*(c_. + d_.*x_)^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && EqQ[PolynomialRemainder[Px, a + b*x, x], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.5 P(x) (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.5 P(x) (a+b x)^m (c+d x)^n.m",
            "rhs": "Int[ExpandIntegrand[1/Sqrt[c + d*x], Px*(c + d*x)^(n + 1/2)/(a + b*x), x], x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(c_. + d_.*x_)^n_./(a_. + b_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && ILtQ[n + 1/2, 0] && GtQ[Expon[Px, x], 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.5 P(x) (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.5 P(x) (a+b x)^m (c+d x)^n.m",
            "rhs": "Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_. + b_.*x_)^m_.*(c_. + d_.*x_)^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) && GtQ[Expon[Px, x], 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.5 P(x) (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.5 P(x) (a+b x)^m (c+d x)^n.m",
            "rhs": "With[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px, a + b*x, x]}, R*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)/((m + 1)*(b*c - a*d)) + 1/((m + 1)*(b*c - a*d))* Int[(a + b*x)^(m + 1)*(c + d*x)^n* ExpandToSum[(m + 1)*(b*c - a*d)*Qx - d*R*(m + n + 2), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && ILtQ[m, -1] && GtQ[Expon[Px, x], 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.5 P(x) (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.5 P(x) (a+b x)^m (c+d x)^n.m",
            "rhs": "With[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px, a + b*x, x]}, R*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)/((m + 1)*(b*c - a*d)) + 1/((m + 1)*(b*c - a*d))* Int[(a + b*x)^(m + 1)*(c + d*x)^n* ExpandToSum[(m + 1)*(b*c - a*d)*Qx - d*R*(m + n + 2), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && LtQ[m, -1] && GtQ[Expon[Px, x], 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.5 P(x) (a+b x)^m (c+d x)^n.m",
       "filename": "1.1.1.5 P(x) (a+b x)^m (c+d x)^n.m",
            "rhs": "With[{q = Expon[Px, x], k = Coeff[Px, x, Expon[Px, x]]}, k*(a + b*x)^(m + q)*(c + d*x)^(n + 1)/(d*b^q*(m + n + q + 1)) + 1/(d*b^q*(m + n + q + 1))*Int[(a + b*x)^m*(c + d*x)^n* ExpandToSum[ d*b^q*(m + n + q + 1)*Px - d*k*(m + n + q + 1)*(a + b*x)^q - k*(b*c - a*d)*(m + q)*(a + b*x)^(q - 1), x], x] /; NeQ[m + n + q + 1, 0]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_. + b_.*x_)^m_.*(c_. + d_.*x_)^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && GtQ[Expon[Px, x], 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.6 P(x) (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.6 P(x) (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_. + b_.*x_)^m_.*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] && EqQ[m, n] && (IntegerQ[m] || GtQ[a, 0] && GtQ[c, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.6 P(x) (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.6 P(x) (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "(a + b*x)^ FracPart[m]*(c + d*x)^FracPart[m]/(a*c + b*d*x^2)^FracPart[m]* Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_*(e_. + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] && EqQ[m, n] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.6 P(x) (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.6 P(x) (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "Int[PolynomialQuotient[Px, a + b*x, x]*(a + b*x)^(m + 1)*(c + d*x)^ n*(e + f*x)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_. + b_.*x_)^m_.*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[PolynomialRemainder[Px, a + b*x, x], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.6 P(x) (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.6 P(x) (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_. + b_.*x_)^m_.*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && IntegersQ[m, n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.6 P(x) (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.6 P(x) (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "With[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px, a + b*x, x]}, b*R*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f)) + 1/((m + 1)*(b*c - a*d)*(b*e - a*f))* Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p* ExpandToSum[(m + 1)*(b*c - a*d)*(b*e - a*f)*Qx + a*d*f*R*(m + 1) - b*R*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*R*(m + n + p + 3)*x, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, p}, x] && PolyQ[Px, x] && ILtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.6 P(x) (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.6 P(x) (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "With[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px, a + b*x, x]}, b*R*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f)) + 1/((m + 1)*(b*c - a*d)*(b*e - a*f))* Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p* ExpandToSum[(m + 1)*(b*c - a*d)*(b*e - a*f)*Qx + a*d*f*R*(m + 1) - b*R*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*R*(m + n + p + 3)*x, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_. + b_.*x_)^m_*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, p}, x] && PolyQ[Px, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.6 P(x) (a+b x)^m (c+d x)^n (e+f x)^p.m",
       "filename": "1.1.1.6 P(x) (a+b x)^m (c+d x)^n (e+f x)^p.m",
            "rhs": "With[{q = Expon[Px, x], k = Coeff[Px, x, Expon[Px, x]]}, k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1)/(d*f* b^(q - 1)*(m + n + p + q + 1)) + 1/(d*f*b^q*(m + n + p + q + 1))* Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p* ExpandToSum[ d*f*b^q*(m + n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a + b*x)^(q - 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m + q + p)))*x), x], x] /; NeQ[m + n + p + q + 1, 0]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_. + b_.*x_)^m_.*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && IntegersQ[2*m, 2*n, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "Coeff[Px, x, n - 1]*(a + b*x^n)^(p + 1)/(b*n*(p + 1)) + Int[(Px - Coeff[Px, x, n - 1]*x^(n - 1))*(a + b*x^n)^p, x] /; FreeQ[{a, b}, x] && PolyQ[Px, x] && IGtQ[p, 1] && IGtQ[n, 1] && NeQ[Coeff[Px, x, n - 1], 0] && NeQ[Px, Coeff[Px, x, n - 1]*x^(n - 1)] && Not[MatchQ[Px, Qx_.*(c_ + d_.*x^m_)^q_",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d}, x] && PolyQ[Qx, x] && IGtQ[q, 1] && IGtQ[m, 1] && NeQ[Coeff[Qx*(a + b*x^n)^p, x, m - 1], 0] && GtQ[m*q, n*p]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "Coeff[Px, x, n - m - 1]*(a + b*x^n)^(p + 1)/(b*n*(p + 1)) + Int[(Px - Coeff[Px, x, n - m - 1]*x^(n - m - 1))* x^m*(a + b*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*x_^m_.*(a_ + b_.*x_^n_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && PolyQ[Px, x] && IGtQ[p, 1] && IGtQ[n - m, 0] && NeQ[Coeff[Px, x, n - m - 1], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "Int[u*x^(m + n*p)*(a + b*x^(q - p))^n, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*x_^m_.*(a_.*x_^p_. + b_.*x_^q_.)^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "Int[u*x^(m + n*p)*(a + b*x^(q - p) + c*x^(r - p))^n, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*x_^m_.*(a_.*x_^p_. + b_.*x_^q_. + c_.*x_^r_.)^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*Px_^p_.*Qx_^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Coeff[Pp, x, p]* Log[RemoveContent[Qq, x]]/(q*Coeff[Qq, x, q]) /; EqQ[p, q - 1] && EqQ[Pp, Simplify[ Coeff[Pp, x, p]/(q*Coeff[Qq, x, q])*D[Qq, x]]]]",
     "rulenumber": 0,
            "lhs": "Int[Pp_/Qq_, x_Symbol]",
        "comment": false,
         "givens": "PolyQ[Pp, x] && PolyQ[Qq, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Coeff[Pp, x, p]*x^(p - q + 1)* Qq^(m + 1)/((p + m*q + 1)*Coeff[Qq, x, q]) /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x, q]*Pp, Coeff[Pp, x, p]* x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]]",
     "rulenumber": 0,
            "lhs": "Int[Pp_*Qq_^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[m, x] && PolyQ[Pp, x] && PolyQ[Qq, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1)/(2*b1* b2*n*(p + 1))",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a1_ + b1_.*x_^n_.)^p_*(a2_ + b2_.*x_^n_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && EqQ[m - 2*n + 1, 0] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "With[{p = Expon[Pp, x], q = Expon[Qq, x], r = Expon[Rr, x]}, Coeff[Pp, x, p]*x^(p - q - r + 1)*Qq^(m + 1)* Rr^(n + 1)/((p + m*q + n*r + 1)*Coeff[Qq, x, q]* Coeff[Rr, x, r]) /; NeQ[p + m*q + n*r + 1, 0] && EqQ[(p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x, r]*Pp, Coeff[Pp, x, p]* x^(p - q - r)*((p - q - r + 1)*Qq*Rr + (m + 1)*x*Rr* D[Qq, x] + (n + 1)*x*Qq*D[Rr, x])]]",
     "rulenumber": 0,
            "lhs": "Int[Pp_*Qq_^m_.*Rr_^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{m, n}, x] && PolyQ[Pp, x] && PolyQ[Qq, x] && PolyQ[Rr, x] && NeQ[m, -1] && NeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "With[{q = Expon[Pq, x], r = Expon[Qr, x]}, Coeff[Qr, x, r]/(q*Coeff[Pq, x, q])* Subst[Int[(a + b*x^n)^p, x], x, Pq] /; EqQ[r, q - 1] && EqQ[Coeff[Qr, x, r]*D[Pq, x], q*Coeff[Pq, x, q]*Qr]]",
     "rulenumber": 0,
            "lhs": "Int[Qr_*(a_. + b_.*Pq_^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n, p}, x] && PolyQ[Pq, x] && PolyQ[Qr, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "Module[{q = Expon[Pq, x], r = Expon[Qr, x]}, Coeff[Qr, x, r]/(q*Coeff[Pq, x, q])* Subst[Int[(a + b*x^n + c*x^(2*n))^p, x], x, Pq] /; EqQ[r, q - 1] && EqQ[Coeff[Qr, x, r]*D[Pq, x], q*Coeff[Pq, x, q]*Qr]]",
     "rulenumber": 0,
            "lhs": "Int[Qr_*(a_. + b_.*Pq_^n_. + c_.*Pq_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && PolyQ[Qr, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "Int[u*x^(n*p)*(a + b*x^(q - p))^n, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_.*x_^p_. + b_.*x_^q_.)^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "Int[u*x^(n*p)*(a + b*x^(q - p) + c*x^(r - p))^n, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_.*x_^p_. + b_.*x_^q_. + c_.*x_^r_.)^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": " B*Sqrt[a+b*x]*Sqrt[e+f*x]*Sqrt[g+h*x]/(f*h*Sqrt[c+d*x]) - B*(b*g-a*h)/(2*f*h)*Int[Sqrt[e+f*x]/(Sqrt[a+b*x]*Sqrt[c+d*x]*Sqrt[g+ h*x]),x] + B*(d*e-c*f)*(d*g-c*h)/(2*d*f*h)*Int[Sqrt[a+b*x]/((c+d*x)^(3/2)*Sqrt[ e+f*x]*Sqrt[g+h*x]),x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_.+b_.*x_]*(A_.+B_.*x_)/(Sqrt[c_.+d_.*x_]*Sqrt[e_.+f_.*x_ ]*Sqrt[g_.+h_.*x_]),x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,g,h,A,B},x] &&  EqQ[2*A*d*f-B*(d*e+c*f),0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "b*B*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]/(d*f*h*Sqrt[a + b*x]) - B*(b*g - a*h)/(2*f*h)* Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[g + h*x]), x] + B*(b*e - a*f)*(b*g - a*h)/(2*d*f*h)* Int[Sqrt[c + d*x]/((a + b*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_. + b_.*x_]*(A_. + B_.*x_)/(Sqrt[c_. + d_.*x_]*Sqrt[e_. + f_.*x_]* Sqrt[g_. + h_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, A, B}, x] && EqQ[2*A*d*f - B*(d*e + c*f), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": " (2*A*d*f-B*(d*e+c*f))/(2*d*f)*Int[Sqrt[a+b*x]/(Sqrt[c+d*x]*Sqrt[e+f*x] *Sqrt[g+h*x]),x] + B/(2*d*f)*Int[(Sqrt[a+b*x]*(d*e+c*f+2*d*f*x))/(Sqrt[c+d*x]*Sqrt[e+f* x]*Sqrt[g+h*x]),x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_.+b_.*x_]*(A_.+B_.*x_)/(Sqrt[c_.+d_.*x_]*Sqrt[e_.+f_.*x_ ]*Sqrt[g_.+h_.*x_]),x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,g,h,A,B},x] &&  NeQ[2*A*d*f-B*(d*e+c*f),0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "B*Sqrt[a + b*x]*Sqrt[e + f*x]*Sqrt[g + h*x]/(f*h*Sqrt[c + d*x]) + B*(d*e - c*f)*(d*g - c*h)/(2*d*f*h)* Int[Sqrt[a + b*x]/((c + d*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x]), x] - B*(b*e - a*f)*(b*g - a*h)/(2*b*f*h)* Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x] + (2*A*b*d*f*h + B*(a*d*f*h - b*(d*f*g + d*e*h + c*f*h)))/(2*b*d*f* h)*Int[Sqrt[ a + b*x]/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_. + b_.*x_]*(A_. + B_.*x_)/(Sqrt[c_. + d_.*x_]*Sqrt[e_. + f_.*x_]* Sqrt[g_. + h_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, A, B}, x] && NeQ[2*A*d*f - B*(d*e + c*f), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "1/(d*f*h*(2*m + 3))* Int[((a + b*x)^(m - 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))* Simp[ a*A*d*f*h*(2*m + 3) + (A*b + a*B)*d*f*h*(2*m + 3)*x + b*B*d*f*h*(2*m + 3)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^ m_.*(A_. + B_.*x_)/(Sqrt[c_. + d_.*x_]*Sqrt[e_. + f_.*x_]* Sqrt[g_. + h_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, A, B}, x] && IntegerQ[2*m] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "(A*b - a*B)/b* Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x] + B/b*Int[ Sqrt[a + b*x]/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*x_)/(Sqrt[a_. + b_.*x_]*Sqrt[c_. + d_.*x_]* Sqrt[e_. + f_.*x_]*Sqrt[g_. + h_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, A, B}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "(A*b^2 - a*b*B)*(a + b*x)^(m + 1)*Sqrt[c + d*x]* Sqrt[e + f*x]* Sqrt[g + h*x]/((m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h)) - 1/(2*(m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h))* Int[((a + b*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]* Sqrt[g + h*x]))* Simp[A*(2*a^2*d*f*h*(m + 1) - 2*a*b*(m + 1)*(d*f*g + d*e*h + c*f*h) + b^2*(2*m + 3)*(d*e*g + c*f*g + c*e*h)) - b*B*(a*(d*e*g + c*f*g + c*e*h) + 2*b*c*e*g*(m + 1)) - 2*((A*b - a*B)*(a*d*f*h*(m + 1) - b*(m + 2)*(d*f*g + d*e*h + c*f*h)))*x + d*f*h*(2*m + 5)*(A*b^2 - a*b*B)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^ m_*(A_. + B_.*x_)/(Sqrt[c_. + d_.*x_]*Sqrt[e_. + f_.*x_]* Sqrt[g_. + h_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, A, B}, x] && IntegerQ[2*m] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "2*C*(a + b*x)^m*Sqrt[c + d*x]*Sqrt[e + f*x]* Sqrt[g + h*x]/(d*f*h*(2*m + 3)) + 1/(d*f*h*(2*m + 3))* Int[((a + b*x)^(m - 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]* Sqrt[g + h*x]))* Simp[a*A*d*f*h*(2*m + 3) - C*(a*(d*e*g + c*f*g + c*e*h) + 2*b*c*e*g*m) + ((A*b + a*B)*d*f*h*(2*m + 3) - C*(2*a*(d*f*g + d*e*h + c*f*h) + b*(2*m + 1)*(d*e*g + c*f*g + c*e*h)))*x + (b*B*d*f*h*(2*m + 3) + 2*C*(a*d*f*h*m - b*(m + 1)*(d*f*g + d*e*h + c*f*h)))*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^ m_.*(A_. + B_.*x_ + C_.*x_^2)/(Sqrt[c_. + d_.*x_]* Sqrt[e_. + f_.*x_]*Sqrt[g_. + h_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, A, B, C}, x] && IntegerQ[2*m] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "2*C*(a + b*x)^m*Sqrt[c + d*x]*Sqrt[e + f*x]* Sqrt[g + h*x]/(d*f*h*(2*m + 3)) + 1/(d*f*h*(2*m + 3))* Int[((a + b*x)^(m - 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]* Sqrt[g + h*x]))* Simp[a*A*d*f*h*(2*m + 3) - C*(a*(d*e*g + c*f*g + c*e*h) + 2*b*c*e*g*m) + (A*b*d*f*h*(2*m + 3) - C*(2*a*(d*f*g + d*e*h + c*f*h) + b*(2*m + 1)*(d*e*g + c*f*g + c*e*h)))*x + 2*C*(a*d*f*h*m - b*(m + 1)*(d*f*g + d*e*h + c*f*h))*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^ m_.*(A_. + C_.*x_^2)/(Sqrt[c_. + d_.*x_]*Sqrt[e_. + f_.*x_]* Sqrt[g_. + h_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, A, C}, x] && IntegerQ[2*m] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "C*Sqrt[a + b*x]*Sqrt[e + f*x]*Sqrt[g + h*x]/(b*f*h*Sqrt[c + d*x]) + C*(d*e - c*f)*(d*g - c*h)/(2*b*d*f*h)* Int[Sqrt[a + b*x]/((c + d*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x]), x] + 1/(2*b*d*f*h)* Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])* Simp[2*A*b*d*f*h - C*(b*d*e*g + a*c*f*h) + (2*b*B*d*f*h - C*(a*d*f*h + b*(d*f*g + d*e*h + c*f*h)))*x, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*x_ + C_.*x_^2)/(Sqrt[a_. + b_.*x_]*Sqrt[c_. + d_.*x_]* Sqrt[e_. + f_.*x_]*Sqrt[g_. + h_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, A, B, C}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "C*Sqrt[a + b*x]*Sqrt[e + f*x]*Sqrt[g + h*x]/(b*f*h*Sqrt[c + d*x]) + C*(d*e - c*f)*(d*g - c*h)/(2*b*d*f*h)* Int[Sqrt[a + b*x]/((c + d*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x]), x] + 1/(2*b*d*f*h)* Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])* Simp[2*A*b*d*f*h - C*(b*d*e*g + a*c*f*h) - C*(a*d*f*h + b*(d*f*g + d*e*h + c*f*h))*x, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + C_.*x_^2)/(Sqrt[a_. + b_.*x_]*Sqrt[c_. + d_.*x_]* Sqrt[e_. + f_.*x_]*Sqrt[g_. + h_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, A, C}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "(A*b^2 - a*b*B + a^2*C)*(a + b*x)^(m + 1)* Sqrt[c + d*x]*Sqrt[e + f*x]* Sqrt[g + h*x]/((m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h)) - 1/(2*(m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h))* Int[((a + b*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]* Sqrt[g + h*x]))* Simp[A*(2*a^2*d*f*h*(m + 1) - 2*a*b*(m + 1)*(d*f*g + d*e*h + c*f*h) + b^2*(2*m + 3)*(d*e*g + c*f*g + c*e*h)) - (b*B - a*C)*(a*(d*e*g + c*f*g + c*e*h) + 2*b*c*e*g*(m + 1)) - 2*((A*b - a*B)*(a*d*f*h*(m + 1) - b*(m + 2)*(d*f*g + d*e*h + c*f*h)) - C*(a^2*(d*f*g + d*e*h + c*f*h) - b^2*c*e*g*(m + 1) + a*b*(m + 1)*(d*e*g + c*f*g + c*e*h)))*x + d*f*h*(2*m + 5)*(A*b^2 - a*b*B + a^2*C)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^ m_*(A_. + B_.*x_ + C_.*x_^2)/(Sqrt[c_. + d_.*x_]* Sqrt[e_. + f_.*x_]*Sqrt[g_. + h_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, A, B, C}, x] && IntegerQ[2*m] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "(A*b^2 + a^2*C)*(a + b*x)^(m + 1)*Sqrt[c + d*x]* Sqrt[e + f*x]* Sqrt[g + h*x]/((m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h)) - 1/(2*(m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h))* Int[((a + b*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]* Sqrt[g + h*x]))* Simp[A*(2*a^2*d*f*h*(m + 1) - 2*a*b*(m + 1)*(d*f*g + d*e*h + c*f*h) + b^2*(2*m + 3)*(d*e*g + c*f*g + c*e*h)) + a*C*(a*(d*e*g + c*f*g + c*e*h) + 2*b*c*e*g*(m + 1)) - 2*(A*b*(a*d*f*h*(m + 1) - b*(m + 2)*(d*f*g + d*e*h + c*f*h)) - C*(a^2*(d*f*g + d*e*h + c*f*h) - b^2*c*e*g*(m + 1) + a*b*(m + 1)*(d*e*g + c*f*g + c*e*h)))*x + d*f*h*(2*m + 5)*(A*b^2 + a^2*C)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^ m_*(A_. + C_.*x_^2)/(Sqrt[c_. + d_.*x_]*Sqrt[e_. + f_.*x_]* Sqrt[g_. + h_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, A, C}, x] && IntegerQ[2*m] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "Int[ExpandIntegrand[ Px*(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_. + b_.*x_)^m_.*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^ p_.*(g_. + h_.*x_)^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n, p, q}, x] && PolyQ[Px, x] && IntegersQ[m, n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "PolynomialRemainder[Px, a + b*x, x]* Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x] + Int[PolynomialQuotient[Px, a + b*x, x]*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_. + b_.*x_)^m_.*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^ p_.*(g_. + h_.*x_)^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n, p, q}, x] && PolyQ[Px, x] && EqQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.1 Linear/1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
       "filename": "1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.m",
            "rhs": "PolynomialRemainder[Px, a + b*x, x]* Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x] + Int[PolynomialQuotient[Px, a + b*x, x]*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_. + b_.*x_)^m_.*(c_. + d_.*x_)^n_.*(e_. + f_.*x_)^ p_.*(g_. + h_.*x_)^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n, p, q}, x] && PolyQ[Px, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.2 Quadratic/1.1.2.x P(x) (a+b x^2)^p.m",
       "filename": "1.1.2.x P(x) (a+b x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.2 Quadratic/1.1.2.x P(x) (a+b x^2)^p.m",
       "filename": "1.1.2.x P(x) (a+b x^2)^p.m",
            "rhs": "Int[x*PolynomialQuotient[Pq, x, x]*(a + b*x^2)^p, x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && EqQ[Coeff[Pq, x, 0], 0] && Not[MatchQ[Pq, x^m_.*u_.",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "IntegerQ[m]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.2 Quadratic/1.1.2.x P(x) (a+b x^2)^p.m",
       "filename": "1.1.2.x P(x) (a+b x^2)^p.m",
            "rhs": "Int[PolynomialQuotient[Px, a + b*x^2, x]*(a + b*x^2)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_ + b_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, p}, x] && PolyQ[Px, x] && EqQ[PolynomialRemainder[Px, a + b*x^2, x], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.2 Quadratic/1.1.2.x P(x) (a+b x^2)^p.m",
       "filename": "1.1.2.x P(x) (a+b x^2)^p.m",
            "rhs": "With[{A = Coeff[Pq, x, 0], Q = PolynomialQuotient[Pq - Coeff[Pq, x, 0], x^2, x]}, A*x*(a + b*x^2)^(p + 1)/a + 1/a*Int[x^2*(a + b*x^2)^p*(a*Q - A*b*(2*p + 3)), x]]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[Pq, x^2] && ILtQ[p + 1/2, 0] && LtQ[Expon[Pq, x] + 2*p + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.2 Quadratic/1.1.2.x P(x) (a+b x^2)^p.m",
       "filename": "1.1.2.x P(x) (a+b x^2)^p.m",
            "rhs": "With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, (a*g - b*f*x)*(a + b*x^2)^(p + 1)/(2*a*b*(p + 1)) + 1/(2*a*(p + 1))* Int[(a + b*x^2)^(p + 1)* ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.2 Quadratic/1.1.2.x P(x) (a+b x^2)^p.m",
       "filename": "1.1.2.x P(x) (a+b x^2)^p.m",
            "rhs": "With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, e*x^(q - 1)*(a + b*x^2)^(p + 1)/(b*(q + 2*p + 1)) + 1/(b*(q + 2*p + 1))* Int[(a + b*x^2)^p* ExpandToSum[ b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && Not[LeQ[p, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.2 Quadratic/1.1.2.y P(x) (c x)^m (a+b x^2)^p.m",
       "filename": "1.1.2.y P(x) (c x)^m (a+b x^2)^p.m",
            "rhs": "1/2*Subst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Pq_*(a_ + b_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.2 Quadratic/1.1.2.y P(x) (c x)^m (a+b x^2)^p.m",
       "filename": "1.1.2.y P(x) (c x)^m (a+b x^2)^p.m",
            "rhs": "1/c*Int[(c*x)^(m + 1)*PolynomialQuotient[Pq, x, x]*(a + b*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*Pq_*(a_ + b_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && EqQ[Coeff[Pq, x, 0], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.2 Quadratic/1.1.2.y P(x) (c x)^m (a+b x^2)^p.m",
       "filename": "1.1.2.y P(x) (c x)^m (a+b x^2)^p.m",
            "rhs": "With[{f = Coeff[P2, x, 0], g = Coeff[P2, x, 1], h = Coeff[P2, x, 2]}, h*(c*x)^(m + 1)*(a + b*x^2)^(p + 1)/(b*c*(m + 2*p + 3)) /; EqQ[g, 0] && EqQ[a*h*(m + 1) - b*f*(m + 2*p + 3), 0]]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*P2_*(a_ + b_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, p}, x] && PolyQ[P2, x, 2] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.2 Quadratic/1.1.2.y P(x) (c x)^m (a+b x^2)^p.m",
       "filename": "1.1.2.y P(x) (c x)^m (a+b x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x^2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*Pq_*(a_ + b_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.2 Quadratic/1.1.2.y P(x) (c x)^m (a+b x^2)^p.m",
       "filename": "1.1.2.y P(x) (c x)^m (a+b x^2)^p.m",
            "rhs": "With[{A = Coeff[Pq, x, 0], Q = PolynomialQuotient[Pq - Coeff[Pq, x, 0], x^2, x]}, A*x^(m + 1)*(a + b*x^2)^(p + 1)/(a*(m + 1)) + 1/(a*(m + 1))* Int[x^(m + 2)*(a + b*x^2)^ p*(a*(m + 1)*Q - A*b*(m + 2*(p + 1) + 1)), x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*Pq_*(a_ + b_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[Pq, x^2] && IntegerQ[m/2] && ILtQ[(m + 1)/2 + p, 0] && LtQ[m + Expon[Pq, x] + 2*p + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.2 Quadratic/1.1.2.y P(x) (c x)^m (a+b x^2)^p.m",
       "filename": "1.1.2.y P(x) (c x)^m (a+b x^2)^p.m",
            "rhs": "With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, (c*x)^m*(a + b*x^2)^(p + 1)*(a*g - b*f*x)/(2*a*b*(p + 1)) + c/(2*a*b*(p + 1))* Int[(c*x)^(m - 1)*(a + b*x^2)^(p + 1)* ExpandToSum[2*a*b*(p + 1)*x*Q - a*g*m + b*f*(m + 2*p + 3)*x, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*Pq_*(a_ + b_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.2 Quadratic/1.1.2.y P(x) (c x)^m (a+b x^2)^p.m",
       "filename": "1.1.2.y P(x) (c x)^m (a+b x^2)^p.m",
            "rhs": "With[{Q = PolynomialQuotient[(c*x)^m*Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 1]}, (a*g - b*f*x)*(a + b*x^2)^(p + 1)/(2*a*b*(p + 1)) + 1/(2*a*(p + 1))* Int[(c*x)^m*(a + b*x^2)^(p + 1)* ExpandToSum[2*a*(p + 1)*(c*x)^(-m)*Q + f*(2*p + 3)*(c*x)^(-m), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*Pq_*(a_ + b_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.2 Quadratic/1.1.2.y P(x) (c x)^m (a+b x^2)^p.m",
       "filename": "1.1.2.y P(x) (c x)^m (a+b x^2)^p.m",
            "rhs": "With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, -(c*x)^(m + 1)*(f + g*x)*(a + b*x^2)^(p + 1)/(2*a*c*(p + 1)) + 1/(2*a*(p + 1))* Int[(c*x)^m*(a + b*x^2)^(p + 1)* ExpandToSum[ 2*a*(p + 1)*Q + f*(m + 2*p + 3) + g*(m + 2*p + 4)*x, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*Pq_*(a_ + b_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && LtQ[p, -1] && Not[GtQ[m, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.2 Quadratic/1.1.2.y P(x) (c x)^m (a+b x^2)^p.m",
       "filename": "1.1.2.y P(x) (c x)^m (a+b x^2)^p.m",
            "rhs": "With[{Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, R*(c*x)^(m + 1)*(a + b*x^2)^(p + 1)/(a*c*(m + 1)) + 1/(a*c*(m + 1))* Int[(c*x)^(m + 1)*(a + b*x^2)^p* ExpandToSum[a*c*(m + 1)*Q - b*R*(m + 2*p + 3)*x, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_*Pq_*(a_ + b_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && LtQ[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.2 Quadratic/1.1.2.y P(x) (c x)^m (a+b x^2)^p.m",
       "filename": "1.1.2.y P(x) (c x)^m (a+b x^2)^p.m",
            "rhs": "With[{q = Expon[Pq, x]}, Coeff[Pq, x, q]/c^q*Int[(c*x)^(m + q)*(a + b*x^2)^p, x] + 1/c^q* Int[(c*x)^m*(a + b*x^2)^p* ExpandToSum[c^q*Pq - Coeff[Pq, x, q]*(c*x)^q, x], x] /; EqQ[q, 1] || EqQ[m + q + 2*p + 1, 0]]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*Pq_*(a_ + b_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && Not[IGtQ[m, 0] && ILtQ[p + 1/2, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.2 Quadratic/1.1.2.y P(x) (c x)^m (a+b x^2)^p.m",
       "filename": "1.1.2.y P(x) (c x)^m (a+b x^2)^p.m",
            "rhs": "With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, f*(c*x)^(m + q - 1)*(a + b*x^2)^(p + 1)/(b* c^(q - 1)*(m + q + 2*p + 1)) + 1/(b*(m + q + 2*p + 1))* Int[(c*x)^m*(a + b*x^2)^p* ExpandToSum[ b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*Pq_*(a_ + b_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && (Not[IGtQ[m, 0]] || IGtQ[p + 1/2, -1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "b^IntPart[p]*(b*x^n)^FracPart[p]/x^(n*FracPart[p])*Int[x^(n*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "1/n*Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "x*(a + b*x^n)^(p + 1)/a",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n, p}, x] && EqQ[1/n + p + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "-x*(a + b*x^n)^(p + 1)/(a*n*(p + 1)) + (n*(p + 1) + 1)/(a*n*(p + 1))*Int[(a + b*x^n)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1], 0] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "Int[x^(n*p)*(b + a*x^(-n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && LtQ[n, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*x^n)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IGtQ[n, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "x*(a + b*x^n)^p/(n*p + 1) + a*n*p/(n*p + 1)*Int[(a + b*x^n)^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || EqQ[n, 2] && IntegerQ[4*p] || EqQ[n, 2] && IntegerQ[3*p] || LtQ[Denominator[p + 1/n], Denominator[p]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "2/(a^(5/4)*Rt[b/a, 2])*EllipticE[1/2*ArcTan[Rt[b/a, 2]*x], 2]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^2)^(5/4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "(1 + b*x^2/a)^(1/4)/(a*(a + b*x^2)^(1/4))* Int[1/(1 + b*x^2/a)^(5/4), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^2)^(5/4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PosQ[a] && PosQ[b/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "1/((a + b*x^2)^(2/3)*(a/(a + b*x^2))^(2/3))* Subst[Int[1/(1 - b*x^2)^(1/3), x], x, x/Sqrt[a + b*x^2]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^2)^(7/6), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "-x*(a + b*x^n)^(p + 1)/(a*n*(p + 1)) + (n*(p + 1) + 1)/(a*n*(p + 1))*Int[(a + b*x^n)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (IntegerQ[2*p] || n == 2 && IntegerQ[4*p] || n == 2 && IntegerQ[3*p] || Denominator[p + 1/n] < Denominator[p])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "1/(3*Rt[a, 3]^2)*Int[1/(Rt[a, 3] + Rt[b, 3]*x), x] + 1/(3*Rt[a, 3]^2)* Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^3), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "With[{r=Numerator[Rt[a/b,5]],  s=Denominator[Rt[a/b,5]]}, r/(5*a)*Int[1/(r+s*x),x] + 2*r/(5*a)*Int[(r-1/4*(1-Sqrt[5])*s*x)/(r^2-1/2*(1-Sqrt[5])*r*s*x+s^ 2*x^2),x] + 2*r/(5*a)*Int[(r-1/4*(1+Sqrt[5])*s*x)/(r^2-1/2*(1+Sqrt[5])*r*s*x+s^ 2*x^2),x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_+b_.*x_^5),x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b},x] && PosQ[a/b] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "With[{r=Numerator[Rt[-a/b,5]],  s=Denominator[Rt[-a/b,5]]}, r/(5*a)*Int[1/(r-s*x),x] + 2*r/(5*a)*Int[(r+1/4*(1-Sqrt[5])*s*x)/(r^2+1/2*(1-Sqrt[5])*r*s*x+s^ 2*x^2),x] + 2*r/(5*a)*Int[(r+1/4*(1+Sqrt[5])*s*x)/(r^2+1/2*(1+Sqrt[5])*r*s*x+s^ 2*x^2),x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_+b_.*x_^5),x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b},x] && NegQ[a/b] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/b, n]], k, u}, u = Int[(r - s*Cos[(2*k - 1)*Pi/n]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*Pi/n]*x + s^2*x^2), x]; r/(a*n)*Int[1/(r + s*x), x] + Dist[2*r/(a*n), Sum[u, {k, 1, (n - 1)/2}], x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IGtQ[(n - 3)/2, 0] && PosQ[a/b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "Module[{r = Numerator[Rt[-a/b, n]], s = Denominator[Rt[-a/b, n]], k, u}, u = Int[(r + s*Cos[(2*k - 1)*Pi/n]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*Pi/n]*x + s^2*x^2), x]; r/(a*n)*Int[1/(r - s*x), x] + Dist[2*r/(a*n), Sum[u, {k, 1, (n - 1)/2}], x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IGtQ[(n - 3)/2, 0] && NegQ[a/b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "1/(Rt[a, 2]*Rt[b, 2])*ArcTan[Rt[b, 2]*x/Rt[a, 2]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "-1/(Rt[-a, 2]*Rt[-b, 2])* ArcTan[Rt[-b, 2]*x/Rt[-a, 2]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "(*Rt[b/a,2]/b*ArcTan[Rt[b/a,2]*x] /; *) Rt[a/b, 2]/a*ArcTan[x/Rt[a/b, 2]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PosQ[a/b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "1/(Rt[a, 2]*Rt[-b, 2])*ArcTanh[Rt[-b, 2]*x/Rt[a, 2]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "-1/(Rt[-a, 2]*Rt[b, 2])* ArcTanh[Rt[b, 2]*x/Rt[-a, 2]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "(*-Rt[-b/a,2]/b*ArcTanh[Rt[-b/a,2]*x] /; *) Rt[-a/b, 2]/a*ArcTanh[x/Rt[-a/b, 2]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && NegQ[a/b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/b, n]], k, u, v}, u = Int[(r - s*Cos[(2*k - 1)*Pi/n]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*Pi/n]*x + s^2*x^2), x] + Int[(r + s*Cos[(2*k - 1)*Pi/n]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*Pi/n]*x + s^2*x^2), x]; 2*r^2/(a*n)*Int[1/(r^2 + s^2*x^2), x] + Dist[2*r/(a*n), Sum[u, {k, 1, (n - 2)/4}], x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && PosQ[a/b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "Module[{r = Numerator[Rt[-a/b, n]], s = Denominator[Rt[-a/b, n]], k, u}, u = Int[(r - s*Cos[(2*k*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x] + Int[(r + s*Cos[(2*k*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x]; 2*r^2/(a*n)*Int[1/(r^2 - s^2*x^2), x] + Dist[2*r/(a*n), Sum[u, {k, 1, (n - 2)/4}], x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && NegQ[a/b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, 1/(2*r)*Int[(r - s*x^2)/(a + b*x^4), x] + 1/(2*r)*Int[(r + s*x^2)/(a + b*x^4), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && (GtQ[a/b, 0] || PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, r/(2*a)*Int[1/(r - s*x^2), x] + r/(2*a)*Int[1/(r + s*x^2), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && Not[GtQ[a/b, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "With[{r = Numerator[Rt[a/b, 4]], s = Denominator[Rt[a/b, 4]]}, r/(2*Sqrt[2]*a)* Int[(Sqrt[2]*r - s*x^(n/4))/(r^2 - Sqrt[2]*r*s*x^(n/4) + s^2*x^(n/2)), x] + r/(2*Sqrt[2]*a)* Int[(Sqrt[2]*r + s*x^(n/4))/(r^2 + Sqrt[2]*r*s*x^(n/4) + s^2*x^(n/2)), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IGtQ[n/4, 1] && GtQ[a/b, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, r/(2*a)*Int[1/(r - s*x^(n/2)), x] + r/(2*a)*Int[1/(r + s*x^(n/2)), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IGtQ[n/4, 1] && Not[GtQ[a/b, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "ArcSinh[Rt[b, 2]*x/Sqrt[a]]/Rt[b, 2]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "ArcSin[Rt[-b, 2]*x/Sqrt[a]]/Rt[-b, 2]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && Not[GtQ[a, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "With[{q=Rt[b/a,3]}, -Sqrt[2]*(1+Sqrt[3])*(1+Sqrt[3]+q*x)^2*Sqrt[(1+q^3*x^3)/(1+Sqrt[3]+ q*x)^4]/(3^(1/4)*q*Sqrt[a+b*x^3])* EllipticF[ArcSin[(-1+Sqrt[3]-q*x)/(1+Sqrt[3]+q*x)],-7-4*Sqrt[3]]] ",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_+b_.*x_^3],x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b},x] && PosQ[a] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "With[{q=Rt[a/b,3]}, 2*Sqrt[2+Sqrt[3]]*(q+x)*Sqrt[(q^2-q*x+x^2)/((1+Sqrt[3])*q+x)^2]/ (3^(1/4)*Sqrt[a+b*x^3]*Sqrt[q*(q+x)/((1+Sqrt[3])*q+x)^2])* EllipticF[ArcSin[((1-Sqrt[3])*q+x)/((1+Sqrt[3])*q+x)],-7-4*Sqrt[3] ]]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_+b_.*x_^3],x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b},x] && PosQ[a] && EqQ[b^2,1] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "With[{q=Rt[b/a,3]}, -2*Sqrt[2+Sqrt[3]]*(1+q*x)*Sqrt[(1-q*x+q^2*x^2)/(1+Sqrt[3]+q*x)^2]/ (3^(1/4)*q*Sqrt[a+b*x^3]*Sqrt[(1+q*x)/(1+Sqrt[3]+q*x)^2])* EllipticF[ArcSin[(-1+Sqrt[3]-q*x)/(1+Sqrt[3]+q*x)],-7-4*Sqrt[3]]] ",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_+b_.*x_^3],x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b},x] && PosQ[a] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, 2*Sqrt[2 + Sqrt[3]]*(s + r*x)* Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/ (3^(1/4)*r*Sqrt[a + b*x^3]* Sqrt[s*(s + r*x)/((1 + Sqrt[3])*s + r*x)^2])* EllipticF[ ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]]]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*x_^3], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PosQ[a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "With[{q=Rt[a/b,3]}, 2*Sqrt[2-Sqrt[3]]*(q+x)*Sqrt[(q^2-q*x+x^2)/((1-Sqrt[3])*q+x)^2]/ (3^(1/4)*Sqrt[a+b*x^3]*Sqrt[-q*(q+x)/((1-Sqrt[3])*q+x)^2])* EllipticF[ArcSin[((1+Sqrt[3])*q+x)/((1-Sqrt[3])*q+x)],-7+4*Sqrt[3] ]]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_+b_.*x_^3],x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b},x] && NegQ[a] && EqQ[b^2,1] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "With[{q=Rt[b/a,3]}, -2*Sqrt[2-Sqrt[3]]*(1+q*x)*Sqrt[(1-q*x+q^2*x^2)/(1-Sqrt[3]+q*x)^2]/ (3^(1/4)*q*Sqrt[a+b*x^3]*Sqrt[-(1+q*x)/(1-Sqrt[3]+q*x)^2])* EllipticF[ArcSin[(1+Sqrt[3]+q*x)/(-1+Sqrt[3]-q*x)],-7+4*Sqrt[3]]] ",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_+b_.*x_^3],x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b},x] && NegQ[a] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, 2*Sqrt[2 - Sqrt[3]]*(s + r*x)* Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/ (3^(1/4)*r*Sqrt[a + b*x^3]* Sqrt[-s*(s + r*x)/((1 - Sqrt[3])*s + r*x)^2])* EllipticF[ ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]]]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*x_^3], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && NegQ[a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "With[{q = Rt[b/a, 4]}, (1 + q^2*x^2)* Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4])* EllipticF[2*ArcTan[q*x], 1/2]]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PosQ[b/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "EllipticF[ArcSin[Rt[-b, 4]*x/Rt[a, 4]], -1]/(Rt[a, 4]*Rt[-b, 4])",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "With[{q = Rt[-a*b, 2]}, Sqrt[-a + q*x^2]* Sqrt[(a + q*x^2)/q]/(Sqrt[2]*Sqrt[-a]*Sqrt[a + b*x^4])* EllipticF[ArcSin[x/Sqrt[(a + q*x^2)/(2*q)]], 1/2] /; IntegerQ[q]]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && LtQ[a, 0] && GtQ[b, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "With[{q = Rt[-a*b, 2]}, Sqrt[(a - q*x^2)/(a + q*x^2)]* Sqrt[(a + q*x^2)/q]/(Sqrt[2]*Sqrt[a + b*x^4]* Sqrt[a/(a + q*x^2)])* EllipticF[ArcSin[x/Sqrt[(a + q*x^2)/(2*q)]], 1/2]]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && LtQ[a, 0] && GtQ[b, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "Sqrt[1 + b*x^4/a]/Sqrt[a + b*x^4]*Int[1/Sqrt[1 + b*x^4/a], x]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && NegQ[b/a] && Not[GtQ[a, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, x*(s + r*x^2)* Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]/ (2*3^(1/4)*s*Sqrt[a + b*x^6]* Sqrt[r*x^2*(s + r*x^2)/(s + (1 + Sqrt[3])*r*x^2)^2])* EllipticF[ ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4]]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*x_^6], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "1/2*Int[(1 - Rt[b/a, 4]*x^2)/Sqrt[a + b*x^8], x] + 1/2*Int[(1 + Rt[b/a, 4]*x^2)/Sqrt[a + b*x^8], x]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*x_^8], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "2*x/(a + b*x^2)^(1/4) - a*Int[1/(a + b*x^2)^(5/4), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^2)^(1/4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "2/(a^(1/4)*Rt[-b/a, 2])*EllipticE[1/2*ArcSin[Rt[-b/a, 2]*x], 2]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^2)^(1/4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "(1 + b*x^2/a)^(1/4)/(a + b*x^2)^(1/4)* Int[1/(1 + b*x^2/a)^(1/4), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^2)^(1/4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PosQ[a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "2*Sqrt[-b*x^2/a]/(b*x)* Subst[Int[x^2/Sqrt[1 - x^4/a], x], x, (a + b*x^2)^(1/4)]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^2)^(1/4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && NegQ[a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "2/(a^(3/4)*Rt[b/a, 2])*EllipticF[1/2*ArcTan[Rt[b/a, 2]*x], 2]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^2)^(3/4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "2/(a^(3/4)*Rt[-b/a, 2])*EllipticF[1/2*ArcSin[Rt[-b/a, 2]*x], 2]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^2)^(3/4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "(1 + b*x^2/a)^(3/4)/(a + b*x^2)^(3/4)* Int[1/(1 + b*x^2/a)^(3/4), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^2)^(3/4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PosQ[a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "2*Sqrt[-b*x^2/a]/(b*x)* Subst[Int[1/Sqrt[1 - x^4/a], x], x, (a + b*x^2)^(1/4)]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^2)^(3/4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && NegQ[a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "3*Sqrt[b*x^2]/(2*b*x)* Subst[Int[x/Sqrt[-a + x^3], x], x, (a + b*x^2)^(1/3)]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^2)^(1/3), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "3*Sqrt[b*x^2]/(2*b*x)* Subst[Int[1/Sqrt[-a + x^3], x], x, (a + b*x^2)^(1/3)]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^2)^(2/3), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "x^3*(1 + a/(b*x^4))^(3/4)/(a + b*x^4)^(3/4)* Int[1/(x^3*(1 + a/(b*x^4))^(3/4)), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^4)^(3/4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "3*x/(2*(a + b*x^2)^(1/6)) - a/2*Int[1/(a + b*x^2)^(7/6), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^2)^(1/6), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "ArcTan[(1 + 2*Rt[b, 3]*x/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]* Rt[b, 3]) - Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3])",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^3)^(1/3), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "a^(p + 1/n)* Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -1/2] && IntegerQ[p + 1/n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "(a/(a + b*x^n))^(p + 1/n)*(a + b*x^n)^(p + 1/n)* Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -1/2] && LtQ[Denominator[p + 1/n], Denominator[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "-Subst[Int[(a + b*x^(-n))^p/x^2, x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, p}, x] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "With[{k = Denominator[n]}, k*Subst[Int[x^(k - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, p}, x] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*x^n)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, -b*x^n/a]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n, p}, x] && Not[IGtQ[p, 0]] && Not[IntegerQ[1/n]] && Not[ILtQ[Simplify[1/n + p], 0]] && (IntegerQ[p] || GtQ[a, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": " x*(a+b*x^n)^(p+1)/a*Hypergeometric2F1[1,1/n+p+1,1/n+1,-b*x^n/a]",
     "rulenumber": 0,
            "lhs": "Int[(a_+b_.*x_^n_)^p_,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,n,p},x] && Not[IGtQ[p,0]] && Not[IntegerQ[1/n]] &&  Not[ILtQ[Simplify[1/n+p],0]] && Not[IntegerQ[p] || GtQ[a,0]] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "a^IntPart[p]*(a + b*x^n)^FracPart[p]/(1 + b*x^n/a)^FracPart[p]* Int[(1 + b*x^n/a)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n, p}, x] && Not[IGtQ[p, 0]] && Not[IntegerQ[1/n]] && Not[ILtQ[Simplify[1/n + p], 0]] && Not[IntegerQ[p] || GtQ[a, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "1/Coefficient[v, x, 1]*Subst[Int[(a + b*x^n)^p, x], x, v]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*v_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && NeQ[v, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "Int[(a1*a2 + b1*b2*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a1_. + b1_.*x_^n_)^p_.*(a2_. + b2_.*x_^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[p] || GtQ[a1, 0] && GtQ[a2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "x*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p/(2*n*p + 1) + 2*a1*a2*n*p/(2*n*p + 1)* Int[(a1 + b1*x^n)^(p - 1)*(a2 + b2*x^n)^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a1_ + b1_.*x_^n_.)^p_.*(a2_ + b2_.*x_^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || Denominator[p + 1/n] < Denominator[p])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "-x*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1)/(2* a1*a2*n*(p + 1)) + (2*n*(p + 1) + 1)/(2*a1*a2*n*(p + 1))* Int[(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a1_ + b1_.*x_^n_.)^p_*(a2_ + b2_.*x_^n_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && LtQ[p, -1] && (IntegerQ[2*p] || Denominator[p + 1/n] < Denominator[p])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "-Subst[ Int[(a1 + b1*x^(-n))^p*(a2 + b2*x^(-n))^p/x^2, x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, p}, x] && EqQ[a2*b1 + a1*b2, 0] && ILtQ[2*n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "With[{k = Denominator[2*n]}, k*Subst[ Int[x^(k - 1)*(a1 + b1*x^(k*n))^p*(a2 + b2*x^(k*n))^p, x], x, x^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, p}, x] && EqQ[a2*b1 + a1*b2, 0] && FractionQ[2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "(a1 + b1*x^n)^ FracPart[p]*(a2 + b2*x^n)^FracPart[p]/(a1*a2 + b1*b2*x^(2*n))^ FracPart[p]*Int[(a1*a2 + b1*b2*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a1_. + b1_.*x_^n_)^p_*(a2_. + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "x/(c*x^q)^(1/q)*Subst[Int[(a + b*x^(n*q))^p, x], x, (c*x^q)^(1/q)]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*(c_.*x_^q_.)^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, p, q}, x] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "With[{k = Denominator[n]}, Subst[Int[(a + b*c^n*x^(n*q))^p, x], x^(1/k), (c*x^q)^(1/k)/(c^(1/k)*(x^(1/k))^(q - 1))]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*(c_.*x_^q_.)^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p, q}, x] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "Subst[Int[(a + b*c^n*x^(n*q))^p, x], x^(n*q), (c*x^q)^n/c^n]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*(c_.*x_^q_.)^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, p, q}, x] && Not[RationalQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "-Subst[Int[(a + b*(d*x^(-q))^n)^p/x^2, x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*(d_.*x_^q_.)^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, n, p}, x] && ILtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": "With[{s = Denominator[q]}, s*Subst[Int[x^(s - 1)*(a + b*(d*x^(q*s))^n)^p, x], x, x^(1/s)]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*(d_.*x_^q_.)^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, n, p}, x] && FractionQ[q]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.1 (a+b x^n)^p.m",
       "filename": "1.1.3.1 (a+b x^n)^p.m",
            "rhs": " Subst[Int[(a+b*x^(n*q))^p,x],x^(n*q),(d*x^q)^n]",
     "rulenumber": 0,
            "lhs": "Int[(a_+b_.*(d_.*x_^q_.)^n_)^p_.,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,d,n,p,q},x] && Not[IntegerQ[n*q]] &&  NeQ[x^(n*q),(d*x^q)^n] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "Int[(c*x)^m*(a1*a2 + b1*b2*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[p] || GtQ[a1, 0] && GtQ[a2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "Log[RemoveContent[a + b*x^n, x]]/(b*n)",
     "rulenumber": 0,
            "lhs": "Int[x_^m_./(a_ + b_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && EqQ[m, n - 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "(a + b*x^n)^(p + 1)/(b*n*(p + 1))",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1)/(2*b1* b2*n*(p + 1))",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a1_ + b1_.*x_^n_.)^p_*(a2_ + b2_.*x_^n_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && EqQ[m, 2*n - 1] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "Int[x^(m + n*p)*(b + a*x^(-n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && IntegerQ[p] && NegQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "(c*x)^(m + 1)*(a + b*x^n)^(p + 1)/(a*c*(m + 1))",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "(c*x)^(m + 1)*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1)/(a1*a2*c*(m + 1))",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && EqQ[(m + 1)/(2*n) + p + 1, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "1/n*Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(a1 + b1*x)^p*(a2 + b2*x)^p, x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IntegerQ[Simplify[(m + 1)/(2*n)]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "c^IntPart[m]*(c*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_*x_)^m_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "c^IntPart[m]*(c*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_*x_)^m_*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IntegerQ[Simplify[(m + 1)/(2*n)]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*(a_ + b_.*x_^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "x^(m + 1)*(a + b*x^n)^(p + 1)/(a*(m + 1)) - b*(m + n*(p + 1) + 1)/(a*(m + 1))* Int[x^(m + n)*(a + b*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n, p}, x] && ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "x^(m + 1)*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1)/(a1* a2*(m + 1)) - b1*b2*(m + 2*n*(p + 1) + 1)/(a1*a2*(m + 1))* Int[x^(m + 2*n)*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && ILtQ[Simplify[(m + 1)/(2*n) + p + 1], 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "-(c*x)^(m + 1)*(a + b*x^n)^(p + 1)/(a*c* n*(p + 1)) + (m + n*(p + 1) + 1)/(a*n*(p + 1))* Int[(c*x)^m*(a + b*x^n)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n, p}, x] && ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "-(c*x)^(m + 1)*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1)/(2*a1*a2*c*n*(p + 1)) + (m + 2*n*(p + 1) + 1)/(2*a1*a2*n*(p + 1))* Int[(c*x)^m*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && ILtQ[Simplify[(m + 1)/(2*n) + p + 1], 0] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{k = GCD[m + 1, n]}, 1/k*Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k] /; k != 1]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{k = GCD[m + 1, 2*n]}, 1/k*Subst[ Int[x^((m + 1)/k - 1)*(a1 + b1*x^(n/k))^p*(a2 + b2*x^(n/k))^p, x], x, x^k] /; k != 1]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "(c*x)^(m + 1)*(a + b*x^n)^p/(c*(m + 1)) - b*n*p/(c^n*(m + 1))*Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] && Not[ILtQ[(m + n*p + n + 1)/n, 0]] && IntBinomialQ[a, b, c, n, m, p, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "(c*x)^(m + 1)*(a1 + b1*x^n)^ p*(a2 + b2*x^n)^p/(c*(m + 1)) - 2*b1*b2*n*p/(c^(2*n)*(m + 1))* Int[(c*x)^(m + 2*n)*(a1 + b1*x^n)^(p - 1)*(a2 + b2*x^n)^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, m}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && GtQ[p, 0] && LtQ[m, -1] && NeQ[m + 2*n*p + 1, 0] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "(c*x)^(m + 1)*(a + b*x^n)^p/(c*(m + n*p + 1)) + a*n*p/(m + n*p + 1)*Int[(c*x)^m*(a + b*x^n)^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m}, x] && IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "(c*x)^(m + 1)*(a1 + b1*x^n)^ p*(a2 + b2*x^n)^p/(c*(m + 2*n*p + 1)) + 2*a1*a2*n*p/(m + 2*n*p + 1)* Int[(c*x)^m*(a1 + b1*x^n)^(p - 1)*(a2 + b2*x^n)^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, m}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && GtQ[p, 0] && NeQ[m + 2*n*p + 1, 0] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "x*(1 + a/(b*x^4))^(1/4)/(b*(a + b*x^4)^(1/4))* Int[1/(x^3*(1 + a/(b*x^4))^(5/4)), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^2/(a_ + b_.*x_^4)^(5/4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PosQ[b/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "x^(m - 3)/(b*(m - 4)*(a + b*x^4)^(1/4)) - a*(m - 3)/(b*(m - 4))*Int[x^(m - 4)/(a + b*x^4)^(5/4), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_/(a_ + b_.*x_^4)^(5/4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PosQ[b/a] && IGtQ[(m - 2)/4, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "x^(m + 1)/(a*(m + 1)*(a + b*x^4)^(1/4)) - b*m/(a*(m + 1))*Int[x^(m + 4)/(a + b*x^4)^(5/4), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_/(a_ + b_.*x_^4)^(5/4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PosQ[b/a] && ILtQ[(m - 2)/4, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "Sqrt[c*x]*(1 + a/(b*x^2))^(1/4)/(b*(a + b*x^2)^(1/4))* Int[1/(x^2*(1 + a/(b*x^2))^(5/4)), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[c_.*x_]/(a_ + b_.*x_^2)^(5/4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && PosQ[b/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "2*c*(c*x)^(m - 1)/(b*(2*m - 3)*(a + b*x^2)^(1/4)) - 2*a*c^2*(m - 1)/(b*(2*m - 3))* Int[(c*x)^(m - 2)/(a + b*x^2)^(5/4), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_/(a_ + b_.*x_^2)^(5/4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && PosQ[b/a] && IntegerQ[2*m] && GtQ[m, 3/2] "
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "(c*x)^(m + 1)/(a*c*(m + 1)*(a + b*x^2)^(1/4)) - b*(2*m + 1)/(2*a*c^2*(m + 1))* Int[(c*x)^(m + 2)/(a + b*x^2)^(5/4), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_/(a_ + b_.*x_^2)^(5/4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && PosQ[b/a] && IntegerQ[2*m] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "-1/(b*x*(a + b*x^4)^(1/4)) - 1/b*Int[1/(x^2*(a + b*x^4)^(1/4)), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^2/(a_ + b_.*x_^4)^(5/4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && NegQ[b/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n)^(p + 1)/(b*n*(p + 1)) - c^n*(m - n + 1)/(b*n*(p + 1))* Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && Not[ILtQ[(m + n*(p + 1) + 1)/n, 0]] && IntBinomialQ[a, b, c, n, m, p, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": " With[{a=BinomialParts[u,x][[1]],b=BinomialParts[u,x][[2]],n= BinomialParts[u,x][[3]]}, c^(n-1)*(c*x)^(m-n+1)*u^(p+1)*v^(p+1)/(b*n*(p+1)) - c^n*(m-n+1)/(b*n*(p+1))*Int[(c*x)^(m-n)*u^(p+1)*v^(p+1),x] /;  IGtQ[n,0] && m+1>n && Not[ILtQ[(m+n*(p+1)+1)/n,0]] && IntBinomialQ[a,b,c,n,m,p,x]]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*u_^p_*v_^p_,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[c,x] && BinomialQ[u*v,x] &&  LtQ[p,-1] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "c^(2*n - 1)*(c*x)^(m - 2*n + 1)*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1)/(2*b1*b2*n*(p + 1)) - c^(2*n)*(m - 2*n + 1)/(2*b1*b2*n*(p + 1))* Int[(c*x)^(m - 2*n)*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && LtQ[p, -1] && m + 1 > 2*n && Not[ILtQ[(m + 2*n*(p + 1) + 1)/(2*n), 0]] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "-(c*x)^(m + 1)*(a + b*x^n)^(p + 1)/(a*c* n*(p + 1)) + (m + n*(p + 1) + 1)/(a*n*(p + 1))* Int[(c*x)^m*(a + b*x^n)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "-(c*x)^(m + 1)*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1)/(2*a1*a2*c*n*(p + 1)) + (m + 2*n*(p + 1) + 1)/(2*a1*a2*n*(p + 1))* Int[(c*x)^m*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, m}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && LtQ[p, -1] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "-1/(3*Rt[a, 3]*Rt[b, 3])* Int[1/(Rt[a, 3] + Rt[b, 3]*x), x] + 1/(3*Rt[a, 3]*Rt[b, 3])* Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[x_/(a_ + b_.*x_^3), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": " With[{r=Numerator[Rt[a/b,5]], s=Denominator[Rt[a/b,5]]}, (-1)^m*r^(m+1)/(5*a*s^m)*Int[1/(r+s*x),x] + 2*r^(m+1)/(5*a*s^m)*Int[(r*Cos[m*Pi/5]-s*Cos[(m+1)*Pi/5]*x)/(r^2-1/ 2*(1+Sqrt[5])*r*s*x+s^2*x^2),x] + 2*r^(m+1)/(5*a*s^m)*Int[(r*Cos[3*m*Pi/5]-s*Cos[3*(m+1)*Pi/5]*x)/(r^ 2-1/2*(1-Sqrt[5])*r*s*x+s^2*x^2),x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_./(a_+b_.*x_^5),x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b},x] && IGtQ[m,0] &&  LtQ[m,4] && PosQ[a/b] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": " With[{r=Numerator[Rt[-a/b,5]], s=Denominator[Rt[-a/b,5]]}, (r^(m+1)/(5*a*s^m))*Int[1/(r-s*x),x] + 2*(-1)^m*r^(m+1)/(5*a*s^m)*Int[(r*Cos[m*Pi/5]+s*Cos[(m+1)*Pi/5]*x)/( r^2+1/2*(1+Sqrt[5])*r*s*x+s^2*x^2),x] + 2*(-1)^m*r^(m+1)/(5*a*s^m)*Int[(r*Cos[3*m*Pi/5]+s*Cos[3*(m+1)*Pi/5]* x)/(r^2+1/2*(1-Sqrt[5])*r*s*x+s^2*x^2),x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_./(a_+b_.*x_^5),x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b},x] &&  IGtQ[m,0] && LtQ[m,4] && NegQ[a/b] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/b, n]], k, u}, u = Int[(r*Cos[(2*k - 1)*m*Pi/n] - s*Cos[(2*k - 1)*(m + 1)*Pi/n]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*Pi/n]*x + s^2*x^2), x]; -(-r)^(m + 1)/(a*n*s^m)*Int[1/(r + s*x), x] + Dist[2*r^(m + 1)/(a*n*s^m), Sum[u, {k, 1, (n - 1)/2}], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_./(a_ + b_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IGtQ[(n - 1)/2, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "Module[{r = Numerator[Rt[-a/b, n]], s = Denominator[Rt[-a/b, n]], k, u}, u = Int[(r*Cos[(2*k - 1)*m*Pi/n] + s*Cos[(2*k - 1)*(m + 1)*Pi/n]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*Pi/n]*x + s^2*x^2), x]; r^(m + 1)/(a*n*s^m)*Int[1/(r - s*x), x] - Dist[2*(-r)^(m + 1)/(a*n*s^m), Sum[u, {k, 1, (n - 1)/2}], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_./(a_ + b_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IGtQ[(n - 1)/2, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && NegQ[a/b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/b, n]], k, u}, u = Int[(r*Cos[(2*k - 1)*m*Pi/n] - s*Cos[(2*k - 1)*(m + 1)*Pi/n]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*Pi/n]*x + s^2*x^2), x] + Int[(r*Cos[(2*k - 1)*m*Pi/n] + s*Cos[(2*k - 1)*(m + 1)*Pi/n]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*Pi/n]*x + s^2*x^2), x]; 2*(-1)^(m/2)*r^(m + 2)/(a*n*s^m)*Int[1/(r^2 + s^2*x^2), x] + Dist[2*r^(m + 1)/(a*n*s^m), Sum[u, {k, 1, (n - 2)/4}], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_./(a_ + b_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "Module[{r = Numerator[Rt[-a/b, n]], s = Denominator[Rt[-a/b, n]], k, u}, u = Int[(r*Cos[2*k*m*Pi/n] - s*Cos[2*k*(m + 1)*Pi/n]*x)/(r^2 - 2*r*s*Cos[2*k*Pi/n]*x + s^2*x^2), x] + Int[(r*Cos[2*k*m*Pi/n] + s*Cos[2*k*(m + 1)*Pi/n]*x)/(r^2 + 2*r*s*Cos[2*k*Pi/n]*x + s^2*x^2), x]; 2*r^(m + 2)/(a*n*s^m)*Int[1/(r^2 - s^2*x^2), x] + Dist[2*r^(m + 1)/(a*n*s^m), Sum[u, {k, 1, (n - 2)/4}], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_./(a_ + b_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && NegQ[a/b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, 1/(2*s)*Int[(r + s*x^2)/(a + b*x^4), x] - 1/(2*s)*Int[(r - s*x^2)/(a + b*x^4), x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^2/(a_ + b_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && (GtQ[a/b, 0] || PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, s/(2*b)*Int[1/(r + s*x^2), x] - s/(2*b)*Int[1/(r - s*x^2), x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^2/(a_ + b_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && Not[GtQ[a/b, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{r = Numerator[Rt[a/b, 4]], s = Denominator[Rt[a/b, 4]]}, s^3/(2*Sqrt[2]*b*r)* Int[x^(m - n/4)/(r^2 - Sqrt[2]*r*s*x^(n/4) + s^2*x^(n/2)), x] - s^3/(2*Sqrt[2]*b*r)* Int[x^(m - n/4)/(r^2 + Sqrt[2]*r*s*x^(n/4) + s^2*x^(n/2)), x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_./(a_ + b_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IGtQ[n/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && GtQ[a/b, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, r/(2*a)*Int[x^m/(r + s*x^(n/2)), x] + r/(2*a)*Int[x^m/(r - s*x^(n/2)), x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_/(a_ + b_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IGtQ[n/4, 0] && IGtQ[m, 0] && LtQ[m, n/2] && Not[GtQ[a/b, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, s/(2*b)*Int[x^(m - n/2)/(r + s*x^(n/2)), x] - s/(2*b)*Int[x^(m - n/2)/(r - s*x^(n/2)), x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_/(a_ + b_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IGtQ[n/4, 0] && IGtQ[m, 0] && LeQ[n/2, m] && LtQ[m, n] && Not[GtQ[a/b, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "Int[PolynomialDivide[x^m, (a + b*x^n), x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_/(a_ + b_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Sqrt[2]*s/(Sqrt[2 + Sqrt[3]]*r)*Int[1/Sqrt[a + b*x^3], x] + 1/r*Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_/Sqrt[a_ + b_.*x_^3], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PosQ[a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, -Sqrt[2]*s/(Sqrt[2 - Sqrt[3]]*r)*Int[1/Sqrt[a + b*x^3], x] + 1/r*Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_/Sqrt[a_ + b_.*x_^3], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && NegQ[a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{q = Rt[b/a, 2]}, 1/q*Int[1/Sqrt[a + b*x^4], x] - 1/q*Int[(1 - q*x^2)/Sqrt[a + b*x^4], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^2/Sqrt[a_ + b_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PosQ[b/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{q = Rt[-b/a, 2]}, 1/q*Int[1/Sqrt[a + b*x^4], x] - 1/q*Int[(1 - q*x^2)/Sqrt[a + b*x^4], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^2/Sqrt[a_ + b_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && LtQ[a, 0] && GtQ[b, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{q = Rt[-b/a, 2]}, -1/q*Int[1/Sqrt[a + b*x^4], x] + 1/q*Int[(1 + q*x^2)/Sqrt[a + b*x^4], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^2/Sqrt[a_ + b_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && NegQ[b/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, (Sqrt[3] - 1)*s^2/(2*r^2)*Int[1/Sqrt[a + b*x^6], x] - 1/(2*r^2)* Int[((Sqrt[3] - 1)*s^2 - 2*r^2*x^4)/Sqrt[a + b*x^6], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^4/Sqrt[a_ + b_.*x_^6], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": " With[{r=Numer[Rt[b/a,3]], s=Denom[Rt[b/a,3]]}, (1+Sqrt[3])*r*x*Sqrt[a+b*x^6]/(2*b*(s+(1+Sqrt[3])*r*x^2)) - 3^(1/4)*s*x*(s+r*x^2)*Sqrt[(s^2-r*s*x^2+r^2*x^4)/(s+(1+Sqrt[3])*r*x^ 2)^2]/ (2*r^2*Sqrt[a+b*x^6]*Sqrt[r*x^2*(s+r*x^2)/(s+(1+Sqrt[3])*r*x^2)^2] )* EllipticE[ArcCos[(s+(1-Sqrt[3])*r*x^2)/(s+(1+Sqrt[3])*r*x^2)],(2+ Sqrt[3])/4] - (1-Sqrt[3])*s*x*(s+r*x^2)*Sqrt[(s^2-r*s*x^2+r^2*x^4)/(s+(1+Sqrt[3])* r*x^2)^2]/ (4*3^(1/4)*r^2*Sqrt[a+b*x^6]*Sqrt[r*x^2*(s+r*x^2)/(s+(1+Sqrt[3])* r*x^2)^2])* EllipticF[ArcCos[(s+(1-Sqrt[3])*r*x^2)/(s+(1+Sqrt[3])*r*x^2)],(2+ Sqrt[3])/4]]",
     "rulenumber": 0,
            "lhs": "Int[x_^4/Sqrt[a_+b_.*x_^6],x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b},x] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "1/(2*Rt[b/a, 4])*Int[(1 + Rt[b/a, 4]*x^2)/Sqrt[a + b*x^8], x] - 1/(2*Rt[b/a, 4])*Int[(1 - Rt[b/a, 4]*x^2)/Sqrt[a + b*x^8], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^2/Sqrt[a_ + b_.*x_^8], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "x^3/(2*(a + b*x^4)^(1/4)) - a/2*Int[x^2/(a + b*x^4)^(5/4), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^2/(a_ + b_.*x_^4)^(1/4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PosQ[b/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "(a + b*x^4)^(3/4)/(2*b*x) + a/(2*b)*Int[1/(x^2*(a + b*x^4)^(1/4)), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^2/(a_ + b_.*x_^4)^(1/4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && NegQ[b/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "-1/(x*(a + b*x^4)^(1/4)) - b*Int[x^2/(a + b*x^4)^(5/4), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(x_^2*(a_ + b_.*x_^4)^(1/4)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PosQ[b/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "x*(1 + a/(b*x^4))^(1/4)/(a + b*x^4)^(1/4)* Int[1/(x^3*(1 + a/(b*x^4))^(1/4)), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(x_^2*(a_ + b_.*x_^4)^(1/4)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && NegQ[b/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "x*Sqrt[c*x]/(a + b*x^2)^(1/4) - a/2*Int[Sqrt[c*x]/(a + b*x^2)^(5/4), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[c_*x_]/(a_ + b_.*x_^2)^(1/4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && PosQ[b/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "c*(a + b*x^2)^(3/4)/(b*Sqrt[c*x]) + a*c^2/(2*b)*Int[1/((c*x)^(3/2)*(a + b*x^2)^(1/4)), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[c_*x_]/(a_ + b_.*x_^2)^(1/4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NegQ[b/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "-2/(c*Sqrt[c*x]*(a + b*x^2)^(1/4)) - b/c^2*Int[Sqrt[c*x]/(a + b*x^2)^(5/4), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((c_.*x_)^(3/2)*(a_ + b_.*x_^2)^(1/4)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && PosQ[b/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "Sqrt[c*x]*(1 + a/(b*x^2))^(1/4)/(c^2*(a + b*x^2)^(1/4))* Int[1/(x^2*(1 + a/(b*x^2))^(1/4)), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((c_.*x_)^(3/2)*(a_ + b_.*x_^2)^(1/4)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NegQ[b/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "-2/(Sqrt[a]*(-b/a)^(3/4))* Subst[Int[Sqrt[1 - 2*x^2]/Sqrt[1 - x^2], x], x, Sqrt[1 - Sqrt[-b/a]*x]/Sqrt[2]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[x_]/Sqrt[a_ + b_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && GtQ[-b/a, 0] && GtQ[a, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "Sqrt[1 + b*x^2/a]/Sqrt[a + b*x^2]*Int[Sqrt[x]/Sqrt[1 + b*x^2/a], x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[x_]/Sqrt[a_ + b_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && GtQ[-b/a, 0] && Not[GtQ[a, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "Sqrt[c*x]/Sqrt[x]*Int[Sqrt[x]/Sqrt[a + b*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[c_*x_]/Sqrt[a_ + b_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && GtQ[-b/a, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n)^(p + 1)/(b*(m + n*p + 1)) - a*c^n*(m - n + 1)/(b*(m + n*p + 1))* Int[(c*x)^(m - n)*(a + b*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n)^(p + 1)/(b*(m + n*p + 1)) - a*c^n*(m - n + 1)/(b*(m + n*p + 1))* Int[(c*x)^(m - n)*(a + b*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, p}, x] && IGtQ[n, 0] && SumSimplerQ[m, -n] && NeQ[m + n*p + 1, 0] && ILtQ[Simplify[(m + 1)/n + p], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "c^(2*n - 1)*(c*x)^(m - 2*n + 1)*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1)/(b1*b2*(m + 2*n*p + 1)) - a1*a2*c^(2*n)*(m - 2*n + 1)/(b1*b2*(m + 2*n*p + 1))* Int[(c*x)^(m - 2*n)*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && GtQ[m, 2*n - 1] && NeQ[m + 2*n*p + 1, 0] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "c^(2*n - 1)*(c*x)^(m - 2*n + 1)*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1)/(b1*b2*(m + 2*n*p + 1)) - a1*a2*c^(2*n)*(m - 2*n + 1)/(b1*b2*(m + 2*n*p + 1))* Int[(c*x)^(m - 2*n)*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, m, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && SumSimplerQ[m, -2*n] && NeQ[m + 2*n*p + 1, 0] && ILtQ[Simplify[(m + 1)/(2*n) + p], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "(c*x)^(m + 1)*(a + b*x^n)^(p + 1)/(a*c*(m + 1)) - b*(m + n*(p + 1) + 1)/(a*c^n*(m + 1))* Int[(c*x)^(m + n)*(a + b*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "(c*x)^(m + 1)*(a + b*x^n)^(p + 1)/(a*c*(m + 1)) - b*(m + n*(p + 1) + 1)/(a*c^n*(m + 1))* Int[(c*x)^(m + n)*(a + b*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, p}, x] && IGtQ[n, 0] && SumSimplerQ[m, n] && ILtQ[Simplify[(m + 1)/n + p], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "(c*x)^(m + 1)*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1)/(a1*a2*c*(m + 1)) - b1*b2*(m + 2*n*(p + 1) + 1)/(a1*a2*c^(2*n)*(m + 1))* Int[(c*x)^(m + 2*n)*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && LtQ[m, -1] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "(c*x)^(m + 1)*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1)/(a1*a2*c*(m + 1)) - b1*b2*(m + 2*n*(p + 1) + 1)/(a1*a2*c^(2*n)*(m + 1))* Int[(c*x)^(m + 2*n)*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, m, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && SumSimplerQ[m, 2*n] && ILtQ[Simplify[(m + 1)/(2*n) + p], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{k = Denominator[m]}, k/c*Subst[Int[x^(k*(m + 1) - 1)*(a + b*x^(k*n)/c^n)^p, x], x, (c*x)^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{k = Denominator[m]}, k/c*Subst[ Int[x^(k*(m + 1) - 1)*(a1 + b1*x^(k*n)/c^n)^ p*(a2 + b2*x^(k*n)/c^n)^p, x], x, (c*x)^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && FractionQ[m] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "a^(p + (m + 1)/n)* Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -1/2] && IntegersQ[m, p + (m + 1)/n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "(a1*a2)^(p + (m + 1)/(2*n))* Subst[ Int[x^m/((1 - b1*x^n)^(p + (m + 1)/(2*n) + 1)*(1 - b2*x^n)^(p + (m + 1)/(2*n) + 1)), x], x, x/((a1 + b1*x^n)^(1/(2*n))*(a2 + b2*x^n)^(1/(2*n)))]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && LtQ[-1, p, 0] && NeQ[p, -1/2] && IntegersQ[m, p + (m + 1)/(2*n)]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "(a/(a + b*x^n))^(p + (m + 1)/n)*(a + b*x^n)^(p + (m + 1)/n)* Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -1/2] && IntegerQ[m] && LtQ[Denominator[p + (m + 1)/n], Denominator[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "(a1/(a1 + b1*x^n))^(p + (m + 1)/(2*n))*(a1 + b1*x^n)^(p + (m + 1)/(2*n))*(a2/(a2 + b2*x^n))^(p + (m + 1)/(2*n))*(a2 + b2*x^n)^(p + (m + 1)/(2*n))* Subst[ Int[x^m/((1 - b1*x^n)^(p + (m + 1)/(2*n) + 1)*(1 - b2*x^n)^(p + (m + 1)/(2*n) + 1)), x], x, x/((a1 + b1*x^n)^(1/(2*n))*(a2 + b2*x^n)^(1/(2*n)))]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && LtQ[-1, p, 0] && NeQ[p, -1/2] && IntegerQ[m] && LtQ[Denominator[p + (m + 1)/(2*n)], Denominator[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "-Subst[Int[(a + b*x^(-n))^p/x^(m + 2), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "-Subst[ Int[(a1 + b1*x^(-n))^p*(a2 + b2*x^(-n))^p/x^(m + 2), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, p}, x] && EqQ[a2*b1 + a1*b2, 0] && ILtQ[2*n, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{k = Denominator[m]}, -k/c* Subst[Int[(a + b*c^(-n)*x^(-k*n))^p/x^(k*(m + 1) + 1), x], x, 1/(c*x)^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && ILtQ[n, 0] && FractionQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{k = Denominator[m]}, -k/c* Subst[Int[(a1 + b1*c^(-n)*x^(-k*n))^p*(a2 + b2*c^(-n)*x^(-k*n))^p/ x^(k*(m + 1) + 1), x], x, 1/(c*x)^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, p}, x] && EqQ[a2*b1 + a1*b2, 0] && ILtQ[2*n, 0] && FractionQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "-1/c*(c*x)^(m + 1)*(1/x)^(m + 1)* Subst[Int[(a + b*x^(-n))^p/x^(m + 2), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, p}, x] && ILtQ[n, 0] && Not[RationalQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "-1/c*(c*x)^(m + 1)*(1/x)^(m + 1)* Subst[Int[(a1 + b1*x^(-n))^p*(a2 + b2*x^(-n))^p/x^(m + 2), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, m, p}, x] && EqQ[a2*b1 + a1*b2, 0] && ILtQ[2*n, 0] && Not[RationalQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{k = Denominator[n]}, k*Subst[Int[x^(k*(m + 1) - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, p}, x] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{k = Denominator[2*n]}, k*Subst[ Int[x^(k*(m + 1) - 1)*(a1 + b1*x^(k*n))^p*(a2 + b2*x^(k*n))^p, x], x, x^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, m, p}, x] && EqQ[a2*b1 + a1*b2, 0] && FractionQ[2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "c^IntPart[m]*(c*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_*x_)^m_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, p}, x] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "c^IntPart[m]*(c*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_*x_)^m_*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, m, p}, x] && EqQ[a2*b1 + a1*b2, 0] && FractionQ[2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "1/(m + 1)* Subst[Int[(a + b*x^Simplify[n/(m + 1)])^p, x], x, x^(m + 1)]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[n/(m + 1)]] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "1/(m + 1)* Subst[Int[(a1 + b1*x^Simplify[n/(m + 1)])^ p*(a2 + b2*x^Simplify[n/(m + 1)])^p, x], x, x^(m + 1)]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IntegerQ[Simplify[2*n/(m + 1)]] && Not[IntegerQ[2*n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "c^IntPart[m]*(c*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_*x_)^m_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n, p}, x] && IntegerQ[Simplify[n/(m + 1)]] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "c^IntPart[m]*(c*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_*x_)^m_*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IntegerQ[Simplify[2*n/(m + 1)]] && Not[IntegerQ[2*n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "x^(m + 1)*(a + b*x^n)^p/(m + 1) - b*n*p/(m + 1)*Int[x^(m + n)*(a + b*x^n)^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && EqQ[(m + 1)/n + p, 0] && GtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "x^(m + 1)*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p/(m + 1) - 2*b1*b2*n*p/(m + 1)* Int[x^(m + 2*n)*(a1 + b1*x^n)^(p - 1)*(a2 + b2*x^n)^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, m, n}, x] && EqQ[a2*b1 + a1*b2, 0] && EqQ[(m + 1)/(2*n) + p, 0] && GtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "c^IntPart[m]*(c*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_*x_)^m_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n}, x] && EqQ[(m + 1)/n + p, 0] && GtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "c^IntPart[m]*(c*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_*x_)^m_*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, m, n}, x] && EqQ[a2*b1 + a1*b2, 0] && EqQ[(m + 1)/(2*n) + p, 0] && GtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "(c*x)^(m + 1)*(a + b*x^n)^p/(c*(m + n*p + 1)) + a*n*p/(m + n*p + 1)*Int[(c*x)^m*(a + b*x^n)^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n}, x] && IntegerQ[p + Simplify[(m + 1)/n]] && GtQ[p, 0] && NeQ[m + n*p + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "(c*x)^(m + 1)*(a1 + b1*x^n)^ p*(a2 + b2*x^n)^p/(c*(m + 2*n*p + 1)) + 2*a1*a2*n*p/(m + 2*n*p + 1)* Int[(c*x)^m*(a1 + b1*x^n)^(p - 1)*(a2 + b2*x^n)^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, m, n}, x] && EqQ[a2*b1 + a1*b2, 0] && IntegerQ[p + Simplify[(m + 1)/(2*n)]] && GtQ[p, 0] && NeQ[m + 2*n*p + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{k = Denominator[p]}, k*a^(p + Simplify[(m + 1)/n])/n* Subst[ Int[x^(k*Simplify[(m + 1)/n] - 1)/(1 - b*x^k)^(p + Simplify[(m + 1)/n] + 1), x], x, x^(n/k)/(a + b*x^n)^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && IntegerQ[p + Simplify[(m + 1)/n]] && LtQ[-1, p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{k = Denominator[p]}, k*(a1*a2)^(p + Simplify[(m + 1)/(2*n)])/(2*n)* Subst[ Int[x^(k*Simplify[(m + 1)/(2*n)] - 1)/(1 - b1*b2*x^k)^(p + Simplify[(m + 1)/(2*n)] + 1), x], x, x^(2*n/k)/((a1 + b1*x^n)^(1/k)*(a2 + b2*x^n)^(1/k))]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, m, n}, x] && EqQ[a2*b1 + a1*b2, 0] && IntegerQ[p + Simplify[(m + 1)/(2*n)]] && LtQ[-1, p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "c^IntPart[m]*(c*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_*x_)^m_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n}, x] && IntegerQ[p + Simplify[(m + 1)/n]] && LtQ[-1, p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "c^IntPart[m]*(c*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_*x_)^m_*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, m, n}, x] && EqQ[a2*b1 + a1*b2, 0] && IntegerQ[p + Simplify[(m + 1)/(2*n)]] && LtQ[-1, p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "-(c*x)^(m + 1)*(a + b*x^n)^(p + 1)/(a*c* n*(p + 1)) + (m + n*(p + 1) + 1)/(a*n*(p + 1))* Int[(c*x)^m*(a + b*x^n)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n}, x] && IntegerQ[p + Simplify[(m + 1)/n]] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "-(c*x)^(m + 1)*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1)/(2*a1*a2*c*n*(p + 1)) + (m + 2*n*(p + 1) + 1)/(2*a1*a2*n*(p + 1))* Int[(c*x)^m*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, m, n}, x] && EqQ[a2*b1 + a1*b2, 0] && IntegerQ[p + Simplify[(m + 1)/(2*n)]] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{mn = Simplify[m - n]}, x^(mn + 1)/(b*(mn + 1)) - a/b*Int[x^mn/(a + b*x^n), x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_./(a_ + b_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && FractionQ[Simplify[(m + 1)/n]] && SumSimplerQ[m, -n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "x^(m + 1)/(a*(m + 1)) - b/a*Int[x^Simplify[m + n]/(a + b*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_/(a_ + b_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && FractionQ[Simplify[(m + 1)/n]] && SumSimplerQ[m, n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "c^IntPart[m]*(c*x)^FracPart[m]/x^FracPart[m]* Int[x^m/(a + b*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_*x_)^m_/(a_ + b_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n}, x] && FractionQ[ Simplify[(m + 1)/n]] && (SumSimplerQ[m, n] || SumSimplerQ[m, -n])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "a^p*(c*x)^(m + 1)/(c*(m + 1))* Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, -b*x^n/a]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n, p}, x] && Not[IGtQ[p, 0]] && (ILtQ[p, 0] || GtQ[a, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": " (c*x)^(m+1)*(a+b*x^n)^(p+1)/(a*c*(m+1))*Hypergeometric2F1[1,(m+1)/n+p+ 1,(m+1)/n+1,-b*x^n/a]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*(a_+b_.*x_^n_)^p_,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,m,n,p},x] && Not[IGtQ[p,0]] &&  Not[ILtQ[p,0] || GtQ[a,0]] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "a^IntPart[p]*(a + b*x^n)^FracPart[p]/(1 + b*x^n/a)^FracPart[p]* Int[(c*x)^m*(1 + b*x^n/a)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n, p}, x] && Not[IGtQ[p, 0]] && Not[ILtQ[p, 0] || GtQ[a, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "(a1 + b1*x^n)^ FracPart[p]*(a2 + b2*x^n)^FracPart[p]/(a1*a2 + b1*b2*x^(2*n))^ FracPart[p]*Int[(c*x)^m*(a1*a2 + b1*b2*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*(a1_ + b1_.*x_^n_)^p_*(a2_ + b2_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "1/c*Subst[Int[(d*x/c)^m*(a + b*x^n)^p, x], x, c*x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_ + b_.*(c_*x_)^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "(d*x)^(m + 1)/(d*((c*x^q)^(1/q))^(m + 1))* Subst[Int[x^m*(a + b*x^(n*q))^p, x], x, (c*x^q)^(1/q)]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_ + b_.*(c_.*x_^q_)^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p, q}, x] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{k = Denominator[n]}, Subst[Int[(d*x)^m*(a + b*c^n*x^(n*q))^p, x], x^(1/k), (c*x^q)^(1/k)/(c^(1/k)*(x^(1/k))^(q - 1))]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_ + b_.*(c_.*x_^q_)^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, p, q}, x] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "Subst[Int[(d*x)^m*(a + b*c^n*x^(n*q))^p, x], x^(n*q), (c*x^q)^n/c^n]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_ + b_.*(c_.*x_^q_)^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p, q}, x] && Not[RationalQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{c = Coefficient[v, x, 0], d = Coefficient[v, x, 1]}, 1/d^(m + 1)* Subst[Int[SimplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v] /; NeQ[c, 0]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*v_^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.2 (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.2 (c x)^m (a+b x^n)^p.m",
            "rhs": "u^m/(Coefficient[v, x, 1]*v^m)* Subst[Int[x^m*(a + b*x^n)^p, x], x, v]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*(a_ + b_.*v_^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n, p}, x] && LinearPairQ[u, v, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "Int[ExpandIntegrand[(a + b*x^n)^p*(c + d*x^n)^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "Int[x^(n*(p + q))*(b + a*x^(-n))^p*(d + c*x^(-n))^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IntegersQ[p, q] && NegQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "-Subst[ Int[(a + b*x^(-n))^p*(c + d*x^(-n))^q/x^2, x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "With[{g = Denominator[n]}, g*Subst[Int[x^(g - 1)*(a + b*x^(g*n))^p*(c + d*x^(g*n))^q, x], x, x^(1/g)]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "Subst[Int[1/(c - (b*c - a*d)*x^n), x], x, x/(a + b*x^n)^(1/n)]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_/(c_ + d_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "-x*(a + b*x^n)^(p + 1)*(c + d*x^n)^q/(a*n*(p + 1)) - c*q/(a*(p + 1))*Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0] && GtQ[q, 0] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "a^p*x/(c^(p + 1)*(c + d*x^n)^(1/n))* Hypergeometric2F1[1/n, -p, 1 + 1/n, -(b*c - a*d)*x^n/(a*(c + d*x^n))]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0] && ILtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "x*(a + b*x^n)^ p/(c*(c*(a + b*x^n)/(a*(c + d*x^n)))^p*(c + d*x^n)^(1/n + p))* Hypergeometric2F1[1/n, -p, 1 + 1/n, -(b*c - a*d)*x^n/(a*(c + d*x^n))]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1)/(a*c)",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 2) + 1, 0] && EqQ[a*d*(p + 1) + b*c*(q + 1), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": " x*(a1+b1*x^(n/2))^(p+1)*(a2+b2*x^(n/2))^(p+1)*(c+d*x^n)^(q+1)/(a1*a2* c)",
     "rulenumber": 0,
            "lhs": "Int[(a1_+b1_.*x_^n2_.)^p_*(a2_+b2_.*x_^n2_.)^p_*(c_+d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1,b1,a2,b2,c,d,n,p,q},x] && EqQ[n2,n/2] &&  EqQ[a2*b1+a1*b2,0] && EqQ[n*(p+q+2)+1,0] &&  EqQ[a1*a2*d*(p+1)+b1*b2*c*(q+1),0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "-b* x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1)/(a* n*(p + 1)*(b*c - a*d)) + (b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a*d))* Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 2) + 1, 0] && (LtQ[p, -1] || Not[LtQ[q, -1]]) && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "c*x*(a + b*x^n)^(p + 1)/a",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[a*d - b*c*(n*(p + 1) + 1), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "c*x*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1)/(a1*a2)",
     "rulenumber": 0,
            "lhs": "Int[(a1_ + b1_.*x_^non2_.)^p_.*(a2_ + b2_.*x_^non2_.)^ p_.*(c_ + d_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, d, n, p}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && EqQ[a1*a2*d - b1*b2*c*(n*(p + 1) + 1), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "-(b*c - a*d)*x*(a + b*x^n)^(p + 1)/(a*b*n*(p + 1)) - (a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1))* Int[(a + b*x^n)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "-(b1*b2*c - a1*a2*d)* x*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1)/(a1*a2*b1* b2*n*(p + 1)) - (a1*a2*d - b1*b2*c*(n*(p + 1) + 1))/(a1*a2*b1*b2*n*(p + 1))* Int[(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a1_ + b1_.*x_^non2_.)^p_.*(a2_ + b2_.*x_^non2_.)^ p_.*(c_ + d_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, d, n}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "c*x/a - (b*c - a*d)/a*Int[1/(b + a*x^(-n)), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_ + d_.*x_^n_)/(a_ + b_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "d*x*(a + b*x^n)^(p + 1)/(b*(n*(p + 1) + 1)) - (a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1))* Int[(a + b*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "d*x*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1)/(b1* b2*(n*(p + 1) + 1)) - (a1*a2*d - b1*b2*c*(n*(p + 1) + 1))/(b1*b2*(n*(p + 1) + 1))* Int[(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a1_ + b1_.*x_^non2_.)^p_.*(a2_ + b2_.*x_^non2_.)^ p_.*(c_ + d_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, d, n, p}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[n*(p + 1) + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "Int[PolynomialDivide[(a + b*x^n)^p, (c + d*x^n)^(-q), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "b/(b*c - a*d)*Int[1/(a + b*x^n), x] - d/(b*c - a*d)*Int[1/(c + d*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_ + b_.*x_^n_)*(c_ + d_.*x_^n_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "With[{q = Rt[b/a, 2]}, q*ArcTanh[Sqrt[3]/(q*x)]/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d) + q*ArcTanh[ Sqrt[3]*(a^(1/3) - 2^(1/3)*(a + b*x^2)^(1/3))/(a^(1/3)*q* x)]/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d) + q*ArcTan[q*x]/(6*2^(2/3)*a^(1/3)*d) - q*ArcTan[(a^(1/3)*q*x)/(a^(1/3) + 2^(1/3)*(a + b*x^2)^(1/3))]/(2*2^(2/3)*a^(1/3)*d)]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_ + b_.*x_^2)^(1/3)*(c_ + d_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c + 3*a*d, 0] && PosQ[b/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "With[{q = Rt[-b/a, 2]}, q*ArcTan[Sqrt[3]/(q*x)]/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d) + q*ArcTan[ Sqrt[3]*(a^(1/3) - 2^(1/3)*(a + b*x^2)^(1/3))/(a^(1/3)*q* x)]/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d) - q*ArcTanh[q*x]/(6*2^(2/3)*a^(1/3)*d) + q*ArcTanh[(a^(1/3)*q*x)/(a^(1/3) + 2^(1/3)*(a + b*x^2)^(1/3))]/(2*2^(2/3)*a^(1/3)*d)]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_ + b_.*x_^2)^(1/3)*(c_ + d_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c + 3*a*d, 0] && NegQ[b/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "With[{q = Rt[b/a, 2]}, q*ArcTan[q*x/3]/(12*Rt[a, 3]*d) + q*ArcTan[(Rt[a, 3] - (a + b*x^2)^(1/3))^2/(3*Rt[a, 3]^2*q* x)]/(12*Rt[a, 3]*d) - q*ArcTanh[(Sqrt[3]*(Rt[a, 3] - (a + b*x^2)^(1/3)))/(Rt[a, 3]*q* x)]/(4*Sqrt[3]*Rt[a, 3]*d)]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_ + b_.*x_^2)^(1/3)*(c_ + d_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c - 9*a*d, 0] && PosQ[b/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "With[{q = Rt[-b/a, 2]}, -q*ArcTanh[q*x/3]/(12*Rt[a, 3]*d) + q*ArcTanh[(Rt[a, 3] - (a + b*x^2)^(1/3))^2/(3*Rt[a, 3]^2*q* x)]/(12*Rt[a, 3]*d) - q*ArcTan[(Sqrt[3]*(Rt[a, 3] - (a + b*x^2)^(1/3)))/(Rt[a, 3]*q* x)]/(4*Sqrt[3]*Rt[a, 3]*d)]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_ + b_.*x_^2)^(1/3)*(c_ + d_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c - 9*a*d, 0] && NegQ[b/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "b/d*Int[1/(a + b*x^2)^(1/3), x] - (b*c - a*d)/d* Int[1/((a + b*x^2)^(1/3)*(c + d*x^2)), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^2)^(2/3)/(c_ + d_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c + 3*a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "With[{q = Rt[b^2/a, 4]}, -b/(2*a*d*q)* ArcTan[(b + q^2*Sqrt[a + b*x^2])/(q^3*x*(a + b*x^2)^(1/4))] - b/(2*a*d*q)* ArcTanh[(b - q^2*Sqrt[a + b*x^2])/(q^3*x*(a + b*x^2)^(1/4))]]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_ + b_.*x_^2)^(1/4)*(c_ + d_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && PosQ[b^2/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "With[{q = Rt[-b^2/a, 4]}, b/(2*Sqrt[2]*a*d*q)*ArcTan[q*x/(Sqrt[2]*(a + b*x^2)^(1/4))] + b/(2*Sqrt[2]*a*d*q)*ArcTanh[q*x/(Sqrt[2]*(a + b*x^2)^(1/4))]]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_ + b_.*x_^2)^(1/4)*(c_ + d_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && NegQ[b^2/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": " With[{q=Rt[-b^2/a,4]}, b/(2*Sqrt[2]*a*d*q)*ArcTan[q*x/(Sqrt[2]*(a+b*x^2)^(1/4))] + b/(4*Sqrt[2]*a*d*q)*Log[(Sqrt[2]*q*x+2*(a+b*x^2)^(1/4))/(Sqrt[2]*q* x-2*(a+b*x^2)^(1/4))]]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_+b_.*x_^2)^(1/4)*(c_+d_.*x_^2)),x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d},x] && EqQ[b*c-2*a*d,0] &&  NegQ[b^2/a] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "2*Sqrt[-b*x^2/a]/x* Subst[Int[x^2/(Sqrt[1 - x^4/a]*(b*c - a*d + d*x^4)), x], x, (a + b*x^2)^(1/4)]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_ + b_.*x_^2)^(1/4)*(c_ + d_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "1/c*Int[1/(a + b*x^2)^(3/4), x] - d/c*Int[x^2/((a + b*x^2)^(3/4)*(c + d*x^2)), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_ + b_.*x_^2)^(3/4)*(c_ + d_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "Sqrt[-b*x^2/a]/(2*x)* Subst[Int[1/(Sqrt[-b*x/a]*(a + b*x)^(3/4)*(c + d*x)), x], x, x^2]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_ + b_.*x_^2)^(3/4)*(c_ + d_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "b/d*Int[(a + b*x^2)^(p - 1), x] - (b*c - a*d)/d* Int[(a + b*x^2)^(p - 1)/(c + d*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^2)^p_./(c_ + d_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "b/(b*c - a*d)*Int[(a + b*x^2)^p, x] - d/(b*c - a*d)*Int[(a + b*x^2)^(p + 1)/(c + d*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^2)^p_/(c_ + d_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && EqQ[Denominator[p], 4] && (EqQ[p, -5/4] || EqQ[p, -7/4])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "a/c*Subst[Int[1/(1 - 4*a*b*x^4), x], x, x/Sqrt[a + b*x^4]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*x_^4]/(c_ + d_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && PosQ[a*b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "With[{q = Rt[-a*b, 4]}, a/(2*c*q)*ArcTan[q*x*(a + q^2*x^2)/(a*Sqrt[a + b*x^4])] + a/(2*c*q)*ArcTanh[q*x*(a - q^2*x^2)/(a*Sqrt[a + b*x^4])]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*x_^4]/(c_ + d_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && NegQ[a*b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "b/d*Int[1/Sqrt[a + b*x^4], x] - (b*c - a*d)/d* Int[1/(Sqrt[a + b*x^4]*(c + d*x^4)), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*x_^4]/(c_ + d_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "Sqrt[a + b*x^4]*Sqrt[a/(a + b*x^4)]* Subst[Int[1/(Sqrt[1 - b*x^4]*(c - (b*c - a*d)*x^4)), x], x, x/(a + b*x^4)^(1/4)]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^4)^(1/4)/(c_ + d_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "b/d*Int[(a + b*x^4)^(p - 1), x] - (b*c - a*d)/d* Int[(a + b*x^4)^(p - 1)/(c + d*x^4), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^4)^p_/(c_ + d_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && (EqQ[p, 3/4] || EqQ[p, 5/4])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "1/(2*c)*Int[1/(Sqrt[a + b*x^4]*(1 - Rt[-d/c, 2]*x^2)), x] + 1/(2*c)*Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-d/c, 2]*x^2)), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_ + b_.*x_^4]*(c_ + d_.*x_^4)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "b/(b*c - a*d)*Int[1/(a + b*x^4)^(3/4), x] - d/(b*c - a*d)*Int[(a + b*x^4)^(1/4)/(c + d*x^4), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_ + b_.*x_^4)^(3/4)*(c_ + d_.*x_^4)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]* Sqrt[c*(a + b*x^2)/(a*(c + d*x^2))])* EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*c/(a*d)]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*x_^2]/(c_ + d_.*x_^2)^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": " a*Sqrt[c+d*x^2]*Sqrt[(c*(a+b*x^2))/(a*(c+d*x^2))]/(c^2*Rt[d/c,2]*Sqrt[ a+b*x^2])*EllipticE[ArcTan[Rt[d/c,2]*x],1-b*c/(a*d)]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_+b_.*x_^2]/(c_+d_.*x_^2)^(3/2),x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d},x] && PosQ[b/a] && PosQ[d/c] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "-x*(a + b*x^n)^(p + 1)*(c + d*x^n)^q/(a*n*(p + 1)) + 1/(a*n*(p + 1))* Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)* Simp[c*(n*(p + 1) + 1) + d*(n*(p + q + 1) + 1)*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && LtQ[0, q, 1] && IntBinomialQ[a, b, c, d, n, p, q, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "(a*d - c*b)* x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)/(a*b*n*(p + 1)) - 1/(a*b*n*(p + 1))* Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)* Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*(p + q) + 1))*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "-b* x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1)/(a* n*(p + 1)*(b*c - a*d)) + 1/(a*n*(p + 1)*(b*c - a*d))* Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q* Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && Not[Not[IntegerQ[p]] && IntegerQ[q] && LtQ[q, -1]] && IntBinomialQ[a, b, c, d, n, p, q, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "Int[ExpandIntegrand[(a + b*x^n)^p*(c + d*x^n)^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegersQ[p, q] && GtQ[p + q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "d*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)/(b*(n*(p + q) + 1)) + 1/(b*(n*(p + q) + 1))* Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)* Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] && Not[IGtQ[p, 1]] && IntBinomialQ[a, b, c, d, n, p, q, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "x*(a + b*x^n)^p*(c + d*x^n)^q/(n*(p + q) + 1) + n/(n*(p + q) + 1)* Int[(a + b*x^n)^(p - 1)*(c + d*x^n)^(q - 1)* Simp[a*c*(p + q) + (q*(b*c - a*d) + a*d*(p + q))*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 0] && GtQ[p, 0] && IntBinomialQ[a, b, c, d, n, p, q, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]* Sqrt[c*(a + b*x^2)/(a*(c + d*x^2))])* EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*c/(a*d)]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_ + b_.*x_^2]*Sqrt[c_ + d_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && Not[SimplerSqrtQ[b/a, d/c]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": " Sqrt[c+d*x^2]*Sqrt[c*(a+b*x^2)/(a*(c+d*x^2))]/(c*Rt[d/c,2]*Sqrt[a+b*x^ 2])*EllipticF[ArcTan[Rt[d/c,2]*x],1-b*c/(a*d)]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_+b_.*x_^2]*Sqrt[c_+d_.*x_^2]),x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d},x] &&  PosQ[d/c] && PosQ[b/a] && Not[SimplerSqrtQ[b/a,d/c]] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2])* EllipticF[ArcSin[Rt[-d/c, 2]*x], b*c/(a*d)]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_ + b_.*x_^2]*Sqrt[c_ + d_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && Not[NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "-1/(Sqrt[c]*Rt[-d/c, 2]*Sqrt[a - b*c/d])* EllipticF[ArcCos[Rt[-d/c, 2]*x], b*c/(b*c - a*d)]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_ + b_.*x_^2]*Sqrt[c_ + d_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a - b*c/d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "Sqrt[1 + d/c*x^2]/Sqrt[c + d*x^2]* Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + d/c*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_ + b_.*x_^2]*Sqrt[c_ + d_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && Not[GtQ[c, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "a*Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x] + b*Int[x^2/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*x_^2]/Sqrt[c_ + d_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "b/d*Int[Sqrt[c + d*x^2]/Sqrt[a + b*x^2], x] - (b*c - a*d)/d* Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*x_^2]/Sqrt[c_ + d_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && PosQ[d/c] && NegQ[b/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.3 (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2])* EllipticE[ArcSin[Rt[-d/c, 2]*x], b*c/(a*d)]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*x_^2]/Sqrt[c_ + d_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]"
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       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "c*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)/(a*e*(m + 1)) + d/e^n*Int[(e*x)^(m + n)*(a + b*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n*(p + 1) + 1, 0] && (IntegerQ[n] || GtQ[e, 0]) && (GtQ[n, 0] && LtQ[m, -1] || LtQ[n, 0] && GtQ[m + n, -1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "(b*c - a*d)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)/(a*b*e*(m + 1)) + d/b*Int[(e*x)^m*(a + b*x^n)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n*(p + 1) + 1, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "c*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)/(a*e*(m + 1)) + (a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1))* Int[(e*x)^(m + n)*(a + b*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] || GtQ[e, 0]) && (GtQ[n, 0] && LtQ[m, -1] || LtQ[n, 0] && GtQ[m + n, -1]) && Not[ILtQ[p, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "c*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1)/(a1*a2*e*(m + 1)) + (a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(a1*a2* e^n*(m + 1))* Int[(e*x)^(m + n)*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a1_ + b1_.*x_^non2_.)^p_.*(a2_ + b2_.*x_^non2_.)^ p_.*(c_ + d_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, d, e, p}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[n] || GtQ[e, 0]) && (GtQ[n, 0] && LtQ[m, -1] || LtQ[n, 0] && GtQ[m + n, -1]) && Not[ILtQ[p, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "(-a)^(m/2 - 1)*(b*c - a*d)* x*(a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1)) + 1/(2*b^(m/2 + 1)*(p + 1))*Int[(a + b*x^2)^(p + 1)* ExpandToSum[ 2*b*(p + 1)*x^2* Together[(b^(m/2)* x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)] - (-a)^(m/2 - 1)*(b*c - a*d), x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*(a_ + b_.*x_^2)^p_*(c_ + d_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "(-a)^(m/2 - 1)*(b*c - a*d)* x*(a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1)) + 1/(2*b^(m/2 + 1)*(p + 1))*Int[x^m*(a + b*x^2)^(p + 1)* ExpandToSum[ 2*b*(p + 1)* Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)* x^(-m + 2))/(a + b*x^2)] - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m), x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*(a_ + b_.*x_^2)^p_*(c_ + d_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ILtQ[m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "-(b*c - a*d)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)/(a*b* e*n*(p + 1)) - (a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b*n*(p + 1))* Int[(e*x)^m*(a + b*x^n)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && (Not[IntegerQ[p + 1/2]] && NeQ[p, -5/4] || Not[RationalQ[m]] || IGtQ[n, 0] && ILtQ[p + 1/2, 0] && LeQ[-1, m, -n*(p + 1)])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "-(b1*b2*c - a1*a2*d)*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1)/(a1*a2*b1*b2*e* n*(p + 1)) - (a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(a1*a2*b1*b2* n*(p + 1))* Int[(e*x)^m*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a1_ + b1_.*x_^non2_.)^p_.*(a2_ + b2_.*x_^non2_.)^ p_.*(c_ + d_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, d, e, m, n}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && LtQ[p, -1] && (Not[IntegerQ[p + 1/2]] && NeQ[p, -5/4] || Not[RationalQ[m]] || IGtQ[n, 0] && ILtQ[p + 1/2, 0] && LeQ[-1, m, -n*(p + 1)])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "d*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1)) - (a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p + 1) + 1))* Int[(e*x)^m*(a + b*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "d*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1)/(b1*b2*e*(m + n*(p + 1) + 1)) - (a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(b1* b2*(m + n*(p + 1) + 1))* Int[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a1_ + b1_.*x_^non2_.)^p_.*(a2_ + b2_.*x_^non2_.)^ p_.*(c_ + d_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "Int[ExpandIntegrand[(e*x)^m*(a + b*x^n)^p/(c + d*x^n), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_/(c_ + d_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] || Not[RationalQ[m]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "c^2*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)/(a*e*(m + 1)) - 1/(a*e^n*(m + 1))* Int[(e*x)^(m + n)*(a + b*x^n)^p* Simp[b*c^2*n*(p + 1) + c*(b*c - 2*a*d)*(m + 1) - a*(m + 1)*d^2*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "-(b*c - a*d)^2*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)/(a* b^2*e*n*(p + 1)) + 1/(a*b^2*n*(p + 1))* Int[(e*x)^m*(a + b*x^n)^(p + 1)* Simp[(b*c - a*d)^2*(m + 1) + b^2*c^2*n*(p + 1) + a*b*d^2*n*(p + 1)*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "d^2*(e*x)^(m + n + 1)*(a + b*x^n)^(p + 1)/(b* e^(n + 1)*(m + n*(p + 2) + 1)) + 1/(b*(m + n*(p + 2) + 1))* Int[(e*x)^m*(a + b*x^n)^p* Simp[b*c^2*(m + n*(p + 2) + 1) + d*((2*b*c - a*d)*(m + n + 1) + 2*b*c*n*(p + 1))*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && NeQ[m + n*(p + 2) + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "With[{k = GCD[m + 1, n]}, 1/k*Subst[ Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k] /; k != 1]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "With[{k = Denominator[m]}, k/e*Subst[ Int[x^(k*(m + 1) - 1)*(a + b*x^(k*n)/e^n)^p*(c + d*x^(k*n)/e^n)^ q, x], x, (e*x)^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "e^(n - 1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^ q/(b*n*(p + 1)) - e^n/(b*n*(p + 1))* Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)* Simp[c*(m - n + 1) + d*(m + n*(q - 1) + 1)*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 0] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "-(c*b - a*d)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)/(a*b*e*n*(p + 1)) + 1/(a*b*n*(p + 1))* Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)* Simp[c*(c*b*n*(p + 1) + (c*b - a*d)*(m + 1)) + d*(c*b*n*(p + 1) + (c*b - a*d)*(m + n*(q - 1) + 1))*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "-(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^ q/(a*e*n*(p + 1)) + 1/(a*n*(p + 1))* Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)* Simp[c*(m + n*(p + 1) + 1) + d*(m + n*(p + q + 1) + 1)*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[0, q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "-a* e^(2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1)/(b*n*(b*c - a*d)*(p + 1)) + e^(2*n)/(b*n*(b*c - a*d)*(p + 1))* Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q* Simp[a*c*(m - 2*n + 1) + (a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "e^(n - 1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1)/(n*(b*c - a*d)*(p + 1)) - e^n/(n*(b*c - a*d)*(p + 1))* Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q* Simp[c*(m - n + 1) + d*(m + n*(p + q + 1) + 1)*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "-b*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1)/(a*e*n*(b*c - a*d)*(p + 1)) + 1/(a*n*(b*c - a*d)*(p + 1))* Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q* Simp[c*b*(m + 1) + n*(b*c - a*d)*(p + 1) + d*b*(m + n*(p + q + 2) + 1)*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "(e*x)^(m + 1)*(a + b*x^n)^ p*(c + d*x^n)^q/(e*(m + 1)) - n/(e^n*(m + 1))* Int[(e*x)^(m + n)*(a + b*x^n)^(p - 1)*(c + d*x^n)^(q - 1)* Simp[b*c*p + a*d*q + b*d*(p + q)*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 0] && LtQ[m, -1] && GtQ[p, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "c*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)/(a* e*(m + 1)) - 1/(a*e^n*(m + 1))* Int[(e*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 2)* Simp[c*(c*b - a*d)*(m + 1) + c*n*(b*c*(p + 1) + a*d*(q - 1)) + d*((c*b - a*d)*(m + 1) + c*b*n*(p + q))*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 1] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^ q/(a*e*(m + 1)) - 1/(a*e^n*(m + 1))* Int[(e*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)* Simp[c*b*(m + 1) + n*(b*c*(p + 1) + a*d*q) + d*(b*(m + 1) + b*n*(p + q + 1))*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[0, q, 1] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "(e*x)^(m + 1)*(a + b*x^n)^ p*(c + d*x^n)^q/(e*(m + n*(p + q) + 1)) + n/(m + n*(p + q) + 1)* Int[(e*x)^m*(a + b*x^n)^(p - 1)*(c + d*x^n)^(q - 1)* Simp[a*c*(p + q) + (q*(b*c - a*d) + a*d*(p + q))*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 0] && GtQ[p, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "d*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)/(b* e*(m + n*(p + q) + 1)) + 1/(b*(m + n*(p + q) + 1))* Int[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 2)* Simp[c*((c*b - a*d)*(m + 1) + c*b*n*(p + q)) + (d*(c*b - a*d)*(m + 1) + d*n*(q - 1)*(b*c - a*d) + c*b*d*n*(p + q))*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "e^(n - 1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^ q/(b*(m + n*(p + q) + 1)) - e^n/(b*(m + n*(p + q) + 1))* Int[(e*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)* Simp[a*c*(m - n + 1) + (a*d*(m - n + 1) - n*q*(b*c - a*d))*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 0] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "e^(2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q) + 1)) - e^(2*n)/(b*d*(m + n*(p + q) + 1))* Int[(e*x)^(m - 2*n)*(a + b*x^n)^p*(c + d*x^n)^q* Simp[a*c*(m - 2*n + 1) + (a*d*(m + n*(q - 1) + 1) + b*c*(m + n*(p - 1) + 1))*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1)/(a*c*e*(m + 1)) - 1/(a*c*e^n*(m + 1))* Int[(e*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q* Simp[(b*c + a*d)*(m + n + 1) + n*(b*c*p + a*d*q) + b*d*(m + n*(p + q + 2) + 1)*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "-a*e^n/(b*c - a*d)* Int[(e*x)^(m - n)/(a + b*x^n), x] + c*e^n/(b*c - a*d)*Int[(e*x)^(m - n)/(c + d*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_./((a_ + b_.*x_^n_)*(c_ + d_.*x_^n_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, m, 2*n - 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "b/(b*c - a*d)*Int[(e*x)^m/(a + b*x^n), x] - d/(b*c - a*d)*Int[(e*x)^m/(c + d*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_./((a_ + b_.*x_^n_)*(c_ + d_.*x_^n_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "e^n/b*Int[(e*x)^(m - n)*(c + d*x^n)^q, x] - a*e^n/b*Int[(e*x)^(m - n)*(c + d*x^n)^q/(a + b*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_*(c_ + d_.*x_^n_)^q_./(a_ + b_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, m, 2*n - 1] && IntBinomialQ[a, b, c, d, e, m, n, -1, q, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "With[{q = Rt[d/c, 3]}, q*ArcTanh[Sqrt[c + d*x^3]/Rt[c, 2]]/(9*2^(2/3)*b*Rt[c, 2]) + q*ArcTan[ Sqrt[c + d*x^3]/(Sqrt[3]*Rt[c, 2])]/(3*2^(2/3)*Sqrt[3]*b* Rt[c, 2]) - q*ArcTan[ Sqrt[3]*Rt[c, 2]*(1 + 2^(1/3)*q*x)/Sqrt[c + d*x^3]]/(3*2^(2/3)* Sqrt[3]*b*Rt[c, 2]) - q*ArcTanh[ Rt[c, 2]*(1 - 2^(1/3)*q*x)/Sqrt[c + d*x^3]]/(3*2^(2/3)*b* Rt[c, 2])]",
     "rulenumber": 0,
            "lhs": "Int[x_/((a_ + b_.*x_^3)*Sqrt[c_ + d_.*x_^3]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[4*b*c - a*d, 0] && PosQ[c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "With[{q = Rt[d/c, 3]}, -q*ArcTan[Sqrt[c + d*x^3]/Rt[-c, 2]]/(9*2^(2/3)*b*Rt[-c, 2]) - q*ArcTanh[ Sqrt[c + d*x^3]/(Sqrt[3]*Rt[-c, 2])]/(3*2^(2/3)*Sqrt[3]*b* Rt[-c, 2]) - q*ArcTanh[ Sqrt[3]*Rt[-c, 2]*(1 + 2^(1/3)*q*x)/ Sqrt[c + d*x^3]]/(3*2^(2/3)*Sqrt[3]*b*Rt[-c, 2]) - q*ArcTan[ Rt[-c, 2]*(1 - 2^(1/3)*q*x)/Sqrt[c + d*x^3]]/(3*2^(2/3)*b* Rt[-c, 2])]",
     "rulenumber": 0,
            "lhs": "Int[x_/((a_ + b_.*x_^3)*Sqrt[c_ + d_.*x_^3]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[4*b*c - a*d, 0] && NegQ[c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "With[{q = Rt[d/c, 3]}, d*q/(4*b)*Int[x^2/((8*c - d*x^3)*Sqrt[c + d*x^3]), x] - q^2/(12*b)*Int[(1 + q*x)/((2 - q*x)*Sqrt[c + d*x^3]), x] + 1/(12*b*c)* Int[(2*c*q^2 - 2*d*x - d*q*x^2)/((4 + 2*q*x + q^2*x^2)* Sqrt[c + d*x^3]), x]]",
     "rulenumber": 0,
            "lhs": "Int[x_/((a_ + b_.*x_^3)*Sqrt[c_ + d_.*x_^3]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[8*b*c + a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "With[{q = Rt[b/a, 3], r = Simplify[(b*c - 10*a*d)/(6*a*d)]}, -q*(2 - r)* ArcTan[(1 - r)*Sqrt[a + b*x^3]/(Sqrt[2]*Rt[a, 2]*r^(3/2))]/(3* Sqrt[2]*Rt[a, 2]*d*r^(3/2)) - q*(2 - r)* ArcTan[Rt[a, 2]* Sqrt[r]*(1 + r)*(1 + q*x)/(Sqrt[2]*Sqrt[a + b*x^3])]/(2* Sqrt[2]*Rt[a, 2]*d*r^(3/2)) - q*(2 - r)* ArcTanh[Rt[a, 2]*(1 - r)* Sqrt[r]*(1 + q*x)/(Sqrt[2]*Sqrt[a + b*x^3])]/(6*Sqrt[2]* Rt[a, 2]*d*Sqrt[r]) - q*(2 - r)* ArcTanh[Rt[a, 2]* Sqrt[r]*(1 + r - 2*q*x)/(Sqrt[2]*Sqrt[a + b*x^3])]/(3*Sqrt[2]* Rt[a, 2]*d*Sqrt[r])]",
     "rulenumber": 0,
            "lhs": "Int[x_/((c_ + d_.*x_^3)*Sqrt[a_ + b_.*x_^3]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b^2*c^2 - 20*a*b*c*d - 8*a^2*d^2, 0] && PosQ[a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "With[{q = Rt[b/a, 3], r = Simplify[(b*c - 10*a*d)/(6*a*d)]}, q*(2 - r)* ArcTanh[(1 - r)*Sqrt[a + b*x^3]/(Sqrt[2]*Rt[-a, 2]*r^(3/2))]/(3* Sqrt[2]*Rt[-a, 2]*d*r^(3/2)) - q*(2 - r)* ArcTanh[Rt[-a, 2]* Sqrt[r]*(1 + r)*(1 + q*x)/(Sqrt[2]*Sqrt[a + b*x^3])]/(2* Sqrt[2]*Rt[-a, 2]*d*r^(3/2)) - q*(2 - r)* ArcTan[Rt[-a, 2]*(1 - r)* Sqrt[r]*(1 + q*x)/(Sqrt[2]*Sqrt[a + b*x^3])]/(6*Sqrt[2]* Rt[-a, 2]*d*Sqrt[r]) - q*(2 - r)* ArcTan[Rt[-a, 2]* Sqrt[r]*(1 + r - 2*q*x)/(Sqrt[2]*Sqrt[a + b*x^3])]/(3*Sqrt[2]* Rt[-a, 2]*d*Sqrt[r])]",
     "rulenumber": 0,
            "lhs": "Int[x_/((c_ + d_.*x_^3)*Sqrt[a_ + b_.*x_^3]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b^2*c^2 - 20*a*b*c*d - 8*a^2*d^2, 0] && NegQ[a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "b/d*Int[x/Sqrt[a + b*x^3], x] - (b*c - a*d)/d* Int[x/((c + d*x^3)*Sqrt[a + b*x^3]), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*Sqrt[a_ + b_.*x_^3]/(c_ + d_.*x_^3), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, a, b}, x] && NeQ[b*c - a*d, 0] && (EqQ[b*c - 4*a*d, 0] || EqQ[b*c + 8*a*d, 0] || EqQ[b^2*c^2 - 20*a*b*c*d - 8*a^2*d^2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, s/(2*b)*Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x] - s/(2*b)*Int[1/((r - s*x^2)*Sqrt[c + d*x^4]), x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^2/((a_ + b_.*x_^4)*Sqrt[c_ + d_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "d/b*Int[x^2/Sqrt[c + d*x^4], x] + (b*c - a*d)/b* Int[x^2/((a + b*x^4)*Sqrt[c + d*x^4]), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^2*Sqrt[c_ + d_.*x_^4]/(a_ + b_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "x*Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2]) - c/b*Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^2/(Sqrt[a_ + b_.*x_^2]*Sqrt[c_ + d_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] && Not[SimplerSqrtQ[b/a, d/c]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "1/b*Int[Sqrt[a + b*x^n]/Sqrt[c + d*x^n], x] - a/b*Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^n_/(Sqrt[a_ + b_.*x_^n_]*Sqrt[c_ + d_.*x_^n_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && (EqQ[n, 2] || EqQ[n, 4]) && Not[EqQ[n, 2] && SimplerSqrtQ[-b/a, -d/c]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "With[{k = Denominator[p]}, k*a^(p + (m + 1)/n)/n* Subst[ Int[x^(k*(m + 1)/n - 1)*(c - (b*c - a*d)*x^k)^ q/(1 - b*x^k)^(p + q + (m + 1)/n + 1), x], x, x^(n/k)/(a + b*x^n)^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && RationalQ[m, p] && IntegersQ[p + (m + 1)/n, q] && LtQ[-1, p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "-Subst[ Int[(a + b*x^(-n))^p*(c + d*x^(-n))^q/x^(m + 2), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "With[{g = Denominator[m]}, -g/e* Subst[Int[(a + b*e^(-n)*x^(-g*n))^p*(c + d*e^(-n)*x^(-g*n))^q/ x^(g*(m + 1) + 1), x], x, 1/(e*x)^(1/g)]]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p, q}, x] && ILtQ[n, 0] && FractionQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "-(e*x)^m*(x^(-1))^m* Subst[Int[(a + b*x^(-n))^p*(c + d*x^(-n))^q/x^(m + 2), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0] && Not[RationalQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "With[{g = Denominator[n]}, g*Subst[ Int[x^(g*(m + 1) - 1)*(a + b*x^(g*n))^p*(c + d*x^(g*n))^q, x], x, x^(1/g)]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*x^n)^p*(c + d*x^n)^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_*x_)^m_*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": " -1/(m+1)*Subst[Int[(a+b*x^Simplify[-n/(m+1)])^p*(c+d*x^Simplify[-n/(m+ 1)])^q/x^2,x],x,x^(-(m+1))]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_+b_.*x_^n_)^p_*(c_+d_.*x_^n_)^q_,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,m,n},x] &&  NeQ[b*c-a*d,0] && NeQ[m,-1] && ILtQ[Simplify[n/(m+1)+1],0] && GeQ[p,-1] && LtQ[p,0] && GeQ[q,-1] && LtQ[q,0] && Not[IntegerQ[n]] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "1/(m + 1)* Subst[Int[(a + b*x^Simplify[n/(m + 1)])^ p*(c + d*x^Simplify[n/(m + 1)])^q, x], x, x^(m + 1)]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ[Simplify[n/(m + 1)]] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*x^n)^p*(c + d*x^n)^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_*x_)^m_.*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ[Simplify[n/(m + 1)]] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "-(c*b - a*d)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)/(a*b*e*n*(p + 1)) + 1/(a*b*n*(p + 1))* Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)* Simp[c*(c*b*n*(p + 1) + (c*b - a*d)*(m + 1)) + d*(c*b*n*(p + 1) + (c*b - a*d)*(m + n*(q - 1) + 1))*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "-(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^ q/(a*e*n*(p + 1)) + 1/(a*n*(p + 1))* Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)* Simp[c*(m + n*(p + 1) + 1) + d*(m + n*(p + q + 1) + 1)*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && LtQ[0, q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "-b*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1)/(a*e*n*(b*c - a*d)*(p + 1)) + 1/(a*n*(b*c - a*d)*(p + 1))* Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q* Simp[c*b*(m + 1) + n*(b*c - a*d)*(p + 1) + d*b*(m + n*(p + q + 2) + 1)*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "(e*x)^(m + 1)*(a + b*x^n)^ p*(c + d*x^n)^q/(e*(m + n*(p + q) + 1)) + n/(m + n*(p + q) + 1)* Int[(e*x)^m*(a + b*x^n)^(p - 1)*(c + d*x^n)^(q - 1)* Simp[a*c*(p + q) + (q*(b*c - a*d) + a*d*(p + q))*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 0] && GtQ[p, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "d*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)/(b* e*(m + n*(p + q) + 1)) + 1/(b*(m + n*(p + q) + 1))* Int[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 2)* Simp[c*((c*b - a*d)*(m + 1) + c*b*n*(p + q)) + (d*(c*b - a*d)*(m + 1) + d*n*(q - 1)*(b*c - a*d) + c*b*d*n*(p + q))*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "-a/(b*c - a*d)*Int[x^(m - n)/(a + b*x^n), x] + c/(b*c - a*d)*Int[x^(m - n)/(c + d*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_/((a_ + b_.*x_^n_)*(c_ + d_.*x_^n_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && (EqQ[m, n] || EqQ[m, 2*n - 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "b/(b*c - a*d)*Int[(e*x)^m/(a + b*x^n), x] - d/(b*c - a*d)*Int[(e*x)^m/(c + d*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_./((a_ + b_.*x_^n_)*(c_ + d_.*x_^n_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, m}, x] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
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       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "a^p*c^q*(e*x)^(m + 1)/(e*(m + 1))* AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -b*x^n/a, -d*x^n/c]",
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       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "a^IntPart[p]*(a + b*x^n)^FracPart[p]/(1 + b*x^n/a)^FracPart[p]* Int[(e*x)^m*(1 + b*x^n/a)^p*(c + d*x^n)^q, x]",
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       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "1/Coefficient[v, x, 1]^(m + 1)* Subst[Int[ SimplifyIntegrand[(x - Coefficient[v, x, 0])^m*(a + b*x^n)^ p*(c + d*x^n)^q, x], x], x, v]",
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       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "u^m/(Coefficient[v, x, 1]*v^m)* Subst[Int[x^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x, v]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p, q}, x] && LinearPairQ[u, v, x]"
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       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "Int[x^(m - n*q)*(a + b*x^n)^p*(d + c*x^n)^q, x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] || Not[IntegerQ[p]])"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "x^(n*FracPart[q])*(c + d*x^(-n))^FracPart[q]/(d + c*x^n)^FracPart[q]* Int[x^(m - n*q)*(a + b*x^n)^p*(d + c*x^n)^q, x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p, q}, x] && EqQ[mn, -n] && Not[IntegerQ[q]] && Not[IntegerQ[p]]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*x^n)^p*(c + d*x^(-n))^q, x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p, q}, x] && EqQ[mn, -n]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x]",
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        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[p] || GtQ[a1, 0] && GtQ[a2, 0])"
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       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n + e*x^(2*n))^q, x]",
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        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, d, e, n, p, q}, x] && EqQ[non2, n/2] && EqQ[n2, 2*n] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[p] || GtQ[a1, 0] && GtQ[a2, 0])"
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       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "(a1 + b1*x^(n/2))^ FracPart[p]*(a2 + b2*x^(n/2))^FracPart[p]/(a1*a2 + b1*b2*x^n)^ FracPart[p]* Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x]",
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        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && Not[EqQ[n, 2] && IGtQ[q, 0]]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
       "filename": "1.1.3.4 (e x)^m (a+b x^n)^p (c+d x^n)^q.m",
            "rhs": "(a1 + b1*x^(n/2))^ FracPart[p]*(a2 + b2*x^(n/2))^FracPart[p]/(a1*a2 + b1*b2*x^n)^ FracPart[p]* Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n + e*x^(2*n))^q, x]",
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        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, d, e, n, p, q}, x] && EqQ[non2, n/2] && EqQ[n2, 2*n] && EqQ[a2*b1 + a1*b2, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "Int[ExpandIntegrand[(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x]",
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            "lhs": "Int[(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^q_.*(e_ + f_.*x_^n_)^r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[p, 0] && IGtQ[q, 0] && IGtQ[r, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "(b*e - a*f)/(b*c - a*d)*Int[1/(a + b*x^n), x] - (d*e - c*f)/(b*c - a*d)*Int[1/(c + d*x^n), x]",
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            "lhs": "Int[(e_ + f_.*x_^n_)/((a_ + b_.*x_^n_)*(c_ + d_.*x_^n_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "f/b*Int[1/Sqrt[c + d*x^n], x] + (b*e - a*f)/b*Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_ + f_.*x_^n_)/((a_ + b_.*x_^n_)*Sqrt[c_ + d_.*x_^n_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "f/b*Int[Sqrt[a + b*x^n]/Sqrt[c + d*x^n], x] + (b*e - a*f)/b*Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]), x]",
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            "lhs": "Int[(e_ + f_.*x_^n_)/(Sqrt[a_ + b_.*x_^n_]*Sqrt[c_ + d_.*x_^n_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && Not[EqQ[n, 2] && (PosQ[b/a] && PosQ[d/c] || NegQ[b/a] && (PosQ[d/c] || GtQ[a, 0] && (Not[GtQ[c, 0]] || SimplerSqrtQ[-b/a, -d/c])))]"
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        "comment": false,
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && GtQ[q, 0]"
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]"
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            "rhs": "f*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^q/(b*(n*(p + q + 1) + 1)) + 1/(b*(n*(p + q + 1) + 1))* Int[(a + b*x^n)^p*(c + d*x^n)^(q - 1)* Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e - a*f) + f*n*q*(b*c - a*d) + b*d*e*n*(p + q + 1))*x^n, x], x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1, 0]"
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            "rhs": "(b*e - a*f)/(b*c - a*d)* Int[1/(a + b*x^4)^(3/4), x] - (d*e - c*f)/(b*c - a*d)* Int[(a + b*x^4)^(1/4)/(c + d*x^4), x]",
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            "lhs": "Int[(e_ + f_.*x_^4)/((a_ + b_.*x_^4)^(3/4)*(c_ + d_.*x_^4)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x]"
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            "rhs": "f/d*Int[(a + b*x^n)^p, x] + (d*e - c*f)/d* Int[(a + b*x^n)^p/(c + d*x^n), x]",
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            "lhs": "Int[(a_ + b_.*x_^n_)^p_*(e_ + f_.*x_^n_)/(c_ + d_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p, n}, x]"
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       "filename": "1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "e*Int[(a + b*x^n)^p*(c + d*x^n)^q, x] + f*Int[x^n*(a + b*x^n)^p*(c + d*x^n)^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^q_.*(e_ + f_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, p, q}, x]"
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         "givens": "FreeQ[{a, b, c, d, e, f}, x] && Not[GtQ[d/c, 0]] && GtQ[c, 0] && GtQ[e, 0] && Not[Not[GtQ[f/e, 0]] && SimplerSqrtQ[-f/e, -d/c]]"
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        "comment": false,
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            "rhs": "c*Sqrt[e + f*x^2]/(a*e*Rt[d/c, 2]*Sqrt[c + d*x^2]* Sqrt[c*(e + f*x^2)/(e*(c + d*x^2))])* EllipticPi[1 - b*c/(a*d), ArcTan[Rt[d/c, 2]*x], 1 - c*f/(d*e)]",
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            "lhs": "Int[Sqrt[c_ + d_.*x_^2]/((a_ + b_.*x_^2)*Sqrt[e_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && PosQ[d/c]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
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            "rhs": "  Sqrt[c+d*x^2]*Sqrt[c*(e+f*x^2)/(e*(c+d*x^2))]/(a*Rt[d/c,2]*Sqrt[e+f* x^2])* EllipticPi[1-b*c/(a*d),ArcTan[Rt[d/c,2]*x],1-c*f/(d*e)]",
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            "lhs": "Int[Sqrt[c_+d_.*x_^2]/((a_+b_.*x_^2)*Sqrt[e_+f_.*x_^2]),x_Symbol]",
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            "rhs": "d/b*Int[1/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x] + (b*c - a*d)/b* Int[1/((a + b*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NegQ[d/c]"
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            "lhs": "Int[Sqrt[e_ + f_.*x_^2]/((a_ + b_.*x_^2)*(c_ + d_.*x_^2)^(3/2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && PosQ[d/c] && PosQ[f/e]"
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            "lhs": "Int[(e_ + f_.*x_^2)^(3/2)/((a_ + b_.*x_^2)*(c_ + d_.*x_^2)^(3/2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && PosQ[d/c] && PosQ[f/e]"
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        "comment": false,
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && LtQ[q, -1] && GtQ[r, 1]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
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        "comment": false,
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        "comment": false,
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         "givens": "FreeQ[{a, b, c, d, e, f, n, q}, x] && ILtQ[p, 0] && LeQ[q, -1]"
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x]"
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            "lhs": "Int[Sqrt[a_ + b_.*x_^2]/(Sqrt[c_ + d_.*x_^2]*Sqrt[e_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "Sqrt[c + d*x^2]* Sqrt[a*(e + f*x^2)/(e*(a + b*x^2))]/(a*Sqrt[e + f*x^2]* Sqrt[a*(c + d*x^2)/(c*(a + b*x^2))])* Subst[ Int[Sqrt[1 - (b*c - a*d)*x^2/c]/Sqrt[1 - (b*e - a*f)*x^2/e], x], x, x/Sqrt[a + b*x^2]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[c_ + d_.*x_^2]/((a_ + b_.*x_^2)^(3/2)*Sqrt[e_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "d*x*Sqrt[a + b*x^2]*Sqrt[e + f*x^2]/(2*f*Sqrt[c + d*x^2]) - c*(d*e - c*f)/(2*f)* Int[Sqrt[a + b*x^2]/((c + d*x^2)^(3/2)*Sqrt[e + f*x^2]), x] + b*c*(d*e - c*f)/(2*d*f)* Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x] - (b*d*e - b*c*f - a*d*f)/(2*d*f)* Int[Sqrt[c + d*x^2]/(Sqrt[a + b*x^2]*Sqrt[e + f*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*x_^2]*Sqrt[c_ + d_.*x_^2]/Sqrt[e_ + f_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && PosQ[(d*e - c*f)/c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]/(2*Sqrt[e + f*x^2]) + e*(b*e - a*f)/(2*f)* Int[Sqrt[c + d*x^2]/(Sqrt[a + b*x^2]*(e + f*x^2)^(3/2)), x] + (b*e - a*f)*(d*e - 2*c*f)/(2*f^2)* Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x] - (b*d*e - b*c*f - a*d*f)/(2*f^2)* Int[Sqrt[e + f*x^2]/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*x_^2]*Sqrt[c_ + d_.*x_^2]/Sqrt[e_ + f_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NegQ[(d*e - c*f)/c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "b/f*Int[Sqrt[c + d*x^2]/(Sqrt[a + b*x^2]*Sqrt[e + f*x^2]), x] - (b*e - a*f)/f* Int[Sqrt[c + d*x^2]/(Sqrt[a + b*x^2]*(e + f*x^2)^(3/2)), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*x_^2]*Sqrt[c_ + d_.*x_^2]/(e_ + f_.*x_^2)^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "With[{u = ExpandIntegrand[(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_*(e_ + f_.*x_^n_)^r_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p, q, r}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "-Subst[ Int[(a + b*x^(-n))^p*(c + d*x^(-n))^q*(e + f*x^(-n))^r/x^2, x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_*(e_ + f_.*x_^n_)^r_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p, q, r}, x] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "Unintegrable[(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^q_.*(e_ + f_.*x_^n_)^r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, p, q, r}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "1/Coefficient[u, x, 1]* Subst[Int[(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x, u]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*u_^n_)^p_.*(c_. + d_.*v_^n_)^q_.*(e_. + f_.*w_^n_)^r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p, n, q, r}, x] && EqQ[u, v] && EqQ[u, w] && LinearQ[u, x] && NeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "Int[(a + b*x^n)^p*(d + c*x^n)^q*(e + f*x^n)^r/x^(n*q), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_^n_.)^p_.*(c_ + d_.*x_^mn_.)^q_.*(e_ + f_.*x_^n_.)^ r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, p, r}, x] && EqQ[mn, -n] && IntegerQ[q]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "Int[x^(n*(p + r))*(b + a*x^(-n))^p*(c + d*x^(-n))^q*(f + e*x^(-n))^r, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_^n_.)^p_.*(c_ + d_.*x_^mn_.)^q_.*(e_ + f_.*x_^n_.)^ r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, q}, x] && EqQ[mn, -n] && IntegerQ[p] && IntegerQ[r]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "x^(n*FracPart[q])*(c + d*x^(-n))^FracPart[q]/(d + c*x^n)^FracPart[q]* Int[(a + b*x^n)^p*(d + c*x^n)^q*(e + f*x^n)^r/x^(n*q), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_^n_.)^p_.*(c_ + d_.*x_^mn_.)^q_*(e_ + f_.*x_^n_.)^ r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, p, q, r}, x] && EqQ[mn, -n] && Not[IntegerQ[q]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "Int[(a + b*x^n)^p*(c + d*x^n)^q*(e1*e2 + f1*f2*x^n)^r, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^q_.*(e1_ + f1_.*x_^n2_.)^ r_.*(e2_ + f2_.*x_^n2_.)^r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e1, f1, e2, f2, n, p, q, r}, x] && EqQ[n2, n/2] && EqQ[e2*f1 + e1*f2, 0] && (IntegerQ[r] || GtQ[e1, 0] && GtQ[e2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.5 (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "(e1 + f1*x^(n/2))^ FracPart[r]*(e2 + f2*x^(n/2))^FracPart[r]/(e1*e2 + f1*f2*x^n)^ FracPart[r]* Int[(a + b*x^n)^p*(c + d*x^n)^q*(e1*e2 + f1*f2*x^n)^r, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^q_.*(e1_ + f1_.*x_^n2_.)^ r_.*(e2_ + f2_.*x_^n2_.)^r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e1, f1, e2, f2, n, p, q, r}, x] && EqQ[n2, n/2] && EqQ[e2*f1 + e1*f2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "g^m/(n*b^(Simplify[(m + 1)/n] - 1))* Subst[Int[(b*x)^(p + Simplify[(m + 1)/n] - 1)*(c + d*x)^ q*(e + f*x)^r, x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^m_.*(b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^q_.*(e_ + f_.*x_^n_)^ r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f, g, m, n, p, q, r}, x] && (IntegerQ[m] || GtQ[g, 0]) && IntegerQ[Simplify[(m + 1)/n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "g^m*b^IntPart[p]*(b*x^n)^FracPart[p]/x^(n*FracPart[p])* Int[x^(m + n*p)*(c + d*x^n)^q*(e + f*x^n)^r, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^m_.*(b_.*x_^n_.)^p_*(c_ + d_.*x_^n_)^ q_.*(e_ + f_.*x_^n_)^r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f, g, m, n, p, q, r}, x] && (IntegerQ[m] || GtQ[g, 0]) && Not[IntegerQ[Simplify[(m + 1)/n]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "g^IntPart[m]*(g*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_*x_)^m_*(b_.*x_^n_.)^p_*(c_ + d_.*x_^n_)^q_.*(e_ + f_.*x_^n_)^ r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f, g, m, n, p, q, r}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^ r, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^m_.*(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^ q_.*(e_ + f_.*x_^n_)^r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "1/n*Subst[Int[(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^q_.*(e_ + f_.*x_^n_)^ r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p, q, r}, x] && EqQ[m - n + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "Int[x^(m + n*(p + q + r))*(b + a*x^(-n))^p*(d + c*x^(-n))^ q*(f + e*x^(-n))^r, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^q_.*(e_ + f_.*x_^n_)^ r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegersQ[p, q, r] && NegQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q*(e + f*x)^ r, x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^q_.*(e_ + f_.*x_^n_)^ r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[Simplify[(m + 1)/n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "g^IntPart[m]*(g*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_*x_)^m_.*(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^ q_.*(e_ + f_.*x_^n_)^r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r}, x] && IntegerQ[Simplify[(m + 1)/n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "With[{k = GCD[m + 1, n]}, 1/k*Subst[ Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^ q*(e + f*x^(n/k))^r, x], x, x^k] /; k != 1]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^q_.*(e_ + f_.*x_^n_)^ r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p, q, r}, x] && IGtQ[n, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "With[{k = Denominator[m]}, k/g*Subst[ Int[x^(k*(m + 1) - 1)*(a + b*x^(k*n)/g^n)^p*(c + d*x^(k*n)/g^n)^ q*(e + f*x^(k*n)/g^n)^r, x], x, (g*x)^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^m_*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^ q_*(e_ + f_.*x_^n_)^r_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, p, q, r}, x] && IGtQ[n, 0] && FractionQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "-(b*e - a*f)*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q/(a*b*g*n*(p + 1)) + 1/(a*b*n*(p + 1))* Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)* Simp[c*(b*e*n*(p + 1) + (b*e - a*f)*(m + 1)) + d*(b*e*n*(p + 1) + (b*e - a*f)*(m + n*q + 1))*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^m_.*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^ q_.*(e_ + f_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 0] && Not[EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "g^(n - 1)*(b*e - a*f)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1)/(b*n*(b*c - a*d)*(p + 1)) - g^n/(b*n*(b*c - a*d)*(p + 1))* Int[(g*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q* Simp[c*(b*e - a*f)*(m - n + 1) + (d*(b*e - a*f)*(m + n*q + 1) - b*n*(c*f - d*e)*(p + 1))*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^m_.*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^ q_*(e_ + f_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, q}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "-(b*e - a*f)*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1)/(a*g*n*(b*c - a*d)*(p + 1)) + 1/(a*n*(b*c - a*d)*(p + 1))* Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q* Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^m_.*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^ q_*(e_ + f_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q/(a*g*(m + 1)) - 1/(a*g^n*(m + 1))* Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)* Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c*(p + 1) + a*d*q) + d*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1))*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^m_*(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^ q_.*(e_ + f_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, p}, x] && IGtQ[n, 0] && GtQ[q, 0] && LtQ[m, -1] && Not[EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "f*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^ q/(b*g*(m + n*(p + q + 1) + 1)) + 1/(b*(m + n*(p + q + 1) + 1))* Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 1)* Simp[c*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1)) + (d*(b*e - a*f)*(m + 1) + f*n*q*(b*c - a*d) + b*e*d*n*(p + q + 1))*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^m_.*(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^ q_.*(e_ + f_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0] && GtQ[q, 0] && Not[EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "f*g^(n - 1)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q + 1) + 1)) - g^n/(b*d*(m + n*(p + q + 1) + 1))* Int[(g*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^q* Simp[a*f* c*(m - n + 1) + (a*f*d*(m + n*q + 1) + b*(f*c*(m + n*p + 1) - e*d*(m + n*(p + q + 1) + 1)))*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^m_.*(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^ q_.*(e_ + f_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && GtQ[m, n - 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1)/(a*c* g*(m + 1)) + 1/(a*c*g^n*(m + 1))*Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q* Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^m_*(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^ q_.*(e_ + f_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*(e + f*x^n)/(c + d*x^n), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^m_.*(a_ + b_.*x_^n_)^ p_*(e_ + f_.*x_^n_)/(c_ + d_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "e*Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x] + f/e^n*Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^m_.*(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^ q_.*(e_ + f_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, p, q}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "e*Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^(r - 1), x] + f/e^n* Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^(r - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^m_.*(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^ q_.*(e_ + f_.*x_^n_)^r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, p, q}, x] && IGtQ[n, 0] && IGtQ[r, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "-Subst[ Int[(a + b*x^(-n))^p*(c + d*x^(-n))^q*(e + f*x^(-n))^r/x^(m + 2), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^q_.*(e_ + f_.*x_^n_)^ r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p, q, r}, x] && ILtQ[n, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "With[{k = Denominator[m]}, -k/g* Subst[Int[(a + b*g^(-n)*x^(-k*n))^p*(c + d*g^(-n)*x^(-k*n))^ q*(e + f*g^(-n)*x^(-k*n))^r/x^(k*(m + 1) + 1), x], x, 1/(g*x)^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^m_*(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^ q_.*(e_ + f_.*x_^n_)^r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, p, q, r}, x] && ILtQ[n, 0] && FractionQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "-(g*x)^m*(x^(-1))^m* Subst[Int[(a + b*x^(-n))^p*(c + d*x^(-n))^q*(e + f*x^(-n))^r/ x^(m + 2), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^m_*(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^ q_.*(e_ + f_.*x_^n_)^r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, p, q, r}, x] && ILtQ[n, 0] && Not[RationalQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "With[{k = Denominator[n]}, k*Subst[ Int[x^(k*(m + 1) - 1)*(a + b*x^(k*n))^p*(c + d*x^(k*n))^ q*(e + f*x^(k*n))^r, x], x, x^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^q_.*(e_ + f_.*x_^n_)^ r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p, q, r}, x] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "g^IntPart[m]*(g*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_*x_)^m_*(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^ q_.*(e_ + f_.*x_^n_)^r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, p, q, r}, x] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "1/(m + 1)* Subst[Int[(a + b*x^Simplify[n/(m + 1)])^ p*(c + d*x^Simplify[n/(m + 1)])^ q*(e + f*x^Simplify[n/(m + 1)])^r, x], x, x^(m + 1)]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^q_.*(e_ + f_.*x_^n_)^ r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[Simplify[n/(m + 1)]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "g^IntPart[m]*(g*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_*x_)^m_.*(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^ q_.*(e_ + f_.*x_^n_)^r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r}, x] && IntegerQ[Simplify[n/(m + 1)]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "-(b*e - a*f)*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q/(a*b*g*n*(p + 1)) + 1/(a*b*n*(p + 1))* Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)* Simp[c*(b*e*n*(p + 1) + (b*e - a*f)*(m + 1)) + d*(b*e*n*(p + 1) + (b*e - a*f)*(m + n*q + 1))*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^m_.*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^ q_.*(e_ + f_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n}, x] && LtQ[p, -1] && GtQ[q, 0] && Not[EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "-(b*e - a*f)*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1)/(a*g*n*(b*c - a*d)*(p + 1)) + 1/(a*n*(b*c - a*d)*(p + 1))* Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q* Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^m_.*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^ q_*(e_ + f_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "f*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^ q/(b*g*(m + n*(p + q + 1) + 1)) + 1/(b*(m + n*(p + q + 1) + 1))* Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 1)* Simp[c*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1)) + (d*(b*e - a*f)*(m + 1) + f*n*q*(b*c - a*d) + b*e*d*n*(p + q + 1))*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^m_.*(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^ q_.*(e_ + f_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && GtQ[q, 0] && Not[EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*(e + f*x^n)/(c + d*x^n), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^m_.*(a_ + b_.*x_^n_)^ p_*(e_ + f_.*x_^n_)/(c_ + d_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "e*Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x] + f*(g*x)^m/x^m*Int[x^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^m_.*(a_ + b_.*x_^n_)^p_*(c_ + d_.*x_^n_)^ q_*(e_ + f_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p, q}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "Int[x^(m - n*q)*(a + b*x^n)^p*(d + c*x^n)^q*(e + f*x^n)^r, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_.)^p_.*(c_ + d_.*x_^mn_.)^ q_.*(e_ + f_.*x_^n_.)^r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p, r}, x] && EqQ[mn, -n] && IntegerQ[q]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "Int[x^(m + n*(p + r))*(b + a*x^(-n))^p*(c + d*x^(-n))^ q*(f + e*x^(-n))^r, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*x_^n_.)^p_.*(c_ + d_.*x_^mn_.)^ q_.*(e_ + f_.*x_^n_.)^r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, q}, x] && EqQ[mn, -n] && IntegerQ[p] && IntegerQ[r]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "x^(n*FracPart[q])*(c + d*x^(-n))^FracPart[q]/(d + c*x^n)^FracPart[q]* Int[x^(m - n*q)*(a + b*x^n)^p*(d + c*x^n)^q*(e + f*x^n)^r, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*x_^n_.)^p_.*(c_ + d_.*x_^mn_.)^ q_*(e_ + f_.*x_^n_.)^r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p, q, r}, x] && EqQ[mn, -n] && Not[IntegerQ[q]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "g^IntPart[m]*(g*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*x^n)^p*(c + d*x^(-n))^q*(e + f*x^n)^r, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_*x_)^m_*(a_ + b_.*x_^n_.)^p_.*(c_ + d_.*x_^mn_.)^ q_.*(e_ + f_.*x_^n_.)^r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r}, x] && EqQ[mn, -n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "Unintegrable[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^m_.*(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^ q_.*(e_ + f_.*x_^n_)^r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "u^m/(Coefficient[v, x, 1]*v^m)* Subst[Int[x^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x, v]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*(a_. + b_.*v_^n_)^p_.*(c_. + d_.*v_^n_)^ q_.*(e_ + f_.*v_^n_)^r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p, q, r}, x] && LinearPairQ[u, v, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e1*e2 + f1*f2*x^n)^r, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^m_.*(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^ q_.*(e1_ + f1_.*x_^n2_.)^r_.*(e2_ + f2_.*x_^n2_.)^r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e1, f1, e2, f2, g, m, n, p, q, r}, x] && EqQ[n2, n/2] && EqQ[e2*f1 + e1*f2, 0] && (IntegerQ[r] || GtQ[e1, 0] && GtQ[e2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
       "filename": "1.1.3.6 (g x)^m (a+b x^n)^p (c+d x^n)^q (e+f x^n)^r.m",
            "rhs": "(e1 + f1*x^(n/2))^ FracPart[r]*(e2 + f2*x^(n/2))^FracPart[r]/(e1*e2 + f1*f2*x^n)^ FracPart[r]* Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e1*e2 + f1*f2*x^n)^r, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^m_.*(a_ + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^ q_.*(e1_ + f1_.*x_^n2_.)^r_.*(e2_ + f2_.*x_^n2_.)^r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e1, f1, e2, f2, g, m, n, p, q, r}, x] && EqQ[n2, n/2] && EqQ[e2*f1 + e1*f2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{n=Denominator[p]}, n/b*Subst[Int[x^(n*p+n-1)*ReplaceAll[Pq,x->-a/b+x^n/b],x],x,(a+b*x)^( 1/n)]]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_+b_.*x_)^p_,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b},x] && PolyQ[Pq,x] && FractionQ[p] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "Int[x*PolynomialQuotient[Pq, x, x]*(a + b*x^n)^p, x] /; FreeQ[{a, b, n, p}, x] && PolyQ[Pq, x] && EqQ[Coeff[Pq, x, 0], 0] && Not[MatchQ[Pq, x^m_.*u_.",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^n_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "IntegerQ[m]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "Int[PolynomialQuotient[Pq, a + b*x^n, x]*(a + b*x^n)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && GeQ[Expon[Pq, x], n] && EqQ[PolynomialRemainder[Pq, a + b*x^n, x], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "Module[{q = Expon[Pq, x], i}, (a + b*x^n)^p* Sum[Coeff[Pq, x, i]*x^(i + 1)/(n*p + i + 1), {i, 0, q}] + a*n*p* Int[(a + b*x^n)^(p - 1)* Sum[Coeff[Pq, x, i]*x^i/(n*p + i + 1), {i, 0, q}], x]]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^n_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[(n - 1)/2, 0] && GtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "Module[{q = Expon[Pq, x], i}, (a*Coeff[Pq, x, q] - b*x*ExpandToSum[Pq - Coeff[Pq, x, q]*x^q, x])*(a + b*x^n)^(p + 1)/(a*b*n*(p + 1)) + 1/(a*n*(p + 1))* Int[Sum[(n*(p + 1) + i + 1)*Coeff[Pq, x, i]*x^i, {i, 0, q - 1}]*(a + b*x^n)^(p + 1), x] /; q == n - 1]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "-x*Pq*(a + b*x^n)^(p + 1)/(a*n*(p + 1)) + 1/(a*n*(p + 1))* Int[ExpandToSum[n*(p + 1)*Pq + D[x*Pq, x], x]*(a + b*x^n)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^n_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[Expon[Pq, x], n - 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{d = Coeff[P4, x, 0], e = Coeff[P4, x, 1], f = Coeff[P4, x, 3], g = Coeff[P4, x, 4]}, -(a*f + 2*a*g*x - b*e*x^2)/(2*a*b*Sqrt[a + b*x^4]) /; EqQ[b*d + a*g, 0]]",
     "rulenumber": 0,
            "lhs": "Int[P4_/(a_ + b_.*x_^4)^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[P4, x, 4] && EqQ[Coeff[P4, x, 2], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{d = Coeff[P6, x, 0], e = Coeff[P6, x, 2], f = Coeff[P6, x, 3], g = Coeff[P6, x, 4], h = Coeff[P6, x, 6]}, -(a*f - 2*b*d*x - 2*a*h*x^3)/(2*a*b*Sqrt[a + b*x^4]) /; EqQ[b*e - 3*a*h, 0] && EqQ[b*d + a*g, 0]]",
     "rulenumber": 0,
            "lhs": "Int[P6_/(a_ + b_.*x_^4)^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[P6, x, 6] && EqQ[Coeff[P6, x, 1], 0] && EqQ[Coeff[P6, x, 5], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x]}, -x* R*(a + b*x^n)^(p + 1)/(a*n*(p + 1)* b^(Floor[(q - 1)/n] + 1)) + 1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))* Int[(a + b*x^n)^(p + 1)* ExpandToSum[a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x]] /; GeQ[q, n]]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^n_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "B^3/b*Int[1/(A^2 - A*B*x + B^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*x_)/(a_ + b_.*x_^3), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B}, x] && EqQ[a*B^3 - b*A^3, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{r = Numerator[Rt[a/b, 3]], s = Denominator[Rt[a/b, 3]]}, -r*(B*r - A*s)/(3*a*s)*Int[1/(r + s*x), x] + r/(3*a*s)* Int[(r*(B*r + 2*A*s) + s*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x]]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*x_)/(a_ + b_.*x_^3), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[a/b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{r = Numerator[Rt[-a/b, 3]], s = Denominator[Rt[-a/b, 3]]}, r*(B*r + A*s)/(3*a*s)*Int[1/(r - s*x), x] - r/(3*a*s)* Int[(r*(B*r - 2*A*s) - s*(B*r + A*s)*x)/(r^2 + r*s*x + s^2*x^2), x]]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*x_)/(a_ + b_.*x_^3), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && NegQ[a/b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2]}, -C^2/b*Int[1/(B - C*x), x] /; EqQ[B^2 - A*C, 0] && EqQ[b*B^3 + a*C^3, 0]]",
     "rulenumber": 0,
            "lhs": "Int[P2_/(a_ + b_.*x_^3), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[P2, x, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2]}, With[{q = a^(1/3)/b^(1/3)}, C/b*Int[1/(q + x), x] + (B + C*q)/b* Int[1/(q^2 - q*x + x^2), x]] /; EqQ[A*b^(2/3) - a^(1/3)*b^(1/3)*B - 2*a^(2/3)*C, 0]]",
     "rulenumber": 0,
            "lhs": "Int[P2_/(a_ + b_.*x_^3), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[P2, x, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2]}, With[{q = (-a)^(1/3)/(-b)^(1/3)}, C/b*Int[1/(q + x), x] + (B + C*q)/b* Int[1/(q^2 - q*x + x^2), x]] /; EqQ[A*(-b)^(2/3) - (-a)^(1/3)*(-b)^(1/3)*B - 2*(-a)^(2/3)*C, 0]]",
     "rulenumber": 0,
            "lhs": "Int[P2_/(a_ + b_.*x_^3), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[P2, x, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2]}, With[{q = (-a)^(1/3)/b^(1/3)}, -C/b* Int[1/(q - x), x] + (B - C*q)/b* Int[1/(q^2 + q*x + x^2), x]] /; EqQ[A*b^(2/3) + (-a)^(1/3)*b^(1/3)*B - 2*(-a)^(2/3)*C, 0]]",
     "rulenumber": 0,
            "lhs": "Int[P2_/(a_ + b_.*x_^3), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[P2, x, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2]}, With[{q = a^(1/3)/(-b)^(1/3)}, -C/b* Int[1/(q - x), x] + (B - C*q)/b* Int[1/(q^2 + q*x + x^2), x]] /; EqQ[A*(-b)^(2/3) + a^(1/3)*(-b)^(1/3)*B - 2*a^(2/3)*C, 0]]",
     "rulenumber": 0,
            "lhs": "Int[P2_/(a_ + b_.*x_^3), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[P2, x, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2]}, With[{q = (a/b)^(1/3)}, C/b*Int[1/(q + x), x] + (B + C*q)/b* Int[1/(q^2 - q*x + x^2), x]] /; EqQ[A - (a/b)^(1/3)*B - 2*(a/b)^(2/3)*C, 0]]",
     "rulenumber": 0,
            "lhs": "Int[P2_/(a_ + b_.*x_^3), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[P2, x, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2]}, With[{q = Rt[a/b, 3]}, C/b*Int[1/(q + x), x] + (B + C*q)/b* Int[1/(q^2 - q*x + x^2), x]] /; EqQ[A - Rt[a/b, 3]*B - 2*Rt[a/b, 3]^2*C, 0]]",
     "rulenumber": 0,
            "lhs": "Int[P2_/(a_ + b_.*x_^3), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[P2, x, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2]}, With[{q = (-a/b)^(1/3)}, -C/b*Int[1/(q - x), x] + (B - C*q)/b* Int[1/(q^2 + q*x + x^2), x]] /; EqQ[A + (-a/b)^(1/3)*B - 2*(-a/b)^(2/3)*C, 0]]",
     "rulenumber": 0,
            "lhs": "Int[P2_/(a_ + b_.*x_^3), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[P2, x, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2]}, With[{q = Rt[-a/b, 3]}, -C/b*Int[1/(q - x), x] + (B - C*q)/b* Int[1/(q^2 + q*x + x^2), x]] /; EqQ[A + Rt[-a/b, 3]*B - 2*Rt[-a/b, 3]^2*C, 0]]",
     "rulenumber": 0,
            "lhs": "Int[P2_/(a_ + b_.*x_^3), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[P2, x, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2]}, Int[(A + B*x)/(a + b*x^3), x] + C*Int[x^2/(a + b*x^3), x] /; EqQ[a*B^3 - b*A^3, 0] || Not[RationalQ[a/b]]]",
     "rulenumber": 0,
            "lhs": "Int[P2_/(a_ + b_.*x_^3), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[P2, x, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2]}, With[{q = (a/b)^(1/3)}, q^2/a*Int[(A + C*q*x)/(q^2 - q*x + x^2), x]] /; EqQ[A - B*(a/b)^(1/3) + C*(a/b)^(2/3), 0]]",
     "rulenumber": 0,
            "lhs": "Int[P2_/(a_ + b_.*x_^3), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[P2, x, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2]}, With[{q = (-a/b)^(1/3)}, q/a*Int[(A*q + (A + B*q)*x)/(q^2 + q*x + x^2), x]] /; EqQ[A + B*(-a/b)^(1/3) + C*(-a/b)^(2/3), 0]]",
     "rulenumber": 0,
            "lhs": "Int[P2_/(a_ + b_.*x_^3), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[P2, x, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2], q = (a/b)^(1/3)}, q*(A - B*q + C*q^2)/(3*a)*Int[1/(q + x), x] + q/(3*a)* Int[(q*(2*A + B*q - C*q^2) - (A - B*q - 2*C*q^2)*x)/(q^2 - q*x + x^2), x] /; NeQ[a*B^3 - b*A^3, 0] && NeQ[A - B*q + C*q^2, 0]]",
     "rulenumber": 0,
            "lhs": "Int[P2_/(a_ + b_.*x_^3), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[P2, x, 2] && GtQ[a/b, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2], q = (-a/b)^(1/3)}, q*(A + B*q + C*q^2)/(3*a)*Int[1/(q - x), x] + q/(3*a)* Int[(q*(2*A - B*q - C*q^2) + (A + B*q - 2*C*q^2)*x)/(q^2 + q*x + x^2), x] /; NeQ[a*B^3 - b*A^3, 0] && NeQ[A + B*q + C*q^2, 0]]",
     "rulenumber": 0,
            "lhs": "Int[P2_/(a_ + b_.*x_^3), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[P2, x, 2] && LtQ[a/b, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{v = Sum[x^ii*(Coeff[Pq, x, ii] + Coeff[Pq, x, n/2 + ii]*x^(n/2))/(a + b*x^n), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]]",
     "rulenumber": 0,
            "lhs": "Int[Pq_/(a_ + b_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && Expon[Pq, x] < n"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{r = Numer[Simplify[(1 - Sqrt[3])*d/c]], s = Denom[Simplify[(1 - Sqrt[3])*d/c]]}, 2*d*s^3*Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x)) - 3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)* Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/ (r^2*Sqrt[a + b*x^3]* Sqrt[s*(s + r*x)/((1 + Sqrt[3])*s + r*x)^2])* EllipticE[ ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]]]",
     "rulenumber": 0,
            "lhs": "Int[(c_ + d_.*x_)/Sqrt[a_ + b_.*x_^3], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && PosQ[a] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, (c*r - (1 - Sqrt[3])*d*s)/r*Int[1/Sqrt[a + b*x^3], x] + d/r*Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x]]",
     "rulenumber": 0,
            "lhs": "Int[(c_ + d_.*x_)/Sqrt[a_ + b_.*x_^3], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && PosQ[a] && NeQ[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{r = Numer[Simplify[(1 + Sqrt[3])*d/c]], s = Denom[Simplify[(1 + Sqrt[3])*d/c]]}, 2*d*s^3*Sqrt[a + b*x^3]/(a*r^2*((1 - Sqrt[3])*s + r*x)) + 3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)* Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/ (r^2*Sqrt[a + b*x^3]* Sqrt[-s*(s + r*x)/((1 - Sqrt[3])*s + r*x)^2])* EllipticE[ ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]]]",
     "rulenumber": 0,
            "lhs": "Int[(c_ + d_.*x_)/Sqrt[a_ + b_.*x_^3], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, (c*r - (1 + Sqrt[3])*d*s)/r*Int[1/Sqrt[a + b*x^3], x] + d/r*Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x]]",
     "rulenumber": 0,
            "lhs": "Int[(c_ + d_.*x_)/Sqrt[a_ + b_.*x_^3], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NegQ[a] && NeQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, (1 + Sqrt[3])*d*s^3*x* Sqrt[a + b*x^6]/(2*a*r^2*(s + (1 + Sqrt[3])*r*x^2)) - 3^(1/4)*d*s*x*(s + r*x^2)* Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]/ (2*r^2* Sqrt[(r*x^2*(s + r*x^2))/(s + (1 + Sqrt[3])*r*x^2)^2]* Sqrt[a + b*x^6])* EllipticE[ ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r* x^2)], (2 + Sqrt[3])/4]]",
     "rulenumber": 0,
            "lhs": "Int[(c_ + d_.*x_^4)/Sqrt[a_ + b_.*x_^6], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[2*Rt[b/a, 3]^2*c - (1 - Sqrt[3])*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{q = Rt[b/a, 3]}, (2*c*q^2 - (1 - Sqrt[3])*d)/(2*q^2)*Int[1/Sqrt[a + b*x^6], x] + d/(2*q^2)*Int[(1 - Sqrt[3] + 2*q^2*x^4)/Sqrt[a + b*x^6], x]]",
     "rulenumber": 0,
            "lhs": "Int[(c_ + d_.*x_^4)/Sqrt[a_ + b_.*x_^6], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[2*Rt[b/a, 3]^2*c - (1 - Sqrt[3])*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "-c*d*x^3*Sqrt[-(c - d*x^2)^2/(c*d*x^2)]* Sqrt[-d^2*(a + b*x^8)/(b*c^2*x^4)]/(Sqrt[2 + Sqrt[2]]*(c - d*x^2)* Sqrt[a + b*x^8])* EllipticF[ ArcSin[1/2* Sqrt[(Sqrt[2]*c^2 + 2*c*d*x^2 + Sqrt[2]*d^2*x^4)/(c*d* x^2)]], -2*(1 - Sqrt[2])]",
     "rulenumber": 0,
            "lhs": "Int[(c_ + d_.*x_^2)/Sqrt[a_ + b_.*x_^8], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[b*c^4 - a*d^4, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "(d + Rt[b/a, 4]*c)/(2*Rt[b/a, 4])* Int[(1 + Rt[b/a, 4]*x^2)/Sqrt[a + b*x^8], x] - (d - Rt[b/a, 4]*c)/(2*Rt[b/a, 4])* Int[(1 - Rt[b/a, 4]*x^2)/Sqrt[a + b*x^8], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_ + d_.*x_^2)/Sqrt[a_ + b_.*x_^8], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c^4 - a*d^4, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "Coeff[Pq, x, 0]*Int[1/(x*Sqrt[a + b*x^n]), x] + Int[ExpandToSum[(Pq - Coeff[Pq, x, 0])/x, x]/Sqrt[a + b*x^n], x]",
     "rulenumber": 0,
            "lhs": "Int[Pq_/(x_*Sqrt[a_ + b_.*x_^n_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && NeQ[Coeff[Pq, x, 0], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "Module[{q = Expon[Pq, x], j, k}, Int[Sum[ x^j*Sum[Coeff[Pq, x, j + k*n/2]*x^(k*n/2), {k, 0, 2*(q - j)/n + 1}]*(a + b*x^n)^p, {j, 0, n/2 - 1}], x]]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && Not[PolyQ[Pq, x^(n/2)]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "Coeff[Pq, x, n - 1]*Int[x^(n - 1)*(a + b*x^n)^p, x] + Int[ExpandToSum[Pq - Coeff[Pq, x, n - 1]*x^(n - 1), x]*(a + b*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && Expon[Pq, x] == n - 1"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "Int[ExpandIntegrand[Pq/(a + b*x^n), x], x]",
     "rulenumber": 0,
            "lhs": "Int[Pq_/(a_ + b_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[Pq, x] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{q = Expon[Pq, x]}, With[{Pqq = Coeff[Pq, x, q]}, Pqq*x^(q - n + 1)*(a + b*x^n)^(p + 1)/(b*(q + n*p + 1)) + 1/(b*(q + n*p + 1))* Int[ExpandToSum[ b*(q + n*p + 1)*(Pq - Pqq*x^q) - a*Pqq*(q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x]] /; NeQ[q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || IntegerQ[p + (q + 1)/(2*n)])]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{q = Expon[Pq, x]}, -Subst[ Int[ExpandToSum[x^q*ReplaceAll[Pq, x -> x^(-1)], x]*(a + b*x^(-n))^p/x^(q + 2), x], x, 1/x]]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{g = Denominator[n]}, g*Subst[ Int[x^(g - 1)*ReplaceAll[Pq, x -> x^g]*(a + b*x^(g*n))^p, x], x, x^(1/g)]]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "A*Int[(a + b*x^n)^p, x] + B*Int[x^m*(a + b*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*x_^m_.)*(a_ + b_.*x_^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B, m, n, p}, x] && EqQ[m - n + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{A = Coeff[P3, x^(n/2), 0], B = Coeff[P3, x^(n/2), 1], C = Coeff[P3, x^(n/2), 2], D = Coeff[P3, x^(n/2), 3]}, -(x*(b*A - a*C + (b*B - a*D)*x^(n/2))*(a + b*x^n)^(p + 1))/(a*b* n*(p + 1)) - 1/(2*a*b*n*(p + 1))* Int[(a + b*x^n)^(p + 1)* Simp[2*a*C - 2*b*A*(n*(p + 1) + 1) + (a*D*(n + 2) - b*B*(n*(2*p + 3) + 2))*x^(n/2), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[P3_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n}, x] && PolyQ[P3, x^(n/2), 3] && ILtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "1/Coeff[v, x, 1]* Subst[Int[SubstFor[v, Pq, x]*(a + b*x^n)^p, x], x, v]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*v_^n_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && PolyQ[Pq, v^n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "Int[Pq*(a1*a2 + b1*b2*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a1_ + b1_.*x_^n_.)^p_.*(a2_ + b2_.*x_^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, n, p}, x] && PolyQ[Pq, x] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[p] || GtQ[a1, 0] && GtQ[a2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "(a1 + b1*x^n)^ FracPart[p]*(a2 + b2*x^n)^FracPart[p]/(a1*a2 + b1*b2*x^(2*n))^ FracPart[p]* Int[Pq*(a1*a2 + b1*b2*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a1_ + b1_.*x_^n_.)^p_.*(a2_ + b2_.*x_^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, n, p}, x] && PolyQ[Pq, x] && EqQ[a2*b1 + a1*b2, 0] && Not[EqQ[n, 1] && LinearQ[Pq, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "e*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(p + 1)/(a*c)",
     "rulenumber": 0,
            "lhs": "Int[(e_ + f_.*x_^n_. + g_.*x_^n2_.)*(a_ + b_.*x_^n_.)^ p_.*(c_ + d_.*x_^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[n2, 2*n] && EqQ[a*c*f - e*(b*c + a*d)*(n*(p + 1) + 1), 0] && EqQ[a*c*g - b*d*e*(2*n*(p + 1) + 1), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "e*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(p + 1)/(a*c)",
     "rulenumber": 0,
            "lhs": "Int[(e_ + g_.*x_^n2_.)*(a_ + b_.*x_^n_.)^p_.*(c_ + d_.*x_^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, g, n, p}, x] && EqQ[n2, 2*n] && EqQ[n*(p + 1) + 1, 0] && EqQ[a*c*g - b*d*e*(2*n*(p + 1) + 1), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "A*Int[(a + b*x^n)^p*(c + d*x^n)^q, x] + B*Int[x^m*(a + b*x^n)^p*(c + d*x^n)^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*x_^m_.)*(a_. + b_.*x_^n_)^p_.*(c_ + d_.*x_^n_)^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, A, B, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m - n + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.7 P(x) (a+b x^n)^p.m",
       "filename": "1.1.3.7 P(x) (a+b x^n)^p.m",
            "rhs": "With[{k = Denominator[n]}, k/d*Subst[ Int[SimplifyIntegrand[ x^(k - 1)*ReplaceAll[Px, x -> x^k/d - c/d]^q*(a + b*x^(k*n))^p, x], x], x, (c + d*x)^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[Px_^q_.*(a_. + b_.*(c_ + d_.*x_)^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, p}, x] && PolynomialQ[Px, x] && IntegerQ[q] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "-(2*a*g + 4*a*h*x^(n/4) - 2*c*f*x^(n/2))/(a*c*n* Sqrt[a + c*x^n])",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(e_ + f_.*x_^q_. + g_.*x_^r_. + h_.*x_^n_.)/(a_ + c_.*x_^n_.)^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, e, f, g, h, m, n}, x] && EqQ[q, n/4] && EqQ[r, 3*n/4] && EqQ[4*m - n + 4, 0] && EqQ[c*e + a*h, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "(d*x)^m/x^m* Int[x^m*(e + f*x^(n/4) + g*x^((3*n)/4) + h*x^n)/(a + c*x^n)^(3/2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_*x_)^ m_.*(e_ + f_.*x_^q_. + g_.*x_^r_. + h_.*x_^n_.)/(a_ + c_.*x_^n_.)^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, h, m, n}, x] && EqQ[4*m - n + 4, 0] && EqQ[q, n/4] && EqQ[r, 3*n/4] && EqQ[c*e + a*h, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{n = Denominator[p]}, n/b*Subst[ Int[x^(n*p + n - 1)*(-a*c/b + c*x^n/b)^m* ReplaceAll[Pq, x -> -a/b + x^n/b], x], x, (a + b*x)^(1/n)]]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_*Pq_*(a_ + b_.*x_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && FractionQ[p] && ILtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "1/(m + 1)* Subst[Int[ SubstFor[x^(m + 1), Pq, x]*(a + b*x^Simplify[n/(m + 1)])^p, x], x, x^(m + 1)]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Pq_*(a_ + b_.*x_^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n, p}, x] && NeQ[m, -1] && IGtQ[Simplify[n/(m + 1)], 0] && PolyQ[Pq, x^(m + 1)]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x^n)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*Pq_*(a_ + b_.*x_^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*SubstFor[x^n, Pq, x]*(a + b*x)^p, x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Pq_*(a_ + b_.*x_^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n, p}, x] && PolyQ[Pq, x^n] && IntegerQ[Simplify[(m + 1)/n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "c^IntPart[m]*(c*x)^FracPart[m]/x^FracPart[m]* Int[x^m*Pq*(a + b*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_*x_)^m_.*Pq_*(a_ + b_.*x_^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n, p}, x] && PolyQ[Pq, x^n] && IntegerQ[Simplify[(m + 1)/n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "Pq*(a + b*x^n)^(p + 1)/(b*n*(p + 1)) - 1/(b*n*(p + 1))*Int[D[Pq, x]*(a + b*x^n)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Pq_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && PolyQ[Pq, x] && EqQ[m - n + 1, 0] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "1/d*Int[(d*x)^(m + 1)*PolynomialQuotient[Pq, x, x]*(a + b*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*Pq_*(a_ + b_.*x_^n_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, m, n, p}, x] && PolyQ[Pq, x] && EqQ[Coeff[Pq, x, 0], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "Module[{u = IntHide[x^m*Pq, x]}, u*(a + b*x^n)^p - b*n*p*Int[ x^(m + n)*(a + b*x^n)^(p - 1)*ExpandToSum[u/x^(m + 1), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Pq_*(a_ + b_.*x_^n_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && GtQ[p, 0] && LtQ[m + Expon[Pq, x] + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "Module[{q = Expon[Pq, x], i}, (c*x)^m*(a + b*x^n)^p* Sum[Coeff[Pq, x, i]*x^(i + 1)/(m + n*p + i + 1), {i, 0, q}] + a*n*p* Int[(c*x)^m*(a + b*x^n)^(p - 1)* Sum[Coeff[Pq, x, i]*x^i/(m + n*p + i + 1), {i, 0, q}], x]]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*Pq_*(a_ + b_.*x_^n_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[(n - 1)/2, 0] && GtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{e = Coeff[P4, x, 0], f = Coeff[P4, x, 1], h = Coeff[P4, x, 4]}, -(f - 2*h*x^3)/(2*b*Sqrt[a + b*x^4]) /; EqQ[b*e - 3*a*h, 0]]",
     "rulenumber": 0,
            "lhs": "Int[x_^2*P4_/(a_ + b_.*x_^4)^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[P4, x, 4] && EqQ[Coeff[P4, x, 2], 0] && EqQ[Coeff[P4, x, 3], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{q = m + Expon[Pq, x]}, Module[{Q = PolynomialQuotient[b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x]}, -x* R*(a + b*x^n)^(p + 1)/(a*n*(p + 1)* b^(Floor[(q - 1)/n] + 1)) + 1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))* Int[(a + b*x^n)^(p + 1)* ExpandToSum[a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x]] /; GeQ[q, n]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Pq_*(a_ + b_.*x_^n_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[a*b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[a*b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x], i}, -x* R*(a + b*x^n)^(p + 1)/(a^2*n*(p + 1)* b^(Floor[(q - 1)/n] + 1)) + 1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))* Int[x^m*(a + b*x^n)^(p + 1)* ExpandToSum[ n*(p + 1)*x^(-m)*Q + Sum[(n*(p + 1) + i + 1)/a*Coeff[R, x, i]*x^(i - m), {i, 0, n - 1}], x], x]]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*Pq_*(a_ + b_.*x_^n_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{g = GCD[m + 1, n]}, 1/g*Subst[ Int[x^((m + 1)/g - 1)* ReplaceAll[Pq, x -> x^(1/g)]*(a + b*x^(n/g))^p, x], x, x^g] /; g != 1]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Pq_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, p}, x] && PolyQ[Pq, x^n] && IGtQ[n, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{v = Sum[(c*x)^(m + ii)*(Coeff[Pq, x, ii] + Coeff[Pq, x, n/2 + ii]*x^(n/2))/(c^ii*(a + b*x^n)), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*Pq_/(a_ + b_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && Expon[Pq, x] < n"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "Coeff[Pq, x, 0]*Int[1/(x*Sqrt[a + b*x^n]), x] + Int[ExpandToSum[(Pq - Coeff[Pq, x, 0])/x, x]/Sqrt[a + b*x^n], x]",
     "rulenumber": 0,
            "lhs": "Int[Pq_/(x_*Sqrt[a_ + b_.*x_^n_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && NeQ[Coeff[Pq, x, 0], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "Module[{q = Expon[Pq, x], j, k}, Int[Sum[(c*x)^(m + j)/c^j* Sum[Coeff[Pq, x, j + k*n/2]*x^(k*n/2), {k, 0, 2*(q - j)/n + 1}]*(a + b*x^n)^p, {j, 0, n/2 - 1}], x]]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*Pq_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && Not[PolyQ[Pq, x^(n/2)]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "Int[ExpandIntegrand[(c*x)^m*Pq/(a + b*x^n), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*Pq_/(a_ + b_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IntegerQ[n] && Not[IGtQ[m, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{Pq0 = Coeff[Pq, x, 0]}, Pq0*(c*x)^(m + 1)*(a + b*x^n)^(p + 1)/(a*c*(m + 1)) + 1/(2*a*c*(m + 1))* Int[(c*x)^(m + 1)* ExpandToSum[ 2*a*(m + 1)*(Pq - Pq0)/x - 2*b*Pq0*(m + n*(p + 1) + 1)*x^(n - 1), x]*(a + b*x^n)^p, x] /; NeQ[Pq0, 0]]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_*Pq_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[m, -1] && LeQ[n - 1, Expon[Pq, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{q = Expon[Pq, x]}, With[{Pqq = Coeff[Pq, x, q]}, Pqq*(c*x)^(m + q - n + 1)*(a + b*x^n)^(p + 1)/(b* c^(q - n + 1)*(m + q + n*p + 1)) + 1/(b*(m + q + n*p + 1))* Int[(c*x)^m* ExpandToSum[ b*(m + q + n*p + 1)*(Pq - Pqq*x^q) - a*Pqq*(m + q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x]] /; NeQ[m + q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || IntegerQ[p + (q + 1)/(2*n)])]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*Pq_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{q = Expon[Pq, x]}, -Subst[ Int[ExpandToSum[x^q*ReplaceAll[Pq, x -> x^(-1)], x]*(a + b*x^(-n))^p/x^(m + q + 2), x], x, 1/x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Pq_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && ILtQ[n, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{g = Denominator[m], q = Expon[Pq, x]}, -g/c* Subst[Int[ ExpandToSum[x^(g*q)*ReplaceAll[Pq, x -> c^(-1)*x^(-g)], x]* (a + b*c^(-n)*x^(-g*n))^p/x^(g*(m + q + 1) + 1), x], x, 1/(c*x)^(1/g)]]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*Pq_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && ILtQ[n, 0] && FractionQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{q = Expon[Pq, x]}, -(c*x)^m*(x^(-1))^m* Subst[Int[ ExpandToSum[x^q*ReplaceAll[Pq, x -> x^(-1)], x]*(a + b*x^(-n))^p/ x^(m + q + 2), x], x, 1/x]]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_*Pq_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && ILtQ[n, 0] && Not[RationalQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "With[{g = Denominator[n]}, g*Subst[ Int[x^(g*(m + 1) - 1)*ReplaceAll[Pq, x -> x^g]*(a + b*x^(g*n))^p, x], x, x^(1/g)]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Pq_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, p}, x] && PolyQ[Pq, x] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "c^IntPart[m]*(c*x)^FracPart[m]/x^FracPart[m]* Int[x^m*Pq*(a + b*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_*x_)^m_*Pq_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "1/(m + 1)* Subst[Int[ ReplaceAll[SubstFor[x^n, Pq, x], x -> x^Simplify[n/(m + 1)]]*(a + b*x^Simplify[n/(m + 1)])^p, x], x, x^(m + 1)]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Pq_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n, p}, x] && PolyQ[Pq, x^n] && IntegerQ[Simplify[n/(m + 1)]] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "c^IntPart[m]*(c*x)^FracPart[m]/x^FracPart[m]* Int[x^m*Pq*(a + b*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_*x_)^m_*Pq_*(a_ + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n, p}, x] && PolyQ[Pq, x^n] && IntegerQ[Simplify[n/(m + 1)]] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x^n)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*Pq_*(a_ + b_.*x_^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) && Not[IGtQ[m, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "u^m/(Coeff[v, x, 1]*v^m)* Subst[Int[x^m*SubstFor[v, Pq, x]*(a + b*x^n)^p, x], x, v]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*Pq_*(a_ + b_.*v_^n_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n, p}, x] && LinearPairQ[u, v, x] && PolyQ[Pq, v^n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "Int[(c*x)^m*Pq*(a1*a2 + b1*b2*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*Pq_*(a1_ + b1_.*x_^n_.)^p_.*(a2_ + b2_.*x_^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, m, n, p}, x] && PolyQ[Pq, x] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[p] || GtQ[a1, 0] && GtQ[a2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "(a1 + b1*x^n)^ FracPart[p]*(a2 + b2*x^n)^FracPart[p]/(a1*a2 + b1*b2*x^(2*n))^ FracPart[p]* Int[(c*x)^m*Pq*(a1*a2 + b1*b2*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*Pq_*(a1_ + b1_.*x_^n_.)^p_.*(a2_ + b2_.*x_^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a1, b1, a2, b2, c, m, n, p}, x] && PolyQ[Pq, x] && EqQ[a2*b1 + a1*b2, 0] && Not[EqQ[n, 1] && LinearQ[Pq, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "e*(h*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(p + 1)/(a*c* h*(m + 1))",
     "rulenumber": 0,
            "lhs": "Int[(h_.*x_)^m_.*(e_ + f_.*x_^n_. + g_.*x_^n2_.)*(a_ + b_.*x_^n_.)^ p_.*(c_ + d_.*x_^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[a*c*f*(m + 1) - e*(b*c + a*d)*(m + n*(p + 1) + 1), 0] && EqQ[a*c*g*(m + 1) - b*d*e*(m + 2*n*(p + 1) + 1), 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
       "filename": "1.1.3.8 P(x) (c x)^m (a+b x^n)^p.m",
            "rhs": "e*(h*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(p + 1)/(a*c* h*(m + 1))",
     "rulenumber": 0,
            "lhs": "Int[(h_.*x_)^m_.*(e_ + g_.*x_^n2_.)*(a_ + b_.*x_^n_.)^ p_.*(c_ + d_.*x_^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, g, h, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[m + n*(p + 1) + 1, 0] && EqQ[a*c*g*(m + 1) - b*d*e*(m + 2*n*(p + 1) + 1), 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.4 Improper/1.1.4.1 (a x^j+b x^n)^p.m",
       "filename": "1.1.4.1 (a x^j+b x^n)^p.m",
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            "rhs": "-2*Sqrt[a*x^j + b*x^n]/(b*(n - 2)*x^(n - 1)) - a*(2*n - j - 2)/(b*(n - 2))* Int[1/(x^(n - j)*Sqrt[a*x^j + b*x^n]), x]",
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            "rhs": " x*(a*x^j+b*x^n)^p/(p*(n-j)*((a*x^j+b*x^n)/(b*x^n))^p)* Hypergeometric2F1[-p,-p,1-p,-a/(b*x^(n-j))]",
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            "rhs": " x*(a*x^j+b*x^n)^p/((j*p+1)*((a*x^j+b*x^n)/(a*x^j))^p)* Hypergeometric2F1[-p,(j*p+1)/(n-j),(j*p+1)/(n-j)+1,-b*x^(n-j)/a]",
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        "comment": false,
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       "filename": "1.1.4.1 (a x^j+b x^n)^p.m",
            "rhs": "(a*x^j + b*x^n)^ FracPart[p]/(x^(j*FracPart[p])*(a + b*x^(n - j))^FracPart[p])* Int[x^(j*p)*(a + b*x^(n - j))^p, x]",
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         "givens": "FreeQ[{a, b, j, n, p}, x] && Not[IntegerQ[p]] && NeQ[n, j] && PosQ[n - j]"
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         "givens": "FreeQ[{a, b, j, n, p}, x] && LinearQ[u, x] && NeQ[u, x]"
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         "givens": "FreeQ[{a, b, c, j, m, n, p}, x] && Not[IntegerQ[p]] && NeQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])"
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            "rhs": "-c^(j - 1)*(c*x)^(m - j + 1)*(a*x^j + b*x^n)^(p + 1)/(a*(n - j)*(p + 1)) + c^j*(m + n*p + n - j + 1)/(a*(n - j)*(p + 1))* Int[(c*x)^(m - j)*(a*x^j + b*x^n)^(p + 1), x]",
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         "givens": "FreeQ[{a, b, c, j, m, n, p}, x] && Not[IntegerQ[p]] && NeQ[n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0]"
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            "rhs": "c^(n - 1)*(c*x)^(m - n + 1)*(a*x^j + b*x^n)^(p + 1)/(b*(m + n*p + 1)) - a*c^(n - j)*(m + j*p - n + j + 1)/(b*(m + n*p + 1))* Int[(c*x)^(m - (n - j))*(a*x^j + b*x^n)^p, x]",
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            "lhs": "Int[(c_.*x_)^m_.*(a_.*x_^j_. + b_.*x_^n_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, p}, x] && Not[IntegerQ[p]] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && GtQ[m + j*p + 1 - n + j, 0] && NeQ[m + n*p + 1, 0]"
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       "filename": "1.1.4.2 (c x)^m (a x^j+b x^n)^p.m",
            "rhs": "c^(j - 1)*(c*x)^(m - j + 1)*(a*x^j + b*x^n)^(p + 1)/(a*(m + j*p + 1)) - b*(m + n*p + n - j + 1)/(a*c^(n - j)*(m + j*p + 1))* Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x]",
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            "lhs": "Int[(c_.*x_)^m_.*(a_.*x_^j_. + b_.*x_^n_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, p}, x] && Not[IntegerQ[p]] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[m + j*p + 1, 0]"
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       "filename": "1.1.4.2 (c x)^m (a x^j+b x^n)^p.m",
            "rhs": "1/(m + 1)* Subst[Int[(a*x^Simplify[j/(m + 1)] + b*x^Simplify[n/(m + 1)])^p, x], x, x^(m + 1)]",
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            "lhs": "Int[x_^m_.*(a_.*x_^j_. + b_.*x_^n_)^p_, x_Symbol]",
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         "givens": "FreeQ[{a, b, j, m, n, p}, x] && Not[IntegerQ[p]] && NeQ[n, j] && IntegerQ[Simplify[j/n]] && NeQ[m, -1] && IntegerQ[Simplify[n/(m + 1)]] && Not[IntegerQ[n]]"
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       "filename": "1.1.4.2 (c x)^m (a x^j+b x^n)^p.m",
            "rhs": "c^IntPart[m]*(c*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a*x^j + b*x^n)^p, x]",
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         "givens": "FreeQ[{a, b, c, j, m, n, p}, x] && Not[IntegerQ[p]] && NeQ[n, j] && IntegerQ[Simplify[j/n]] && NeQ[m, -1] && IntegerQ[Simplify[n/(m + 1)]] && Not[IntegerQ[n]]"
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       "filename": "1.1.4.2 (c x)^m (a x^j+b x^n)^p.m",
            "rhs": "(c*x)^(m + 1)*(a*x^j + b*x^n)^p/(c*p*(n - j)) + a/c^j*Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p - 1), x]",
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            "lhs": "Int[(c_.*x_)^m_.*(a_.*x_^j_. + b_.*x_^n_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, j, m, n}, x] && IGtQ[p + 1/2, 0] && NeQ[n, j] && EqQ[Simplify[m + j*p + 1], 0] && (IntegerQ[j] || GtQ[c, 0])"
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       "filename": "1.1.4.2 (c x)^m (a x^j+b x^n)^p.m",
            "rhs": "-2/(n - j)* Subst[Int[1/(1 - a*x^2), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_./Sqrt[a_.*x_^j_. + b_.*x_^n_.], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.4 Improper/1.1.4.2 (c x)^m (a x^j+b x^n)^p.m",
       "filename": "1.1.4.2 (c x)^m (a x^j+b x^n)^p.m",
            "rhs": "-c^(j - 1)*(c*x)^(m - j + 1)*(a*x^j + b*x^n)^(p + 1)/(a*(n - j)*(p + 1)) + c^j*(m + n*p + n - j + 1)/(a*(n - j)*(p + 1))* Int[(c*x)^(m - j)*(a*x^j + b*x^n)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*(a_.*x_^j_. + b_.*x_^n_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, j, m, n}, x] && ILtQ[p + 1/2, 0] && NeQ[n, j] && EqQ[Simplify[m + j*p + 1], 0] && (IntegerQ[j] || GtQ[c, 0])"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.4 Improper/1.1.4.2 (c x)^m (a x^j+b x^n)^p.m",
       "filename": "1.1.4.2 (c x)^m (a x^j+b x^n)^p.m",
            "rhs": "c^IntPart[m]*(c*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a*x^j + b*x^n)^p, x]",
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            "lhs": "Int[(c_*x_)^m_.*(a_.*x_^j_. + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, j, m, n, p}, x] && IntegerQ[p + 1/2] && NeQ[n, j] && EqQ[Simplify[m + j*p + 1], 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.4 Improper/1.1.4.2 (c x)^m (a x^j+b x^n)^p.m",
       "filename": "1.1.4.2 (c x)^m (a x^j+b x^n)^p.m",
            "rhs": " (a*x^j+b*x^n)^(p+1)/(b*p*(n-j)*x^(n+j*p))*Hypergeometric2F1[1,1,1-p,- a/(b*x^(n-j))]",
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            "lhs": "Int[x_^m_.*(a_.*x_^j_.+b_.*x_^n_.)^p_,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,j,m,n,p},x] && NeQ[n,j] &&  EqQ[m+j*p+1,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.4 Improper/1.1.4.2 (c x)^m (a x^j+b x^n)^p.m",
       "filename": "1.1.4.2 (c x)^m (a x^j+b x^n)^p.m",
            "rhs": " (a*x^j+b*x^n)^(p+1)/(b*(p-1)*(n-j)*x^(2*n+j*(p-1)))*Hypergeometric2F1[ 1,2,2-p,-a/(b*x^(n-j))]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_.*x_^j_.+b_.*x_^n_.)^p_,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,j,m,n,p},x] && NeQ[n,j] &&  EqQ[m+n+(p-1)*j+1,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.4 Improper/1.1.4.2 (c x)^m (a x^j+b x^n)^p.m",
       "filename": "1.1.4.2 (c x)^m (a x^j+b x^n)^p.m",
            "rhs": " (x^(m-j+1)*(a*x^j+b*x^n)^(p+1))/(a*(m+j*p+1))*Hypergeometric2F1[1,(m+ n*p+1)/(n-j)+1,(m+j*p+1)/(n-j)+1,-b*x^(n-j)/a]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_.*x_^j_.+b_.*x_^n_.)^p_,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,j,m,n,p},x] && NeQ[n,j] && NeQ[m+j*p+1,0] &&  NeQ[m+n+(p-1)*j+1,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.4 Improper/1.1.4.2 (c x)^m (a x^j+b x^n)^p.m",
       "filename": "1.1.4.2 (c x)^m (a x^j+b x^n)^p.m",
            "rhs": "c^IntPart[m]*(c*x)^FracPart[m]*(a*x^j + b*x^n)^FracPart[p]/ (x^(FracPart[m] + j*FracPart[p])*(a + b*x^(n - j))^ FracPart[p])* Int[x^(m + j*p)*(a + b*x^(n - j))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*(a_.*x_^j_. + b_.*x_^n_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, j, m, n, p}, x] && Not[IntegerQ[p]] && NeQ[n, j] && PosQ[n - j]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.4 Improper/1.1.4.2 (c x)^m (a x^j+b x^n)^p.m",
       "filename": "1.1.4.2 (c x)^m (a x^j+b x^n)^p.m",
            "rhs": "u^m/(Coefficient[v, x, 1]*v^m)* Subst[Int[x^m*(a*x^j + b*x^n)^p, x], x, v]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*(a_.*v_^j_. + b_.*v_^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, j, m, n, p}, x] && LinearPairQ[u, v, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.4 Improper/1.1.4.3 (e x)^m (a x^j+b x^k)^p (c+d x^n)^q.m",
       "filename": "1.1.4.3 (e x)^m (a x^j+b x^k)^p (c+d x^n)^q.m",
            "rhs": "1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(a*x^Simplify[j/n] + b*x^Simplify[k/n])^p*(c + d*x)^q, x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_.*x_^j_ + b_.*x_^k_.)^p_*(c_ + d_.*x_^n_)^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, j, k, m, n, p, q}, x] && Not[IntegerQ[p]] && NeQ[k, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[k/n]] && IntegerQ[Simplify[(m + 1)/n]] && NeQ[n^2, 1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.4 Improper/1.1.4.3 (e x)^m (a x^j+b x^k)^p (c+d x^n)^q.m",
       "filename": "1.1.4.3 (e x)^m (a x^j+b x^k)^p (c+d x^n)^q.m",
            "rhs": "e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a*x^j + b*x^k)^p*(c + d*x^n)^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_*x_)^m_.*(a_.*x_^j_ + b_.*x_^k_.)^p_*(c_ + d_.*x_^n_.)^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, j, k, m, n, p, q}, x] && Not[IntegerQ[p]] && NeQ[k, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[k/n]] && IntegerQ[Simplify[(m + 1)/n]] && NeQ[n^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.4 Improper/1.1.4.3 (e x)^m (a x^j+b x^k)^p (c+d x^n)^q.m",
       "filename": "1.1.4.3 (e x)^m (a x^j+b x^k)^p (c+d x^n)^q.m",
            "rhs": "c*e^(j - 1)*(e*x)^(m - j + 1)*(a*x^j + b*x^(j + n))^(p + 1)/(a*(m + j*p + 1))",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_.*x_^j_. + b_.*x_^jn_.)^p_*(c_ + d_.*x_^n_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, j, m, n, p}, x] && EqQ[jn, j + n] && Not[IntegerQ[p]] && NeQ[b*c - a*d, 0] && EqQ[a*d*(m + j*p + 1) - b*c*(m + n + p*(j + n) + 1), 0] && (GtQ[e, 0] || IntegersQ[j]) && NeQ[m + j*p + 1, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.4 Improper/1.1.4.3 (e x)^m (a x^j+b x^k)^p (c+d x^n)^q.m",
       "filename": "1.1.4.3 (e x)^m (a x^j+b x^k)^p (c+d x^n)^q.m",
            "rhs": "-e^(j - 1)*(b*c - a*d)*(e*x)^(m - j + 1)*(a*x^j + b*x^(j + n))^(p + 1)/(a*b* n*(p + 1)) - e^j*(a*d*(m + j*p + 1) - b*c*(m + n + p*(j + n) + 1))/(a*b* n*(p + 1))* Int[(e*x)^(m - j)*(a*x^j + b*x^(j + n))^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_.*x_^j_. + b_.*x_^jn_.)^p_*(c_ + d_.*x_^n_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, j, m, n}, x] && EqQ[jn, j + n] && Not[IntegerQ[p]] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[j, 0] && LeQ[j, m] && (GtQ[e, 0] || IntegerQ[j])"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.4 Improper/1.1.4.3 (e x)^m (a x^j+b x^k)^p (c+d x^n)^q.m",
       "filename": "1.1.4.3 (e x)^m (a x^j+b x^k)^p (c+d x^n)^q.m",
            "rhs": "c*e^(j - 1)*(e*x)^(m - j + 1)*(a*x^j + b*x^(j + n))^(p + 1)/(a*(m + j*p + 1)) + (a*d*(m + j*p + 1) - b*c*(m + n + p*(j + n) + 1))/(a* e^n*(m + j*p + 1))* Int[(e*x)^(m + n)*(a*x^j + b*x^(j + n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_.*x_^j_. + b_.*x_^jn_.)^p_*(c_ + d_.*x_^n_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, j, p}, x] && EqQ[jn, j + n] && Not[IntegerQ[p]] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && (LtQ[m + j*p, -1] || IntegersQ[m - 1/2, p - 1/2] && LtQ[p, 0] && LtQ[m, -n*p - 1]) && (GtQ[e, 0] || IntegersQ[j, n]) && NeQ[m + j*p + 1, 0] && NeQ[m - n + j*p + 1, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.4 Improper/1.1.4.3 (e x)^m (a x^j+b x^k)^p (c+d x^n)^q.m",
       "filename": "1.1.4.3 (e x)^m (a x^j+b x^k)^p (c+d x^n)^q.m",
            "rhs": "d*e^(j - 1)*(e*x)^(m - j + 1)*(a*x^j + b*x^(j + n))^(p + 1)/(b*(m + n + p*(j + n) + 1)) - (a*d*(m + j*p + 1) - b*c*(m + n + p*(j + n) + 1))/(b*(m + n + p*(j + n) + 1))* Int[(e*x)^m*(a*x^j + b*x^(j + n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_.*x_^j_. + b_.*x_^jn_.)^p_*(c_ + d_.*x_^n_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, j, m, n, p}, x] && EqQ[jn, j + n] && Not[IntegerQ[p]] && NeQ[b*c - a*d, 0] && NeQ[m + n + p*(j + n) + 1, 0] && (GtQ[e, 0] || IntegerQ[j])"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.4 Improper/1.1.4.3 (e x)^m (a x^j+b x^k)^p (c+d x^n)^q.m",
       "filename": "1.1.4.3 (e x)^m (a x^j+b x^k)^p (c+d x^n)^q.m",
            "rhs": "1/(m + 1)* Subst[Int[(a*x^Simplify[j/(m + 1)] + b*x^Simplify[k/(m + 1)])^ p*(c + d*x^Simplify[n/(m + 1)])^q, x], x, x^(m + 1)]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_.*x_^j_ + b_.*x_^k_.)^p_*(c_ + d_.*x_^n_.)^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, j, k, m, n, p, q}, x] && Not[IntegerQ[p]] && NeQ[k, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[k/n]] && NeQ[m, -1] && IntegerQ[Simplify[n/(m + 1)]] && Not[IntegerQ[n]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.4 Improper/1.1.4.3 (e x)^m (a x^j+b x^k)^p (c+d x^n)^q.m",
       "filename": "1.1.4.3 (e x)^m (a x^j+b x^k)^p (c+d x^n)^q.m",
            "rhs": "e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a*x^j + b*x^k)^p*(c + d*x^n)^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_*x_)^m_.*(a_.*x_^j_ + b_.*x_^k_.)^p_*(c_ + d_.*x_^n_.)^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, j, k, m, n, p, q}, x] && Not[IntegerQ[p]] && NeQ[k, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[k/n]] && NeQ[m, -1] && IntegerQ[Simplify[n/(m + 1)]] && Not[IntegerQ[n]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.4 Improper/1.1.4.3 (e x)^m (a x^j+b x^k)^p (c+d x^n)^q.m",
       "filename": "1.1.4.3 (e x)^m (a x^j+b x^k)^p (c+d x^n)^q.m",
            "rhs": "e^IntPart[m]*(e*x)^FracPart[m]*(a*x^j + b*x^(j + n))^FracPart[p]/ (x^(FracPart[m] + j*FracPart[p])*(a + b*x^n)^FracPart[p])* Int[x^(m + j*p)*(a + b*x^n)^p*(c + d*x^n)^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_.*x_^j_. + b_.*x_^jn_.)^p_*(c_ + d_.*x_^n_.)^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, j, m, n, p, q}, x] && EqQ[jn, j + n] && Not[IntegerQ[p]] && NeQ[b*c - a*d, 0] && Not[EqQ[n, 1] && EqQ[j, 1]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.4 Improper/1.1.4.4 P(x) (c x)^m (a x^j+b x^n)^p.m",
       "filename": "1.1.4.4 P(x) (c x)^m (a x^j+b x^n)^p.m",
            "rhs": "With[{d = Denominator[n]}, d*Subst[ Int[x^(d - 1)* ReplaceAll[SubstFor[x^n, Pq, x], x -> x^(d*n)]*(a*x^(d*j) + b*x^(d*n))^p, x], x, x^(1/d)]]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_.*x_^j_. + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, j, n, p}, x] && PolyQ[Pq, x^n] && Not[IntegerQ[p]] && NeQ[n, j] && RationalQ[j, n] && IntegerQ[j/n] && LtQ[-1, n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.4 Improper/1.1.4.4 P(x) (c x)^m (a x^j+b x^n)^p.m",
       "filename": "1.1.4.4 P(x) (c x)^m (a x^j+b x^n)^p.m",
            "rhs": "1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)* SubstFor[x^n, Pq, x]*(a*x^Simplify[j/n] + b*x)^p, x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Pq_*(a_.*x_^j_. + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, j, m, n, p}, x] && PolyQ[Pq, x^n] && Not[IntegerQ[p]] && NeQ[n, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[(m + 1)/n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.4 Improper/1.1.4.4 P(x) (c x)^m (a x^j+b x^n)^p.m",
       "filename": "1.1.4.4 P(x) (c x)^m (a x^j+b x^n)^p.m",
            "rhs": "c^(Sign[m]*Quotient[m, Sign[m]])*(c*x)^Mod[m, Sign[m]]/ x^Mod[m, Sign[m]]*Int[x^m*Pq*(a*x^j + b*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_*x_)^m_.*Pq_*(a_.*x_^j_. + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, j, n, p}, x] && PolyQ[Pq, x^n] && Not[IntegerQ[p]] && NeQ[n, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[(m + 1)/n]] && RationalQ[m] && GtQ[m^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.4 Improper/1.1.4.4 P(x) (c x)^m (a x^j+b x^n)^p.m",
       "filename": "1.1.4.4 P(x) (c x)^m (a x^j+b x^n)^p.m",
            "rhs": "(c*x)^m/x^m*Int[x^m*Pq*(a*x^j + b*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_*x_)^m_.*Pq_*(a_.*x_^j_. + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, j, m, n, p}, x] && PolyQ[Pq, x^n] && Not[IntegerQ[p]] && NeQ[n, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[(m + 1)/n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.4 Improper/1.1.4.4 P(x) (c x)^m (a x^j+b x^n)^p.m",
       "filename": "1.1.4.4 P(x) (c x)^m (a x^j+b x^n)^p.m",
            "rhs": "With[{g = GCD[m + 1, n]}, 1/g*Subst[ Int[x^((m + 1)/g - 1)* ReplaceAll[Pq, x -> x^(1/g)]*(a*x^(j/g) + b*x^(n/g))^p, x], x, x^g] /; NeQ[g, 1]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Pq_*(a_.*x_^j_. + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, p}, x] && PolyQ[Pq, x^n] && Not[IntegerQ[p]] && IGtQ[j, 0] && IGtQ[n, 0] && IGtQ[j/n, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.4 Improper/1.1.4.4 P(x) (c x)^m (a x^j+b x^n)^p.m",
       "filename": "1.1.4.4 P(x) (c x)^m (a x^j+b x^n)^p.m",
            "rhs": "With[{q = Expon[Pq, x]}, With[{Pqq = Coeff[Pq, x, q]}, Pqq*(c*x)^(m + q - n + 1)*(a*x^j + b*x^n)^(p + 1)/(b* c^(q - n + 1)*(m + q + n*p + 1)) + Int[(c*x)^m* ExpandToSum[ Pq - Pqq*x^q - a*Pqq*(m + q - n + 1)*x^(q - n)/(b*(m + q + n*p + 1)), x]*(a*x^j + b*x^n)^p, x]] /; GtQ[q, n - 1] && NeQ[m + q + n*p + 1, 0] && (IntegerQ[2*p] || IntegerQ[p + (q + 1)/(2*n)])]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*Pq_*(a_.*x_^j_. + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && Not[IntegerQ[p]] && IGtQ[j, 0] && IGtQ[n, 0] && LtQ[j, n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.4 Improper/1.1.4.4 P(x) (c x)^m (a x^j+b x^n)^p.m",
       "filename": "1.1.4.4 P(x) (c x)^m (a x^j+b x^n)^p.m",
            "rhs": "1/(m + 1)*Subst[ Int[ ReplaceAll[SubstFor[x^n, Pq, x], x -> x^Simplify[n/(m + 1)]]*(a*x^Simplify[j/(m + 1)] + b*x^Simplify[n/(m + 1)])^p, x], x, x^(m + 1)]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Pq_*(a_.*x_^j_. + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, j, m, n, p}, x] && PolyQ[Pq, x^n] && Not[IntegerQ[p]] && NeQ[n, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[n/(m + 1)]] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.4 Improper/1.1.4.4 P(x) (c x)^m (a x^j+b x^n)^p.m",
       "filename": "1.1.4.4 P(x) (c x)^m (a x^j+b x^n)^p.m",
            "rhs": "c^(Sign[m]*Quotient[m, Sign[m]])*(c*x)^Mod[m, Sign[m]]/ x^Mod[m, Sign[m]]*Int[x^m*Pq*(a*x^j + b*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_*x_)^m_*Pq_*(a_.*x_^j_. + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, j, n, p}, x] && PolyQ[Pq, x^n] && Not[IntegerQ[p]] && NeQ[n, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[n/(m + 1)]] && Not[IntegerQ[n]] && GtQ[m^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.4 Improper/1.1.4.4 P(x) (c x)^m (a x^j+b x^n)^p.m",
       "filename": "1.1.4.4 P(x) (c x)^m (a x^j+b x^n)^p.m",
            "rhs": "(c*x)^m/x^m*Int[x^m*Pq*(a*x^j + b*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_*x_)^m_*Pq_*(a_.*x_^j_. + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, j, m, n, p}, x] && PolyQ[Pq, x^n] && Not[IntegerQ[p]] && NeQ[n, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[n/(m + 1)]] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.4 Improper/1.1.4.4 P(x) (c x)^m (a x^j+b x^n)^p.m",
       "filename": "1.1.4.4 P(x) (c x)^m (a x^j+b x^n)^p.m",
            "rhs": "Int[ExpandIntegrand[(c*x)^m*Pq*(a*x^j + b*x^n)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*Pq_*(a_.*x_^j_. + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, j, m, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) && Not[IntegerQ[p]] && NeQ[n, j]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.1 Binomial products/1.1.4 Improper/1.1.4.4 P(x) (c x)^m (a x^j+b x^n)^p.m",
       "filename": "1.1.4.4 P(x) (c x)^m (a x^j+b x^n)^p.m",
            "rhs": "Int[ExpandIntegrand[Pq*(a*x^j + b*x^n)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_.*x_^j_. + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, j, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) && Not[IntegerQ[p]] && NeQ[n, j]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.1 (a+b x+c x^2)^p.m",
       "filename": "1.2.1.1 (a+b x+c x^2)^p.m",
            "rhs": "2*(a + b*x + c*x^2)^(p + 1)/((2*p + 1)*(b + 2*c*x))",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.1 (a+b x+c x^2)^p.m",
       "filename": "1.2.1.1 (a+b x+c x^2)^p.m",
            "rhs": "(b/2 + c*x)/Sqrt[a + b*x + c*x^2]* Int[1/(b/2 + c*x), x]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*x_ + c_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.1 (a+b x+c x^2)^p.m",
       "filename": "1.2.1.1 (a+b x+c x^2)^p.m",
            "rhs": "(b + 2*c*x)*(a + b*x + c*x^2)^p/(2*c*(2*p + 1))",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && NeQ[p, -1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.1 (a+b x+c x^2)^p.m",
       "filename": "1.2.1.1 (a+b x+c x^2)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, 1/c^p* Int[Simp[b/2 - q/2 + c*x, x]^p*Simp[b/2 + q/2 + c*x, x]^p, x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && PerfectSquareQ[b^2 - 4*a*c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.1 (a+b x+c x^2)^p.m",
       "filename": "1.2.1.1 (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*x + c*x^2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && (EqQ[a, 0] || Not[PerfectSquareQ[b^2 - 4*a*c]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.1 (a+b x+c x^2)^p.m",
       "filename": "1.2.1.1 (a+b x+c x^2)^p.m",
            "rhs": "(b + 2*c*x)*(a + b*x + c*x^2)^p/(2*c*(2*p + 1)) - p*(b^2 - 4*a*c)/(2*c*(2*p + 1))* Int[(a + b*x + c*x^2)^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.1 (a+b x+c x^2)^p.m",
       "filename": "1.2.1.1 (a+b x+c x^2)^p.m",
            "rhs": "-2*(b + 2*c*x)/((b^2 - 4*a*c)* Sqrt[a + b*x + c*x^2])",
     "rulenumber": 0,
            "lhs": "Int[1/(a_. + b_.*x_ + c_.*x_^2)^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.1 (a+b x+c x^2)^p.m",
       "filename": "1.2.1.1 (a+b x+c x^2)^p.m",
            "rhs": "(b + 2*c*x)*(a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)) - 2*c*(2*p + 3)/((p + 1)*(b^2 - 4*a*c))* Int[(a + b*x + c*x^2)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.1 (a+b x+c x^2)^p.m",
       "filename": "1.2.1.1 (a+b x+c x^2)^p.m",
            "rhs": "Log[x]/b - Log[RemoveContent[b + c*x, x]]/b",
     "rulenumber": 0,
            "lhs": "Int[1/(b_.*x_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.1 (a+b x+c x^2)^p.m",
       "filename": "1.2.1.1 (a+b x+c x^2)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, c/q*Int[1/Simp[b/2 - q/2 + c*x, x], x] - c/q*Int[1/Simp[b/2 + q/2 + c*x, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_. + b_.*x_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[b^2 - 4*a*c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.1 (a+b x+c x^2)^p.m",
       "filename": "1.2.1.1 (a+b x+c x^2)^p.m",
            "rhs": "With[{q = 1 - 4*Simplify[a*c/b^2]}, -2/b*Subst[Int[1/(q - x^2), x], x, 1 + 2*c*x/b] /; RationalQ[q] && (EqQ[q^2, 1] || Not[RationalQ[b^2 - 4*a*c]])]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.1 (a+b x+c x^2)^p.m",
       "filename": "1.2.1.1 (a+b x+c x^2)^p.m",
            "rhs": "-2* Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_. + b_.*x_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.1 (a+b x+c x^2)^p.m",
       "filename": "1.2.1.1 (a+b x+c x^2)^p.m",
            "rhs": "1/(2*c*(-4*c/(b^2 - 4*a*c))^p)* Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.1 (a+b x+c x^2)^p.m",
       "filename": "1.2.1.1 (a+b x+c x^2)^p.m",
            "rhs": "2*Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[b_.*x_ + c_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.1 (a+b x+c x^2)^p.m",
       "filename": "1.2.1.1 (a+b x+c x^2)^p.m",
            "rhs": "2*Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*x_ + c_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.1 (a+b x+c x^2)^p.m",
       "filename": "1.2.1.1 (a+b x+c x^2)^p.m",
            "rhs": "(b*x + c*x^2)^p/(-c*(b*x + c*x^2)/(b^2))^p* Int[(-c*x/b - c^2*x^2/b^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c}, x] && RationalQ[p] && 3 <= Denominator[p] <= 4"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.1 (a+b x+c x^2)^p.m",
       "filename": "1.2.1.1 (a+b x+c x^2)^p.m",
            "rhs": " (a+b*x+c*x^2)^p/(-c*(a+b*x+c*x^2)/(b^2-4*a*c))^p*Int[(-a*c/(b^2-4*a*c) -b*c*x/(b^2-4*a*c)-c^2*x^2/(b^2-4*a*c))^p,x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.+b_.*x_+c_.*x_^2)^p_,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c},x] &&  NeQ[b^2-4*a*c,0] && RationalQ[p] && 3<=Denominator[p]<=4 *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.1 (a+b x+c x^2)^p.m",
       "filename": "1.2.1.1 (a+b x+c x^2)^p.m",
            "rhs": "With[{d = Denominator[p]}, d*Sqrt[(b + 2*c*x)^2]/(b + 2*c*x)* Subst[Int[x^(d*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4*c*x^d], x], x, (a + b*x + c*x^2)^(1/d)] /; 3 <= d <= 4]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && RationalQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.1 (a+b x+c x^2)^p.m",
       "filename": "1.2.1.1 (a+b x+c x^2)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, -(a + b*x + c*x^2)^(p + 1)/(q*(p + 1)*((q - b - 2*c*x)/(2*q))^(p + 1))* Hypergeometric2F1[-p, p + 1, p + 2, (b + q + 2*c*x)/(2*q)]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && NeQ[b^2 - 4*a*c, 0] && Not[IntegerQ[4*p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.1 (a+b x+c x^2)^p.m",
       "filename": "1.2.1.1 (a+b x+c x^2)^p.m",
            "rhs": "1/Coefficient[u, x, 1]*Subst[Int[(a + b*x + c*x^2)^p, x], x, u]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*u_ + c_.*u_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && LinearQ[u, x] && NeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": " 1/c^p*Int[(d+e*x)^m*(b/2+c*x)^(2*p),x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.+e_.*x_)^m_.*(a_+b_.*x_+c_.*x_^2)^p_.,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,m},x] &&  EqQ[b^2-4*a*c,0] && IntegerQ[p] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "Int[(d + e*x)^(m + p)*(a/d + c/e*x)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_.*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "Int[(d + e*x)^(m + p)*(a/d + c/e*x)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_.*(a_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || GtQ[a, 0] && GtQ[d, 0] && IntegerQ[m + p])"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "d*Log[RemoveContent[a + b*x + c*x^2, x]]/b",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)/(a_. + b_.*x_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "d*(a + b*x + c*x^2)^(p + 1)/(b*(p + 1))",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "Int[(d + e*x)^(p + 1)*(a/d + c/e*x)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && IGtQ[p, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x)*(a + b*x + c*x^2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || EqQ[a, 0])"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, (c*d - e*(b/2 - q/2))/q* Int[1/(b/2 - q/2 + c*x), x] - (c*d - e*(b/2 + q/2))/q* Int[1/(b/2 + q/2 + c*x), x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)/(a_ + b_.*x_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*c]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "With[{q = Rt[-a*c, 2]}, (e/2 + c*d/(2*q))*Int[1/(-q + c*x), x] + (e/2 - c*d/(2*q))* Int[1/(q + c*x), x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)/(a_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NiceSqrtQ[-a*c]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "(* (d-b*e/(2*c))*Int[1/(a+b*x+c*x^2),x] + *) (2*c* d - b*e)/(2*c)*Int[1/(a + b*x + c*x^2), x] + e/(2*c)*Int[(b + 2*c*x)/(a + b*x + c*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)/(a_ + b_.*x_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && Not[NiceSqrtQ[b^2 - 4*a*c]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "d*Int[1/(a + c*x^2), x] + e*Int[x/(a + c*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)/(a_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && Not[NiceSqrtQ[-a*c]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "-2*(b*d - 2*a*e + (2*c*d - b*e)*x)/((b^2 - 4*a*c)* Sqrt[a + b*x + c*x^2])",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)/(a_. + b_.*x_ + c_.*x_^2)^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "(-a*e + c*d*x)/(a*c*Sqrt[a + c*x^2])",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)/(a_ + c_.*x_^2)^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "(b*d - 2*a*e + (2*c*d - b*e)*x)/((p + 1)*(b^2 - 4*a*c))*(a + b*x + c*x^2)^(p + 1) - (2*p + 3)*(2*c*d - b*e)/((p + 1)*(b^2 - 4*a*c))* Int[(a + b*x + c*x^2)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "(a*e - c*d*x)/(2*a*c*(p + 1))*(a + c*x^2)^(p + 1) + d*(2*p + 3)/(2*a*(p + 1))*Int[(a + c*x^2)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && LtQ[p, -1] && NeQ[p, -3/2]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "e*(a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1)) + (2*c*d - b*e)/(2*c)* Int[(a + b*x + c*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "e*(a + c*x^2)^(p + 1)/(2*c*(p + 1)) + d*Int[(a + c*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)*(a_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "e^m/c^(m/2)*Int[(a + b*x + c*x^2)^(p + m/2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(a_ + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]] && EqQ[2*c*d - b*e, 0] && IntegerQ[m/2]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "e^(m - 1)/c^((m - 1)/2)* Int[(d + e*x)*(a + b*x + c*x^2)^(p + (m - 1)/2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(a_ + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]] && EqQ[2*c*d - b*e, 0] && IntegerQ[(m - 1)/2]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "(a + b*x + c*x^2)^p/(d + e*x)^(2*p)* Int[(d + e*x)^(m + 2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(a_ + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]] && EqQ[2*c*d - b*e, 0] && Not[IntegerQ[m]]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "(a + b*x + c*x^2)^ FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p]))* Int[ ExpandLinearProduct[(b/2 + c*x)^(2*p), (d + e*x)^m, b/2, c, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(a_ + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]] && NeQ[2*c*d - b*e, 0] && IGtQ[m, 0] && EqQ[m - 2*p + 1, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "(a + b*x + c*x^2)^ FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p]))* Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(a_ + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]] && NeQ[2*c*d - b*e, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "1/e^p*Int[(e*x)^(m + p)*(b + c*x)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, e, m}, x] && IntegerQ[p]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "e*(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)/(c*(p + 1))",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[p]] && EqQ[m + p, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "e*(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)/(c*(p + 1))",
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "e*m*d^(m - 1)*(a + c*x^2)^(p + 1)/(2*c*(p + 1)) + Int[((d + e*x)^m - e*m*d^(m - 1)*x)*(a + c*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 1] && IGtQ[m, 0] && LeQ[m, p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x)^m*(a + c*x^2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(a_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || EqQ[a, 0] && IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": " With[{q=Rt[(c*d^2-b*d*e+a*e^2)/c,2]}, 1/2*Int[(d+q+e*x)/(Sqrt[d+e*x]*(a+b*x+c*x^2)),x] + 1/2*Int[(d-q+e*x)/(Sqrt[d+e*x]*(a+b*x+c*x^2)),x]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[d_.+e_.*x_]/(a_.+b_.*x_+c_.*x_^2),x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] &&  NeQ[2*c*d-b*e,0] && LtQ[b^2-4*a*c,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": " With[{q=Rt[(c*d^2+a*e^2)/c,2]}, 1/2*Int[(d+q+e*x)/(Sqrt[d+e*x]*(a+c*x^2)),x] + 1/2*Int[(d-q+e*x)/(Sqrt[d+e*x]*(a+c*x^2)),x]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[d_+e_.*x_]/(a_+c_.*x_^2),x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,c,d,e},x]  && NeQ[c*d^2+a*e^2,0] && LtQ[-a*c,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": " With[{q=Rt[b^2-4*a*c,2]}, (2*c*d-b*e+e*q)/q*Int[1/(Sqrt[d+e*x]*(b-q+2*c*x)),x] - (2*c*d-b*e-e*q)/q*Int[1/(Sqrt[d+e*x]*(b+q+2*c*x)),x]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[d_.+e_.*x_]/(a_.+b_.*x_+c_.*x_^2),x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] &&  NeQ[2*c*d-b*e,0] (* && Not[LtQ[b^2-4*a*c,0]] *) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": " With[{q=Rt[-a*c,2]}, (c*d+e*q)/(2*q)*Int[1/(Sqrt[d+e*x]*(-q+c*x)),x] - (c*d-e*q)/(2*q)*Int[1/(Sqrt[d+e*x]*(+q+c*x)),x]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[d_+e_.*x_]/(a_+c_.*x_^2),x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2,0] (* && Not[LtQ[-a*c,0]] *) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "2*e*Subst[ Int[x^2/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[d_. + e_.*x_]/(a_. + b_.*x_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "2*e*Subst[Int[x^2/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[d_ + e_.*x_]/(a_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "Int[PolynomialDivide[(d + e*x)^m, a + b*x + c*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_/(a_. + b_.*x_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IGtQ[m, 1] && (NeQ[d, 0] || GtQ[m, 2])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "Int[PolynomialDivide[(d + e*x)^m, a + c*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_/(a_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[m, 1] && (NeQ[d, 0] || GtQ[m, 2])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "e*(d + e*x)^(m - 1)/(c*(m - 1)) + 1/c*Int[(d + e*x)^(m - 2)* Simp[c*d^2 - a*e^2 + e*(2*c*d - b*e)*x, x]/(a + b*x + c*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_/(a_. + b_.*x_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && GtQ[m, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "e*(d + e*x)^(m - 1)/(c*(m - 1)) + 1/c*Int[(d + e*x)^(m - 2)* Simp[c*d^2 - a*e^2 + 2*c*d*e*x, x]/(a + c*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_/(a_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "e^2/(c*d^2 - b*d*e + a*e^2)*Int[1/(d + e*x), x] + 1/(c*d^2 - b*d*e + a*e^2)* Int[(c*d - b*e - c*e*x)/(a + b*x + c*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((d_. + e_.*x_)*(a_. + b_.*x_ + c_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "e^2/(c*d^2 + a*e^2)*Int[1/(d + e*x), x] + 1/(c*d^2 + a*e^2)*Int[(c*d - c*e*x)/(a + c*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((d_ + e_.*x_)*(a_ + c_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": " With[{q=Rt[(c*d^2-b*d*e+a*e^2)/c,2]}, 1/(2*q)*Int[(d+q+e*x)/(Sqrt[d+e*x]*(a+b*x+c*x^2)),x] - 1/(2*q)*Int[(d-q+e*x)/(Sqrt[d+e*x]*(a+b*x+c*x^2)),x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[d_.+e_.*x_]*(a_.+b_.*x_+c_.*x_^2)),x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] &&  NeQ[2*c*d-b*e,0] && LtQ[b^2-4*a*c,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": " With[{q=Rt[(c*d^2+a*e^2)/c,2]}, 1/(2*q)*Int[(d+q+e*x)/(Sqrt[d+e*x]*(a+c*x^2)),x] - 1/(2*q)*Int[(d-q+e*x)/(Sqrt[d+e*x]*(a+c*x^2)),x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[d_+e_.*x_]*(a_+c_.*x_^2)),x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,c,d,e},x] && NeQ[c*d^2+a*e^2,0] && LtQ[-a*c,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": " With[{q=Rt[b^2-4*a*c,2]}, 2*c/q*Int[1/(Sqrt[d+e*x]*(b-q+2*c*x)),x] - 2*c/q*Int[1/(Sqrt[d+e*x]*(b+q+2*c*x)),x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[d_.+e_.*x_]*(a_.+b_.*x_+c_.*x_^2)),x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e},x] &&  NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && NeQ[2*c*d-b*e,0] (* &&  Not[LtQ[b^2-4*a*c,0]] *) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": " With[{q=Rt[-a*c,2]}, c/(2*q)*Int[1/(Sqrt[d+e*x]*(-q+c*x)),x] - c/(2*q)*Int[1/(Sqrt[d+e*x]*(q+c*x)),x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[d_+e_.*x_]*(a_+c_.*x_^2)),x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,c,d,e},x] &&  NeQ[c*d^2+a*e^2,0] (* && Not[LtQ[-a*c,0]] *) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "2*e*Subst[ Int[1/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[d_. + e_.*x_]*(a_. + b_.*x_ + c_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "2*e*Subst[Int[1/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[d_ + e_.*x_]*(a_ + c_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "e*(d + e*x)^(m + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2)) + 1/(c*d^2 - b*d*e + a*e^2)* Int[(d + e*x)^(m + 1)* Simp[c*d - b*e - c*e*x, x]/(a + b*x + c*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_/(a_. + b_.*x_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "e*(d + e*x)^(m + 1)/((m + 1)*(c*d^2 + a*e^2)) + c/(c*d^2 + a*e^2)* Int[(d + e*x)^(m + 1)*(d - e*x)/(a + c*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_/(a_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x)^m, 1/(a + b*x + c*x^2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_/(a_. + b_.*x_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && Not[IntegerQ[m]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x)^m, 1/(a + c*x^2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_/(a_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[m]]"
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       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^ FracPart[p]*(a + b*x + c*x^2)^FracPart[p]/(a*d + c*e*x^3)^ FracPart[p]*Int[(d + e*x)^(m - p)*(a*d + c*e*x^3)^p, x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*d + a*e, 0] && EqQ[c*d + b*e, 0] && IGtQ[m - p + 1, 0] && Not[IntegerQ[p]]"
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       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "Int[(d + e*x)^m/(Sqrt[b*x]*Sqrt[1 + c/b*x]), x]",
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            "lhs": "Int[(d_. + e_.*x_)^m_/Sqrt[b_.*x_ + c_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4] && LtQ[c, 0] && RationalQ[b]"
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       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "Sqrt[x]*Sqrt[b + c*x]/Sqrt[b*x + c*x^2]* Int[(d + e*x)^m/(Sqrt[x]*Sqrt[b + c*x]), x]",
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            "lhs": "Int[(d_. + e_.*x_)^m_/Sqrt[b_.*x_ + c_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]"
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       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "2*Subst[Int[x^(2*m + 1)/Sqrt[a + b*x^2 + c*x^4], x], x, Sqrt[x]]",
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            "lhs": "Int[x_^m_/Sqrt[a_ + b_.*x_ + c_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[m^2, 1/4]"
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       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "(e*x)^m/x^m*Int[x^m/Sqrt[a + b*x + c*x^2], x]",
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            "lhs": "Int[(e_*x_)^m_/Sqrt[a_ + b_.*x_ + c_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[m^2, 1/4]"
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       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m* Sqrt[-c*(a + b*x + c*x^2)/(b^2 - 4*a*c)]/ (c* Sqrt[a + b*x + c*x^2]*(2*c*(d + e*x)/(2*c*d - b*e - e*Rt[b^2 - 4*a*c, 2]))^m)* Subst[ Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]* x^2/(2*c*d - b*e - e*Rt[b^2 - 4*a*c, 2]))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]]",
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            "lhs": "Int[(d_. + e_.*x_)^m_/Sqrt[a_. + b_.*x_ + c_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]"
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       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "2*a*Rt[-c/a, 2]*(d + e*x)^m* Sqrt[1 + c*x^2/a]/(c* Sqrt[a + c*x^2]*(c*(d + e*x)/(c*d - a*e*Rt[-c/a, 2]))^m)* Subst[ Int[(1 + 2*a*e*Rt[-c/a, 2]*x^2/(c*d - a*e*Rt[-c/a, 2]))^m/ Sqrt[1 - x^2], x], x, Sqrt[(1 - Rt[-c/a, 2]*x)/2]]",
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            "lhs": "Int[(d_ + e_.*x_)^m_/Sqrt[a_ + c_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m^2, 1/4]"
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       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "-(d + e*x)^(m + 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^ p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)) + p*(b^2 - 4*a*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))* Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x]",
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            "lhs": "Int[(d_. + e_.*x_)^m_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m + 2*p + 2, 0] && GtQ[p, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "-(d + e*x)^(m + 1)*(-2*a*e + (2*c*d)*x)*(a + c*x^2)^ p/(2*(m + 1)*(c*d^2 + a*e^2)) - 4*a*c*p/(2*(m + 1)*(c*d^2 + a*e^2))* Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1), x]",
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            "lhs": "Int[(d_ + e_.*x_)^m_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2, 0] && GtQ[p, 0]"
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       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)* x)*(a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)) - 2*(2*p + 3)*(c*d^2 - b*d*e + a*e^2)/((p + 1)*(b^2 - 4*a*c))* Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1), x]",
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            "lhs": "Int[(d_. + e_.*x_)^m_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m + 2*p + 2, 0] && LtQ[p, -1]"
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  {
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       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^(m - 1)*(a*e - c*d*x)*(a + c*x^2)^(p + 1)/(2*a*c*(p + 1)) + (2*p + 3)*(c*d^2 + a*e^2)/(2*a*c*(p + 1))* Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1), x]",
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            "lhs": "Int[(d_ + e_.*x_)^m_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2, 0] && LtQ[p, -1]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "-2* Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]]",
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            "lhs": "Int[1/((d_. + e_.*x_)*Sqrt[a_. + b_.*x_ + c_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]"
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       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "-Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x, (a*e - c*d*x)/Sqrt[a + c*x^2]]",
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            "lhs": "Int[1/((d_ + e_.*x_)*Sqrt[a_ + c_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "-(b - Rt[b^2 - 4*a*c, 2] + 2*c*x)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p/ ((m + 1)*(2*c*d - b*e + e*Rt[b^2 - 4*a*c, 2])* ((2*c*d - b*e + e*Rt[b^2 - 4*a*c, 2])*(b + Rt[b^2 - 4*a*c, 2] + 2*c*x)/((2*c*d - b*e - e*Rt[b^2 - 4*a*c, 2])*(b - Rt[b^2 - 4*a*c, 2] + 2*c*x)))^p)* Hypergeometric2F1[m + 1, -p, m + 2, -4*c* Rt[b^2 - 4*a*c, 2]*(d + e* x)/((2*c*d - b*e - e*Rt[b^2 - 4*a*c, 2])*(b - Rt[b^2 - 4*a*c, 2] + 2*c*x))]",
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            "lhs": "Int[(d_. + e_.*x_)^m_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && Not[IntegerQ[p]] && EqQ[m + 2*p + 2, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "(Rt[-a*c, 2] - c*x)*(d + e*x)^(m + 1)*(a + c*x^2)^ p/ ((m + 1)*(c*d + e*Rt[-a*c, 2])*((c*d + e*Rt[-a*c, 2])*(Rt[-a*c, 2] + c*x)/((c*d - e*Rt[-a*c, 2])*(-Rt[-a*c, 2] + c*x)))^p)* Hypergeometric2F1[m + 1, -p, m + 2, 2*c*Rt[-a*c, 2]*(d + e*x)/((c*d - e*Rt[-a*c, 2])*(Rt[-a*c, 2] - c*x))]",
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            "lhs": "Int[(d_ + e_.*x_)^m_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]] && EqQ[m + 2*p + 2, 0]"
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       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^ m*(b + 2*c*x)*(a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)) + m*(2*c*d - b*e)/((p + 1)*(b^2 - 4*a*c))* Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m + 2*p + 3, 0] && LtQ[p, -1]"
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         "givens": "FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 3, 0] && LtQ[p, -1]"
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       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "e*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2)) + (2*c*d - b*e)/(2*(c*d^2 - b*d*e + a*e^2))* Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x]",
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         "givens": "FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m + 2*p + 3, 0]"
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       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "e*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1)/((m + 1)*(c*d^2 + a*e^2)) + c*d/(c*d^2 + a*e^2)*Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x]",
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            "lhs": "Int[(d_ + e_.*x_)^m_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 3, 0]"
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       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
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            "lhs": "Int[(d_. + e_.*x_)^m_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] && Not[ILtQ[m + 2*p + 1, 0]] && IntQuadraticQ[a, b, c, d, e, m, p, x]"
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            "lhs": "Int[(d_ + e_.*x_)^m_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] && Not[ILtQ[m + 2*p + 1, 0]] && IntQuadraticQ[a, 0, c, d, e, m, p, x]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^(m + 1)*(a + b*x + c*x^2)^ p/(e*(m + 2*p + 1)) - p/(e*(m + 2*p + 1))* Int[(d + e*x)^m* Simp[b*d - 2*a*e + (2*c*d - b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && (Not[RationalQ[m]] || LtQ[m, 1]) && Not[ILtQ[m + 2*p, 0]] && IntQuadraticQ[a, b, c, d, e, m, p, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^(m + 1)*(a + c*x^2)^ p/(e*(m + 2*p + 1)) + 2*p/(e*(m + 2*p + 1))* Int[(d + e*x)^m*Simp[a*e - c*d*x, x]*(a + c*x^2)^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && (Not[RationalQ[m]] || LtQ[m, 1]) && Not[ILtQ[m + 2*p, 0]] && IntQuadraticQ[a, 0, c, d, e, m, p, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^ m*(b + 2*c*x)*(a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)) - 1/((p + 1)*(b^2 - 4*a*c))* Int[(d + e*x)^(m - 1)*(b*e*m + 2*c*d*(2*p + 3) + 2*c*e*(m + 2*p + 3)*x)*(a + b*x + c*x^2)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m, 0] && (LtQ[m, 1] || ILtQ[m + 2*p + 3, 0] && NeQ[m, 2]) && IntQuadraticQ[a, b, c, d, e, m, p, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "-x*(d + e*x)^m*(a + c*x^2)^(p + 1)/(2*a*(p + 1)) + 1/(2*a*(p + 1))* Int[(d + e*x)^(m - 1)*(d*(2*p + 3) + e*(m + 2*p + 3)*x)*(a + c*x^2)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && (LtQ[m, 1] || ILtQ[m + 2*p + 3, 0] && NeQ[m, 2]) && IntQuadraticQ[a, 0, c, d, e, m, p, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)* x)*(a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)) + 1/((p + 1)*(b^2 - 4*a*c))* Int[(d + e*x)^(m - 2)* Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]* (a + b*x + c*x^2)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d, e, m, p, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^(m - 1)*(a*e - c*d*x)*(a + c*x^2)^(p + 1)/(2*a*c*(p + 1)) + 1/((p + 1)*(-2*a*c))* Int[(d + e*x)^(m - 2)* Simp[a*e^2*(m - 1) - c*d^2*(2*p + 3) - d*c*e*(m + 2*p + 2)*x, x]*(a + c*x^2)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)* x)*(a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) + 1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))* Int[(d + e*x)^m* Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]* (a + b*x + c*x^2)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "-(d + e*x)^(m + 1)*(a*e + c*d*x)*(a + c*x^2)^(p + 1)/(2*a*(p + 1)*(c*d^2 + a*e^2)) + 1/(2*a*(p + 1)*(c*d^2 + a*e^2))* Int[(d + e*x)^m* Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "e*(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1)) + 1/(c*(m + 2*p + 1))* Int[(d + e*x)^(m - 2)* Simp[c*d^2*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* (a + b*x + c*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "e*(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)/(c*(m + 2*p + 1)) + 1/(c*(m + 2*p + 1))* Int[(d + e*x)^(m - 2)* Simp[c*d^2*(m + 2*p + 1) - a*e^2*(m - 1) + 2*c*d*e*(m + p)*x, x]*(a + c*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, 0, c, d, e, m, p, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "e*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2)) + 1/((m + 1)*(c*d^2 - b*d*e + a*e^2))* Int[(d + e*x)^(m + 1)* Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, x]*(a + b*x + c*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && NeQ[m, -1] && (LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x] || SumSimplerQ[m, 1] && IntegerQ[p] || ILtQ[Simplify[m + 2*p + 3], 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "e*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1)/((m + 1)*(c*d^2 + a*e^2)) + c/((m + 1)*(c*d^2 + a*e^2))* Int[(d + e*x)^(m + 1)* Simp[d*(m + 1) - e*(m + 2*p + 3)*x, x]*(a + c*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[m, -1] && (LtQ[m, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x] || SumSimplerQ[m, 1] && IntegerQ[p] || ILtQ[Simplify[m + 2*p + 3], 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "d*Int[1/((d^2 - e^2*x^2)*(a + c*x^2)^(1/4)), x] - e*Int[x/((d^2 - e^2*x^2)*(a + c*x^2)^(1/4)), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((d_ + e_.*x_)*(a_ + c_.*x_^2)^(1/4)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "d*Int[1/((d^2 - e^2*x^2)*(a + c*x^2)^(3/4)), x] - e*Int[x/((d^2 - e^2*x^2)*(a + c*x^2)^(3/4)), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((d_ + e_.*x_)*(a_ + c_.*x_^2)^(3/4)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "1/(-4*c/(b^2 - 4*a*c))^p* Subst[Int[ Simp[1 - x^2/(b^2 - 4*a*c), x]^p/Simp[2*c*d - b*e + e*x, x], x], x, b + 2*c*x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_ + c_.*x_^2)^p_/(d_. + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && GtQ[4*a - b^2/c, 0] && IntegerQ[4*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "(a + b*x + c*x^2)^ p/(-c*(a + b*x + c*x^2)/(b^2 - 4*a*c))^p* Int[(-a*c/(b^2 - 4*a*c) - b*c*x/(b^2 - 4*a*c) - c^2*x^2/(b^2 - 4*a*c))^p/(d + e*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_ + c_.*x_^2)^p_/(d_. + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && Not[GtQ[4*a - b^2/c, 0]] && IntegerQ[4*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "With[{q = Rt[3*c*e^2*(2*c*d - b*e), 3]}, -Sqrt[3]*c*e* ArcTan[1/Sqrt[3] + 2*(c*d - b*e - c*e*x)/(Sqrt[3]*q*(a + b*x + c*x^2)^(1/3))]/ q^2 - 3*c*e*Log[d + e*x]/(2*q^2) + 3*c*e* Log[c*d - b*e - c*e*x - q*(a + b*x + c*x^2)^(1/3)]/(2*q^2)]",
     "rulenumber": 0,
            "lhs": "Int[1/((d_. + e_.*x_)*(a_ + b_.*x_ + c_.*x_^2)^(1/3)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && EqQ[c^2*d^2 - b*c*d*e + b^2*e^2 - 3*a*c*e^2, 0] && PosQ[c*e^2*(2*c*d - b*e)]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "With[{q = Rt[6*c^2*e^2/d^2, 3]}, -Sqrt[3]*c*e* ArcTan[1/Sqrt[3] + 2*c*(d - e*x)/(Sqrt[3]*d*q*(a + c*x^2)^(1/3))]/(d^2*q^2) - 3*c*e*Log[d + e*x]/(2*d^2*q^2) + 3*c*e*Log[c*d - c*e*x - d*q*(a + c*x^2)^(1/3)]/(2*d^2*q^2)]",
     "rulenumber": 0,
            "lhs": "Int[1/((d_ + e_.*x_)*(a_ + c_.*x_^2)^(1/3)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - 3*a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "With[{q = Rt[-3*c*e^2*(2*c*d - b*e), 3]}, -Sqrt[3]*c*e* ArcTan[1/Sqrt[3] - 2*(c*d - b*e - c*e*x)/(Sqrt[3]*q*(a + b*x + c*x^2)^(1/3))]/ q^2 - 3*c*e*Log[d + e*x]/(2*q^2) + 3*c*e* Log[c*d - b*e - c*e*x + q*(a + b*x + c*x^2)^(1/3)]/(2*q^2)]",
     "rulenumber": 0,
            "lhs": "Int[1/((d_. + e_.*x_)*(a_ + b_.*x_ + c_.*x_^2)^(1/3)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && EqQ[c^2*d^2 - b*c*d*e + b^2*e^2 - 3*a*c*e^2, 0] && NegQ[c*e^2*(2*c*d - b*e)]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": " With[{q=Rt[-6*c^2*d*e^2,3]}, -Sqrt[3]*c*e*ArcTan[1/Sqrt[3]-2*(c*d-c*e*x)/(Sqrt[3]*q*(a+c*x^2)^(1/ 3))]/q^2 - 3*c*e*Log[d+e*x]/(2*q^2) + 3*c*e*Log[c*d-c*e*x+q*(a+c*x^2)^(1/3)]/(2*q^2)]",
     "rulenumber": 0,
            "lhs": "Int[1/((d_+e_.*x_)*(a_+c_.*x_^2)^(1/3)),x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,c,d,e},x] && EqQ[c*d^2-3*a*e^2,0] && NegQ[c^2*d*e^2] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "a^(1/3)*Int[1/((d + e*x)*(1 - 3*e*x/d)^(1/3)*(1 + 3*e*x/d)^(1/3)), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((d_ + e_.*x_)*(a_ + c_.*x_^2)^(1/3)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + 9*a*e^2, 0] && GtQ[a, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "(1 + c*x^2/a)^(1/3)/(a + c*x^2)^(1/3)* Int[1/((d + e*x)*(1 + c*x^2/a)^(1/3)), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((d_ + e_.*x_)*(a_ + c_.*x_^2)^(1/3)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + 9*a*e^2, 0] && Not[GtQ[a, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, (b + q + 2*c*x)^(1/ 3)*(b - q + 2*c*x)^(1/3)/(a + b*x + c*x^2)^(1/3)* Int[1/((d + e*x)*(b + q + 2*c*x)^(1/3)*(b - q + 2*c*x)^(1/3)), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/((d_. + e_.*x_)*(a_ + b_.*x_ + c_.*x_^2)^(1/3)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c^2*d^2 - b*c*d*e - 2*b^2*e^2 + 9*a*c*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "Int[(d + e*x)^m*(Rt[a, 2] + Rt[-c, 2]*x)^p*(Rt[a, 2] - Rt[-c, 2]*x)^ p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]] && GtQ[a, 0] && LtQ[c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[(a + c*x^2)^ p, (d/(d^2 - e^2*x^2) - e*x/(d^2 - e^2*x^2))^(-m), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, p}, x] && NeQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]] && ILtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, -(1/(d + e*x))^(2*p)*(a + b*x + c*x^2)^ p/(e*(e*(b - q + 2*c*x)/(2*c*(d + e*x)))^ p*(e*(b + q + 2*c*x)/(2*c*(d + e*x)))^p)* Subst[ Int[x^(-m - 2*(p + 1))*Simp[1 - (d - e*(b - q)/(2*c))*x, x]^p* Simp[1 - (d - e*(b + q)/(2*c))*x, x]^p, x], x, 1/(d + e*x)]]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && Not[IntegerQ[p]] && ILtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, (a + b*x + c*x^2)^ p/(e*(1 - (d + e*x)/(d - e*(b - q)/(2*c)))^ p*(1 - (d + e*x)/(d - e*(b + q)/(2*c)))^p)* Subst[ Int[x^m*Simp[1 - x/(d - e*(b - q)/(2*c)), x]^p* Simp[1 - x/(d - e*(b + q)/(2*c)), x]^p, x], x, d + e*x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "With[{q = Rt[-a*c, 2]}, (a + c*x^2)^ p/(e*(1 - (d + e*x)/(d + e*q/c))^p*(1 - (d + e*x)/(d - e*q/c))^ p)* Subst[ Int[x^m*Simp[1 - x/(d + e*q/c), x]^p* Simp[1 - x/(d - e*q/c), x]^p, x], x, d + e*x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "1/Coefficient[u, x, 1]* Subst[Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x, u]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*u_)^m_.*(a_ + b_.*u_ + c_.*u_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, p}, x] && LinearQ[u, x] && NeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.2 (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "1/Coefficient[u, x, 1]* Subst[Int[(d + e*x)^m*(a + c*x^2)^p, x], x, u]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*u_)^m_.*(a_ + c_.*u_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m, p}, x] && LinearQ[u, x] && NeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "g*(e*x)^m*(b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(f_ + g_.*x_)*(b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, e, f, g, m, p}, x] && EqQ[b*g*(m + p + 1) - c*f*(m + 2*p + 2), 0] && NeQ[m + 2*p + 2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "f*Int[x^m*(a + c*x^2)^p, x] + g*Int[x^(m + 1)*(a + c*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(f_ + g_.*x_)*(a_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, f, g, p}, x] && IntegerQ[m] && Not[IntegerQ[2*p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(f_. + g_.*x_)*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (GtQ[p, 0] || EqQ[a, 0] && IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[(e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(f_. + g_.*x_)*(a_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "-f* g*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1)/(b*(p + 1)*(e*f - d*g))",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*(f_ + g_.*x_)*(a_ + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && EqQ[m + 2*p + 3, 0] && EqQ[2*c*f - b*g, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1)) - e*g*m/(2*c*(p + 1))* Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*(f_. + g_.*x_)*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[2*c*f - b*g, 0] && LtQ[p, -1] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "-2* c*(e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1)/((p + 1)*(2*c*d - b*e)^2) + (2*c*f - b*g)/(2*c*d - b*e)* Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*(f_. + g_.*x_)*(a_ + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && EqQ[m + 2*p + 3, 0] && NeQ[2*c*f - b*g, 0] && NeQ[2*c*d - b*e, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "(a + b*x + c*x^2)^ FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p]))* Int[(d + e*x)^m*(f + g*x)*(b/2 + c*x)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*(f_. + g_.*x_)*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*(f_. + g_.*x_)*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || EqQ[a, 0] && IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*(f_. + g_.*x_)*(a_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "e*g*x/c + 1/c*Int[(c*d*f - a*e*g + (c*e*f + c*d*g - b*e*g)*x)/(a + b*x + c*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)*(f_ + g_.*x_)/(a_. + b_.*x_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "e*g*x/c + 1/c*Int[(c*d*f - a*e*g + c*(e*f + d*g)*x)/(a + c*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)*(f_ + g_.*x_)/(a_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1)/(2* c^2*(p + 1)*(2*p + 3))",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)*(f_. + g_.*x_)*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3), 0] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "((e*f + d*g)*(2*p + 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1)/(2*c*(p + 1)*(2*p + 3))",
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            "lhs": "Int[(d_. + e_.*x_)*(f_. + g_.*x_)*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, p}, x] && EqQ[a*e*g - c*d*f*(2*p + 3), 0] && NeQ[p, -1]"
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       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "-(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (b^2*e*g - b*c*(e*f + d*g) + 2*c*(c*d*f - a*e*g))*x)*(a + b*x + c*x^2)^(p + 1)/(c*(p + 1)*(b^2 - 4*a*c)) - (b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(c*(p + 1)*(b^2 - 4*a*c))*Int[(a + b*x + c*x^2)^(p + 1), x]",
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            "lhs": "Int[(d_. + e_.*x_)*(f_. + g_.*x_)*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]"
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       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "(a*(e*f + d*g) - (c*d*f - a*e*g)* x)*(a + c*x^2)^(p + 1)/(2*a*c*(p + 1)) - (a*e*g - c*d*f*(2*p + 3))/(2*a*c*(p + 1))* Int[(a + c*x^2)^(p + 1), x]",
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            "lhs": "Int[(d_. + e_.*x_)*(f_. + g_.*x_)*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && LtQ[p, -1]"
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            "rhs": "-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1)/(2* c^2*(p + 1)*(2*p + 3)) + (b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3))* Int[(a + b*x + c*x^2)^p, x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && Not[LeQ[p, -1]]"
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            "rhs": "((e*f + d*g)*(2*p + 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1)/(2*c*(p + 1)*(2*p + 3)) - (a*e*g - c*d*f*(2*p + 3))/(c*(2*p + 3))*Int[(a + c*x^2)^p, x]",
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        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, p}, x] && Not[LeQ[p, -1]]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
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            "rhs": "1/e^p*Int[(e*x)^(m + p)*(f + g*x)*(b + c*x)^p, x]",
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            "lhs": "Int[(e_.*x_)^m_.*(f_. + g_.*x_)*(b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, e, f, g, m}, x] && IntegerQ[p]"
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            "rhs": "Int[(d + e*x)^(m + p)*(f + g*x)*(a/d + c/e*x)^p, x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]"
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       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "Int[(d + e*x)^(m + p)*(f + g*x)*(a/d + c/e*x)^p, x]",
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            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)*(a_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, m}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || GtQ[a, 0] && GtQ[d, 0] && EqQ[m + p, 0])"
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       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "d^m*e^m*Int[(f + g*x)*(a + b*x + c*x^2)^(m + p)/(a*e + c*d*x)^m, x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[2*p]] && ILtQ[m, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "d^m*e^m*Int[x*(a + c*x^2)^(m + p)/(a*e + c*d*x)^m, x]",
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        "comment": false,
         "givens": "FreeQ[{a, c, d, e, p}, x] && EqQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]] && ILtQ[m, 0] && EqQ[m, -1] && Not[ILtQ[p - 1/2, 0]]"
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       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))",
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            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g), 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "g*(d + e*x)^m*(a + c*x^2)^(p + 1)/(c*(m + 2*p + 2))",
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            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && EqQ[m*(d*g + e*f) + 2*e*f*(p + 1), 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "(g*(c*d - b*e) + c*e*f)*(d + e*x)^ m*(a + b*x + c*x^2)^(p + 1)/(c*(p + 1)*(2*c*d - b*e)) - e*(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g))/(c*(p + 1)*(2*c*d - b*e))* Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x]",
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            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0]"
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  {
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       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "(d*g + e*f)*(d + e*x)^ m*(a + c*x^2)^(p + 1)/(2*c*d*(p + 1)) - e*(m*(d*g + e*f) + 2*e*f*(p + 1))/(2*c*d*(p + 1))* Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1), x]",
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            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && EqQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0]"
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  {
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            "rhs": "(g*(c*d - b*e) + c*e*f)*(d + e*x)^ m*(a + b*x + c*x^2)^(p + 1)/(c*(p + 1)*(2*c*d - b*e)) - e*(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g))/(c*(p + 1)*(2*c*d - b*e))* Int[(d + e*x)^Simplify[m - 1]*(a + b*x + c*x^2)^ Simplify[p + 1], x]",
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            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && SumSimplerQ[p, 1] && SumSimplerQ[m, -1] && NeQ[p, -1]"
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       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "(d*g + e*f)*(d + e*x)^ m*(a + c*x^2)^(p + 1)/(2*c*d*(p + 1)) - e*(m*(d*g + e*f) + 2*e*f*(p + 1))/(2*c*d*(p + 1))* Int[(d + e*x)^Simplify[m - 1]*(a + c*x^2)^Simplify[p + 1], x]",
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            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && SumSimplerQ[p, 1] && SumSimplerQ[m, -1] && NeQ[p, -1] && Not[IGtQ[m, 0]]"
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  {
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       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "(d*g - e*f)*(d + e*x)^ m*(a + b*x + c*x^2)^(p + 1)/((2*c*d - b*e)*(m + p + 1)) + (m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1))* Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x]",
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            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (LtQ[m, -1] && Not[IGtQ[m + p + 1, 0]] || LtQ[m, 0] && LtQ[p, -1] || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0]"
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  {
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       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "(d*g - e*f)*(d + e*x)^ m*(a + c*x^2)^(p + 1)/(2*c*d*(m + p + 1)) + (m*(g*c*d + c*e*f) + 2*e*c*f*(p + 1))/(e*(2*c*d)*(m + p + 1))* Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x]",
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            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)*(a_ + c_.*x_^2)^p_, x_Symbol]",
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         "givens": "FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (LtQ[m, -1] && Not[IGtQ[m + p + 1, 0]] || LtQ[m, 0] && LtQ[p, -1] || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0]"
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  {
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       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) + (m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g))/(c* e*(m + 2*p + 2))*Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x]",
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            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[m + 2*p + 2, 0] && (NeQ[m, 2] || EqQ[d, 0])"
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       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "g*(d + e*x)^m*(a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) + (m*(d*g + e*f) + 2*e*f*(p + 1))/(e*(m + 2*p + 2))* Int[(d + e*x)^m*(a + c*x^2)^p, x]",
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         "givens": "FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && NeQ[m + 2*p + 2, 0] && NeQ[m, 2]"
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            "rhs": "x^2*(a*g - c*f*x)*(a + c*x^2)^(p + 1)/(2*a*c*(p + 1)) - 1/(2*a*c*(p + 1))* Int[x*Simp[2*a*g - c*f*(2*p + 5)*x, x]*(a + c*x^2)^(p + 1), x]",
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            "lhs": "Int[x_^2*(f_ + g_.*x_)*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, f, g}, x] && EqQ[a*g^2 + f^2*c, 0] && LtQ[p, -2]"
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            "rhs": "1/c*Int[(f + g*x)*(a + c*x^2)^(p + 1), x] - a/c*Int[(f + g*x)*(a + c*x^2)^p, x]",
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            "lhs": "Int[x_^2*(f_ + g_.*x_)*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, f, g, p}, x] && EqQ[a*g^2 + f^2*c, 0]"
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       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "Int[(d + e*x)^m*(f + g*x)^(p + 1)*(a/f + c/g*x)^p, x]",
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         "givens": "FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*f^2 - b*f*g + a*g^2, 0] && IntegerQ[p]"
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            "rhs": "Int[(d + e*x)^m*(f + g*x)^(p + 1)*(a/f + c/g*x)^p, x]",
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        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, m}, x] && EqQ[c*f^2 + a*g^2, 0] && (IntegerQ[p] || GtQ[a, 0] && GtQ[f, 0] && EqQ[p, -1])"
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            "rhs": "Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)/(a + b*x + c*x^2), x], x]",
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         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]"
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            "rhs": "Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)/(a + c*x^2), x], x]",
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       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "-(e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))",
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         "givens": "FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Simplify[m + 2*p + 3], 0] && EqQ[b*(e*f + d*g) - 2*(c*d*f + a*e*g), 0]"
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       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "-(e*f - d*g)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 + a*e^2))",
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            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)*(a_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[Simplify[m + 2*p + 3], 0] && EqQ[c*d*f + a*e*g, 0]"
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       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^ m*(a + b*x + c*x^2)^(p + 1)*(b*f - 2*a*g + (2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c)) - m*(b*(e*f + d*g) - 2*(c*d*f + a*e*g))/((p + 1)*(b^2 - 4*a*c))* Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Simplify[m + 2*p + 3], 0] && LtQ[p, -1]"
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       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^ m*(a + c*x^2)^(p + 1)*(a*g - c*f*x)/(2*a*c*(p + 1)) - m*(c*d*f + a*e*g)/(2*a*c*(p + 1))* Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1), x]",
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        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[Simplify[m + 2*p + 3], 0] && LtQ[p, -1]"
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  {
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       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "-(e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)) - (b*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2))* Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x]",
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            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Simplify[m + 2*p + 3], 0]"
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       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "-(e*f - d*g)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 + a*e^2)) + (c*d*f + a*e*g)/(c*d^2 + a*e^2)* Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x]",
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            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)*(a_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[Simplify[m + 2*p + 3], 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "f*Int[(e*x)^m*(a + c*x^2)^p, x] + g/e*Int[(e*x)^(m + 1)*(a + c*x^2)^p, x]",
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            "lhs": "Int[(e_.*x_)^m_*(f_ + g_.*x_)*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, e, f, g, p}, x] && Not[RationalQ[m]] && Not[IGtQ[p, 0]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^ FracPart[p]*(a + b*x + c*x^2)^FracPart[p]/(a*d + c*e*x^3)^ FracPart[p]*Int[(f + g*x)*(a*d + c*e*x^3)^p, x]",
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            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)*(a_ + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[m, p] && EqQ[b*d + a*e, 0] && EqQ[c*d + b*e, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "-(d + e*x)^(m + 1)*(a + b*x + c*x^2)^ p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2))* ((d*g - e*f*(m + 2))*(c*d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x) - p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2))* Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1)* Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*(m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x]",
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            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] && Not[ILtQ[m + 2*p + 3, 0]]"
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  {
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       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "-(d + e*x)^(m + 1)*(a + c*x^2)^ p/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2))* ((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*p*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e^2) + 2*c*d*p*(e*f - d*g))*x) - p/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2))* Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1)* Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x]",
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            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)*(a_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] && Not[ILtQ[m + 2*p + 3, 0]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + b*x + c*x^2)^ p/(e^2*(m + 1)*(m + 2*p + 2)) + p/(e^2*(m + 1)*(m + 2*p + 2))* Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)* Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x]",
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            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || IntegerQ[p] && Not[RationalQ[m]]) && NeQ[m, -1] && Not[ILtQ[m + 2*p + 1, 0]] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])"
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       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + c*x^2)^ p/(e^2*(m + 1)*(m + 2*p + 2)) + p/(e^2*(m + 1)*(m + 2*p + 2))* Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)* Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x]",
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        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || IntegerQ[p] && Not[RationalQ[m]]) && NeQ[m, -1] && Not[ILtQ[m + 2*p + 1, 0]] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])"
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       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*(a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) - p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))* Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)* Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] || Not[RationalQ[m]] || GeQ[m, -1] && LtQ[m, 0]) && Not[ILtQ[m + 2*p, 0]] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])"
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       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*(a + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) + 2*p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))* Int[(d + e*x)^m*(a + c*x^2)^(p - 1)* Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g* m - (c^2*f*d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x]",
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        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] || Not[RationalQ[m]] || GeQ[m, -1] && LtQ[m, 0]) && Not[ILtQ[m + 2*p, 0]] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])"
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       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "Int[(a + b*x + c*x^2)^p*ExpandIntegrand[(d + e*x)^m*(f + g*x), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(f_ + g_.*x_)*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[p, -1] && IGtQ[m, 0] && RationalQ[a, b, c, d, e, f, g]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "Int[(a + c*x^2)^p*ExpandIntegrand[(d + e*x)^m*(f + g*x), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(f_ + g_.*x_)*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && ILtQ[p, -1] && IGtQ[m, 0] && RationalQ[a, c, d, e, f, g]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "-(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c*(b*e*f + b*d*g + 2*a*e*g))*x)/ (c*(p + 1)*(b^2 - 4*a*c)) - 1/(c*(p + 1)*(b^2 - 4*a*c))* Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)* Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g] || Not[ILtQ[m + 2*p + 3, 0]])"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*(a*(e*f + d*g) - (c*d*f - a*e*g)*x)/(2*a*c*(p + 1)) - 1/(2*a*c*(p + 1))*Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)* Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^2*f*(2*p + 3) + e*(a*e*g*m - c*d*f*(m + 2*p + 2))*x, x], x]",
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            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)*(a_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && (EqQ[d, 0] || EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g] || Not[ILtQ[m + 2*p + 3, 0]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^ m*(a + b*x + c*x^2)^(p + 1)*(f*b - 2*a*g + (2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c)) + 1/((p + 1)*(b^2 - 4*a*c))* Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)* Simp[g*(2*a*e*m + b*d*(2*p + 3)) - f*(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)*(m + 2*p + 3)*x, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^ m*(a + c*x^2)^(p + 1)*(a*g - c*f*x)/(2*a*c*(p + 1)) - 1/(2*a*c*(p + 1))* Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)* Simp[a*e*g*m - c*d*f*(2*p + 3) - c*e*f*(m + 2*p + 3)*x, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))* x)*(a + b*x + c*x^2)^(p + 1)/ ((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) + 1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))* Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)* Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "-(d + e*x)^(m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1)/(2*a* c*(p + 1)*(c*d^2 + a*e^2)) + 1/(2*a*c*(p + 1)*(c*d^2 + a*e^2))* Int[(d + e*x)^m*(a + c*x^2)^(p + 1)* Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "g*(d + e*x)^m/(c*m) + 1/c*Int[(d + e*x)^(m - 1)* Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x]/(a + b*x + c*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)/(a_. + b_.*x_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[m] && GtQ[m, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "g*(d + e*x)^m/(c*m) + 1/c*Int[(d + e*x)^(m - 1)* Simp[c*d*f - a*e*g + (g*c*d + c*e*f)*x, x]/(a + c*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)/(a_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && FractionQ[m] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "2*Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)/(Sqrt[d_. + e_.*x_]*(a_. + b_.*x_ + c_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "2*Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)/(Sqrt[d_. + e_.*x_]*(a_ + c_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "(e*f - d*g)*(d + e*x)^(m + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2)) + 1/(c*d^2 - b*d*e + a*e^2)* Int[(d + e*x)^(m + 1)* Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x]/(a + b*x + c*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)/(a_. + b_.*x_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[m] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "(e*f - d*g)*(d + e*x)^(m + 1)/((m + 1)*(c*d^2 + a*e^2)) + 1/(c*d^2 + a*e^2)* Int[(d + e*x)^(m + 1)* Simp[c*d*f + a*e*g - c*(e*f - d*g)*x, x]/(a + c*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)/(a_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] && FractionQ[m] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x)^m, (f + g*x)/(a + b*x + c*x^2), x], x]",
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            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)/(a_. + b_.*x_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && Not[RationalQ[m]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x)^m, (f + g*x)/(a + c*x^2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)/(a_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Not[RationalQ[m]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) + 1/(c*(m + 2*p + 2))*Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p* Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m*(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && Not[IGtQ[m, 0] && EqQ[f, 0]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "g*(d + e*x)^m*(a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) + 1/(c*(m + 2*p + 2))*Int[(d + e*x)^(m - 1)*(a + c*x^2)^p* Simp[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x]",
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            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)*(a_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && Not[IGtQ[m, 0] && EqQ[f, 0]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "(e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2)) + 1/((m + 1)*(c*d^2 - b*d*e + a*e^2))* Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p* Simp[(c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "(e*f - d*g)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1)/((m + 1)*(c*d^2 + a*e^2)) + 1/((m + 1)*(c*d^2 + a*e^2))* Int[(d + e*x)^(m + 1)*(a + c*x^2)^p* Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x]",
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            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)*(a_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])"
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       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "(e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2)) + 1/((m + 1)*(c*d^2 - b*d*e + a*e^2))* Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p* Simp[(c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x]",
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            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[Simplify[m + 2*p + 3], 0] && NeQ[m, -1]"
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       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "(e*f - d*g)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1)/((m + 1)*(c*d^2 + a*e^2)) + 1/((m + 1)*(c*d^2 + a*e^2))* Int[(d + e*x)^(m + 1)*(a + c*x^2)^p* Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x]",
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        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && ILtQ[Simplify[m + 2*p + 3], 0] && NeQ[m, -1]"
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       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "4*f*(a - d)/(b*d - a*e)* Subst[Int[1/(4*(a - d) - x^2), x], x, (2*(a - d) + (b - e)*x)/Sqrt[a + b*x + c*x^2]]",
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            "lhs": "Int[(f_ + g_.*x_)/((d_. + e_.*x_)*Sqrt[a_. + b_.*x_ + c_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[4*c*(a - d) - (b - e)^2, 0] && EqQ[e*f*(b - e) - 2*g*(b*d - a*e), 0] && NeQ[b*d - a*e, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "2*Subst[Int[(f + g*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x, Sqrt[x]]",
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            "lhs": "Int[(f_ + g_.*x_)/(Sqrt[x_]*Sqrt[a_ + b_.*x_ + c_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, f, g}, x] && NeQ[b^2 - 4*a*c, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "2*Subst[Int[(f + g*x^2)/Sqrt[a + c*x^4], x], x, Sqrt[x]]",
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            "lhs": "Int[(f_ + g_.*x_)/(Sqrt[x_]*Sqrt[a_ + c_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, f, g}, x]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "Sqrt[x]/Sqrt[e*x]*Int[(f + g*x)/(Sqrt[x]*Sqrt[a + b*x + c*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)/(Sqrt[e_*x_]*Sqrt[a_ + b_.*x_ + c_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "Sqrt[x]/Sqrt[e*x]*Int[(f + g*x)/(Sqrt[x]*Sqrt[a + c*x^2]), x]",
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            "lhs": "Int[(f_ + g_.*x_)/(Sqrt[e_*x_]*Sqrt[a_ + c_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, e, f, g}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "g/e*Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x] + (e*f - d*g)/e* Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IGtQ[m, 0]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.3 (d+e x)^m (f+g x) (a+b x+c x^2)^p.m",
            "rhs": "g/e*Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x] + (e*f - d*g)/e* Int[(d + e*x)^m*(a + c*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)*(a_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && Not[IGtQ[m, 0]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "c*x^(m + 2)*(f + g*x)^(n + 1)/(g*(m + n + 3))",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(f_ + g_.*x_)^n_.*(b_.*x_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, f, g, m, n}, x] && EqQ[c*f*(m + 2) - b*g*(m + n + 3), 0] && NeQ[m + n + 3, 0]"
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       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "(a + b*x + c*x^2)^ FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p]))* Int[(d + e*x)^m*(f + g*x)^n*(b/2 + c*x)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "Int[(d + e*x)^(m + p)*(f + g*x)^n*(a/d + c/e*x)^p, x]",
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            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && Not[IGtQ[n, 0]]"
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       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "Int[(d + e*x)^(m + p)*(f + g*x)^n*(a/d + c/e*x)^p, x]",
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        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || GtQ[a, 0] && GtQ[d, 0] && EqQ[m + p, 0])"
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       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "Int[x^n*(a/d + c*x/e)*(a + b*x + c*x^2)^(p - 1), x]",
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            "lhs": "Int[x_^n_.*(a_. + b_.*x_ + c_.*x_^2)^p_/(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[p]] && (Not[IntegerQ[n]] || Not[IntegerQ[2*p]] || IGtQ[n, 2] || GtQ[p, 0] && NeQ[n, 2])"
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       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "Int[x^n*(a/d + c*x/e)*(a + c*x^2)^(p - 1), x]",
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        "comment": false,
         "givens": "FreeQ[{a, c, d, e, n, p}, x] && EqQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]] && (Not[IntegerQ[n]] || Not[IntegerQ[2*p]] || IGtQ[n, 2] || GtQ[p, 0] && NeQ[n, 2])"
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       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "Int[(a/d + c*x/e)^(-m)*(f + g*x)^n*(a + b*x + c*x^2)^(m + p), x]",
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            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[p]] && ILtQ[m, 0] && IntegerQ[n] && (LtQ[n, 0] || GtQ[p, 0])"
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       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "d^(2*m)/a^m*Int[(f + g*x)^n*(a + c*x^2)^(m + p)/(d - e*x)^m, x]",
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            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_ + c_.*x_^2)^p_, x_Symbol]",
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         "givens": "FreeQ[{a, c, d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]] && EqQ[f, 0] && ILtQ[m, -1] && Not[IGtQ[n, 0] && ILtQ[m + n, 0] && Not[GtQ[p, 1]]]"
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       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "d^(2*m)/a^m*Int[(f + g*x)^n*(a + c*x^2)^(m + p)/(d - e*x)^m, x]",
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            "lhs": "Int[(d_ + e_.*x_)^m_*(f_ + g_.*x_)^n_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]] && ILtQ[m, 0] && IntegerQ[n]"
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       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "-(2*c*d - b*e)*(f + g*x)^ n*(a + b*x + c*x^2)^(p + 1)/(e*p*(b^2 - 4*a*c)*(d + e*x)) - 1/(d*e*p*(b^2 - 4*a*c))*Int[(f + g*x)^(n - 1)*(a + b*x + c*x^2)^p* Simp[b*(a*e*g*n - c*d*f*(2*p + 1)) - 2*a*c*(d*g*n - e*f*(2*p + 1)) - c*g*(b*d - 2*a*e)*(n + 2*p + 1)*x, x], x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[p]] && IGtQ[n, 0] && ILtQ[n + 2*p, 0]"
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       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "d*(f + g*x)^n*(a + c*x^2)^(p + 1)/(2*a*e*p*(d + e*x)) - 1/(2*d*e*p)* Int[(f + g*x)^(n - 1)*(a + c*x^2)^p* Simp[d*g*n - e*f*(2*p + 1) - e*g*(n + 2*p + 1)*x, x], x]",
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            "lhs": "Int[(f_. + g_.*x_)^n_*(a_ + c_.*x_^2)^p_/(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]] && IGtQ[n, 0] && ILtQ[n + 2*p, 0]"
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       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "(f + g*x)^(n + 1)*(a + b*x + c*x^2)^ p*(c*d - b*e - c*e*x)/(p*(2*c*d - b*e)*(e*f - d*g)) + 1/(p*(2*c*d - b*e)*(e*f - d*g))* Int[(f + g*x)^n*(a + b*x + c*x^2)^ p*(b*e*g*(n + p + 1) + c*e*f*(2*p + 1) - c*d*g*(n + 2*p + 1) + c*e*g*(n + 2*p + 2)*x), x]",
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            "lhs": "Int[(f_. + g_.*x_)^n_*(a_. + b_.*x_ + c_.*x_^2)^p_/(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[p]] && ILtQ[n, 0] && ILtQ[n + 2*p, 0] && Not[IGtQ[n, 0]]"
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       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "d*(f + g*x)^(n + 1)*(a + c*x^2)^(p + 1)/(2*a* p*(e*f - d*g)*(d + e*x)) + 1/(p*(2*c*d)*(e*f - d*g))* Int[(f + g*x)^n*(a + c*x^2)^ p*(c*e*f*(2*p + 1) - c*d*g*(n + 2*p + 1) + c*e*g*(n + 2*p + 2)*x), x]",
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            "lhs": "Int[(f_. + g_.*x_)^n_*(a_ + c_.*x_^2)^p_/(d_ + e_.*x_), x_Symbol]",
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         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]] && ILtQ[n, 0] && ILtQ[n + 2*p, 0] && Not[IGtQ[n, 0]]"
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       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "-e*(d + e*x)^(m - 1)*(f + g*x)^ n*(a + b*x + c*x^2)^(p + 1)/(c*(m - n - 1))",
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            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[p]] && EqQ[m + p, 0] && EqQ[c*e*f + c*d*g - b*e*g, 0] && NeQ[m - n - 1, 0]"
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       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "-e*(d + e*x)^(m - 1)*(f + g*x)^ n*(a + c*x^2)^(p + 1)/(c*(m - n - 1))",
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            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, m, n, p}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]] && EqQ[m + p, 0] && EqQ[e*f + d*g, 0] && NeQ[m - n - 1, 0]"
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       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "-e^2*(d + e*x)^(m - 1)*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^(p + 1)/((n + 1)*(c*e*f + c*d*g - b*e*g))",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[p]] && EqQ[m + p, 0] && EqQ[m - n - 2, 0]"
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       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "-e^2*(d + e*x)^(m - 1)*(f + g*x)^(n + 1)*(a + c*x^2)^(p + 1)/(c*(n + 1)*(e*f + d*g))",
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            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, m, n, p}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]] && EqQ[m + p, 0] && EqQ[m - n - 2, 0]"
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       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^ m*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^p/(g*(n + 1)) + c*m/(e*g*(n + 1))* Int[(d + e*x)^(m + 1)*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^(p - 1), x]",
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            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[p]] && EqQ[m + p, 0] && GtQ[p, 0] && LtQ[n, -1] && Not[IntegerQ[n + p] && LeQ[n + p + 2, 0]]"
  },
  {
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       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^ m*(f + g*x)^(n + 1)*(a + c*x^2)^p/(g*(n + 1)) + c*m/(e*g*(n + 1))* Int[(d + e*x)^(m + 1)*(f + g*x)^(n + 1)*(a + c*x^2)^(p - 1), x]",
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            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]] && EqQ[m + p, 0] && GtQ[p, 0] && LtQ[n, -1] && Not[IntegerQ[n + p] && LeQ[n + p + 2, 0]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "-(d + e*x)^m*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^ p/(g*(m - n - 1)) - m*(c*e*f + c*d*g - b*e*g)/(e^2*g*(m - n - 1))* Int[(d + e*x)^(m + 1)*(f + g*x)^n*(a + b*x + c*x^2)^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[p]] && EqQ[m + p, 0] && GtQ[p, 0] && NeQ[m - n - 1, 0] && Not[IGtQ[n, 0]] && Not[IntegerQ[n + p] && LtQ[n + p + 2, 0]] && RationalQ[n]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "-(d + e*x)^m*(f + g*x)^(n + 1)*(a + c*x^2)^ p/(g*(m - n - 1)) - c*m*(e*f + d*g)/(e^2*g*(m - n - 1))* Int[(d + e*x)^(m + 1)*(f + g*x)^n*(a + c*x^2)^(p - 1), x]",
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        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]] && EqQ[m + p, 0] && GtQ[p, 0] && NeQ[m - n - 1, 0] && Not[IGtQ[n, 0]] && Not[IntegerQ[n + p] && LtQ[n + p + 2, 0]] && RationalQ[n]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "e*(d + e*x)^(m - 1)*(f + g*x)^ n*(a + b*x + c*x^2)^(p + 1)/(c*(p + 1)) - e*g*n/(c*(p + 1))* Int[(d + e*x)^(m - 1)*(f + g*x)^(n - 1)*(a + b*x + c*x^2)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[p]] && EqQ[m + p, 0] && LtQ[p, -1] && GtQ[n, 0]"
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  {
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       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "e*(d + e*x)^(m - 1)*(f + g*x)^n*(a + c*x^2)^(p + 1)/(c*(p + 1)) - e*g*n/(c*(p + 1))* Int[(d + e*x)^(m - 1)*(f + g*x)^(n - 1)*(a + c*x^2)^(p + 1), x]",
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            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]] && EqQ[m + p, 0] && LtQ[p, -1] && GtQ[n, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "e^2*(d + e*x)^(m - 1)*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^(p + 1)/((p + 1)*(c*e*f + c*d*g - b*e*g)) + e^2*g*(m - n - 2)/((p + 1)*(c*e*f + c*d*g - b*e*g))* Int[(d + e*x)^(m - 1)*(f + g*x)^n*(a + b*x + c*x^2)^(p + 1), x]",
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            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[p]] && EqQ[m + p, 0] && LtQ[p, -1] && RationalQ[n]"
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  {
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       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "e^2*(d + e*x)^(m - 1)*(f + g*x)^(n + 1)*(a + c*x^2)^(p + 1)/(c*(p + 1)*(e*f + d*g)) + e^2*g*(m - n - 2)/(c*(p + 1)*(e*f + d*g))* Int[(d + e*x)^(m - 1)*(f + g*x)^n*(a + c*x^2)^(p + 1), x]",
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            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]] && EqQ[m + p, 0] && LtQ[p, -1] && RationalQ[n]"
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       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "-e*(d + e*x)^(m - 1)*(f + g*x)^ n*(a + b*x + c*x^2)^(p + 1)/(c*(m - n - 1)) - n*(c*e*f + c*d*g - b*e*g)/(c*e*(m - n - 1))* Int[(d + e*x)^m*(f + g*x)^(n - 1)*(a + b*x + c*x^2)^p, x]",
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            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[p]] && EqQ[m + p, 0] && GtQ[n, 0] && NeQ[m - n - 1, 0] && (IntegerQ[2*p] || IntegerQ[n])"
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       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "-e*(d + e*x)^(m - 1)*(f + g*x)^ n*(a + c*x^2)^(p + 1)/(c*(m - n - 1)) - n*(e*f + d*g)/(e*(m - n - 1))* Int[(d + e*x)^m*(f + g*x)^(n - 1)*(a + c*x^2)^p, x]",
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            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]] && EqQ[m + p, 0] && GtQ[n, 0] && NeQ[m - n - 1, 0] && (IntegerQ[2*p] || IntegerQ[n])"
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            "rhs": "-e^2*(d + e*x)^(m - 1)*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^(p + 1)/((n + 1)*(c*e*f + c*d*g - b*e*g)) - c*e*(m - n - 2)/((n + 1)*(c*e*f + c*d*g - b*e*g))* Int[(d + e*x)^m*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^p, x]",
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            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[p]] && EqQ[m + p, 0] && LtQ[n, -1] && IntegerQ[2*p]"
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  {
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       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "-e^2*(d + e*x)^(m - 1)*(f + g*x)^(n + 1)*(a + c*x^2)^(p + 1)/((n + 1)*(c*e*f + c*d*g)) - e*(m - n - 2)/((n + 1)*(e*f + d*g))* Int[(d + e*x)^m*(f + g*x)^(n + 1)*(a + c*x^2)^p, x]",
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            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]] && EqQ[m + p, 0] && LtQ[n, -1] && IntegerQ[2*p]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "2*e^2*Subst[Int[1/(c*(e*f + d*g) - b*e*g + e^2*g*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]]",
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            "lhs": "Int[Sqrt[d_ + e_.*x_]/((f_. + g_.*x_)*Sqrt[a_. + b_.*x_ + c_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]"
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       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "2*e^2*Subst[Int[1/(c*(e*f + d*g) + e^2*g*x^2), x], x, Sqrt[a + c*x^2]/Sqrt[d + e*x]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[d_ + e_.*x_]/((f_. + g_.*x_)*Sqrt[a_ + c_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "e^2*(d + e*x)^(m - 2)*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^(p + 1)/(c* g*(n + p + 2))",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[p]] && EqQ[m + p - 1, 0] && EqQ[b*e*g*(n + 1) + c*e*f*(p + 1) - c*d*g*(2*n + p + 3), 0] && NeQ[n + p + 2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "e^2*(d + e*x)^(m - 2)*(f + g*x)^(n + 1)*(a + c*x^2)^(p + 1)/(c* g*(n + p + 2))",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, m, n, p}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]] && EqQ[m + p - 1, 0] && EqQ[e*f*(p + 1) - d*g*(2*n + p + 3), 0] && NeQ[n + p + 2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "e^2*(e*f - d*g)*(d + e*x)^(m - 2)*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^(p + 1)/(g*(n + 1)*(c*e*f + c*d*g - b*e*g)) - e*(b*e*g*(n + 1) + c*e*f*(p + 1) - c*d*g*(2*n + p + 3))/(g*(n + 1)*(c*e*f + c*d*g - b*e*g))* Int[(d + e*x)^(m - 1)*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[p]] && EqQ[m + p - 1, 0] && LtQ[n, -1] && IntegerQ[2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "e^2*(e*f - d*g)*(d + e*x)^(m - 2)*(f + g*x)^(n + 1)*(a + c*x^2)^(p + 1)/(c* g*(n + 1)*(e*f + d*g)) - e*(e*f*(p + 1) - d*g*(2*n + p + 3))/(g*(n + 1)*(e*f + d*g))* Int[(d + e*x)^(m - 1)*(f + g*x)^(n + 1)*(a + c*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]] && EqQ[m + p - 1, 0] && LtQ[n, -1] && IntegerQ[2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "e^2*(d + e*x)^(m - 2)*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^(p + 1)/(c* g*(n + p + 2)) - (b*e*g*(n + 1) + c*e*f*(p + 1) - c*d*g*(2*n + p + 3))/(c* g*(n + p + 2))* Int[(d + e*x)^(m - 1)*(f + g*x)^n*(a + b*x + c*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[p]] && EqQ[m + p - 1, 0] && Not[LtQ[n, -1]] && IntegerQ[2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "e^2*(d + e*x)^(m - 2)*(f + g*x)^(n + 1)*(a + c*x^2)^(p + 1)/(c* g*(n + p + 2)) - (e*f*(p + 1) - d*g*(2*n + p + 3))/(g*(n + p + 2))* Int[(d + e*x)^(m - 1)*(f + g*x)^n*(a + c*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, m, n, p}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]] && EqQ[m + p - 1, 0] && Not[LtQ[n, -1]] && IntegerQ[2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[p]] && ILtQ[m, 0] && (ILtQ[n, 0] || IGtQ[n, 0] && ILtQ[p + 1/2, 0]) && Not[IGtQ[n, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[ 1/Sqrt[a + c*x^2], (d + e*x)^m*(f + g*x)^n*(a + c*x^2)^(p + 1/2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && IntegerQ[p - 1/2] && ILtQ[m, 0] && ILtQ[n, 0] && Not[IGtQ[n, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]] && ILtQ[m, 0] && (ILtQ[n, 0] || IGtQ[n, 0] && ILtQ[p + 1/2, 0]) && Not[IGtQ[n, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "With[{Q = PolynomialQuotient[(f + g*x)^n, a*e + c*d*x, x], h = PolynomialRemainder[(f + g*x)^n, a*e + c*d*x, x]}, h*(2*c*d - b*e)*(d + e*x)^ m*(a + b*x + c*x^2)^(p + 1)/(e*(p + 1)*(b^2 - 4*a*c)) + 1/((p + 1)*(b^2 - 4*a*c))* Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)* ExpandToSum[ d*e*(p + 1)*(b^2 - 4*a*c)*Q - h*(2*c*d - b*e)*(m + 2*p + 2), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*(f_. + g_.*x_)^n_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[p + 1/2, 0] && IGtQ[m, 0] && IGtQ[n, 0] && Not[IGtQ[n, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "With[{Q = PolynomialQuotient[(f + g*x)^n, a*e + c*d*x, x], h = PolynomialRemainder[(f + g*x)^n, a*e + c*d*x, x]}, -d*h*(d + e*x)^m*(a + c*x^2)^(p + 1)/(2*a*e*(p + 1)) + d/(2*a*(p + 1))* Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)* ExpandToSum[2*a*e*(p + 1)*Q + h*(m + 2*p + 2), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*(f_. + g_.*x_)^n_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && ILtQ[p + 1/2, 0] && IGtQ[m, 0] && IGtQ[n, 0] && Not[IGtQ[n, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*x + c*x^2)^p, (d + e*x)^m*(f + g*x)^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[p]] && EqQ[m + n + 2*p + 1, 0] && ILtQ[m, 0] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[(a + c*x^2)^p, (d + e*x)^m*(f + g*x)^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]] && EqQ[m + n + 2*p + 1, 0] && ILtQ[m, 0] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": " g^n*(d+e*x)^(m+n-1)*(a+b*x+c*x^2)^(p+1)/(c*e^(n-1)*(m+n+2*p+1)) + 1/(c*e^n*(m+n+2*p+1))*Int[(d+e*x)^m*(a+b*x+c*x^2)^p* ExpandToSum[c*e^n*(m+n+2*p+1)*(f+g*x)^n-c*g^n*(m+n+2*p+1)*(d+e*x)^ n+e*g^n*(m+p+n)*(d+e*x)^(n-2)*(b*d-2*a*e+(2*c*d-b*e)*x),x],x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.+e_.*x_)^m_.*(f_.+g_.*x_)^n_*(a_.+b_.*x_+c_.*x_^2)^p_,x_ Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e,f,g,m,p},x] && NeQ[e*f-d*g,0] && NeQ[b^2-4*a*c,0] &&  EqQ[c*d^2-b*d*e+a*e^2,0] && Not[IntegerQ[p]] && NeQ[m+n+2*p+1,0] && IGtQ[n,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "  g^n*(d+e*x)^(m+n-1)*(a+c*x^2)^(p+1)/(c*e^(n-1)*(m+n+2*p+1)) + 1/(c*e^n*(m+n+2*p+1))*Int[(d+e*x)^m*(a+c*x^2)^p* ExpandToSum[c*e^n*(m+n+2*p+1)*(f+g*x)^n-c*g^n*(m+n+2*p+1)*(d+e*x)^ n-2*e*g^n*(m+p+n)*(d+e*x)^(n-2)*(a*e-c*d*x),x],x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.+e_.*x_)^m_.*(f_.+g_.*x_)^n_*(a_+c_.*x_^2)^p_,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,c,d,e,f,g,m,p},x] && NeQ[e*f-d*g,0] && EqQ[c*d^2+a*e^2,0] &&  Not[IntegerQ[p]] && NeQ[m+n+2*p+1,0] && IGtQ[n,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "(e*x)^m*(b*x + c*x^2)^p/(x^(m + p)*(b + c*x)^p)* Int[x^(m + p)*(f + g*x)^n*(b + c*x)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_*(f_. + g_.*x_)^n_*(b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, e, f, g, m, n}, x] && Not[IntegerQ[p]] && Not[IGtQ[n, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "Int[(d + e*x)^(m + p)*(f + g*x)^n*(a/d + c/e*x)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]] && GtQ[a, 0] && GtQ[d, 0] && Not[IGtQ[m, 0]] && Not[IGtQ[n, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": " (*(a+b*x+c*x^2)^p/((d+e*x)^p*(a*e+c*d*x)^p)*Int[(d+e*x)^(m+p)*(f+g*x)^ n*(a*e+c*d*x)^p,x] /; *) (a + b*x + c*x^2)^ FracPart[p]/((d + e*x)^FracPart[p]*(a/d + (c*x)/e)^FracPart[p])* Int[(d + e*x)^(m + p)*(f + g*x)^n*(a/d + c/e*x)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[p]] && Not[IGtQ[m, 0]] && Not[IGtQ[n, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "(a + c*x^2)^ FracPart[p]/((d + e*x)^FracPart[p]*(a/d + (c*x)/e)^FracPart[p])* Int[(d + e*x)^(m + p)*(f + g*x)^n*(a/d + c/e*x)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]] && Not[IGtQ[m, 0]] && Not[IGtQ[n, 0]]"
  },
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && (EqQ[p, 1] && IntegersQ[m, n] || ILtQ[m, 0] && ILtQ[n, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p] && (EqQ[p, 1] && IntegersQ[m, n] || ILtQ[m, 0] && ILtQ[n, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "(c*d^2 - b*d*e + a*e^2)/(e*(e*f - d*g))* Int[(a + b*x + c*x^2)^(p - 1)/(d + e*x), x] - 1/(e*(e*f - d*g))* Int[Simp[c*d*f - b*e*f + a*e*g - c*(e*f - d*g)*x, x]*(a + b*x + c*x^2)^(p - 1)/(f + g*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_ + c_.*x_^2)^p_/((d_. + e_.*x_)*(f_. + g_.*x_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[p] && GtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "(c*d^2 + a*e^2)/(e*(e*f - d*g))* Int[(a + c*x^2)^(p - 1)/(d + e*x), x] - 1/(e*(e*f - d*g))* Int[Simp[c*d*f + a*e*g - c*(e*f - d*g)*x, x]*(a + c*x^2)^(p - 1)/(f + g*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + c_.*x_^2)^p_/((d_. + e_.*x_)*(f_. + g_.*x_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && FractionQ[p] && GtQ[p, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "With[{q = Denominator[m]}, q/e*Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + g*x^q/e)^n* ((c*d^2 - b*d*e + a*e^2)/e^2 - (2*c*d - b*e)*x^q/e^2 + c*x^(2*q)/e^2)^p, x], x, (d + e*x)^(1/q)]]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n, p] && FractionQ[m]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "With[{q = Denominator[m]}, q/e*Subst[ Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + g*x^q/e)^ n*((c*d^2 + a*e^2)/e^2 - 2*c*d*x^q/e^2 + c*x^(2*q)/e^2)^p, x], x, (d + e*x)^(1/q)]]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegersQ[n, p] && FractionQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "Int[(d*f + e*g*x^2)^m*(a + b*x + c*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(f_ + g_.*x_)^n_*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[m - n, 0] && EqQ[e*f + d*g, 0] && (IntegerQ[m] || GtQ[d, 0] && GtQ[f, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "Int[(d*f + e*g*x^2)^m*(a + c*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(f_ + g_.*x_)^n_*(a_. + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, m, n, p}, x] && EqQ[m - n, 0] && EqQ[e*f + d*g, 0] && (IntegerQ[m] || GtQ[d, 0] && GtQ[f, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^ FracPart[m]*(f + g*x)^FracPart[m]/(d*f + e*g*x^2)^FracPart[m]* Int[(d*f + e*g*x^2)^m*(a + b*x + c*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(f_ + g_.*x_)^n_*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[m - n, 0] && EqQ[e*f + d*g, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^ FracPart[m]*(f + g*x)^FracPart[m]/(d*f + e*g*x^2)^FracPart[m]* Int[(d*f + e*g*x^2)^m*(a + c*x^2)^p, x]",
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            "lhs": "Int[(d_ + e_.*x_)^m_*(f_ + g_.*x_)^n_*(a_. + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, m, n, p}, x] && EqQ[m - n, 0] && EqQ[e*f + d*g, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "g/c^2*Int[ Simp[2*c*e*f + c*d*g - b*e*g + c*e*g*x, x]*(d + e*x)^(m - 1)*(f + g*x)^(n - 2), x] + 1/c^2* Int[Simp[ c^2*d*f^2 - 2*a*c*e*f*g - a*c*d*g^2 + a*b*e*g^2 + (c^2*e*f^2 + 2*c^2*d*f*g - 2*b*c*e*f*g - b*c*d*g^2 + b^2*e*g^2 - a*c*e*g^2)*x, x]* (d + e*x)^(m - 1)*(f + g*x)^(n - 2)/(a + b*x + c*x^2), x]",
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            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)^n_/(a_. + b_.*x_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && GtQ[m, 0] && GtQ[n, 1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "g/c*Int[Simp[2*e*f + d*g + e*g*x, x]*(d + e*x)^(m - 1)*(f + g*x)^(n - 2), x] + 1/c*Int[ Simp[c*d*f^2 - 2*a*e*f*g - a*d*g^2 + (c*e*f^2 + 2*c*d*f*g - a*e*g^2)*x, x]*(d + e*x)^(m - 1)*(f + g*x)^(n - 2)/(a + c*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)^n_/(a_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && GtQ[m, 0] && GtQ[n, 1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "e*g/c*Int[(d + e*x)^(m - 1)*(f + g*x)^(n - 1), x] + 1/c*Int[ Simp[c*d*f - a*e*g + (c*e*f + c*d*g - b*e*g)*x, x]*(d + e*x)^(m - 1)*(f + g*x)^(n - 1)/(a + b*x + c*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)^n_/(a_. + b_.*x_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && GtQ[m, 0] && GtQ[n, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "e*g/c*Int[(d + e*x)^(m - 1)*(f + g*x)^(n - 1), x] + 1/c*Int[ Simp[c*d*f - a*e*g + (c*e*f + c*d*g)*x, x]*(d + e*x)^(m - 1)*(f + g*x)^(n - 1)/(a + c*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)^n_/(a_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && GtQ[m, 0] && GtQ[n, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
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            "rhs": "-g*(e*f - d*g)/(c*f^2 - b*f*g + a*g^2)* Int[(d + e*x)^(m - 1)*(f + g*x)^n, x] + 1/(c*f^2 - b*f*g + a*g^2)* Int[ Simp[c*d*f - b*d*g + a*e*g + c*(e*f - d*g)*x, x]*(d + e*x)^(m - 1)*(f + g*x)^(n + 1)/(a + b*x + c*x^2), x]",
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            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)^n_/(a_. + b_.*x_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && GtQ[m, 0] && LtQ[n, -1]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "-g*(e*f - d*g)/(c*f^2 + a*g^2)* Int[(d + e*x)^(m - 1)*(f + g*x)^n, x] + 1/(c*f^2 + a*g^2)* Int[ Simp[c*d*f + a*e*g + c*(e*f - d*g)*x, x]*(d + e*x)^(m - 1)*(f + g*x)^(n + 1)/(a + c*x^2), x]",
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            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)^n_/(a_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && GtQ[m, 0] && LtQ[n, -1]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[ 1/(Sqrt[d + e*x]*Sqrt[f + g*x]), (d + e*x)^(m + 1/2)/(a + b*x + c*x^2), x], x]",
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            "lhs": "Int[(d_. + e_.*x_)^m_/(Sqrt[f_. + g_.*x_]*(a_. + b_.*x_ + c_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[m + 1/2, 0]"
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       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[ 1/(Sqrt[d + e*x]*Sqrt[f + g*x]), (d + e*x)^(m + 1/2)/(a + c*x^2), x], x]",
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            "lhs": "Int[(d_. + e_.*x_)^m_/(Sqrt[f_. + g_.*x_]*(a_. + c_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[m + 1/2, 0]"
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       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n, 1/(a + b*x + c*x^2), x], x]",
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            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)^n_/(a_. + b_.*x_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[m]] && Not[IntegerQ[n]]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n, 1/(a + c*x^2), x], x]",
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            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)^n_/(a_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[m]] && Not[IntegerQ[n]]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1)/(c* e*(m + 2*p + 3))",
     "rulenumber": 0,
            "lhs": "Int[x_^2*(d_. + e_.*x_)^m_.*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*e*(m + p + 2) + 2*c*d*(p + 1), 0] && EqQ[b*d*(p + 1) + a*e*(m + 1), 0] && NeQ[m + 2*p + 3, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1)/(c* e*(m + 2*p + 3))",
     "rulenumber": 0,
            "lhs": "Int[x_^2*(d_. + e_.*x_)^m_.*(a_. + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m, p}, x] && EqQ[d*(p + 1), 0] && EqQ[a*(m + 1), 0] && NeQ[m + 2*p + 3, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^ FracPart[p]*(a + b*x + c*x^2)^FracPart[p]/(a*d + c*e*x^3)^ FracPart[p]*Int[(g*x)^n*(a*d + c*e*x^3)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^n_*(d_. + e_.*x_)^m_*(a_ + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, g, m, n, p}, x] && EqQ[m - p, 0] && EqQ[b*d + a*e, 0] && EqQ[c*d + b*e, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^(m + 1)*Sqrt[f + g*x]* Sqrt[a + b*x + c*x^2]/(e*(m + 1)) - 1/(2*e*(m + 1))* Int[(d + e*x)^(m + 1)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])* Simp[b*f + a*g + 2*(c*f + b*g)*x + 3*c*g*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*Sqrt[f_. + g_.*x_]* Sqrt[a_. + b_.*x_ + c_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[2*m] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^(m + 1)*Sqrt[f + g*x]* Sqrt[a + c*x^2]/(e*(m + 1)) - 1/(2*e*(m + 1))* Int[(d + e*x)^(m + 1)/(Sqrt[f + g*x]*Sqrt[a + c*x^2])* Simp[a*g + 2*c*f*x + 3*c*g*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*Sqrt[f_. + g_.*x_]*Sqrt[a_ + c_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[2*m] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "2*(d + e*x)^(m + 1)*Sqrt[f + g*x]* Sqrt[a + b*x + c*x^2]/(e*(2*m + 5)) - 1/(e*(2*m + 5))* Int[(d + e*x)^m/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])* Simp[b*d*f - 3*a*e*f + a*d*g + 2*(c*d*f - b*e*f + b*d*g - a*e*g)* x - (c*e*f - 3*c*d*g + b*e*g)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*Sqrt[f_. + g_.*x_]* Sqrt[a_. + b_.*x_ + c_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[2*m] && Not[LtQ[m, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "2*(d + e*x)^(m + 1)*Sqrt[f + g*x]*Sqrt[a + c*x^2]/(e*(2*m + 5)) + 1/(e*(2*m + 5))*Int[(d + e*x)^m/(Sqrt[f + g*x]*Sqrt[a + c*x^2])* Simp[3*a*e*f - a*d*g - 2*(c*d*f - a*e*g)*x + (c*e*f - 3*c*d*g)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*Sqrt[f_. + g_.*x_]*Sqrt[a_ + c_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[2*m] && Not[LtQ[m, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "2*(d + e*x)^m*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]/(g*(2*m + 3)) - 1/(g*(2*m + 3))* Int[(d + e*x)^(m - 1)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])* Simp[b*d*f + 2*a*(e*f*m - d*g*(m + 1)) + (2*c*d*f - 2*a*e*g + b*(e*f - d*g)*(2*m + 1))* x - (b*e*g + 2*c*(d*g*m - e*f*(m + 1)))*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.* Sqrt[a_. + b_.*x_ + c_.*x_^2]/Sqrt[f_. + g_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[2*m] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "2*(d + e*x)^m*Sqrt[f + g*x]*Sqrt[a + c*x^2]/(g*(2*m + 3)) - 1/(g*(2*m + 3))* Int[(d + e*x)^(m - 1)/(Sqrt[f + g*x]*Sqrt[a + c*x^2])* Simp[2*a*(e*f*m - d*g*(m + 1)) + (2*c*d*f - 2*a*e*g)* x - (2*c*(d*g*m - e*f*(m + 1)))*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*Sqrt[a_ + c_.*x_^2]/Sqrt[f_. + g_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[2*m] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "(c*d^2 - b*d*e + a*e^2)/e^2* Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x] - 1/e^2* Int[(c*d - b*e - c*e*x)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_. + b_.*x_ + c_.*x_^2]/((d_. + e_.*x_)*Sqrt[f_. + g_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "(c*d^2 + a*e^2)/e^2* Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[a + c*x^2]), x] - 1/e^2*Int[(c*d - c*e*x)/(Sqrt[f + g*x]*Sqrt[a + c*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + c_.*x_^2]/((d_. + e_.*x_)*Sqrt[f_. + g_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^(m + 1)*Sqrt[f + g*x]* Sqrt[a + b*x + c*x^2]/((m + 1)*(e*f - d*g)) - 1/(2*(m + 1)*(e*f - d*g))* Int[(d + e*x)^(m + 1)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])* Simp[b*f + a*g*(2*m + 3) + 2*(c*f + b*g*(m + 2))*x + c*g*(2*m + 5)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.* Sqrt[a_. + b_.*x_ + c_.*x_^2]/Sqrt[f_. + g_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[2*m] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "(d + e*x)^(m + 1)*Sqrt[f + g*x]* Sqrt[a + c*x^2]/((m + 1)*(e*f - d*g)) - 1/(2*(m + 1)*(e*f - d*g))* Int[(d + e*x)^(m + 1)/(Sqrt[f + g*x]*Sqrt[a + c*x^2])* Simp[a*g*(2*m + 3) + 2*(c*f)*x + c*g*(2*m + 5)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*Sqrt[a_ + c_.*x_^2]/Sqrt[f_. + g_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[2*m] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, Sqrt[2]*Sqrt[2*c*f - g*(b + q)]*Sqrt[b - q + 2*c*x]*(d + e*x)* Sqrt[(e*f - d*g)*(b + q + 2*c*x)/((2*c*f - g*(b + q))*(d + e*x))]* Sqrt[(e*f - d*g)*(2*a + (b + q)*x)/((b*f + q*f - 2*a*g)*(d + e*x))]/ (g*Sqrt[2*c*d - e*(b + q)]*Sqrt[2*a*c/(b + q) + c*x]* Sqrt[a + b*x + c*x^2])* EllipticPi[e*(2*c*f - g*(b + q))/(g*(2*c*d - e*(b + q))), ArcSin[Sqrt[2*c*d - e*(b + q)]* Sqrt[f + g*x]/(Sqrt[2*c*f - g*(b + q)]*Sqrt[d + e*x])], (b*d + q*d - 2*a*e)*(2*c*f - g*(b + q))/((b*f + q*f - 2*a*g)*(2*c*d - e*(b + q)))]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[d_. + e_.*x_]/(Sqrt[f_. + g_.*x_]* Sqrt[a_. + b_.*x_ + c_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "With[{q = Rt[-4*a*c, 2]}, Sqrt[2]*Sqrt[2*c*f - g*q]*Sqrt[-q + 2*c*x]*(d + e*x)* Sqrt[(e*f - d*g)*(q + 2*c*x)/((2*c*f - g*q)*(d + e*x))]* Sqrt[(e*f - d*g)*(2*a + q*x)/((q*f - 2*a*g)*(d + e*x))]/ (g*Sqrt[2*c*d - e*q]*Sqrt[2*a*c/q + c*x]*Sqrt[a + c*x^2])* EllipticPi[e*(2*c*f - g*q)/(g*(2*c*d - e*q)), ArcSin[Sqrt[2*c*d - e*q]* Sqrt[f + g*x]/(Sqrt[2*c*f - g*q]*Sqrt[d + e*x])], (q*d - 2*a*e)*(2*c*f - g*q)/((q*f - 2*a*g)*(2*c*d - e*q))]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[d_. + e_.*x_]/(Sqrt[f_. + g_.*x_]*Sqrt[a_ + c_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "e/g*Int[Sqrt[d + e*x]*Sqrt[f + g*x]/Sqrt[a + b*x + c*x^2], x] - (e*f - d*g)/g* Int[Sqrt[d + e*x]/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^(3/2)/(Sqrt[f_. + g_.*x_]* Sqrt[a_. + b_.*x_ + c_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "e/g*Int[Sqrt[d + e*x]*Sqrt[f + g*x]/Sqrt[a + c*x^2], x] - (e*f - d*g)/g* Int[Sqrt[d + e*x]/(Sqrt[f + g*x]*Sqrt[a + c*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^(3/2)/(Sqrt[f_. + g_.*x_]*Sqrt[a_ + c_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "2*e^2*(d + e*x)^(m - 2)*Sqrt[f + g*x]* Sqrt[a + b*x + c*x^2]/(c*g*(2*m - 1)) - 1/(c*g*(2*m - 1))* Int[(d + e*x)^(m - 3)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])* Simp[b*d*e^2*f + a*e^2*(d*g + 2*e*f*(m - 2)) - c*d^3*g*(2*m - 1) + e*(e*(2*b*d*g + e*(b*f + a*g)*(2*m - 3)) + c*d*(2*e*f - 3*d*g*(2*m - 1)))*x + 2*e^2*(c*e*f - 3*c*d*g + b*e*g)*(m - 1)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^ m_/(Sqrt[f_. + g_.*x_]*Sqrt[a_. + b_.*x_ + c_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[2*m] && GeQ[m, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "2*e^2*(d + e*x)^(m - 2)*Sqrt[f + g*x]* Sqrt[a + c*x^2]/(c*g*(2*m - 1)) - 1/(c*g*(2*m - 1))* Int[(d + e*x)^(m - 3)/(Sqrt[f + g*x]*Sqrt[a + c*x^2])* Simp[a*e^2*(d*g + 2*e*f*(m - 2)) - c*d^3*g*(2*m - 1) + e*(e*(a*e*g*(2*m - 3)) + c*d*(2*e*f - 3*d*g*(2*m - 1)))*x + 2*e^2*(c*e*f - 3*c*d*g)*(m - 1)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_/(Sqrt[f_. + g_.*x_]*Sqrt[a_ + c_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[2*m] && GeQ[m, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "With[{q = Rt[-c/a, 2]}, 1/Sqrt[a]* Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/((d_. + e_.*x_)*Sqrt[f_. + g_.*x_]*Sqrt[a_ + c_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && GtQ[a, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "With[{q = Rt[-c/a, 2]}, Sqrt[1 + c*x^2/a]/Sqrt[a + c*x^2]* Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/((d_. + e_.*x_)*Sqrt[f_. + g_.*x_]*Sqrt[a_ + c_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && Not[GtQ[a, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x]/Sqrt[a + b*x + c*x^2]* Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[b - q + 2*c*x]* Sqrt[b + q + 2*c*x]), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/((d_. + e_.*x_)*Sqrt[f_. + g_.*x_]* Sqrt[a_. + b_.*x_ + c_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "-2*(d + e*x)* Sqrt[(e*f - d*g)^2*(a + b*x + c*x^2)/((c*f^2 - b*f*g + a*g^2)*(d + e*x)^2)]/((e*f - d*g)* Sqrt[a + b*x + c*x^2])* Subst[ Int[ 1/Sqrt[1 - (2*c*d*f - b*e*f - b*d*g + 2*a*e*g)* x^2/(c*f^2 - b*f*g + a*g^2) + (c*d^2 - b*d*e + a*e^2)* x^4/(c*f^2 - b*f*g + a*g^2)], x], x, Sqrt[f + g*x]/Sqrt[d + e*x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[d_. + e_.*x_]*Sqrt[f_. + g_.*x_]* Sqrt[a_. + b_.*x_ + c_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "-2*(d + e*x)* Sqrt[(e*f - d*g)^2*(a + c*x^2)/((c*f^2 + a*g^2)*(d + e*x)^2)]/((e* f - d*g)*Sqrt[a + c*x^2])* Subst[ Int[ 1/Sqrt[1 - (2*c*d*f + 2*a*e*g)* x^2/(c*f^2 + a*g^2) + (c*d^2 + a*e^2)*x^4/(c*f^2 + a*g^2)], x], x, Sqrt[f + g*x]/Sqrt[d + e*x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[d_. + e_.*x_]*Sqrt[f_. + g_.*x_]*Sqrt[a_ + c_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "-g/(e*f - d*g)* Int[1/(Sqrt[d + e*x]*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x] + e/(e*f - d*g)* Int[Sqrt[f + g*x]/((d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((d_. + e_.*x_)^(3/2)*Sqrt[f_. + g_.*x_]* Sqrt[a_. + b_.*x_ + c_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "-g/(e*f - d*g)* Int[1/(Sqrt[d + e*x]*Sqrt[f + g*x]*Sqrt[a + c*x^2]), x] + e/(e*f - d*g)* Int[Sqrt[f + g*x]/((d + e*x)^(3/2)*Sqrt[a + c*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((d_. + e_.*x_)^(3/2)*Sqrt[f_. + g_.*x_]*Sqrt[a_ + c_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "e^2*(d + e*x)^(m + 1)*Sqrt[f + g*x]* Sqrt[a + b*x + c*x^2]/((m + 1)*(e*f - d*g)*(c*d^2 - b*d*e + a*e^2)) + 1/(2*(m + 1)*(e*f - d*g)*(c*d^2 - b*d*e + a*e^2))* Int[(d + e*x)^(m + 1)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])* Simp[2*d*(c*e*f - c*d*g + b*e*g)*(m + 1) - e^2*(b*f + a*g)*(2*m + 3) + 2*e*(c*d*g*(m + 1) - e*(c*f + b*g)*(m + 2))*x - c*e^2*g*(2*m + 5)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^ m_/(Sqrt[f_. + g_.*x_]*Sqrt[a_. + b_.*x_ + c_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[2*m] && LeQ[m, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "e^2*(d + e*x)^(m + 1)*Sqrt[f + g*x]* Sqrt[a + c*x^2]/((m + 1)*(e*f - d*g)*(c*d^2 + a*e^2)) + 1/(2*(m + 1)*(e*f - d*g)*(c*d^2 + a*e^2))* Int[(d + e*x)^(m + 1)/(Sqrt[f + g*x]*Sqrt[a + c*x^2])* Simp[2*d*(c*e*f - c*d*g)*(m + 1) - a*e^2*g*(2*m + 3) + 2*e*(c*d*g*(m + 1) - c*e*f*(m + 2))*x - c*e^2*g*(2*m + 5)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_/(Sqrt[f_. + g_.*x_]*Sqrt[a_ + c_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[2*m] && LeQ[m, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "0",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[d_.+e_.*x_]*Sqrt[f_.+g_.*x_]/Sqrt[a_.+b_.*x_+c_.*x_^2],x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,g},x] && NeQ[e*f-d*g,0] &&  NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "2*e*(d + e*x)^(m - 1)*Sqrt[f + g*x]* Sqrt[a + b*x + c*x^2]/(c*(2*m + 1)) - 1/(c*(2*m + 1))* Int[(d + e*x)^(m - 2)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])* Simp[e*(b*d*f + a*(d*g + 2*e*f*(m - 1))) - c*d^2*f*(2*m + 1) + (a*e^2*g*(2*m - 1) - c*d*(4*e*f*m + d*g*(2*m + 1)) + b*e*(2*d*g + e*f*(2*m - 1)))*x + e*(2*b*e*g*m - c*(e*f + d*g*(4*m - 1)))*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_* Sqrt[f_. + g_.*x_]/Sqrt[a_. + b_.*x_ + c_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[2*m] && GtQ[m, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "2*e*(d + e*x)^(m - 1)*Sqrt[f + g*x]*Sqrt[a + c*x^2]/(c*(2*m + 1)) - 1/(c*(2*m + 1))* Int[(d + e*x)^(m - 2)/(Sqrt[f + g*x]*Sqrt[a + c*x^2])* Simp[a*e*(d*g + 2*e*f*(m - 1)) - c*d^2*f*(2*m + 1) + (a*e^2*g*(2*m - 1) - c*d*(4*e*f*m + d*g*(2*m + 1)))*x - c*e*(e*f + d*g*(4*m - 1))*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*Sqrt[f_. + g_.*x_]/Sqrt[a_ + c_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[2*m] && GtQ[m, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "g/e*Int[1/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x] + (e*f - d*g)/e* Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[f_. + g_.*x_]/((d_. + e_.*x_)*Sqrt[a_. + b_.*x_ + c_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "g/e*Int[1/(Sqrt[f + g*x]*Sqrt[a + c*x^2]), x] + (e*f - d*g)/e* Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[a + c*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[f_. + g_.*x_]/((d_. + e_.*x_)*Sqrt[a_ + c_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "e*(d + e*x)^(m + 1)*Sqrt[f + g*x]* Sqrt[a + b*x + c*x^2]/((m + 1)*(c*d^2 - b*d*e + a*e^2)) + 1/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))* Int[(d + e*x)^(m + 1)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])* Simp[2*c*d*f*(m + 1) - e*(a*g + b*f*(2*m + 3)) - 2*(b*e*g*(2 + m) - c*(d*g*(m + 1) - e*f*(m + 2)))*x - c*e*g*(2*m + 5)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_* Sqrt[f_. + g_.*x_]/Sqrt[a_. + b_.*x_ + c_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[2*m] && LeQ[m, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "e*(d + e*x)^(m + 1)*Sqrt[f + g*x]* Sqrt[a + c*x^2]/((m + 1)*(c*d^2 + a*e^2)) + 1/(2*(m + 1)*(c*d^2 + a*e^2))* Int[(d + e*x)^(m + 1)/(Sqrt[f + g*x]*Sqrt[a + c*x^2])* Simp[2*c*d*f*(m + 1) - e*(a*g) + 2*c*(d*g*(m + 1) - e*f*(m + 2))*x - c*e*g*(2*m + 5)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*Sqrt[f_. + g_.*x_]/Sqrt[a_ + c_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[2*m] && LeQ[m, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && (IGtQ[m, 0] || EqQ[m, -2] && EqQ[p, 1] && EqQ[2*c*d - b*e, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && (IGtQ[m, 0] || EqQ[m, -2] && EqQ[p, 1] && EqQ[d, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "With[{Qx = PolynomialQuotient[(a + b*x + c*x^2)^p, d + e*x, x], R = PolynomialRemainder[(a + b*x + c*x^2)^p, d + e*x, x]}, R*(d + e*x)^(m + 1)*(f + g*x)^(n + 1)/((m + 1)*(e*f - d*g)) + 1/((m + 1)*(e*f - d*g))* Int[(d + e*x)^(m + 1)*(f + g*x)^n* ExpandToSum[(m + 1)*(e*f - d*g)*Qx - g*R*(m + n + 2), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "With[{Qx = PolynomialQuotient[(a + c*x^2)^p, d + e*x, x], R = PolynomialRemainder[(a + c*x^2)^p, d + e*x, x]}, R*(d + e*x)^(m + 1)*(f + g*x)^(n + 1)/((m + 1)*(e*f - d*g)) + 1/((m + 1)*(e*f - d*g))* Int[(d + e*x)^(m + 1)*(f + g*x)^n* ExpandToSum[(m + 1)*(e*f - d*g)*Qx - g*R*(m + n + 2), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "c^p*(d + e*x)^(m + 2*p)*(f + g*x)^(n + 1)/(g* e^(2*p)*(m + n + 2*p + 1)) + 1/(g*e^(2*p)*(m + n + 2*p + 1))*Int[(d + e*x)^m*(f + g*x)^n* ExpandToSum[ g*(m + n + 2*p + 1)*(e^(2*p)*(a + b*x + c*x^2)^p - c^p*(d + e*x)^(2*p)) - c^p*(e*f - d*g)*(m + 2*p)*(d + e*x)^(2*p - 1), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && NeQ[m + n + 2*p + 1, 0] && (IntegerQ[n] || Not[IntegerQ[m]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "c^p*(d + e*x)^(m + 2*p)*(f + g*x)^(n + 1)/(g* e^(2*p)*(m + n + 2*p + 1)) + 1/(g*e^(2*p)*(m + n + 2*p + 1))*Int[(d + e*x)^m*(f + g*x)^n* ExpandToSum[ g*(m + n + 2*p + 1)*(e^(2*p)*(a + c*x^2)^p - c^p*(d + e*x)^(2*p)) - c^p*(e*f - d*g)*(m + 2*p)*(d + e*x)^(2*p - 1), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && NeQ[m + n + 2*p + 1, 0] && (IntegerQ[n] || Not[IntegerQ[m]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "(c*d^2 - b*d*e + a*e^2)/(e*(e*f - d*g))* Int[(f + g*x)^(n + 1)*(a + b*x + c*x^2)^(p - 1)/(d + e*x), x] - 1/(e*(e*f - d*g))* Int[(f + g*x)^ n*(c*d*f - b*e*f + a*e*g - c*(e*f - d*g)*x)*(a + b*x + c*x^2)^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^n_*(a_. + b_.*x_ + c_.*x_^2)^p_/(d_. + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[n]] && Not[IntegerQ[p]] && GtQ[p, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "(c*d^2 + a*e^2)/(e*(e*f - d*g))* Int[(f + g*x)^(n + 1)*(a + c*x^2)^(p - 1)/(d + e*x), x] - 1/(e*(e*f - d*g))* Int[(f + g*x)^ n*(c*d*f + a*e*g - c*(e*f - d*g)*x)*(a + c*x^2)^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^n_*(a_ + c_.*x_^2)^p_/(d_. + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[n]] && Not[IntegerQ[p]] && GtQ[p, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "e*(e*f - d*g)/(c*d^2 - b*d*e + a*e^2)* Int[(f + g*x)^(n - 1)*(a + b*x + c*x^2)^(p + 1)/(d + e*x), x] + 1/(c*d^2 - b*d*e + a*e^2)* Int[(f + g*x)^(n - 1)*(c*d*f - b*e*f + a*e*g - c*(e*f - d*g)*x)*(a + b*x + c*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^n_*(a_. + b_.*x_ + c_.*x_^2)^p_/(d_. + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[n]] && Not[IntegerQ[p]] && LtQ[p, -1] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "e*(e*f - d*g)/(c*d^2 + a*e^2)* Int[(f + g*x)^(n - 1)*(a + c*x^2)^(p + 1)/(d + e*x), x] + 1/(c*d^2 + a*e^2)* Int[(f + g*x)^(n - 1)*(c*d*f + a*e*g - c*(e*f - d*g)*x)*(a + c*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^n_*(a_ + c_.*x_^2)^p_/(d_. + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[n]] && Not[IntegerQ[p]] && LtQ[p, -1] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[ 1/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), (f + g*x)^(n + 1/2)/(d + e*x), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^n_/((d_. + e_.*x_)*Sqrt[a_. + b_.*x_ + c_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[n + 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[ 1/(Sqrt[f + g*x]*Sqrt[a + c*x^2]), (f + g*x)^(n + 1/2)/(d + e*x), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^n_/((d_. + e_.*x_)*Sqrt[a_ + c_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[n + 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "d*(g*x)^n/x^n*Int[(x^n*(a + c*x^2)^p)/(d^2 - e^2*x^2), x] - e*(g*x)^n/x^n*Int[(x^(n + 1)*(a + c*x^2)^p)/(d^2 - e^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^n_.*(a_ + c_.*x_^2)^p_/(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, g, n, p}, x] && NeQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]] && Not[IntegersQ[n, 2*p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && (IntegerQ[p] || ILtQ[m, 0] && ILtQ[n, 0]) && Not[IGtQ[m, 0] || IGtQ[n, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || ILtQ[m, 0] && ILtQ[n, 0]) && Not[IGtQ[m, 0] || IGtQ[n, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "(g*x)^n/x^n* Int[ExpandIntegrand[ x^n*(a + c*x^2)^ p, (d/(d^2 - e^2*x^2) - e*x/(d^2 - e^2*x^2))^(-m), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^n_.*(d_ + e_.*x_)^m_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, g, n, p}, x] && NeQ[c*d^2 + a*e^2, 0] && ILtQ[m, 0] && Not[IntegerQ[p]] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "Unintegrable[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && Not[IGtQ[m, 0] || IGtQ[n, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "Unintegrable[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*(f_. + g_.*x_)^n_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, m, n, p}, x] && Not[IGtQ[m, 0] || IGtQ[n, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "1/Coefficient[u, x, 1]* Subst[Int[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x, u]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*u_)^m_.*(f_. + g_.*u_)^n_.*(a_ + b_.*u_ + c_.*u_^2)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && LinearQ[u, x] && NeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
       "filename": "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.m",
            "rhs": "1/Coefficient[u, x, 1]* Subst[Int[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x, u]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*u_)^m_.*(f_. + g_.*u_)^n_.*(a_ + c_.*u_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, m, n, p}, x] && LinearQ[u, x] && NeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "(c/f)^p*Int[(d + e*x + f*x^2)^(p + q), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_.*(d_ + e_.*x_ + f_.*x_^2)^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p, q}, x] && EqQ[c*d - a*f, 0] && EqQ[b*d - a*e, 0] && (IntegerQ[p] || GtQ[c/f, 0]) && (Not[IntegerQ[q]] || LeafCount[d + e*x + f*x^2] <= LeafCount[a + b*x + c*x^2])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "a^IntPart[p]*(a + b*x + c*x^2)^ FracPart[p]/(d^IntPart[p]*(d + e*x + f*x^2)^FracPart[p])* Int[(d + e*x + f*x^2)^(p + q), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_*(d_ + e_.*x_ + f_.*x_^2)^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p, q}, x] && EqQ[c*d - a*f, 0] && EqQ[b*d - a*e, 0] && Not[IntegerQ[p]] && Not[IntegerQ[q]] && Not[GtQ[c/f, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "(a + b*x + c*x^2)^ FracPart[p]/((4*c)^IntPart[p]*(b + 2*c*x)^(2*FracPart[p]))* Int[(b + 2*c*x)^(2*p)*(d + e*x + f*x^2)^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_*(d_ + e_.*x_ + f_.*x_^2)^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p, q}, x] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "(a + b*x + c*x^2)^ FracPart[p]/((4*c)^IntPart[p]*(b + 2*c*x)^(2*FracPart[p]))* Int[(b + 2*c*x)^(2*p)*(d + f*x^2)^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_*(d_ + f_.*x_^2)^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, p, q}, x] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "  1/(2^(2*p+2*q+1)*c*(-c/(b^2-4*a*c))^p*(-f/(e^2-4*d*f))^q)* Subst[Int[(1-x^2/(b^2-4*a*c))^p*(1+e*x^2/(b*(4*c*d-b*e)))^q,x],x, b+2*c*x]",
     "rulenumber": 0,
            "lhs": "Int[(a_+b_.*x_+c_.*x_^2)^p_*(d_.+e_.*x_+f_.*x_^2)^q_,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,p,q},x] && NeQ[b^2-4*a*c,0] &&  NeQ[e^2-4*d*f,0] && EqQ[c*e-b*f,0] && (IntegerQ[p] || GtQ[-c/(b^2-4*a*c),0]) && (IntegerQ[q] ||  GtQ[-f/(e^2-4*d*f),0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "  (d+e*x+f*x^2)^q/(2^(2*p+2*q+1)*c*(-c/(b^2-4*a*c))^p*(-f*(d+e*x+f*x^2) /(e^2-4*d*f))^q)* Subst[Int[(1-x^2/(b^2-4*a*c))^p*(1+e*x^2/(b*(4*c*d-b*e)))^q,x],x, b+2*c*x]",
     "rulenumber": 0,
            "lhs": "Int[(a_+b_.*x_+c_.*x_^2)^p_*(d_.+e_.*x_+f_.*x_^2)^q_,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,p,q},x] && NeQ[b^2-4*a*c,0] &&  NeQ[e^2-4*d*f,0] && EqQ[c*e-b*f,0] && (IntegerQ[p] || GtQ[-c/(b^2-4*a*c),0]) && Not[IntegerQ[q] ||  GtQ[-f/(e^2-4*d*f),0]] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "  (a+b*x+c*x^2)^p*(d+e*x+f*x^2)^q/(2^(2*p+2*q+1)*c*(-c*(a+b*x+c*x^2)/( b^2-4*a*c))^p*(-f*(d+e*x+f*x^2)/(e^2-4*d*f))^q)* Subst[Int[(1-x^2/(b^2-4*a*c))^p*(1+e*x^2/(b*(4*c*d-b*e)))^q,x],x, b+2*c*x]",
     "rulenumber": 0,
            "lhs": "Int[(a_+b_.*x_+c_.*x_^2)^p_*(d_.+e_.*x_+f_.*x_^2)^q_,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,p,q},x] && NeQ[b^2-4*a*c,0] &&  NeQ[e^2-4*d*f,0] && EqQ[c*e-b*f,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "(b + 2*c*x)*(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^ q/((b^2 - 4*a*c)*(p + 1)) - (1/((b^2 - 4*a*c)*(p + 1)))* Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)* Simp[2*c*d*(2*p + 3) + b*e*q + (2*b*f*q + 2*c*e*(2*p + q + 3))*x + 2*c*f*(2*p + 2*q + 3)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_ + c_.*x_^2)^p_*(d_. + e_.*x_ + f_.*x_^2)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[q, 0] && Not[IGtQ[q, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "(b + 2*c*x)*(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^ q/((b^2 - 4*a*c)*(p + 1)) - (1/((b^2 - 4*a*c)*(p + 1)))* Int[(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^(q - 1)* Simp[2*c*d*(2*p + 3) + (2*b*f*q)*x + 2*c*f*(2*p + 2*q + 3)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_ + c_.*x_^2)^p_*(d_. + f_.*x_^2)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[q, 0] && Not[IGtQ[q, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "(2*c*x)*(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^ q/((-4*a*c)*(p + 1)) - (1/((-4*a*c)*(p + 1)))* Int[(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)* Simp[2*c*d*(2*p + 3) + (2*c*e*(2*p + q + 3))*x + 2*c*f*(2*p + 2*q + 3)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + c_.*x_^2)^p_*(d_. + e_.*x_ + f_.*x_^2)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f}, x] && NeQ[e^2 - 4*d*f] && LtQ[p, -1] && GtQ[q, 0] && Not[IGtQ[q, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "(2*a*c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a*f) + c*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*x)*(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)/ ((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)) - (1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)))* Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q* Simp[2*c*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(a*f*(p + 1) - c*d*(p + 2)) - e*(b^2*c*e - 2*a*c^2*e - b^3*f - b*c*(c*d - 3*a*f))*(p + q + 2) + (2* f*(2*a*c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a*f))*(p + q + 2) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(b*f*(p + 1) - c*e*(2*p + q + 4)))*x + c*f*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(2*p + 2*q + 5)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_ + c_.*x_^2)^p_*(d_. + e_.*x_ + f_.*x_^2)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] && Not[Not[IntegerQ[p]] && ILtQ[q, -1]] && Not[IGtQ[q, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "(b^3*f + b*c*(c*d - 3*a*f) + c*(2*c^2*d + b^2*f - c*(2*a*f))*x)*(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^(q + 1)/ ((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)) - (1/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)))* Int[(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^q* Simp[2*c*(b^2*d*f + (c*d - a*f)^2)*(p + 1) - (2*c^2*d + b^2*f - c*(2*a*f))*(a*f*(p + 1) - c*d*(p + 2)) + (2* f*(b^3*f + b*c*(c*d - 3*a*f))*(p + q + 2) - (2*c^2*d + b^2*f - c*(2*a*f))*(b*f*(p + 1)))*x + c*f*(2*c^2*d + b^2*f - c*(2*a*f))*(2*p + 2*q + 5)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_ + c_.*x_^2)^p_*(d_. + f_.*x_^2)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, q}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[b^2*d*f + (c*d - a*f)^2, 0] && Not[Not[IntegerQ[p]] && ILtQ[q, -1]] && Not[IGtQ[q, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "(2*a*c^2*e + c*(2*c^2*d - c*(2*a*f))*x)*(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)/ ((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p + 1)) - (1/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p + 1)))* Int[(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q* Simp[2*c*((c*d - a*f)^2 - (-a*e)*(c*e))*(p + 1) - (2*c^2*d - c*(2*a*f))*(a*f*(p + 1) - c*d*(p + 2)) - e*(-2*a*c^2*e)*(p + q + 2) + (2* f*(2*a*c^2*e)*(p + q + 2) - (2*c^2*d - c*(2*a*f))*(-c* e*(2*p + q + 4)))*x + c*f*(2*c^2*d - c*(2*a*f))*(2*p + 2*q + 5)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + c_.*x_^2)^p_*(d_. + e_.*x_ + f_.*x_^2)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, q}, x] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[a*c*e^2 + (c*d - a*f)^2, 0] && Not[Not[IntegerQ[p]] && ILtQ[q, -1]] && Not[IGtQ[q, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "(b*f*(3*p + 2*q) - c*e*(2*p + q) + 2*c*f*(p + q)*x)*(a + b*x + c*x^2)^(p - 1)*(d + e*x + f*x^2)^(q + 1)/(2* f^2*(p + q)*(2*p + 2*q + 1)) - 1/(2*f^2*(p + q)*(2*p + 2*q + 1))* Int[(a + b*x + c*x^2)^(p - 2)*(d + e*x + f*x^2)^q* Simp[(b*d - a*e)*(c*e - b*f)*(1 - p)*(2*p + q) - (p + q)*(b^2*d*f*(1 - p) - a*(f*(b*e - 2*a*f)*(2*p + 2*q + 1) + c*(2*d*f - e^2*(2*p + q)))) + (2*(c*d - a*f)*(c*e - b*f)*(1 - p)*(2*p + q) - (p + q)*((b^2 - 4*a*c)*e*f*(1 - p) + b*(c*(e^2 - 4*d*f)*(2*p + q) + f*(2*c*d - b*e + 2*a*f)*(2*p + 2*q + 1))))*x + ((c*e - b*f)^2*(1 - p)*p + c*(p + q)*(f*(b*e - 2*a*f)*(4*p + 2*q - 1) - c*(2*d*f*(1 - 2*p) + e^2*(3*p + q - 1))))*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_ + c_.*x_^2)^p_*(d_. + e_.*x_ + f_.*x_^2)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && GtQ[p, 1] && NeQ[p + q, 0] && NeQ[2*p + 2*q + 1, 0] && Not[IGtQ[p, 0]] && Not[IGtQ[q, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "(b*(3*p + 2*q) + 2*c*(p + q)*x)*(a + b*x + c*x^2)^(p - 1)*(d + f*x^2)^(q + 1)/(2* f*(p + q)*(2*p + 2*q + 1)) - 1/(2*f*(p + q)*(2*p + 2*q + 1))* Int[(a + b*x + c*x^2)^(p - 2)*(d + f*x^2)^q* Simp[b^2* d*(p - 1)*(2*p + q) - (p + q)*(b^2*d*(1 - p) - 2*a*(c*d - a*f*(2*p + 2*q + 1))) - (2*b*(c*d - a*f)*(1 - p)*(2*p + q) - 2*(p + q)* b*(2*c*d*(2*p + q) - (c*d + a*f)*(2*p + 2*q + 1)))*x + (b^2*f*p*(1 - p) + 2*c*(p + q)*(c*d*(2*p - 1) - a*f*(4*p + 2*q - 1)))*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_ + c_.*x_^2)^p_*(d_. + f_.*x_^2)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, q}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 1] && NeQ[p + q, 0] && NeQ[2*p + 2*q + 1, 0] && Not[IGtQ[p, 0]] && Not[IGtQ[q, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "-c*(e*(2*p + q) - 2*f*(p + q)*x)*(a + c*x^2)^(p - 1)*(d + e*x + f*x^2)^(q + 1)/(2* f^2*(p + q)*(2*p + 2*q + 1)) - 1/(2*f^2*(p + q)*(2*p + 2*q + 1))* Int[(a + c*x^2)^(p - 2)*(d + e*x + f*x^2)^q* Simp[-a*c*e^2*(1 - p)*(2*p + q) + a*(p + q)*(-2*a*f^2*(2*p + 2*q + 1) + c*(2*d*f - e^2*(2*p + q))) + (2*(c*d - a*f)*(c*e)*(1 - p)*(2*p + q) + 4*a*c*e*f*(1 - p)*(p + q))*x + (p*c^2*e^2*(1 - p) - c*(p + q)*(2*a*f^2*(4*p + 2*q - 1) + c*(2*d*f*(1 - 2*p) + e^2*(3*p + q - 1))))*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + c_.*x_^2)^p_*(d_. + e_.*x_ + f_.*x_^2)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, q}, x] && NeQ[e^2 - 4*d*f, 0] && GtQ[p, 1] && NeQ[p + q, 0] && NeQ[2*p + 2*q + 1, 0] && Not[IGtQ[p, 0]] && Not[IGtQ[q, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "With[{q = c^2*d^2 - b*c*d*e + a*c*e^2 + b^2*d*f - 2*a*c*d*f - a*b*e*f + a^2*f^2}, 1/q*Int[(c^2*d - b*c*e + b^2*f - a*c*f - (c^2*e - b*c*f)*x)/(a + b*x + c*x^2), x] + 1/q* Int[(c*e^2 - c*d*f - b*e*f + a*f^2 + (c*e*f - b*f^2)*x)/(d + e*x + f*x^2), x] /; NeQ[q, 0]]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_ + b_.*x_ + c_.*x_^2)*(d_ + e_.*x_ + f_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "With[{q = c^2*d^2 + b^2*d*f - 2*a*c*d*f + a^2*f^2}, 1/q*Int[(c^2*d + b^2*f - a*c*f + b*c*f*x)/(a + b*x + c*x^2), x] - 1/q*Int[(c*d*f - a*f^2 + b*f^2*x)/(d + f*x^2), x] /; NeQ[q, 0]]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_ + b_.*x_ + c_.*x_^2)*(d_ + f_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f}, x] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "-2*e* Subst[Int[1/(e*(b*e - 4*a*f) - (b*d - a*e)*x^2), x], x, (e + 2*f*x)/Sqrt[d + e*x + f*x^2]]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_ + b_.*x_ + c_.*x_^2)*Sqrt[d_. + e_.*x_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[c*e - b*f, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "  With[{k=Rt[(a/c-d/f)^2+(b/c-e/f)*(b*d/(c*f)-a*e/(c*f)),2]}, -2*(c*d-a*f+c*f*k+(c*e-b*f)*x)*Sqrt[(d+e*x+f*x^2)*((c*f*k)/(c*d-a*f+ c*f*k+(c*e-b*f)*x))^2]/(c*Sqrt[d+e*x+f*x^2])* Subst[Int[(1-x)/( (b*d-a*e-b*f*k-(c*d-a*f-c*f*k)^2/(c*e-b*f)+(b*d-a*e+b*f*k-(a*f- c*d-c*f*k)^2/(c*e-b*f))*x^2)* Sqrt[-f*((b*d-a*e-c*e*k)/(c*e-b*f)-(c*d-a*f-c*f*k)^2/(c*e-b*f)^ 2)-f*((b*d-a*e+c*e*k)/(c*e-b*f)-(a*f-c*d-c*f*k)^2/(c*e-b*f)^2)*x^2]), x],x, (c*d-a*f-c*f*k+(c*e-b*f)*x)/(c*d-a*f+c*f*k+(c*e-b*f)*x)]]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_+b_.*x_+c_.*x_^2)*Sqrt[d_.+e_.*x_+f_.*x_^2]),x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e,f},x] && RationalQ[a,b,c,d,e,f] && NeQ[b^2-4*a*c,0] &&  NeQ[e^2-4*d*f,0] && NeQ[c*e-b*f,0] && LtQ[b^2-4*a*c,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": " With[{k=Rt[(a/c-d/f)^2+a*e^2/(c*f^2),2]}, -2*(c*d-a*f+c*f*k+c*e*x)*Sqrt[(d+e*x+f*x^2)*((c*f*k)/(c*d-a*f+c*f*k+ c*e*x))^2]/(c*Sqrt[d+e*x+f*x^2])* Subst[Int[(1-x)/( (-a*e-(c*d-a*f-c*f*k)^2/(c*e)+(-a*e-(a*f-c*d-c*f*k)^2/(c*e))*x^ 2)* Sqrt[-f*((-a*e-c*e*k)/(c*e)-(c*d-a*f-c*f*k)^2/(c*e)^2)-f*((-a*e+ c*e*k)/(c*e)-(a*f-c*d-c*f*k)^2/(c*e)^2)*x^2]),x],x, (c*d-a*f-c*f*k+(c*e)*x)/(c*d-a*f+c*f*k+(c*e)*x)]]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_+c_.*x_^2)*Sqrt[d_.+e_.*x_+f_.*x_^2]),x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,c,d,e,f},x] && RationalQ[a,c,d,e,f] && NeQ[e^2-4*d*f,0] &&  LtQ[-a*c,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": " With[{k=Rt[(a/c-d/f)^2+b^2*d/(c^2*f),2]}, -2*(c*d-a*f+c*f*k-b*f*x)*Sqrt[(d+f*x^2)*((c*f*k)/(c*d-a*f+c*f*k-b*f* x))^2]/(c*Sqrt[d+f*x^2])* Subst[Int[(1-x)/( (b*d-b*f*k+(c*d-a*f-c*f*k)^2/(b*f)+(b*d+b*f*k+(a*f-c*d-c*f*k)^2/ (b*f))*x^2)* Sqrt[-f*(-d/f-(c*d-a*f-c*f*k)^2/(b*f)^2)-f*(-d/f-(a*f-c*d-c*f*k) ^2/(b*f)^2)*x^2]),x],x, (c*d-a*f-c*f*k+(-b*f)*x)/(c*d-a*f+c*f*k+(-b*f)*x)]]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_+b_.*x_+c_.*x_^2)*Sqrt[d_.+f_.*x_^2]),x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,f},x] && RationalQ[a,b,c,d,f] && NeQ[b^2-4*a*c,0] &&  LtQ[b^2-4*a*c,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, 2*c/q*Int[1/((b - q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x] - 2*c/q*Int[1/((b + q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_ + b_.*x_ + c_.*x_^2)*Sqrt[d_. + e_.*x_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && NeQ[c*e - b*f, 0] && PosQ[b^2 - 4*a*c]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "1/2*Int[1/((a - Rt[-a*c, 2]*x)*Sqrt[d + e*x + f*x^2]), x] + 1/2*Int[1/((a + Rt[-a*c, 2]*x)*Sqrt[d + e*x + f*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_ + c_.*x_^2)*Sqrt[d_. + e_.*x_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f}, x] && NeQ[e^2 - 4*d*f, 0] && PosQ[-a*c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, 2*c/q*Int[1/((b - q + 2*c*x)*Sqrt[d + f*x^2]), x] - 2*c/q*Int[1/((b + q + 2*c*x)*Sqrt[d + f*x^2]), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_ + b_.*x_ + c_.*x_^2)*Sqrt[d_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "With[{q = Rt[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 2]}, 1/(2*q)* Int[(c*d - a*f + q + (c*e - b*f)*x)/((a + b*x + c*x^2)* Sqrt[d + e*x + f*x^2]), x] - 1/(2*q)* Int[(c*d - a*f - q + (c*e - b*f)*x)/((a + b*x + c*x^2)* Sqrt[d + e*x + f*x^2]), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_. + b_.*x_ + c_.*x_^2)*Sqrt[d_. + e_.*x_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && NeQ[c*e - b*f, 0] && NegQ[b^2 - 4*a*c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "With[{q = Rt[(c*d - a*f)^2 + a*c*e^2, 2]}, 1/(2*q)* Int[(c*d - a*f + q + c*e*x)/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x] - 1/(2*q)* Int[(c*d - a*f - q + c*e*x)/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_. + c_.*x_^2)*Sqrt[d_. + e_.*x_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f}, x] && NeQ[e^2 - 4*d*f, 0] && NegQ[-a*c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "With[{q = Rt[(c*d - a*f)^2 + b^2*d*f, 2]}, 1/(2*q)* Int[(c*d - a*f + q + (-b*f)*x)/((a + b*x + c*x^2)* Sqrt[d + f*x^2]), x] - 1/(2*q)* Int[(c*d - a*f - q + (-b*f)*x)/((a + b*x + c*x^2)* Sqrt[d + f*x^2]), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_. + b_.*x_ + c_.*x_^2)*Sqrt[d_. + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "c/f*Int[1/Sqrt[a + b*x + c*x^2], x] - 1/f*Int[(c*d - a*f + (c*e - b*f)*x)/(Sqrt[ a + b*x + c*x^2]*(d + e*x + f*x^2)), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*x_ + c_.*x_^2]/(d_ + e_.*x_ + f_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "c/f*Int[1/Sqrt[a + b*x + c*x^2], x] - 1/f*Int[(c*d - a*f - b*f*x)/(Sqrt[a + b*x + c*x^2]*(d + f*x^2)), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*x_ + c_.*x_^2]/(d_ + f_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f}, x] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "c/f*Int[1/Sqrt[a + c*x^2], x] - 1/f*Int[(c*d - a*f + c*e*x)/(Sqrt[a + c*x^2]*(d + e*x + f*x^2)), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + c_.*x_^2]/(d_ + e_.*x_ + f_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f}, x] && NeQ[e^2 - 4*d*f, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "With[{r = Rt[b^2 - 4*a*c, 2]}, Sqrt[b + r + 2*c*x]*Sqrt[2*a + (b + r)*x]/Sqrt[a + b*x + c*x^2]* Int[1/(Sqrt[b + r + 2*c*x]*Sqrt[2*a + (b + r)*x]* Sqrt[d + e*x + f*x^2]), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_ + b_.*x_ + c_.*x_^2]*Sqrt[d_ + e_.*x_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "With[{r = Rt[b^2 - 4*a*c, 2]}, Sqrt[b + r + 2*c*x]*Sqrt[2*a + (b + r)*x]/Sqrt[a + b*x + c*x^2]* Int[1/(Sqrt[b + r + 2*c*x]*Sqrt[2*a + (b + r)*x]*Sqrt[d + f*x^2]), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_ + b_.*x_ + c_.*x_^2]*Sqrt[d_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f}, x] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "Unintegrable[(a + b*x + c*x^2)^p*(d + e*x + f*x^2)^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_ + c_.*x_^2)^p_*(d_. + e_.*x_ + f_.*x_^2)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p, q}, x] && Not[IGtQ[p, 0]] && Not[IGtQ[q, 0]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "Unintegrable[(a + c*x^2)^p*(d + e*x + f*x^2)^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + c_.*x_^2)^p_*(d_. + e_.*x_ + f_.*x_^2)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, p, q}, x] && Not[IGtQ[p, 0]] && Not[IGtQ[q, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "1/Coefficient[u, x, 1]* Subst[Int[(a + b*x + c*x^2)^p*(d + e*x + f*x^2)^q, x], x, u]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*u_ + c_.*u_^2)^p_.*(d_. + e_.*u_ + f_.*u_^2)^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p, q}, x] && LinearQ[u, x] && NeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.5 (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "1/Coefficient[u, x, 1]* Subst[Int[(a + c*x^2)^p*(d + e*x + f*x^2)^q, x], x, u]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + c_.*u_^2)^p_.*(d_. + e_.*u_ + f_.*u_^2)^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, p, q}, x] && LinearQ[u, x] && NeQ[u, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "(c/f)^p* Int[(g + h*x)^m*(d + e*x + f*x^2)^(p + q), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)^m_.*(a_ + b_.*x_ + c_.*x_^2)^ p_*(d_ + e_.*x_ + f_.*x_^2)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, p, q}, x] && EqQ[c*d - a*f, 0] && EqQ[b*d - a*e, 0] && (IntegerQ[p] || GtQ[c/f, 0]) && (Not[IntegerQ[q]] || LeafCount[d + e*x + f*x^2] <= LeafCount[a + b*x + c*x^2])"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "a^IntPart[p]*(a + b*x + c*x^2)^ FracPart[p]/(d^IntPart[p]*(d + e*x + f*x^2)^FracPart[p])* Int[(g + h*x)^m*(d + e*x + f*x^2)^(p + q), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)^m_.*(a_ + b_.*x_ + c_.*x_^2)^ p_*(d_ + e_.*x_ + f_.*x_^2)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, p, q}, x] && EqQ[c*d - a*f, 0] && EqQ[b*d - a*e, 0] && Not[IntegerQ[p]] && Not[IntegerQ[q]] && Not[GtQ[c/f, 0]]"
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         "givens": " FreeQ[{a,b,c,d,f,g,h,m,q},x] && NeQ[b^2-4*a*c,0] &&  EqQ[c*g^2-b*g*h+a*h^2,0] && IntegerQ[p] *)"
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       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
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         "givens": " FreeQ[{a,c,d,f,g,h,m,q},x] && EqQ[c*g^2+a*h^2,0] && IntegerQ[p] *)"
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        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e,f,g,h,m,q},x] && EqQ[c*g^2-b*g*h+a*h^2,0] &&  Not[IntegerQ[p]] *)"
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        "comment": false,
         "givens": " FreeQ[{a,c,d,e,f,g,h,m,q},x] && NeQ[e^2-4*d*f,0] &&  EqQ[c*g^2+a*h^2,0] && Not[IntegerQ[p]] *)"
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            "rhs": " (a+b*x+c*x^2)^FracPart[p]/((g+h*x)^FracPart[p]*(a/g+(c*x)/h)^FracPart[ p])*Int[(g+h*x)^(m+p)*(a/g+c/h*x)^p*(d+f*x^2)^q,x]",
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        "comment": false,
         "givens": " FreeQ[{a,b,c,d,f,g,h,m,q},x] && NeQ[b^2-4*a*c,0] &&  EqQ[c*g^2-b*g*h+a*h^2,0] && Not[IntegerQ[p]] *)"
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            "rhs": "  (a+c*x^2)^FracPart[p]/((g+h*x)^FracPart[p]*(a/g+(c*x)/h)^FracPart[p] )*Int[(g+h*x)^(m+p)*(a/g+c/h*x)^p*(d+f*x^2)^q,x]",
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        "comment": false,
         "givens": " FreeQ[{a,c,d,f,g,h,m,q},x] && EqQ[c*g^2+a*h^2,0] && Not[IntegerQ[p]] *)"
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       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "Int[(a/e + c/f*x)^p*(e*x + f*x^2)^(p + q), x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, e, f, q}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*e^2 - b*e*f + a*f^2, 0] && IntegerQ[p]"
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        "comment": false,
         "givens": "FreeQ[{a, c, e, f, q}, x] && EqQ[c*e^2 + a*f^2, 0] && IntegerQ[p]"
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            "rhs": "Sqrt[3]*h* ArcTan[1/Sqrt[3] - 2^(2/3)*(1 - 3*h*x/g)^(2/3)/(Sqrt[ 3]*(1 + 3*h*x/g)^(1/3))]/(2^(2/3)*a^(1/3)*f) + h*Log[d + f*x^2]/(2^(5/3)*a^(1/3)*f) - 3*h*Log[(1 - 3*h*x/g)^(2/3) + 2^(1/3)*(1 + 3*h*x/g)^(1/3)]/(2^(5/3)*a^(1/3)*f)",
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         "givens": "FreeQ[{a, c, d, f, g, h}, x] && EqQ[c*d + 3*a*f, 0] && EqQ[c*g^2 + 9*a*h^2, 0] && GtQ[a, 0]"
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       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "(1 + c*x^2/a)^(1/3)/(a + c*x^2)^(1/3)* Int[(g + h*x)/((1 + c*x^2/a)^(1/3)*(d + f*x^2)), x]",
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       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "g*Int[(a + c*x^2)^p*(d + f*x^2)^q, x] + h*Int[x*(a + c*x^2)^p*(d + f*x^2)^q, x]",
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            "lhs": "Int[(g_ + h_.*x_)*(a_ + c_.*x_^2)^p_*(d_ + f_.*x_^2)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, f, g, h, p, q}, x]"
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       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "Int[ExpandIntegrand[(a + b*x + c*x^2)^p*(d + e*x + f*x^2)^ q*(g + h*x), x], x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && IGtQ[p, 0] && IntegerQ[q]"
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            "rhs": "Int[ExpandIntegrand[(a + c*x^2)^p*(d + e*x + f*x^2)^q*(g + h*x), x], x]",
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        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, h}, x] && NeQ[e^2 - 4*d*f, 0] && IntegersQ[p, q] && (GtQ[p, 0] || GtQ[q, 0])"
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       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "(g*b - 2*a*h - (b*h - 2*g*c)*x)*(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q/((b^2 - 4*a*c)*(p + 1)) - 1/((b^2 - 4*a*c)*(p + 1))* Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)* Simp[e*q*(g*b - 2*a*h) - d*(b*h - 2*g*c)*(2*p + 3) + (2*f*q*(g*b - 2*a*h) - e*(b*h - 2*g*c)*(2*p + q + 3))* x - f*(b*h - 2*g*c)*(2*p + 2*q + 3)*x^2, x], x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[q, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "(a*h - g*c*x)*(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^ q/(2*a*c*(p + 1)) + 2/(4*a*c*(p + 1))* Int[(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)* Simp[g*c*d*(2*p + 3) - a*(h*e*q) + (g*c*e*(2*p + q + 3) - a*(2*h*f*q))*x + g*c*f*(2*p + 2*q + 3)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + c_.*x_^2)^p_*(d_ + e_.*x_ + f_.*x_^2)^q_*(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, h}, x] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "(g*b - 2*a*h - (b*h - 2*g*c)*x)*(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^ q/((b^2 - 4*a*c)*(p + 1)) - 1/((b^2 - 4*a*c)*(p + 1))* Int[(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^(q - 1)* Simp[-d*(b*h - 2*g*c)*(2*p + 3) + (2*f*q*(g*b - 2*a*h))*x - f*(b*h - 2*g*c)*(2*p + 2*q + 3)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_*(d_ + f_.*x_^2)^q_*(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1))* (g* c*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(g*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - h*(b*c*d - 2*a*c*e + a*b*f))*x) + 1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1))* Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q* Simp[(b*h - 2*g*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) + (b^2*(g*f) - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) - a*(-h*c*e)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((g*c)*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (2* f*((g*c)*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (b^2*g*f - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) - a*(-h*c*e)))* (b*f*(p + 1) - c*e*(2*p + q + 4)))*x - c*f*(b^2*(g*f) - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) + a*h*c*e))*(2*p + 2*q + 5)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_*(d_ + e_.*x_ + f_.*x_^2)^ q_*(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] && Not[Not[IntegerQ[p]] && ILtQ[q, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p + 1))* (g*c*(2*a*c*e) + (-a*h)*(2*c^2*d - c*(2*a*f)) + c*(g*(2*c^2*d - c*(2*a*f)) - h*(-2*a*c*e))*x) + 1/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p + 1))* Int[(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q* Simp[(-2*g*c)*((c*d - a*f)^2 - (-a*e)*(c*e))*(p + 1) + (2*(g*c*(c*d - a*f) - a*(-h*c*e)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((g*c)*(2*a*c*e) + (-a*h)*(2*c^2*d - c*(+2*a*f)))*(p + q + 2) - (2* f*((g*c)*(2*a*c*e) + (-a*h)*(2*c^2*d + -c*(+2*a*f)))*(p + q + 2) - (2*(g*c*(c*d - a*f) - a*(-h*c*e)))*(-c* e*(2*p + q + 4)))*x - c*f*(2*(g*c*(c*d - a*f) - a*(-h*c*e)))*(2*p + 2*q + 5)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + c_.*x_^2)^p_*(d_ + e_.*x_ + f_.*x_^2)^q_*(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, h, q}, x] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[a*c*e^2 + (c*d - a*f)^2, 0] && Not[Not[IntegerQ[p]] && ILtQ[q, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^(q + 1)/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1))* ((g*c)*(-b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(2*a*f)) + c*(g*(2*c^2*d + b^2*f - c*(2*a*f)) - h*(b*c*d + a*b*f))*x) + 1/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1))* Int[(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^q* Simp[(b*h - 2*g*c)*((c*d - a*f)^2 - (b*d)*(-b*f))*(p + 1) + (b^2*(g*f) - b*(h*c*d + a*h*f) + 2*(g*c*(c*d - a*f)))*(a*f*(p + 1) - c*d*(p + 2)) - (2* f*((g*c)*(-b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(2*a*f)))*(p + q + 2) - (b^2*(g*f) - b*(h*c*d + a*h*f) + 2*(g*c*(c*d - a*f)))* (b*f*(p + 1)))*x - c*f*(b^2*(g*f) - b*(h*c*d + a*h*f) + 2*(g*c*(c*d - a*f)))*(2*p + 2*q + 5)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_*(d_ + f_.*x_^2)^q_*(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, g, h, q}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[b^2*d*f + (c*d - a*f)^2, 0] && Not[Not[IntegerQ[p]] && ILtQ[q, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "h*(a + b*x + c*x^2)^p*(d + e*x + f*x^2)^(q + 1)/(2*f*(p + q + 1)) - (1/(2*f*(p + q + 1)))* Int[(a + b*x + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q* Simp[h*p*(b*d - a*e) + a*(h*e - 2*g*f)*(p + q + 1) + (2*h*p*(c*d - a*f) + b*(h*e - 2*g*f)*(p + q + 1))*x + (h*p*(c*e - b*f) + c*(h*e - 2*g*f)*(p + q + 1))*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_*(d_ + e_.*x_ + f_.*x_^2)^ q_*(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && GtQ[p, 0] && NeQ[p + q + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "h*(a + c*x^2)^p*(d + e*x + f*x^2)^(q + 1)/(2*f*(p + q + 1)) + (1/(2*f*(p + q + 1)))* Int[(a + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q* Simp[a*h*e*p - a*(h*e - 2*g*f)*(p + q + 1) - 2*h*p*(c*d - a*f)*x - (h*c*e*p + c*(h*e - 2*g*f)*(p + q + 1))* x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + c_.*x_^2)^p_*(d_ + e_.*x_ + f_.*x_^2)^q_*(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, h, q}, x] && NeQ[e^2 - 4*d*f, 0] && GtQ[p, 0] && NeQ[p + q + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "h*(a + b*x + c*x^2)^p*(d + f*x^2)^(q + 1)/(2*f*(p + q + 1)) - (1/(2*f*(p + q + 1)))* Int[(a + b*x + c*x^2)^(p - 1)*(d + f*x^2)^q* Simp[h*p*(b*d) + a*(-2*g*f)*(p + q + 1) + (2*h*p*(c*d - a*f) + b*(-2*g*f)*(p + q + 1))*x + (h*p*(-b*f) + c*(-2*g*f)*(p + q + 1))*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_*(d_ + f_.*x_^2)^q_*(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, g, h, q}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && NeQ[p + q + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "With[{q = Simplify[ c^2*d^2 - b*c*d*e + a*c*e^2 + b^2*d*f - 2*a*c*d*f - a*b*e*f + a^2*f^2]}, 1/q*Int[ Simp[g*c^2*d - g*b*c*e + a*h*c*e + g*b^2*f - a*b*h*f - a*g*c*f + c*(h*c*d - g*c*e + g*b*f - a*h*f)*x, x]/(a + b*x + c*x^2), x] + 1/q* Int[Simp[-h*c*d*e + g*c*e^2 + b*h*d*f - g*c*d*f - g*b*e*f + a*g*f^2 - f*(h*c*d - g*c*e + g*b*f - a*h*f)*x, x]/(d + e*x + f*x^2), x] /; NeQ[q, 0]]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)/((a_ + b_.*x_ + c_.*x_^2)*(d_ + e_.*x_ + f_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "With[{q = Simplify[c^2*d^2 + b^2*d*f - 2*a*c*d*f + a^2*f^2]}, 1/q*Int[ Simp[g*c^2*d + g*b^2*f - a*b*h*f - a*g*c*f + c*(h*c*d + g*b*f - a*h*f)*x, x]/(a + b*x + c*x^2), x] + 1/q* Int[Simp[ b*h*d*f - g*c*d*f + a*g*f^2 - f*(h*c*d + g*b*f - a*h*f)*x, x]/(d + f*x^2), x] /; NeQ[q, 0]]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)/((a_ + b_.*x_ + c_.*x_^2)*(d_ + f_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "-2*g* Subst[Int[1/(b*d - a*e - b*x^2), x], x, Sqrt[d + e*x + f*x^2]]",
     "rulenumber": 0,
            "lhs": "Int[(g_ + h_.*x_)/((a_ + b_.*x_ + c_.*x_^2)* Sqrt[d_. + e_.*x_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[c*e - b*f, 0] && EqQ[h*e - 2*g*f, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "-(h*e - 2*g*f)/(2*f)* Int[1/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x] + h/(2*f)* Int[(e + 2*f*x)/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)/((a_ + b_.*x_ + c_.*x_^2)* Sqrt[d_. + e_.*x_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[c*e - b*f, 0] && NeQ[h*e - 2*g*f, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "-2*e* Subst[Int[(1 - d*x^2)/(c*e - b*f - e*(2*c*d - b*e + 2*a*f)*x^2 + d^2*(c*e - b*f)*x^4), x], x, (1 + (e + Sqrt[e^2 - 4*d*f])*x/(2*d))/Sqrt[d + e*x + f*x^2]]",
     "rulenumber": 0,
            "lhs": "Int[x_/((a_ + b_.*x_ + c_.*x_^2)*Sqrt[d_ + e_.*x_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[b*d - a*e, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "g*Subst[Int[1/(a + (c*d - a*f)*x^2), x], x, x/Sqrt[d + e*x + f*x^2]]",
     "rulenumber": 0,
            "lhs": "Int[(g_ + h_.*x_)/((a_ + b_.*x_ + c_.*x_^2)* Sqrt[d_ + e_.*x_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[b*d - a*e, 0] && EqQ[2*h*d - g*e, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "-(2*h*d - g*e)/e* Int[1/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x] + h/e*Int[(2*d + e*x)/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_ + h_.*x_)/((a_ + b_.*x_ + c_.*x_^2)* Sqrt[d_ + e_.*x_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[b*d - a*e, 0] && NeQ[2*h*d - g*e, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "-2*g*(g*b - 2*a*h)* Subst[ Int[1/Simp[g*(g*b - 2*a*h)*(b^2 - 4*a*c) - (b*d - a*e)*x^2, x], x], x, Simp[g*b - 2*a*h - (b*h - 2*g*c)*x, x]/ Sqrt[d + e*x + f*x^2]]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)/((a_. + b_.*x_ + c_.*x_^2)* Sqrt[d_. + e_.*x_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && NeQ[b*d - a*e, 0] && EqQ[h^2*(b*d - a*e) - 2*g*h*(c*d - a*f) + g^2*(c*e - b*f), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "-2*a*g*h* Subst[Int[1/Simp[2*a^2*g*h*c + a*e*x^2, x], x], x, Simp[a*h - g*c*x, x]/Sqrt[d + e*x + f*x^2]]",
     "rulenumber": 0,
            "lhs": "Int[(g_ + h_.*x_)/((a_ + c_.*x_^2)*Sqrt[d_. + e_.*x_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, h}, x] && EqQ[a*h^2*e + 2*g*h*(c*d - a*f) - g^2*c*e, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "-2*g*(g*b - 2*a*h)* Subst[Int[1/Simp[g*(g*b - 2*a*h)*(b^2 - 4*a*c) - b*d*x^2, x], x], x, Simp[g*b - 2*a*h - (b*h - 2*g*c)*x, x]/Sqrt[d + f*x^2]]",
     "rulenumber": 0,
            "lhs": "Int[(g_ + h_.*x_)/((a_. + b_.*x_ + c_.*x_^2)*Sqrt[d_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[b*h^2*d - 2*g*h*(c*d - a*f) - g^2*b*f, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, (2*c*g - h*(b - q))/q* Int[1/((b - q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x] - (2*c*g - h*(b + q))/q* Int[1/((b + q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x]]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)/((a_ + b_.*x_ + c_.*x_^2)* Sqrt[d_. + e_.*x_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && PosQ[b^2 - 4*a*c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "With[{q = Rt[-a*c, 2]}, (h/2 + c*g/(2*q))*Int[1/((-q + c*x)*Sqrt[d + e*x + f*x^2]), x] + (h/2 - c*g/(2*q))*Int[1/((q + c*x)*Sqrt[d + e*x + f*x^2]), x]]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)/((a_ + c_.*x_^2)*Sqrt[d_. + e_.*x_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, h}, x] && NeQ[e^2 - 4*d*f, 0] && PosQ[-a*c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, (2*c*g - h*(b - q))/q* Int[1/((b - q + 2*c*x)*Sqrt[d + f*x^2]), x] - (2*c*g - h*(b + q))/q* Int[1/((b + q + 2*c*x)*Sqrt[d + f*x^2]), x]]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)/((a_ + b_.*x_ + c_.*x_^2)*Sqrt[d_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "With[{q = Rt[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 2]}, 1/(2*q)* Int[Simp[ h*(b*d - a*e) - g*(c*d - a*f - q) - (g*(c*e - b*f) - h*(c*d - a*f + q))*x, x]/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x] - 1/(2*q)* Int[Simp[ h*(b*d - a*e) - g*(c*d - a*f + q) - (g*(c*e - b*f) - h*(c*d - a*f - q))*x, x]/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x]]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)/((a_. + b_.*x_ + c_.*x_^2)* Sqrt[d_. + e_.*x_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && NeQ[b*d - a*e, 0] && NegQ[b^2 - 4*a*c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "With[{q = Rt[(c*d - a*f)^2 + a*c*e^2, 2]}, 1/(2*q)* Int[Simp[-a*h*e - g*(c*d - a*f - q) + (h*(c*d - a*f + q) - g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x] - 1/(2*q)* Int[Simp[-a*h*e - g*(c*d - a*f + q) + (h*(c*d - a*f - q) - g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x]]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)/((a_ + c_.*x_^2)*Sqrt[d_. + e_.*x_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, h}, x] && NeQ[e^2 - 4*d*f, 0] && NegQ[-a*c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "With[{q = Rt[(c*d - a*f)^2 + b^2*d*f, 2]}, 1/(2*q)* Int[Simp[ h*b*d - g*(c*d - a*f - q) + (h*(c*d - a*f + q) + g*b*f)*x, x]/((a + b*x + c*x^2)*Sqrt[d + f*x^2]), x] - 1/(2*q)* Int[Simp[ h*b*d - g*(c*d - a*f + q) + (h*(c*d - a*f - q) + g*b*f)*x, x]/((a + b*x + c*x^2)*Sqrt[d + f*x^2]), x]]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)/((a_. + b_.*x_ + c_.*x_^2)*Sqrt[d_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "With[{s = Rt[b^2 - 4*a*c, 2], t = Rt[e^2 - 4*d*f, 2]}, Sqrt[b + s + 2*c*x]*Sqrt[2*a + (b + s)*x]*Sqrt[e + t + 2*f*x]* Sqrt[2*d + (e + t)*x]/(Sqrt[a + b*x + c*x^2]* Sqrt[d + e*x + f*x^2])* Int[(g + h*x)/(Sqrt[b + s + 2*c*x]*Sqrt[2*a + (b + s)*x]* Sqrt[e + t + 2*f*x]*Sqrt[2*d + (e + t)*x]), x]]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)/(Sqrt[a_ + b_.*x_ + c_.*x_^2]* Sqrt[d_ + e_.*x_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "With[{s = Rt[b^2 - 4*a*c, 2], t = Rt[-4*d*f, 2]}, Sqrt[b + s + 2*c*x]*Sqrt[2*a + (b + s)*x]*Sqrt[t + 2*f*x]* Sqrt[2*d + t*x]/(Sqrt[a + b*x + c*x^2]*Sqrt[d + f*x^2])* Int[(g + h*x)/(Sqrt[b + s + 2*c*x]*Sqrt[2*a + (b + s)*x]* Sqrt[t + 2*f*x]*Sqrt[2*d + t*x]), x]]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)/(Sqrt[a_ + b_.*x_ + c_.*x_^2]*Sqrt[d_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "With[{q = (-9*c*h^2/(2*c*g - b*h)^2)^(1/3)}, Sqrt[3]*h*q* ArcTan[1/Sqrt[3] - 2^(2/3)*(1 - (3*h*(b + 2*c*x))/(2*c*g - b*h))^(2/3)/(Sqrt[ 3]*(1 + (3*h*(b + 2*c*x))/(2*c*g - b*h))^(1/3))]/f + h*q*Log[d + e*x + f*x^2]/(2*f) - 3*h*q* Log[(1 - 3*h*(b + 2*c*x)/(2*c*g - b*h))^(2/3) + 2^(1/3)*(1 + 3*h*(b + 2*c*x)/(2*c*g - b*h))^(1/3)]/(2*f)]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)/((a_. + b_.*x_ + c_.*x_^2)^(1/3)*(d_. + e_.*x_ + f_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h}, x] && EqQ[c*e - b*f, 0] && EqQ[c^2*d - f*(b^2 - 3*a*c), 0] && EqQ[c^2*g^2 - b*c*g*h - 2*b^2*h^2 + 9*a*c*h^2, 0] && GtQ[-9*c*h^2/(2*c*g - b*h)^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "With[{q = -c/(b^2 - 4*a*c)}, (q*(a + b*x + c*x^2))^(1/3)/(a + b*x + c*x^2)^(1/3)* Int[(g + h*x)/((q*a + b*q*x + c*q*x^2)^(1/3)*(d + e*x + f*x^2)), x]]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)/((a_. + b_.*x_ + c_.*x_^2)^(1/3)*(d_. + e_.*x_ + f_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h}, x] && EqQ[c*e - b*f, 0] && EqQ[c^2*d - f*(b^2 - 3*a*c), 0] && EqQ[c^2*g^2 - b*c*g*h - 2*b^2*h^2 + 9*a*c*h^2, 0] && Not[GtQ[4*a - b^2/c, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "Unintegrable[(a + b*x + c*x^2)^p*(d + e*x + f*x^2)^q*(g + h*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_ + c_.*x_^2)^p_*(d_. + e_.*x_ + f_.*x_^2)^ q_*(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, p, q}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "Unintegrable[(a + c*x^2)^p*(d + e*x + f*x^2)^q*(g + h*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + c_.*x_^2)^p_*(d_. + e_.*x_ + f_.*x_^2)^q_*(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, h, p, q}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "1/Coefficient[u, x, 1]* Subst[Int[(g + h*x)^m*(a + b*x + c*x^2)^p*(d + e*x + f*x^2)^q, x], x, u]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*u_)^m_.*(a_. + b_.*u_ + c_.*u_^2)^ p_.*(d_. + e_.*u_ + f_.*u_^2)^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, p, q}, x] && LinearQ[u, x] && NeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
       "filename": "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.m",
            "rhs": "1/Coefficient[u, x, 1]* Subst[Int[(g + h*x)^m*(a + c*x^2)^p*(d + e*x + f*x^2)^q, x], x, u]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*u_)^m_.*(a_. + c_.*u_^2)^p_.*(d_. + e_.*u_ + f_.*u_^2)^ q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, h, m, p, q}, x] && LinearQ[u, x] && NeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "(c/f)^p* Int[(d + e*x + f*x^2)^(p + q)*(A + B*x + C*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_.*(d_ + e_.*x_ + f_.*x_^2)^ q_.*(A_. + B_.*x_ + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, p, q}, x] && EqQ[c*d - a*f, 0] && EqQ[b*d - a*e, 0] && (IntegerQ[p] || GtQ[c/f, 0]) && (Not[IntegerQ[q]] || LeafCount[d + e*x + f*x^2] <= LeafCount[a + b*x + c*x^2])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "(c/f)^p* Int[(d + e*x + f*x^2)^(p + q)*(A + C*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_.*(d_ + e_.*x_ + f_.*x_^2)^ q_.*(A_. + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C, p, q}, x] && EqQ[c*d - a*f, 0] && EqQ[b*d - a*e, 0] && (IntegerQ[p] || GtQ[c/f, 0]) && (Not[IntegerQ[q]] || LeafCount[d + e*x + f*x^2] <= LeafCount[a + b*x + c*x^2])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "a^IntPart[p]*(a + b*x + c*x^2)^ FracPart[p]/(d^IntPart[p]*(d + e*x + f*x^2)^FracPart[p])* Int[(d + e*x + f*x^2)^(p + q)*(A + B*x + C*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_.*(d_ + e_.*x_ + f_.*x_^2)^ q_.*(A_. + B_.*x_ + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, p, q}, x] && EqQ[c*d - a*f, 0] && EqQ[b*d - a*e, 0] && Not[IntegerQ[p]] && Not[IntegerQ[q]] && Not[GtQ[c/f, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "a^IntPart[p]*(a + b*x + c*x^2)^ FracPart[p]/(d^IntPart[p]*(d + e*x + f*x^2)^FracPart[p])* Int[(d + e*x + f*x^2)^(p + q)*(A + C*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_.*(d_ + e_.*x_ + f_.*x_^2)^ q_.*(A_. + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C, p, q}, x] && EqQ[c*d - a*f, 0] && EqQ[b*d - a*e, 0] && Not[IntegerQ[p]] && Not[IntegerQ[q]] && Not[GtQ[c/f, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "(a + b*x + c*x^2)^ FracPart[p]/((4*c)^IntPart[p]*(b + 2*c*x)^(2*FracPart[p]))* Int[(b + 2*c*x)^(2*p)*(d + e*x + f*x^2)^q*(A + B*x + C*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_.*(d_. + e_.*x_ + f_.*x_^2)^ q_.*(A_. + B_.*x_ + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, p, q}, x] && EqQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "(a + b*x + c*x^2)^ FracPart[p]/((4*c)^IntPart[p]*(b + 2*c*x)^(2*FracPart[p]))* Int[(b + 2*c*x)^(2*p)*(d + e*x + f*x^2)^q*(A + C*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_.*(d_. + e_.*x_ + f_.*x_^2)^ q_.*(A_. + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C, p, q}, x] && EqQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "(a + b*x + c*x^2)^ FracPart[p]/((4*c)^IntPart[p]*(b + 2*c*x)^(2*FracPart[p]))* Int[(b + 2*c*x)^(2*p)*(d + f*x^2)^q*(A + B*x + C*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_.*(d_. + f_.*x_^2)^ q_.*(A_. + B_.*x_ + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, A, B, C, p, q}, x] && EqQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "(a + b*x + c*x^2)^ FracPart[p]/((4*c)^IntPart[p]*(b + 2*c*x)^(2*FracPart[p]))* Int[(b + 2*c*x)^(2*p)*(d + f*x^2)^q*(A + C*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_.*(d_. + f_.*x_^2)^ q_.*(A_. + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, A, C, p, q}, x] && EqQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "(A*b*c - 2*a*B*c + a*b*C - (c*(b*B - 2*A*c) - C*(b^2 - 2*a*c))*x)*(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^ q/(c*(b^2 - 4*a*c)*(p + 1)) - 1/(c*(b^2 - 4*a*c)*(p + 1))* Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)* Simp[e*q*(A*b*c - 2*a*B*c + a*b*C) - d*(c*(b*B - 2*A*c)*(2*p + 3) + C*(2*a*c - b^2*(p + 2))) + (2*f*q*(A*b*c - 2*a*B*c + a*b*C) - e*(c*(b*B - 2*A*c)*(2*p + q + 3) + C*(2*a*c*(q + 1) - b^2*(p + q + 2))))*x - f*(c*(b*B - 2*A*c)*(2*p + 2*q + 3) + C*(2*a*c*(2*q + 1) - b^2*(p + 2*q + 2)))*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_*(d_ + e_.*x_ + f_.*x_^2)^ q_*(A_. + B_.*x_ + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[q, 0] && Not[IGtQ[q, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "(A*b*c + a*b*C + (2*A*c^2 + C*(b^2 - 2*a*c))*x)*(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q/(c*(b^2 - 4*a*c)*(p + 1)) - 1/(c*(b^2 - 4*a*c)*(p + 1))* Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)* Simp[A*c*(2*c*d*(2*p + 3) + b*e*q) - C*(2*a*c*d - b^2*d*(p + 2) - a*b*e*q) + (C*(2*a*b*f*q - 2*a*c*e*(q + 1) + b^2*e*(p + q + 2)) + 2*A*c*(b*f*q + c*e*(2*p + q + 3)))* x - f*(-2*A*c^2*(2*p + 2*q + 3) + C*(2*a*c*(2*q + 1) - b^2*(p + 2*q + 2)))*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_*(d_ + e_.*x_ + f_.*x_^2)^ q_*(A_. + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[q, 0] && Not[IGtQ[q, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "(a*B - (A*c - a*C)*x)*(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q/(2*a*c*(p + 1)) - 2/((-4*a*c)*(p + 1))* Int[(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)* Simp[A*c*d*(2*p + 3) - a*(C*d + B*e*q) + (A*c*e*(2*p + q + 3) - a*(2*B*f*q + C*e*(q + 1)))*x - f*(a*C*(2*q + 1) - A*c*(2*p + 2*q + 3))*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + c_.*x_^2)^p_*(d_ + e_.*x_ + f_.*x_^2)^ q_*(A_. + B_.*x_ + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, A, B, C}, x] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[q, 0] && Not[IGtQ[q, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "-(A*c - a*C)* x*(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q/(2*a*c*(p + 1)) + 2/(4*a*c*(p + 1))* Int[(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)* Simp[A*c*d*(2*p + 3) - a*C*d + (A*c*e*(2*p + q + 3) - a*C*e*(q + 1))*x - f*(a*C*(2*q + 1) - A*c*(2*p + 2*q + 3))*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + c_.*x_^2)^p_*(d_ + e_.*x_ + f_.*x_^2)^q_*(A_. + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, A, C}, x] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[q, 0] && Not[IGtQ[q, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "(A*b*c - 2*a*B*c + a*b*C - (c*(b*B - 2*A*c) - C*(b^2 - 2*a*c))*x)*(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^q/(c*(b^2 - 4*a*c)*(p + 1)) - 1/(c*(b^2 - 4*a*c)*(p + 1))* Int[(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^(q - 1)* Simp[-d*(c*(b*B - 2*A*c)*(2*p + 3) + C*(2*a*c - b^2*(p + 2))) + (2*f*q*(A*b*c - 2*a*B*c + a*b*C))*x - f*(c*(b*B - 2*A*c)*(2*p + 2*q + 3) + C*(2*a*c*(2*q + 1) - b^2*(p + 2*q + 2)))*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_*(d_ + f_.*x_^2)^ q_*(A_. + B_.*x_ + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, A, B, C}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[q, 0] && Not[IGtQ[q, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "(A*b*c + a*b*C + (2*A*c^2 + C*(b^2 - 2*a*c))*x)*(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^q/(c*(b^2 - 4*a*c)*(p + 1)) - 1/(c*(b^2 - 4*a*c)*(p + 1))* Int[(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^(q - 1)* Simp[A*c*(2*c*d*(2*p + 3)) - C*(2*a*c*d - b^2*d*(p + 2)) + (C*(2*a*b*f*q) + 2*A*c*(b*f*q))*x - f*(-2*A*c^2*(2*p + 2*q + 3) + C*(2*a*c*(2*q + 1) - b^2*(p + 2*q + 2)))*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_*(d_ + f_.*x_^2)^q_*(A_. + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, A, C}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[q, 0] && Not[IGtQ[q, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1))* ((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - B*(b*c*d - 2*a*c*e + a*b*f) + C*(b^2*d - a*b*e - 2*a*(c*d - a*f)))*x) + 1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1))* Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q* Simp[(b*B - 2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) + (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(a* f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (2* f*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))* (b*f*(p + 1) - c*e*(2*p + q + 4)))*x - c*f*(b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_*(d_ + e_.*x_ + f_.*x_^2)^ q_*(A_. + B_.*x_ + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] && Not[Not[IntegerQ[p]] && ILtQ[q, -1]] && Not[IGtQ[q, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1))* ((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + C*(b^2*d - a*b*e - 2*a*(c*d - a*f)))*x) + 1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1))* Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q* Simp[(-2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) + (b^2*(C*d + A*f) - b*(+A*c*e + a*C*e) + 2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (2* f*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A* b)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) - b*(A*c*e + a*C*e) + 2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))* (b*f*(p + 1) - c*e*(2*p + q + 4)))*x - c*f*(b^2*(C*d + A*f) - b*(A*c*e + a*C*e) + 2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(2*p + 2*q + 5)* x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_*(d_ + e_.*x_ + f_.*x_^2)^ q_*(A_. + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] && Not[Not[IntegerQ[p]] && ILtQ[q, -1]] && Not[IGtQ[q, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p + 1))* ((A*c - a*C)*(2*a*c*e) + (-a*B)*(2*c^2*d - c*(2*a*f)) + c*(A*(2*c^2*d - c*(2*a*f)) - B*(-2*a*c*e) + C*(-2*a*(c*d - a*f)))*x) + 1/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p + 1))* Int[(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q* Simp[(-2*A*c - 2*a*C)*((c*d - a*f)^2 - (-a*e)*(c*e))*(p + 1) + (2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(a* f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e) + (-a*B)*(2*c^2*d - c*(+2*a*f)))*(p + q + 2) - (2* f*((A*c - a*C)*(2*a*c*e) + (-a* B)*(2*c^2*d + -c*(+2*a*f)))*(p + q + 2) - (2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))* (-c*e*(2*p + q + 4)))*x - c*f*(2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + c_.*x_^2)^p_*(d_ + e_.*x_ + f_.*x_^2)^ q_*(A_. + B_.*x_ + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, A, B, C, q}, x] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[a*c*e^2 + (c*d - a*f)^2, 0] && Not[Not[IntegerQ[p]] && ILtQ[q, -1]] && Not[IGtQ[q, 0]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p + 1))* ((A*c - a*C)*(2*a*c*e) + c*(A*(2*c^2*d - c*(2*a*f)) + C*(-2*a*(c*d - a*f)))*x) + 1/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p + 1))* Int[(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q* Simp[(-2*A*c - 2*a*C)*((c*d - a*f)^2 - (-a*e)*(c*e))*(p + 1) + (2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(a* f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e))*(p + q + 2) - (2* f*((A*c - a*C)*(2*a*c*e))*(p + q + 2) - (2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(-c* e*(2*p + q + 4)))*x - c*f*(2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(2*p + 2*q + 5)* x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + c_.*x_^2)^p_*(d_ + e_.*x_ + f_.*x_^2)^q_*(A_. + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, A, C, q}, x] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[a*c*e^2 + (c*d - a*f)^2, 0] && Not[Not[IntegerQ[p]] && ILtQ[q, -1]] && Not[IGtQ[q, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^(q + 1)/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1))* ((A*c - a*C)*(-b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(2*a*f)) - B*(b*c*d + a*b*f) + C*(b^2*d - 2*a*(c*d - a*f)))*x) + 1/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1))* Int[(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^q* Simp[(b*B - 2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d)*(-b*f))*(p + 1) + (b^2*(C*d + A*f) - b*(B*c*d + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(a*f*(p + 1) - c*d*(p + 2)) - (2* f*((A*c - a*C)*(-b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) - b*(B*c*d + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))* (b*f*(p + 1)))*x - c*f*(b^2*(C*d + A*f) - b*(B*c*d + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(2*p + 2*q + 5)* x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_*(d_ + f_.*x_^2)^ q_*(A_. + B_.*x_ + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, A, B, C, q}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[b^2*d*f + (c*d - a*f)^2, 0] && Not[Not[IntegerQ[p]] && ILtQ[q, -1]] && Not[IGtQ[q, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^(q + 1)/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1))* ((A*c - a*C)*(-b*(c*d + a*f)) + (A*b)*(2*c^2*d + b^2*f - c*(2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(2*a*f)) + C*(b^2*d - 2*a*(c*d - a*f)))*x) + 1/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1))* Int[(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^q* Simp[(-2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d)*(-b*f))*(p + 1) + (b^2*(C*d + A*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(a*f*(p + 1) - c*d*(p + 2)) - (2* f*((A*c - a*C)*(-b*(c*d + a*f)) + (A*b)*(2*c^2*d + b^2*f - c*(2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))* (b*f*(p + 1)))*x - c*f*(b^2*(C*d + A*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(2*p + 2*q + 5)* x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_*(d_ + f_.*x_^2)^q_*(A_. + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, A, C, q}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[b^2*d*f + (c*d - a*f)^2, 0] && Not[Not[IntegerQ[p]] && ILtQ[q, -1]] && Not[IGtQ[q, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "(B*c*f*(2*p + 2*q + 3) + C*(b*f*p - c*e*(2*p + q + 2)) + 2*c*C*f*(p + q + 1)*x)*(a + b*x + c*x^2)^p* (d + e*x + f*x^2)^(q + 1)/(2*c* f^2*(p + q + 1)*(2*p + 2*q + 3)) - (1/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)))* Int[(a + b*x + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q* Simp[p*(b*d - a*e)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(b^2*C*d*f*p + a*c*(C*(2*d*f - e^2*(2*p + q + 2)) + f*(B*e - 2*A*f)*(2*p + 2*q + 3))) + (2* p*(c*d - a*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*e*f*p*(b^2 - 4*a*c) - b*c*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C*d - B*e + 2*A*f)*(2*p + 2*q + 3))))*x + (p*(c*e - b*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*f^2*p*(b^2 - 4*a*c) - c^2*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C*d - B*e + 2*A*f)*(2*p + 2*q + 3))))*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_*(d_ + e_.*x_ + f_.*x_^2)^ q_*(A_. + B_.*x_ + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && GtQ[p, 0] && NeQ[p + q + 1, 0] && NeQ[2*p + 2*q + 3, 0] && Not[IGtQ[p, 0]] && Not[IGtQ[q, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "(C*(b*f*p - c*e*(2*p + q + 2)) + 2*c*C*f*(p + q + 1)*x)*(a + b*x + c*x^2)^p* (d + e*x + f*x^2)^(q + 1)/(2*c* f^2*(p + q + 1)*(2*p + 2*q + 3)) - (1/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)))* Int[(a + b*x + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q* Simp[p*(b*d - a*e)*(C*(c*e - b*f)*(q + 1) - c*(C*e)*(2*p + 2*q + 3)) + (p + q + 1)*(b^2*C*d*f*p + a*c*(C*(2*d*f - e^2*(2*p + q + 2)) + f*(-2*A*f)*(2*p + 2*q + 3))) + (2* p*(c*d - a*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e)*(2*p + 2*q + 3)) + (p + q + 1)*(C*e*f*p*(b^2 - 4*a*c) - b*c*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C*d + 2*A*f)*(2*p + 2*q + 3))))*x + (p*(c*e - b*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e)*(2*p + 2*q + 3)) + (p + q + 1)*(C*f^2*p*(b^2 - 4*a*c) - c^2*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C*d + 2*A*f)*(2*p + 2*q + 3))))*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_*(d_ + e_.*x_ + f_.*x_^2)^ q_*(A_. + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && GtQ[p, 0] && NeQ[p + q + 1, 0] && NeQ[2*p + 2*q + 3, 0] && Not[IGtQ[p, 0]] && Not[IGtQ[q, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "(B*c*f*(2*p + 2*q + 3) + C*(-c*e*(2*p + q + 2)) + 2*c*C*f*(p + q + 1)*x)*(a + c*x^2)^p* (d + e*x + f*x^2)^(q + 1)/(2*c* f^2*(p + q + 1)*(2*p + 2*q + 3)) - (1/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)))* Int[(a + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q* Simp[p*(-a*e)*(C*(c*e)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(a* c*(C*(2*d*f - e^2*(2*p + q + 2)) + f*(B*e - 2*A*f)*(2*p + 2*q + 3))) + (2* p*(c*d - a*f)*(C*(c*e)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*e*f*p*(-4*a*c)))*x + (p*(c*e)*(C*(c*e)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*f^2*p*(-4*a*c) - c^2*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C*d - B*e + 2*A*f)*(2*p + 2*q + 3))))*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + c_.*x_^2)^p_*(d_ + e_.*x_ + f_.*x_^2)^ q_*(A_. + B_.*x_ + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, A, B, C, q}, x] && NeQ[e^2 - 4*d*f, 0] && GtQ[p, 0] && NeQ[p + q + 1, 0] && NeQ[2*p + 2*q + 3, 0] && Not[IGtQ[p, 0]] && Not[IGtQ[q, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "(C*(-c*e*(2*p + q + 2)) + 2*c*C*f*(p + q + 1)*x)*(a + c*x^2)^ p*(d + e*x + f*x^2)^(q + 1)/(2*c* f^2*(p + q + 1)*(2*p + 2*q + 3)) - (1/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)))* Int[(a + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q* Simp[p*(-a*e)*(C*(c*e)*(q + 1) - c*(C*e)*(2*p + 2*q + 3)) + (p + q + 1)*(a* c*(C*(2*d*f - e^2*(2*p + q + 2)) + f*(-2*A*f)*(2*p + 2*q + 3))) + (2* p*(c*d - a*f)*(C*(c*e)*(q + 1) - c*(C*e)*(2*p + 2*q + 3)) + (p + q + 1)*(C*e*f* p*(-4*a*c)))*x + (p*(c*e)*(C*(c*e)*(q + 1) - c*(C*e)*(2*p + 2*q + 3)) + (p + q + 1)*(C*f^2*p*(-4*a*c) - c^2*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C*d + 2*A*f)*(2*p + 2*q + 3))))*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + c_.*x_^2)^p_*(d_ + e_.*x_ + f_.*x_^2)^q_*(A_. + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, A, C, q}, x] && NeQ[e^2 - 4*d*f, 0] && GtQ[p, 0] && NeQ[p + q + 1, 0] && NeQ[2*p + 2*q + 3, 0] && Not[IGtQ[p, 0]] && Not[IGtQ[q, 0]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "(B*c*f*(2*p + 2*q + 3) + C*(b*f*p) + 2*c*C*f*(p + q + 1)*x)*(a + b*x + c*x^2)^p* (d + f*x^2)^(q + 1)/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)) - (1/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)))* Int[(a + b*x + c*x^2)^(p - 1)*(d + f*x^2)^q* Simp[p*(b*d)*(C*(-b*f)*(q + 1) - c*(-B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(b^2*C*d*f*p + a*c*(C*(2*d*f) + f*(-2*A*f)*(2*p + 2*q + 3))) + (2* p*(c*d - a*f)*(C*(-b*f)*(q + 1) - c*(-B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(-b* c*(C*(-4*d*f)*(2*p + q + 2) + f*(2*C*d + 2*A*f)*(2*p + 2*q + 3))))*x + (p*(-b*f)*(C*(-b*f)*(q + 1) - c*(-B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*f^2*p*(b^2 - 4*a*c) - c^2*(C*(-4*d*f)*(2*p + q + 2) + f*(2*C*d + 2*A*f)*(2*p + 2*q + 3))))*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_*(d_ + f_.*x_^2)^ q_*(A_. + B_.*x_ + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, A, B, C, q}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && NeQ[p + q + 1, 0] && NeQ[2*p + 2*q + 3, 0] && Not[IGtQ[p, 0]] && Not[IGtQ[q, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "(C*(b*f*p) + 2*c*C*f*(p + q + 1)*x)*(a + b*x + c*x^2)^p* (d + f*x^2)^(q + 1)/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)) - (1/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)))* Int[(a + b*x + c*x^2)^(p - 1)*(d + f*x^2)^q* Simp[p*(b*d)*(C*(-b*f)*(q + 1)) + (p + q + 1)*(b^2*C*d*f*p + a*c*(C*(2*d*f) + f*(-2*A*f)*(2*p + 2*q + 3))) + (2*p*(c*d - a*f)*(C*(-b*f)*(q + 1)) + (p + q + 1)*(-b* c*(C*(-4*d*f)*(2*p + q + 2) + f*(2*C*d + 2*A*f)*(2*p + 2*q + 3))))*x + (p*(-b*f)*(C*(-b*f)*(q + 1)) + (p + q + 1)*(C*f^2*p*(b^2 - 4*a*c) - c^2*(C*(-4*d*f)*(2*p + q + 2) + f*(2*C*d + 2*A*f)*(2*p + 2*q + 3))))*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_ + c_.*x_^2)^p_*(d_ + f_.*x_^2)^q_*(A_. + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, A, C, q}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && NeQ[p + q + 1, 0] && NeQ[2*p + 2*q + 3, 0] && Not[IGtQ[p, 0]] && Not[IGtQ[q, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "With[{q = c^2*d^2 - b*c*d*e + a*c*e^2 + b^2*d*f - 2*a*c*d*f - a*b*e*f + a^2*f^2}, 1/q*Int[(A*c^2*d - a*c*C*d - A*b*c*e + a*B*c*e + A*b^2*f - a*b*B*f - a*A*c*f + a^2*C*f + c*(B*c*d - b*C*d - A*c*e + a*C*e + A*b*f - a*B*f)*x)/(a + b*x + c*x^2), x] + 1/q* Int[(c*C*d^2 - B*c*d*e + A*c*e^2 + b*B*d*f - A*c*d*f - a*C*d*f - A*b*e*f + a*A*f^2 - f*(B*c*d - b*C*d - A*c*e + a*C*e + A*b*f - a*B*f)*x)/(d + e*x + f*x^2), x] /; NeQ[q, 0]]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*x_ + C_.*x_^2)/((a_ + b_.*x_ + c_.*x_^2)*(d_ + e_.*x_ + f_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "With[{q = c^2*d^2 - b*c*d*e + a*c*e^2 + b^2*d*f - 2*a*c*d*f - a*b*e*f + a^2*f^2}, 1/q*Int[(A*c^2*d - a*c*C*d - A*b*c*e + A*b^2*f - a*A*c*f + a^2*C*f + c*(-b*C*d - A*c*e + a*C*e + A*b*f)*x)/(a + b*x + c*x^2), x] + 1/q* Int[(c*C*d^2 + A*c*e^2 - A*c*d*f - a*C*d*f - A*b*e*f + a*A*f^2 - f*(-b*C*d - A*c*e + a*C*e + A*b*f)*x)/(d + e*x + f*x^2), x] /; NeQ[q, 0]]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + C_.*x_^2)/((a_ + b_.*x_ + c_.*x_^2)*(d_ + e_.*x_ + f_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "With[{q = c^2*d^2 + b^2*d*f - 2*a*c*d*f + a^2*f^2}, 1/q*Int[(A*c^2*d - a*c*C*d + A*b^2*f - a*b*B*f - a*A*c*f + a^2*C*f + c*(B*c*d - b*C*d + A*b*f - a*B*f)*x)/(a + b*x + c*x^2), x] + 1/q* Int[(c*C*d^2 + b*B*d*f - A*c*d*f - a*C*d*f + a*A*f^2 - f*(B*c*d - b*C*d + A*b*f - a*B*f)*x)/(d + f*x^2), x] /; NeQ[q, 0]]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*x_ + C_.*x_^2)/((a_ + b_.*x_ + c_.*x_^2)*(d_ + f_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, A, B, C}, x] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "With[{q = c^2*d^2 + b^2*d*f - 2*a*c*d*f + a^2*f^2}, 1/q*Int[(A*c^2*d - a*c*C*d + A*b^2*f - a*A*c*f + a^2*C*f + c*(-b*C*d + A*b*f)*x)/(a + b*x + c*x^2), x] + 1/q* Int[(c*C*d^2 - A*c*d*f - a*C*d*f + a*A*f^2 - f*(-b*C*d + A*b*f)*x)/(d + f*x^2), x] /; NeQ[q, 0]]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + C_.*x_^2)/((a_ + b_.*x_ + c_.*x_^2)*(d_ + f_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, A, C}, x] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "C/c*Int[1/Sqrt[d + e*x + f*x^2], x] + 1/c*Int[(A*c - a*C + (B*c - b*C)*x)/((a + b*x + c*x^2)* Sqrt[d + e*x + f*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*x_ + C_.*x_^2)/((a_ + b_.*x_ + c_.*x_^2)* Sqrt[d_. + e_.*x_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "C/c*Int[1/Sqrt[d + e*x + f*x^2], x] + 1/c*Int[(A*c - a*C - b*C*x)/((a + b*x + c*x^2)* Sqrt[d + e*x + f*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + C_.*x_^2)/((a_ + b_.*x_ + c_.*x_^2)* Sqrt[d_. + e_.*x_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "C/c*Int[1/Sqrt[d + e*x + f*x^2], x] + 1/c*Int[(A*c - a*C + B*c*x)/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*x_ + C_.*x_^2)/((a_ + c_.*x_^2)* Sqrt[d_. + e_.*x_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, A, B, C}, x] && NeQ[e^2 - 4*d*f, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "C/c*Int[1/Sqrt[d + e*x + f*x^2], x] + (A*c - a*C)/c* Int[1/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + C_.*x_^2)/((a_ + c_.*x_^2)*Sqrt[d_. + e_.*x_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, A, C}, x] && NeQ[e^2 - 4*d*f, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "C/c*Int[1/Sqrt[d + f*x^2], x] + 1/c*Int[(A*c - a*C + (B*c - b*C)*x)/((a + b*x + c*x^2)* Sqrt[d + f*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*x_ + C_.*x_^2)/((a_ + b_.*x_ + c_.*x_^2)* Sqrt[d_. + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, A, B, C}, x] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "C/c*Int[1/Sqrt[d + f*x^2], x] + 1/c*Int[(A*c - a*C - b*C*x)/((a + b*x + c*x^2)*Sqrt[d + f*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + C_.*x_^2)/((a_ + b_.*x_ + c_.*x_^2)*Sqrt[d_. + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, A, C}, x] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "1/Coefficient[u, x, 1]* Subst[Int[(a + b*x + c*x^2)^p*(d + e*x + f*x^2)^ q*(A + B*x + C*x^2), x], x, u]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*u_ + c_.*u_^2)^p_.*(d_. + e_.*u_ + f_.*u_^2)^ q_.*(A_. + B_.*u_ + C_.*u_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, p, q}, x] && LinearQ[u, x] && NeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "1/Coefficient[u, x, 1]* Subst[Int[(a + b*x + c*x^2)^p*(d + e*x + f*x^2)^q*(A + B*x), x], x, u]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*u_ + c_.*u_^2)^p_.*(d_. + e_.*u_ + f_.*u_^2)^ q_.*(A_. + B_.*u_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, p, q}, x] && LinearQ[u, x] && NeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "1/Coefficient[u, x, 1]* Subst[Int[(a + b*x + c*x^2)^p*(d + e*x + f*x^2)^q*(A + C*x^2), x], x, u]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*u_ + c_.*u_^2)^p_.*(d_. + e_.*u_ + f_.*u_^2)^ q_.*(A_. + C_.*u_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C, p, q}, x] && LinearQ[u, x] && NeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "1/Coefficient[u, x, 1]* Subst[Int[(a + c*x^2)^p*(d + e*x + f*x^2)^q*(A + B*x + C*x^2), x], x, u]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + c_.*u_^2)^p_.*(d_. + e_.*u_ + f_.*u_^2)^ q_.*(A_. + B_.*u_ + C_.*u_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, A, B, C, p, q}, x] && LinearQ[u, x] && NeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "1/Coefficient[u, x, 1]* Subst[Int[(a + c*x^2)^p*(d + e*x + f*x^2)^q*(A + B*x), x], x, u]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + c_.*u_^2)^p_.*(d_. + e_.*u_ + f_.*u_^2)^q_.*(A_. + B_.*u_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, A, B, C, p, q}, x] && LinearQ[u, x] && NeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
       "filename": "1.2.1.7 (a+b x+c x^2)^p (d+e x+f x^2)^q (A+B x+C x^2).m",
            "rhs": "1/Coefficient[u, x, 1]* Subst[Int[(a + c*x^2)^p*(d + e*x + f*x^2)^q*(A + C*x^2), x], x, u]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + c_.*u_^2)^p_.*(d_. + e_.*u_ + f_.*u_^2)^ q_.*(A_. + C_.*u_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, A, C, p, q}, x] && LinearQ[u, x] && NeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.8 P(x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.8 P(x) (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.8 P(x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.8 P(x) (a+b x+c x^2)^p.m",
            "rhs": "Int[x*PolynomialQuotient[Pq, x, x]*(a + b*x + c*x^2)^p, x] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && EqQ[Coeff[Pq, x, 0], 0] && Not[MatchQ[Pq, x^m_.*u_.",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "IntegerQ[m]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.8 P(x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.8 P(x) (a+b x+c x^2)^p.m",
            "rhs": "(a + b*x + c*x^2)^ FracPart[p]/((4*c)^IntPart[p]*(b + 2*c*x)^(2*FracPart[p]))* Int[Pq*(b + 2*c*x)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && EqQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.8 P(x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.8 P(x) (a+b x+c x^2)^p.m",
            "rhs": "With[{Q = PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, (b*f - 2*a*g + (2*c*f - b*g)* x)*(a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)) + 1/((p + 1)*(b^2 - 4*a*c))* Int[(a + b*x + c*x^2)^(p + 1)* ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (2*p + 3)*(2*c*f - b*g), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.8 P(x) (a+b x+c x^2)^p.m",
       "filename": "1.2.1.8 P(x) (a+b x+c x^2)^p.m",
            "rhs": "With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, e*x^(q - 1)*(a + b*x + c*x^2)^(p + 1)/(c*(q + 2*p + 1)) + 1/(c*(q + 2*p + 1))*Int[(a + b*x + c*x^2)^p* ExpandToSum[ c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && Not[LeQ[p, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "Int[(d + e*x)^(m + 1)* PolynomialQuotient[Pq, d + e*x, x]*(a + b*x + c*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*Pq_*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[PolynomialRemainder[Pq, d + e*x, x], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "Int[(d + e*x)^(m + 1)* PolynomialQuotient[Pq, d + e*x, x]*(a + c*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_.*Pq_*(a_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[PolynomialRemainder[Pq, d + e*x, x], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "With[{f = Coeff[P2, x, 0], g = Coeff[P2, x, 1], h = Coeff[P2, x, 2]}, h*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1)/(c* e*(m + 2*p + 3)) /; EqQ[b*e*h*(m + p + 2) + 2*c*d*h*(p + 1) - c*e*g*(m + 2*p + 3), 0] && EqQ[b*d*h*(p + 1) + a*e*h*(m + 1) - c*e*f*(m + 2*p + 3), 0]]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*P2_*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[P2, x, 2] && NeQ[m + 2*p + 3, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "With[{f = Coeff[P2, x, 0], g = Coeff[P2, x, 1], h = Coeff[P2, x, 2]}, h*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1)/(c* e*(m + 2*p + 3)) /; EqQ[2*d*h*(p + 1) - e*g*(m + 2*p + 3), 0] && EqQ[a*h*(m + 1) - c*f*(m + 2*p + 3), 0]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_.*P2_*(a_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m, p}, x] && PolyQ[P2, x, 2] && NeQ[m + 2*p + 3, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*Pq_*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + c*x^2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_.*Pq_*(a_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "(a + b*x + c*x^2)^ FracPart[p]/((4*c)^IntPart[p]*(b + 2*c*x)^(2*FracPart[p]))* Int[(d + e*x)^m*Pq*(b + 2*c*x)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*Pq_*(a_ + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "e*Int[(e*x)^(m - 1)* PolynomialQuotient[Pq, b + c*x, x]*(b*x + c*x^2)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Pq_*(b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, e, m, p}, x] && PolyQ[Pq, x] && EqQ[PolynomialRemainder[Pq, b + c*x, x], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "d*e*Int[(d + e*x)^(m - 1)* PolynomialQuotient[Pq, a*e + c*d*x, x]*(a + b*x + c*x^2)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_.*Pq_*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[PolynomialRemainder[Pq, a*e + c*d*x, x], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "d*e*Int[(d + e*x)^(m - 1)* PolynomialQuotient[Pq, a*e + c*d*x, x]*(a + c*x^2)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_.*Pq_*(a_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] && EqQ[PolynomialRemainder[Pq, a*e + c*d*x, x], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "With[{Q = PolynomialQuotient[Pq, a*e + c*d*x, x], f = PolynomialRemainder[Pq, a*e + c*d*x, x]}, f*(2*c*d - b*e)*(d + e*x)^ m*(a + b*x + c*x^2)^(p + 1)/(e*(p + 1)*(b^2 - 4*a*c)) + 1/((p + 1)*(b^2 - 4*a*c))* Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)* ExpandToSum[ d*e*(p + 1)*(b^2 - 4*a*c)*Q - f*(2*c*d - b*e)*(m + 2*p + 2), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*Pq_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[p + 1/2, 0] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "With[{Q = PolynomialQuotient[Pq, a*e + c*d*x, x], f = PolynomialRemainder[Pq, a*e + c*d*x, x]}, -d*f*(d + e*x)^m*(a + c*x^2)^(p + 1)/(2*a*e*(p + 1)) + d/(2*a*(p + 1))* Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)* ExpandToSum[2*a*e*(p + 1)*Q + f*(m + 2*p + 2), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_.*Pq_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] && ILtQ[p + 1/2, 0] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*x + c*x^2)^p, (d + e*x)^m*Pq, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*Pq_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + Expon[Pq, x] + 2*p + 1, 0] && ILtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "Int[ExpandIntegrand[(a + c*x^2)^p, (d + e*x)^m*Pq, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*Pq_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] && EqQ[m + Expon[Pq, x] + 2*p + 1, 0] && ILtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, f*(d + e*x)^(m + q - 1)*(a + b*x + c*x^2)^(p + 1)/(c* e^(q - 1)*(m + q + 2*p + 1)) + 1/(c*e^q*(m + q + 2*p + 1))*Int[(d + e*x)^m*(a + b*x + c*x^2)^p* ExpandToSum[ c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q + e*f*(m + p + q)*(d + e*x)^(q - 2)*(b*d - 2*a*e + (2*c*d - b*e)*x), x], x] /; NeQ[m + q + 2*p + 1, 0]]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*Pq_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, f*(d + e*x)^(m + q - 1)*(a + c*x^2)^(p + 1)/(c* e^(q - 1)*(m + q + 2*p + 1)) + 1/(c*e^q*(m + q + 2*p + 1))*Int[(d + e*x)^m*(a + c*x^2)^p* ExpandToSum[ c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - 2*e*f*(m + p + q)*(d + e*x)^(q - 2)*(a*e - c*d*x), x], x] /; NeQ[m + q + 2*p + 1, 0]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_.*Pq_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] && Not[IGtQ[m, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "Int[(d + e*x)^(m + p)*(a/d + c/e*x)^p*Pq, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*Pq_*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "Int[(d + e*x)^(m + p)*(a/d + c/e*x)^p*Pq, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_.*Pq_*(a_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "(a + b*x + c*x^2)^ FracPart[p]/((d + e*x)^FracPart[p]*(a/d + (c*x)/e)^FracPart[p])* Int[(d + e*x)^(m + p)*(a/d + c/e*x)^p*Pq, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*Pq_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[p]] && Not[IGtQ[m, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "(a + c*x^2)^ FracPart[p]/((d + e*x)^FracPart[p]*(a/d + (c*x)/e)^FracPart[p])* Int[(d + e*x)^(m + p)*(a/d + c/e*x)^p*Pq, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_.*Pq_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]] && Not[IGtQ[m, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "With[{Q = PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, (d + e*x)^ m*(a + b*x + c*x^2)^(p + 1)*(f*b - 2*a*g + (2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c)) + 1/((p + 1)*(b^2 - 4*a*c))* Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)* ExpandToSum[(p + 1)*(b^2 - 4*a*c)*(d + e*x)*Q + g*(2*a*e*m + b*d*(2*p + 3)) - f*(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)*(m + 2*p + 3)*x, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*Pq_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && (IntegerQ[p] || Not[IntegerQ[m]] || Not[RationalQ[a, b, c, d, e]]) && Not[IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0])]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "With[{Q = PolynomialQuotient[Pq, a + c*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + c*x^2, x], x, 1]}, (d + e*x)^m*(a + c*x^2)^(p + 1)*(a*g - c*f*x)/(2*a*c*(p + 1)) + 1/(2*a*c*(p + 1))*Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)* ExpandToSum[ 2*a*c*(p + 1)*(d + e*x)*Q - a*e*g*m + c*d*f*(2*p + 3) + c*e*f*(m + 2*p + 3)*x, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_.*Pq_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && Not[IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0])]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "With[{Q = PolynomialQuotient[(d + e*x)^m*Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, (b*f - 2*a*g + (2*c*f - b*g)* x)*(a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)) + 1/((p + 1)*(b^2 - 4*a*c))* Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)* ExpandToSum[(p + 1)*(b^2 - 4*a*c)*(d + e*x)^(-m)* Q - (2*p + 3)*(2*c*f - b*g)*(d + e*x)^(-m), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*Pq_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "With[{Q = PolynomialQuotient[(d + e*x)^m*Pq, a + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 1]}, (a*g - c*f*x)*(a + c*x^2)^(p + 1)/(2*a*c*(p + 1)) + 1/(2*a*c*(p + 1))*Int[(d + e*x)^m*(a + c*x^2)^(p + 1)* ExpandToSum[ 2*a*c*(p + 1)*(d + e*x)^(-m)*Q + c*f*(2*p + 3)*(d + e*x)^(-m), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_.*Pq_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "With[{Q = PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, (d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)/ ((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) + 1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))* Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)* ExpandToSum[(p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*Q + f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*Pq_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && Not[IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0])]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "With[{Q = PolynomialQuotient[Pq, a + c*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + c*x^2, x], x, 1]}, -(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1)*(a*(e*f - d*g) + (c*d*f + a*e*g)*x)/(2* a*(p + 1)*(c*d^2 + a*e^2)) + 1/(2*a*(p + 1)*(c*d^2 + a*e^2))* Int[(d + e*x)^m*(a + c*x^2)^(p + 1)* ExpandToSum[ 2*a*(p + 1)*(c*d^2 + a*e^2)*Q + c*d^2*f*(2*p + 3) - a*e*(d*g*m - e*f*(m + 2*p + 3)) + e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_.*Pq_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && Not[IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0])]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "With[{Q = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, (e*R*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)) + 1/((m + 1)*(c*d^2 - b*d*e + a*e^2))* Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p* ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Q + c*d*R*(m + 1) - b*e*R*(m + p + 2) - c*e*R*(m + 2*p + 3)*x, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*Pq_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "With[{Q = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, (e*R*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1)*(c*d^2 + a*e^2)) + 1/((m + 1)*(c*d^2 + a*e^2))* Int[(d + e*x)^(m + 1)*(a + c*x^2)^p* ExpandToSum[(m + 1)*(c*d^2 + a*e^2)*Q + c*d*R*(m + 1) - c*e*R*(m + 2*p + 3)*x, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*Pq_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "Module[{q = Expon[Pq, x], k}, Int[x^m* Sum[Coeff[Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2)^p, x] + Int[ x^(m + 1)* Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0, (q - 1)/2}]*(a + b*x^2)^p, x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Pq_*(a_ + b_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && Not[PolyQ[Pq, x^2]] && IGtQ[m, -2] && Not[IntegerQ[2*p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, f*(d + e*x)^(m + q - 1)*(a + b*x + c*x^2)^(p + 1)/(c* e^(q - 1)*(m + q + 2*p + 1)) + 1/(c*e^q*(m + q + 2*p + 1))* Int[(d + e*x)^m*(a + b*x + c*x^2)^p* ExpandToSum[ c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*Pq_*(a_. + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0])]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, f*(d + e*x)^(m + q - 1)*(a + c*x^2)^(p + 1)/(c* e^(q - 1)*(m + q + 2*p + 1)) + 1/(c*e^q*(m + q + 2*p + 1))* Int[(d + e*x)^m*(a + c*x^2)^p* ExpandToSum[ c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(m + q + p)*x), x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_.*Pq_*(a_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && Not[EqQ[d, 0] && True] && Not[IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0])]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "With[{q = Expon[Pq, x]}, Coeff[Pq, x, q]/e^q* Int[(d + e*x)^(m + q)*(a + b*x + c*x^2)^p, x] + 1/e^q* Int[(d + e*x)^m*(a + b*x + c*x^2)^p* ExpandToSum[e^q*Pq - Coeff[Pq, x, q]*(d + e*x)^q, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*Pq_*(a_. + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0])]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
       "filename": "1.2.1.9 P(x) (d+e x)^m (a+b x+c x^2)^p.m",
            "rhs": "With[{q = Expon[Pq, x]}, Coeff[Pq, x, q]/e^q*Int[(d + e*x)^(m + q)*(a + c*x^2)^p, x] + 1/e^q* Int[(d + e*x)^m*(a + c*x^2)^p* ExpandToSum[e^q*Pq - Coeff[Pq, x, q]*(d + e*x)^q, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_.*Pq_*(a_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && Not[IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0])]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.1 (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.1 (a+b x^2+c x^4)^p.m",
            "rhs": " 1/c^p*Int[(b/2+c*x^2)^(2*p),x]",
     "rulenumber": 0,
            "lhs": "Int[(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,p},x] &&  EqQ[b^2-4*a*c,0] && IntegerQ[p] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.1 (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.1 (a+b x^2+c x^4)^p.m",
            "rhs": " 2*x/(3*a*(a+b*x^2+c*x^4)^(1/4)) +  x*(2*a+b*x^2)/(6*a*(a+b*x^2+c*x^4)^(5/4))",
     "rulenumber": 0,
            "lhs": "Int[1/(a_+b_.*x_^2+c_.*x_^4)^(5/4),x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c},x] &&  EqQ[b^2-4*a*c,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.1 (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.1 (a+b x^2+c x^4)^p.m",
            "rhs": "(a + b*x^2 + c*x^4)^p/(b + 2*c*x^2)^(2*p)* Int[(b + 2*c*x^2)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.1 (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.1 (a+b x^2+c x^4)^p.m",
            "rhs": "a^IntPart[p]*(a + b*x^2 + c*x^4)^ FracPart[p]/(1 + 2*c*x^2/b)^(2*FracPart[p])* Int[(1 + 2*c*x^2/b)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[2*p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.1 (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.1 (a+b x^2+c x^4)^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*x^2 + c*x^4)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.1 (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.1 (a+b x^2+c x^4)^p.m",
            "rhs": "x*(a + b*x^2 + c*x^4)^p/(4*p + 1) + 2*p/(4*p + 1)*Int[(2*a + b*x^2)*(a + b*x^2 + c*x^4)^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.1 (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.1 (a+b x^2+c x^4)^p.m",
            "rhs": "-x*(b^2 - 2*a*c + b*c*x^2)*(a + b*x^2 + c*x^4)^(p + 1)/(2* a*(p + 1)*(b^2 - 4*a*c)) + 1/(2*a*(p + 1)*(b^2 - 4*a*c))* Int[(b^2 - 2*a*c + 2*(p + 1)*(b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.1 (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.1 (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, c/q*Int[1/(b/2 - q/2 + c*x^2), x] - c/q*Int[1/(b/2 + q/2 + c*x^2), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.1 (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.1 (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, 1/(2*c*q*r)*Int[(r - x)/(q - r*x + x^2), x] + 1/(2*c*q*r)*Int[(r + x)/(q + r*x + x^2), x]]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.1 (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.1 (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, 2*Sqrt[-c]* Int[1/(Sqrt[b + q + 2*c*x^2]*Sqrt[-b + q - 2*c*x^2]), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.1 (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.1 (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[c/a, 4]}, (1 + q^2*x^2)* Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q* Sqrt[a + b*x^2 + c*x^4])* EllipticF[2*ArcTan[q*x], 1/2 - b*q^2/(4*c)]]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] && GtQ[c/a, 0] && LtQ[b/a, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.1 (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.1 (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, Sqrt[-2*a - (b - q)*x^2]* Sqrt[(2*a + (b + q)*x^2)/q]/(2*Sqrt[-a]* Sqrt[a + b*x^2 + c*x^4])* EllipticF[ ArcSin[x/ Sqrt[(2*a + (b + q)*x^2)/(2*q)]], (b + q)/(2*q)] /; IntegerQ[q]]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.1 (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.1 (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, Sqrt[(2*a + (b - q)*x^2)/(2*a + (b + q)*x^2)]* Sqrt[(2*a + (b + q)*x^2)/q]/(2*Sqrt[a + b*x^2 + c*x^4]* Sqrt[a/(2*a + (b + q)*x^2)])* EllipticF[ ArcSin[x/Sqrt[(2*a + (b + q)*x^2)/(2*q)]], (b + q)/(2*q)]]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.1 (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.1 (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, (2*a + (b + q)*x^2)* Sqrt[(2*a + (b - q)*x^2)/(2*a + (b + q)*x^2)]/(2*a* Rt[(b + q)/(2*a), 2]*Sqrt[a + b*x^2 + c*x^4])* EllipticF[ArcTan[Rt[(b + q)/(2*a), 2]*x], 2*q/(b + q)] /; PosQ[(b + q)/a] && Not[PosQ[(b - q)/a] && SimplerSqrtQ[(b - q)/(2*a), (b + q)/(2*a)]]]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.1 (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.1 (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, (2*a + (b - q)*x^2)* Sqrt[(2*a + (b + q)*x^2)/(2*a + (b - q)*x^2)]/(2*a* Rt[(b - q)/(2*a), 2]*Sqrt[a + b*x^2 + c*x^4])* EllipticF[ ArcTan[Rt[(b - q)/(2*a), 2]*x], -2*q/(b - q)] /; PosQ[(b - q)/a]]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.1 (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.1 (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, Sqrt[1 + (b + q)*x^2/(2*a)]* Sqrt[1 + (b - q)*x^2/(2*a)]/(Rt[-(b + q)/(2*a), 2]* Sqrt[a + b*x^2 + c*x^4])* EllipticF[ ArcSin[Rt[-(b + q)/(2*a), 2]*x], (b - q)/(b + q)] /; NegQ[(b + q)/a] && Not[NegQ[(b - q)/a] && SimplerSqrtQ[-(b - q)/(2*a), -(b + q)/(2*a)]]]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.1 (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.1 (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, Sqrt[1 + (b - q)*x^2/(2*a)]* Sqrt[1 + (b + q)*x^2/(2*a)]/(Rt[-(b - q)/(2*a), 2]* Sqrt[a + b*x^2 + c*x^4])* EllipticF[ ArcSin[Rt[-(b - q)/(2*a), 2]*x], (b + q)/(b - q)] /; NegQ[(b - q)/a]]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.1 (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.1 (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[c/a, 4]}, (1 + q^2*x^2)* Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q* Sqrt[a + b*x^2 + c*x^4])* EllipticF[2*ArcTan[q*x], 1/2 - b*q^2/(4*c)]]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.1 (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.1 (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, Sqrt[1 + 2*c*x^2/(b - q)]* Sqrt[1 + 2*c*x^2/(b + q)]/Sqrt[a + b*x^2 + c*x^4]* Int[1/(Sqrt[1 + 2*c*x^2/(b - q)]*Sqrt[1 + 2*c*x^2/(b + q)]), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[c/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.1 (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.1 (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, a^IntPart[p]*(a + b*x^2 + c*x^4)^ FracPart[ p]/((1 + 2*c*x^2/(b + q))^FracPart[p]*(1 + 2*c*x^2/(b - q))^ FracPart[p])* Int[(1 + 2*c*x^2/(b + q))^p*(1 + 2*c*x^2/(b - q))^p, x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.1 (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.1 (a+b x^2+c x^4)^p.m",
            "rhs": "With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1], c = Coeff[P4, x, 2], d = Coeff[P4, x, 3], e = Coeff[P4, x, 4]}, Subst[ Int[SimplifyIntegrand[(a + d^4/(256*e^3) - b*d/(8*e) + (c - 3*d^2/(8*e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2, 0] && NeQ[d, 0]]",
     "rulenumber": 0,
            "lhs": "Int[P4_^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[p, x] && PolyQ[P4, x, 4] && NeQ[p, 2] && NeQ[p, 3]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": " 1/d^(2*p)*Int[(d*x)^(m+2*p)*(b+c*x^2)^p,x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(b_.*x_^2+c_.*x_^4)^p_.,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b,c,d,m},x] &&  IntegerQ[p] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": " (b*x^2+c*x^4)^p/((d*x)^(2*p)*(b+c*x^2)^p)*Int[(d*x)^(m+2*p)*(b+c*x^2)^ p,x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(b_.*x_^2+c_.*x_^4)^p_,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b,c,d,m,p},x] && Not[IntegerQ[p]] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "1/2*Subst[Int[(a + b*x + c*x^2)^p, x], x, x^2]",
     "rulenumber": 0,
            "lhs": "Int[x_*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "Int[ExpandIntegrand[(d*x)^m*(a + b*x^2 + c*x^4)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && Not[IntegerQ[(m + 1)/2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": " 1/c^p*Int[(d*x)^m*(b/2+c*x^2)^(2*p),x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,m,p},x] &&  EqQ[b^2-4*a*c,0] && IntegerQ[p] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "2*(d*x)^(m + 1)*(a + b*x^2 + c*x^4)^(p + 1)/(d*(m + 3)*(2*a + b*x^2)) - (d*x)^(m + 1)*(a + b*x^2 + c*x^4)^(p + 1)/(2*a* d*(m + 3)*(p + 1))",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]] && EqQ[m + 4*p + 5, 0] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "(d*x)^(m + 1)*(a + b*x^2 + c*x^4)^(p + 1)/(4*a* d*(p + 1)*(2*p + 1)) - (d*x)^(m + 1)*(2*a + b*x^2)*(a + b*x^2 + c*x^4)^ p/(4*a*d*(2*p + 1))",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]] && EqQ[m + 4*p + 5, 0] && NeQ[p, -1/2]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "1/2*Subst[Int[x^((m - 1)/2)*(a + b*x + c*x^2)^p, x], x, x^2]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2] && IntegerQ[(m - 1)/2] && (GtQ[m, 0] || LtQ[0, 4*p, -m - 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": " c*(a+b*x^2+c*x^4)^(p+1)/(b/2+c*x^2)^(2*(p+1))*Int[(d*x)^m*(b/2+c*x^2)^ (2*p),x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,m,p},x] && EqQ[b^2-4*a*c,0] &&  IntegerQ[p-1/2] && IGeQ[m,2*p] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "(a + b*x^2 + c*x^4)^ FracPart[p]/(c^IntPart[p]*(b/2 + c*x^2)^(2*FracPart[p]))* Int[(d*x)^m*(b/2 + c*x^2)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "a^IntPart[p]*(a + b*x^2 + c*x^4)^ FracPart[p]/(1 + 2*c*x^2/b)^(2*FracPart[p])* Int[(d*x)^m*(1 + 2*c*x^2/b)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[2*p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "1/2*Subst[Int[x^((m - 1)/2)*(a + b*x + c*x^2)^p, x], x, x^2]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "With[{k = Denominator[m]}, k/d*Subst[ Int[x^(k*(m + 1) - 1)*(a + b*x^(2*k)/d^2 + c*x^(4*k)/d^4)^p, x], x, (d*x)^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && FractionQ[m] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "d*(d*x)^(m - 1)*(a + b*x^2 + c*x^4)^ p*(2*b*p + c*(m + 4*p - 1)*x^2)/(c*(m + 4*p + 1)*(m + 4*p - 1)) - 2*p*d^2/(c*(m + 4*p + 1)*(m + 4*p - 1))* Int[(d*x)^(m - 2)*(a + b*x^2 + c*x^4)^(p - 1)* Simp[a*b*(m - 1) - (2*a*c*(m + 4*p - 1) - b^2*(m + 2*p - 1))* x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && GtQ[m, 1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "(d*x)^(m + 1)*(a + b*x^2 + c*x^4)^p/(d*(m + 1)) - 2*p/(d^2*(m + 1))* Int[(d*x)^(m + 2)*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && LtQ[m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "(d*x)^(m + 1)*(a + b*x^2 + c*x^4)^ p/(d*(m + 4*p + 1)) + 2*p/(m + 4*p + 1)* Int[(d*x)^m*(2*a + b*x^2)*(a + b*x^2 + c*x^4)^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && NeQ[m + 4*p + 1, 0] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "d*(d*x)^(m - 1)*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^(p + 1)/(2*(p + 1)*(b^2 - 4*a*c)) - d^2/(2*(p + 1)*(b^2 - 4*a*c))* Int[(d*x)^(m - 2)*(b*(m - 1) + 2*c*(m + 4*p + 5)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[m, 1] && LeQ[m, 3] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "-d^3*(d*x)^(m - 3)*(2*a + b*x^2)*(a + b*x^2 + c*x^4)^(p + 1)/(2*(p + 1)*(b^2 - 4*a*c)) + d^4/(2*(p + 1)*(b^2 - 4*a*c))* Int[(d*x)^(m - 4)*(2*a*(m - 3) + b*(m + 4*p + 3)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[m, 3] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "-(d*x)^(m + 1)*(b^2 - 2*a*c + b*c*x^2)*(a + b*x^2 + c*x^4)^(p + 1)/(2*a* d*(p + 1)*(b^2 - 4*a*c)) + 1/(2*a*(p + 1)*(b^2 - 4*a*c))* Int[(d*x)^m*(a + b*x^2 + c*x^4)^(p + 1)* Simp[b^2*(m + 2*p + 3) - 2*a*c*(m + 4*p + 5) + b*c*(m + 4*p + 7)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "d^3*(d*x)^(m - 3)*(a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 1)) - d^4/(c*(m + 4*p + 1))* Int[(d*x)^(m - 4)* Simp[a*(m - 3) + b*(m + 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "(d*x)^(m + 1)*(a + b*x^2 + c*x^4)^(p + 1)/(a* d*(m + 1)) - 1/(a*d^2*(m + 1))* Int[(d*x)^(m + 2)*(b*(m + 2*p + 3) + c*(m + 4*p + 5)*x^2)*(a + b*x^2 + c*x^4)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "(d*x)^(m + 1)/(a*d*(m + 1)) - 1/(a*d^2)*Int[(d*x)^(m + 2)*(b + c*x^2)/(a + b*x^2 + c*x^4), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_/(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "Int[PolynomialDivide[x^m, (a + b*x^2 + c*x^4), x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_/(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 5]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "d^3*(d*x)^(m - 3)/(c*(m - 3)) - d^4/c*Int[(d*x)^(m - 4)*(a + b*x^2)/(a + b*x^2 + c*x^4), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_/(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[m, 3]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[a/c, 2]}, 1/2*Int[(q + x^2)/(a + b*x^2 + c*x^4), x] - 1/2*Int[(q - x^2)/(a + b*x^2 + c*x^4), x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^2/(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && LtQ[b^2 - 4*a*c, 0] && PosQ[a*c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, 1/(2*c*r)*Int[x^(m - 3)*(q + r*x)/(q + r*x + x^2), x] - 1/(2*c*r)*Int[x^(m - 3)*(q - r*x)/(q - r*x + x^2), x]]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_./(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 3] && LtQ[m, 4] && NegQ[b^2 - 4*a*c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, 1/(2*c*r)*Int[x^(m - 1)/(q - r*x + x^2), x] - 1/(2*c*r)*Int[x^(m - 1)/(q + r*x + x^2), x]]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_./(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 1] && LtQ[m, 3] && NegQ[b^2 - 4*a*c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, d^2/2*(b/q + 1)*Int[(d*x)^(m - 2)/(b/2 + q/2 + c*x^2), x] - d^2/2*(b/q - 1)*Int[(d*x)^(m - 2)/(b/2 - q/2 + c*x^2), x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_/(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, c/q*Int[(d*x)^m/(b/2 - q/2 + c*x^2), x] - c/q*Int[(d*x)^m/(b/2 + q/2 + c*x^2), x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_./(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, 2*Sqrt[-c]* Int[x^2/(Sqrt[b + q + 2*c*x^2]*Sqrt[-b + q - 2*c*x^2]), x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^2/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[c/a, 2]}, 1/q*Int[1/Sqrt[a + b*x^2 + c*x^4], x] - 1/q*Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^2/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] && GtQ[c/a, 0] && LtQ[b/a, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, -(b - q)/(2*c)*Int[1/Sqrt[a + b*x^2 + c*x^4], x] + 1/(2*c)*Int[(b - q + 2*c*x^2)/Sqrt[a + b*x^2 + c*x^4], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^2/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, x*(b + q + 2*c*x^2)/(2*c*Sqrt[a + b*x^2 + c*x^4]) - Rt[(b + q)/(2*a), 2]*(2*a + (b + q)*x^2)* Sqrt[(2*a + (b - q)*x^2)/(2*a + (b + q)*x^2)]/(2*c* Sqrt[a + b*x^2 + c*x^4])* EllipticE[ArcTan[Rt[(b + q)/(2*a), 2]*x], 2*q/(b + q)] /; PosQ[(b + q)/a] && Not[PosQ[(b - q)/a] && SimplerSqrtQ[(b - q)/(2*a), (b + q)/(2*a)]]]",
     "rulenumber": 0,
            "lhs": "Int[x_^2/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, x*(b - q + 2*c*x^2)/(2*c*Sqrt[a + b*x^2 + c*x^4]) - Rt[(b - q)/(2*a), 2]*(2*a + (b - q)*x^2)* Sqrt[(2*a + (b + q)*x^2)/(2*a + (b - q)*x^2)]/(2*c* Sqrt[a + b*x^2 + c*x^4])* EllipticE[ArcTan[Rt[(b - q)/(2*a), 2]*x], -2*q/(b - q)] /; PosQ[(b - q)/a]]",
     "rulenumber": 0,
            "lhs": "Int[x_^2/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, -(b + q)/(2*c)*Int[1/Sqrt[a + b*x^2 + c*x^4], x] + 1/(2*c)*Int[(b + q + 2*c*x^2)/Sqrt[a + b*x^2 + c*x^4], x] /; NegQ[(b + q)/a] && Not[NegQ[(b - q)/a] && SimplerSqrtQ[-(b - q)/(2*a), -(b + q)/(2*a)]]]",
     "rulenumber": 0,
            "lhs": "Int[x_^2/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, -(b - q)/(2*c)*Int[1/Sqrt[a + b*x^2 + c*x^4], x] + 1/(2*c)*Int[(b - q + 2*c*x^2)/Sqrt[a + b*x^2 + c*x^4], x] /; NegQ[(b - q)/a]]",
     "rulenumber": 0,
            "lhs": "Int[x_^2/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[c/a, 2]}, 1/q*Int[1/Sqrt[a + b*x^2 + c*x^4], x] - 1/q*Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^2/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, Sqrt[1 + 2*c*x^2/(b - q)]* Sqrt[1 + 2*c*x^2/(b + q)]/Sqrt[a + b*x^2 + c*x^4]* Int[x^2/(Sqrt[1 + 2*c*x^2/(b - q)]*Sqrt[1 + 2*c*x^2/(b + q)]), x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^2/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[c/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "a^IntPart[p]*(a + b*x^2 + c*x^4)^FracPart[p]/ ((1 + 2*c*x^2/(b + Rt[b^2 - 4*a*c, 2]))^ FracPart[p]*(1 + 2*c*x^2/(b - Rt[b^2 - 4*a*c, 2]))^FracPart[p])* Int[(d*x)^m*(1 + 2*c*x^2/(b + Sqrt[b^2 - 4*a*c]))^ p*(1 + 2*c*x^2/(b - Sqrt[b^2 - 4*a*c]))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "u^m/(Coefficient[v, x, 1]*v^m)* Subst[Int[x^m*(a + b*x^2 + c*x^(2*2))^p, x], x, v]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*(a_. + b_.*v_^2 + c_.*v_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, p}, x] && LinearPairQ[u, v, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "-2*(c*d - b*e)*(b*x^2 + c*x^4)^(1/4)/(b*c*x) + e/c*Int[(b*x^2 + c*x^4)^(1/4)/x^2, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/(b_.*x_^2 + c_.*x_^4)^(3/4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "e*(b*x^2 + c*x^4)^(p + 1)/(c*(4*p + 3)*x)",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)*(b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, p}, x] && Not[IntegerQ[p]] && NeQ[4*p + 3, 0] && EqQ[b*e*(2*p + 1) - c*d*(4*p + 3), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "e*(b*x^2 + c*x^4)^(p + 1)/(c*(4*p + 3)* x) - ((b*e*(2*p + 1) - c*d*(4*p + 3))/(c*(4*p + 3)))* Int[(b*x^2 + c*x^4)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)*(b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, p}, x] && Not[IntegerQ[p]] && NeQ[4*p + 3, 0] && NeQ[b*e*(2*p + 1) - c*d*(4*p + 3), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "(b*x^2 + c*x^4)^ FracPart[p]/(x^(2*FracPart[p])*(b + c*x^2)^FracPart[p])* Int[x^(2*p)*(d + e*x^2)^q*(b + c*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_.*(b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, p, q}, x] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": " 1/c^p*Int[(d+e*x^2)^q*(b/2+c*x^2)^(2*p),x]",
     "rulenumber": 0,
            "lhs": "Int[(d_+e_.*x_^2)^q_.*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e,p,q},x] && EqQ[b^2-4*a*c,0] && IntegerQ[p] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "(a + b*x^2 + c*x^4)^p/(d + e*x^2)^(2*p)* Int[(d + e*x^2)^(q + 2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p, q}, x] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]] && EqQ[2*c*d - b*e, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "(a + b*x^2 + c*x^4)^ FracPart[p]/(c^IntPart[p]*(b/2 + c*x^2)^(2*FracPart[p]))* Int[(d + e*x^2)^q*(b/2 + c*x^2)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p, q}, x] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[(d + e*x^2)^(p + q)*(a/d + c/e*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[(d + e*x^2)^(p + q)*(a/d + c/e*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_.*(a_ + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, q}, x] && EqQ[c*d^2 + a*e^2, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "(a + b*x^2 + c*x^4)^ FracPart[ p]/((d + e*x^2)^FracPart[p]*(a/d + c*x^2/e)^FracPart[p])* Int[(d + e*x^2)^(p + q)*(a/d + c/e*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "(a + c*x^4)^ FracPart[ p]/((d + e*x^2)^FracPart[p]*(a/d + c*x^2/e)^FracPart[p])* Int[(d + e*x^2)^(p + q)*(a/d + c/e*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_ + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, p, q}, x] && EqQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x^2)^q*(a + c*x^4)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_.*(a_ + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "a^p*x*(d + e*x^2)^(q + 1)/d + 1/d*Int[ x^2*(d + e*x^2)^ q*(d*PolynomialQuotient[(a + b*x^2 + c*x^4)^p - a^p, x^2, x] - e*a^p*(2*q + 3)), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && ILtQ[q + 1/2, 0] && LtQ[4*p + 2*q + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "a^p*x*(d + e*x^2)^(q + 1)/d + 1/d*Int[ x^2*(d + e*x^2)^ q*(d*PolynomialQuotient[(a + c*x^4)^p - a^p, x^2, x] - e*a^p*(2*q + 3)), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_ + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && ILtQ[q + 1/2, 0] && LtQ[4*p + 2*q + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[ PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x, 0]}, -R*x*(d + e*x^2)^(q + 1)/(2*d*(q + 1)) + 1/(2*d*(q + 1))* Int[(d + e*x^2)^(q + 1)* ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{Qx = PolynomialQuotient[(a + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + c*x^4)^p, d + e*x^2, x], x, 0]}, -R*x*(d + e*x^2)^(q + 1)/(2*d*(q + 1)) + 1/(2*d*(q + 1))* Int[(d + e*x^2)^(q + 1)* ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_ + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "c^p*x^(4*p - 1)*(d + e*x^2)^(q + 1)/(e*(4*p + 2*q + 1)) + 1/(e*(4*p + 2*q + 1))* Int[(d + e*x^2)^q* ExpandToSum[ e*(4*p + 2*q + 1)*(a + b*x^2 + c*x^4)^p - d*c^p*(4*p - 1)*x^(4*p - 2) - e*c^p*(4*p + 2*q + 1)*x^(4*p), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && Not[LtQ[q, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "c^p*x^(4*p - 1)*(d + e*x^2)^(q + 1)/(e*(4*p + 2*q + 1)) + 1/(e*(4*p + 2*q + 1))* Int[(d + e*x^2)^q* ExpandToSum[ e*(4*p + 2*q + 1)*(a + c*x^4)^p - d*c^p*(4*p - 1)*x^(4*p - 2) - e*c^p*(4*p + 2*q + 1)*x^(4*p), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_ + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, q}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && Not[LtQ[q, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[2*d/e - b/c, 2]}, e/(2*c)*Int[1/Simp[d/e + q*x + x^2, x], x] + e/(2*c)*Int[1/Simp[d/e - q*x + x^2, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && (GtQ[2*d/e - b/c, 0] || Not[LtQ[2*d/e - b/c, 0]] && EqQ[d - e*Rt[a/c, 2], 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[2*d/e, 2]}, e/(2*c)*Int[1/Simp[d/e + q*x + x^2, x], x] + e/(2*c)*Int[1/Simp[d/e - q*x + x^2, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/(a_ + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, (e/2 + (2*c*d - b*e)/(2*q))* Int[1/(b/2 - q/2 + c*x^2), x] + (e/2 - (2*c*d - b*e)/(2*q))* Int[1/(b/2 + q/2 + c*x^2), x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && GtQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[-2*d/e - b/c, 2]}, e/(2*c*q)*Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x] + e/(2*c*q)*Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && Not[GtQ[b^2 - 4*a*c, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[-2*d/e, 2]}, e/(2*c*q)*Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x] + e/(2*c*q)*Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/(a_ + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, (e/2 + (2*c*d - b*e)/(2*q))* Int[1/(b/2 - q/2 + c*x^2), x] + (e/2 - (2*c*d - b*e)/(2*q))* Int[1/(b/2 + q/2 + c*x^2), x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[-a*c, 2]}, (e/2 + c*d/(2*q))*Int[1/(-q + c*x^2), x] + (e/2 - c*d/(2*q))* Int[1/(q + c*x^2), x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/(a_ + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[-a*c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[a*c, 2]}, (d*q + a*e)/(2*a*c)* Int[(q + c*x^2)/(a + c*x^4), x] + (d*q - a*e)/(2*a*c)* Int[(q - c*x^2)/(a + c*x^4), x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/(a_ + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[-a*c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, 1/(2*c*q*r)*Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x] + 1/(2*c*q*r)*Int[(d*r + (d - e*q)*x)/(q + r*x + x^2), x]]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x^2)^q/(a + b*x^2 + c*x^4), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_/(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[q]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x^2)^q/(a + c*x^4), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_/(a_ + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[q]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "e^2/(c*d^2 - b*d*e + a*e^2)*Int[(d + e*x^2)^q, x] + 1/(c*d^2 - b*d*e + a*e^2)* Int[(d + e*x^2)^(q + 1)*(c*d - b*e - c*e*x^2)/(a + b*x^2 + c*x^4), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_/(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[q]] && LtQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "e^2/(c*d^2 + a*e^2)*Int[(d + e*x^2)^q, x] + c/(c*d^2 + a*e^2)* Int[(d + e*x^2)^(q + 1)*(d - e*x^2)/(a + c*x^4), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_/(a_ + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[q]] && LtQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{r = Rt[b^2 - 4*a*c, 2]}, 2*c/r*Int[(d + e*x^2)^q/(b - r + 2*c*x^2), x] - 2*c/r*Int[(d + e*x^2)^q/(b + r + 2*c*x^2), x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_/(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[q]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{r = Rt[-a*c, 2]}, -c/(2*r)*Int[(d + e*x^2)^q/(r - c*x^2), x] - c/(2*r)*Int[(d + e*x^2)^q/(r + c*x^2), x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_/(a_ + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, q}, x] && NeQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[q]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "x*(2*b*e*p + c*d*(4*p + 3) + c*e*(4*p + 1)*x^2)*(a + b*x^2 + c*x^4)^ p/(c*(4*p + 1)*(4*p + 3)) + 2*p/(c*(4*p + 1)*(4*p + 3))* Int[Simp[ 2*a*c*d*(4*p + 3) - a*b*e + (2*a*c*e*(4*p + 1) + b*c*d*(4*p + 3) - b^2*e*(2*p + 1))*x^2, x]* (a + b*x^2 + c*x^4)^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && FractionQ[p] && IntegerQ[2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "x*(d*(4*p + 3) + e*(4*p + 1)*x^2)*(a + c*x^4)^ p/((4*p + 1)*(4*p + 3)) + 2*p/((4*p + 1)*(4*p + 3))* Int[Simp[2*a*d*(4*p + 3) + (2*a*e*(4*p + 1))*x^2, x]*(a + c*x^4)^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)*(a_ + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && FractionQ[p] && IntegerQ[2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*(a + b*x^2 + c*x^4)^(p + 1)/(2* a*(p + 1)*(b^2 - 4*a*c)) + 1/(2*a*(p + 1)*(b^2 - 4*a*c))* Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]* (a + b*x^2 + c*x^4)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && IntegerQ[2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "-x*(d + e*x^2)*(a + c*x^4)^(p + 1)/(4*a*(p + 1)) + 1/(4*a*(p + 1))* Int[Simp[d*(4*p + 5) + e*(4*p + 7)*x^2, x]*(a + c*x^4)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)*(a_ + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntegerQ[2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, 2*Sqrt[-c]* Int[(d + e*x^2)/(Sqrt[b + q + 2*c*x^2]*Sqrt[-b + q - 2*c*x^2]), x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[-a*c, 2]}, Sqrt[-c]*Int[(d + e*x^2)/(Sqrt[q + c*x^2]*Sqrt[q - c*x^2]), x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/Sqrt[a_ + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && GtQ[a, 0] && LtQ[c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[c/a, 4]}, -d*x*Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q^2*x^2)) + d*(1 + q^2*x^2)* Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q* Sqrt[a + b*x^2 + c*x^4])* EllipticE[2*ArcTan[q*x], 1/2 - b*q^2/(4*c)] /; EqQ[e + d*q^2, 0]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0] && GtQ[c/a, 0] && LtQ[b/a, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[c/a, 2]}, (e + d*q)/q*Int[1/Sqrt[a + b*x^2 + c*x^4], x] - e/q*Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x] /; NeQ[e + d*q, 0]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0] && GtQ[c/a, 0] && LtQ[b/a, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, e*x*(b + q + 2*c*x^2)/(2*c*Sqrt[a + b*x^2 + c*x^4]) - e*q*Sqrt[(2*a + (b - q)*x^2)/(2*a + (b + q)*x^2)]* Sqrt[(2*a + (b + q)*x^2)/q]/(2*c*Sqrt[a + b*x^2 + c*x^4]* Sqrt[a/(2*a + (b + q)*x^2)])* EllipticE[ ArcSin[x/ Sqrt[(2*a + (b + q)*x^2)/(2*q)]], (b + q)/(2*q)] /; EqQ[2*c*d - e*(b - q), 0]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[-a*c, 2]}, e*x*(q + c*x^2)/(c*Sqrt[a + c*x^4]) - Sqrt[2]*e*q*Sqrt[-a + q*x^2]* Sqrt[(a + q*x^2)/q]/(Sqrt[-a]*c*Sqrt[a + c*x^4])* EllipticE[ArcSin[x/Sqrt[(a + q*x^2)/(2*q)]], 1/2] /; EqQ[c*d + e*q, 0] && IntegerQ[q]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/Sqrt[a_ + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && LtQ[a, 0] && GtQ[c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[-a*c, 2]}, e*x*(q + c*x^2)/(c*Sqrt[a + c*x^4]) - Sqrt[2]*e*q*Sqrt[(a - q*x^2)/(a + q*x^2)]* Sqrt[(a + q*x^2)/q]/(c*Sqrt[a + c*x^4]*Sqrt[a/(a + q*x^2)])* EllipticE[ArcSin[x/Sqrt[(a + q*x^2)/(2*q)]], 1/2] /; EqQ[c*d + e*q, 0]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/Sqrt[a_ + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && LtQ[a, 0] && GtQ[c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, (2*c*d - e*(b - q))/(2*c)*Int[1/Sqrt[a + b*x^2 + c*x^4], x] + e/(2*c)*Int[(b - q + 2*c*x^2)/Sqrt[a + b*x^2 + c*x^4], x] /; NeQ[2*c*d - e*(b - q), 0]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[-a*c, 2]}, (c*d + e*q)/c*Int[1/Sqrt[a + c*x^4], x] - e/c*Int[(q - c*x^2)/Sqrt[a + c*x^4], x] /; NeQ[c*d + e*q, 0]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/Sqrt[a_ + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && LtQ[a, 0] && GtQ[c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, d*Int[1/Sqrt[a + b*x^2 + c*x^4], x] + e*Int[x^2/Sqrt[a + b*x^2 + c*x^4], x] /; PosQ[(b + q)/a] || PosQ[(b - q)/a]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "d*Int[1/Sqrt[a + c*x^4], x] + e*Int[x^2/Sqrt[a + c*x^4], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/Sqrt[a_ + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && GtQ[-a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, -a*e*Rt[-(b + q)/(2*a), 2]*Sqrt[1 + (b + q)*x^2/(2*a)]* Sqrt[1 + (b - q)*x^2/(2*a)]/(c*Sqrt[a + b*x^2 + c*x^4])* EllipticE[ ArcSin[Rt[-(b + q)/(2*a), 2]*x], (b - q)/(b + q)] /; NegQ[(b + q)/a] && EqQ[2*c*d - e*(b + q), 0] && Not[SimplerSqrtQ[-(b - q)/(2*a), -(b + q)/(2*a)]]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, (2*c*d - e*(b + q))/(2*c)*Int[1/Sqrt[a + b*x^2 + c*x^4], x] + e/(2*c)*Int[(b + q + 2*c*x^2)/Sqrt[a + b*x^2 + c*x^4], x] /; NegQ[(b + q)/a] && NeQ[2*c*d - e*(b + q), 0] && Not[SimplerSqrtQ[-(b - q)/(2*a), -(b + q)/(2*a)]]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, -a*e*Rt[-(b - q)/(2*a), 2]*Sqrt[1 + (b - q)*x^2/(2*a)]* Sqrt[1 + (b + q)*x^2/(2*a)]/(c*Sqrt[a + b*x^2 + c*x^4])* EllipticE[ ArcSin[Rt[-(b - q)/(2*a), 2]*x], (b + q)/(b - q)] /; NegQ[(b - q)/a] && EqQ[2*c*d - e*(b - q), 0]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, (2*c*d - e*(b - q))/(2*c)*Int[1/Sqrt[a + b*x^2 + c*x^4], x] + e/(2*c)*Int[(b - q + 2*c*x^2)/Sqrt[a + b*x^2 + c*x^4], x] /; NegQ[(b - q)/a] && NeQ[2*c*d - e*(b - q), 0]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[c/a, 4]}, -d*x*Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q^2*x^2)) + d*(1 + q^2*x^2)* Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q* Sqrt[a + b*x^2 + c*x^4])* EllipticE[2*ArcTan[q*x], 1/2 - b*q^2/(4*c)] /; EqQ[e + d*q^2, 0]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[c/a, 4]}, -d*x*Sqrt[a + c*x^4]/(a*(1 + q^2*x^2)) + d*(1 + q^2*x^2)* Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4])* EllipticE[2*ArcTan[q*x], 1/2] /; EqQ[e + d*q^2, 0]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/Sqrt[a_ + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && PosQ[c/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[c/a, 2]}, (e + d*q)/q*Int[1/Sqrt[a + b*x^2 + c*x^4], x] - e/q*Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x] /; NeQ[e + d*q, 0]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[c/a, 2]}, (e + d*q)/q*Int[1/Sqrt[a + c*x^4], x] - e/q*Int[(1 - q*x^2)/Sqrt[a + c*x^4], x] /; NeQ[e + d*q, 0]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/Sqrt[a_ + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && PosQ[c/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "d/Sqrt[a]*Int[Sqrt[1 + e*x^2/d]/Sqrt[1 - e*x^2/d], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/Sqrt[a_ + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NegQ[c/a] && EqQ[c*d^2 + a*e^2, 0] && GtQ[a, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Sqrt[1 + c*x^4/a]/Sqrt[a + c*x^4]* Int[(d + e*x^2)/Sqrt[1 + c*x^4/a], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/Sqrt[a_ + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NegQ[c/a] && EqQ[c*d^2 + a*e^2, 0] && Not[GtQ[a, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[-c/a, 2]}, (d*q - e)/q*Int[1/Sqrt[a + c*x^4], x] + e/q*Int[(1 + q*x^2)/Sqrt[a + c*x^4], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/Sqrt[a_ + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NegQ[c/a] && NeQ[c*d^2 + a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, Sqrt[1 + 2*c*x^2/(b - q)]* Sqrt[1 + 2*c*x^2/(b + q)]/Sqrt[a + b*x^2 + c*x^4]* Int[(d + e*x^2)/(Sqrt[1 + 2*c*x^2/(b - q)]* Sqrt[1 + 2*c*x^2/(b + q)]), x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[c/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x^2)*(a + b*x^2 + c*x^4)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x^2)*(a + c*x^4)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)*(a_ + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": " e^2*x*Sqrt[a+b*x^2+c*x^4]/(3*c) + 2*(3*c*d-b*e)/(3*c)*Int[(d+e*x^2)/Sqrt[a+b*x^2+c*x^4],x] - (3*c*d^2-2*b*d*e+a*e^2)/(3*c)*Int[1/Sqrt[a+b*x^2+c*x^4],x]",
     "rulenumber": 0,
            "lhs": "Int[(d_+e_.*x_^2)^2/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": " e^2*x*Sqrt[a+c*x^4]/(3*c) + 2*d*Int[(d+e*x^2)/Sqrt[a+c*x^4],x] - (3*c*d^2+a*e^2)/(3*c)*Int[1/Sqrt[a+c*x^4],x]",
     "rulenumber": 0,
            "lhs": "Int[(d_+e_.*x_^2)^2/Sqrt[a_+c_.*x_^4],x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,c,d,e},x] &&  NeQ[c*d^2+a*e^2,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": " e^2*x*(d+e*x^2)^(q-2)*Sqrt[a+b*x^2+c*x^4]/(c*(2*q-1)) + 2*(q-1)*(3*c*d-b*e)/(c*(2*q-1))*Int[(d+e*x^2)^(q-1)/Sqrt[a+b*x^2+c* x^4],x] - (2*q-3)*(3*c*d^2-2*b*d*e+a*e^2)/(c*(2*q-1))*Int[(d+e*x^2)^(q-2)/ Sqrt[a+b*x^2+c*x^4],x] + 2*d*(q-2)*(c*d^2-b*d*e+a*e^2)/(c*(2*q-1))*Int[(d+e*x^2)^(q-3)/Sqrt[ a+b*x^2+c*x^4],x]",
     "rulenumber": 0,
            "lhs": "Int[(d_+e_.*x_^2)^q_/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e},x] && NeQ[b^2-4*a*c,0] &&  IGtQ[q,2] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": " e^2*x*(d+e*x^2)^(q-2)*Sqrt[a+c*x^4]/(c*(2*q-1)) + 6*d*(q-1)/(2*q-1)*Int[(d+e*x^2)^(q-1)/Sqrt[a+c*x^4],x] - (2*q-3)*(3*c*d^2+a*e^2)/(c*(2*q-1))*Int[(d+e*x^2)^(q-2)/Sqrt[a+c*x^ 4],x] + 2*d*(q-2)*(c*d^2+a*e^2)/(c*(2*q-1))*Int[(d+e*x^2)^(q-3)/Sqrt[a+c*x^ 4],x]",
     "rulenumber": 0,
            "lhs": "Int[(d_+e_.*x_^2)^q_/Sqrt[a_+c_.*x_^4],x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,c,d,e},x] && IGtQ[q,2] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{f = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[ PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 2]}, x*(a + b*x^2 + c*x^4)^(p + 1)*(a*b*g - f*(b^2 - 2*a*c) - c*(b*f - 2*a*g)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c)) + 1/(2*a*(p + 1)*(b^2 - 4*a*c))*Int[(a + b*x^2 + c*x^4)^(p + 1)* ExpandToSum[ 2*a*(p + 1)*(b^2 - 4*a*c)* PolynomialQuotient[(d + e*x^2)^q, a + b*x^2 + c*x^4, x] + b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)*x^2, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 1] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "e^q*x^(2*q - 3)*(a + b*x^2 + c*x^4)^(p + 1)/(c*(4*p + 2*q + 1)) + 1/(c*(4*p + 2*q + 1))*Int[(a + b*x^2 + c*x^4)^p* ExpandToSum[ c*(4*p + 2*q + 1)*(d + e*x^2)^q - a*(2*q - 3)*e^q*x^(2*q - 4) - b*(2*p + 2*q - 1)*e^q*x^(2*q - 2) - c*(4*p + 2*q + 1)*e^q*x^(2*q), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "e^q*x^(2*q - 3)*(a + c*x^4)^(p + 1)/(c*(4*p + 2*q + 1)) + 1/(c*(4*p + 2*q + 1))*Int[(a + c*x^4)^p* ExpandToSum[ c*(4*p + 2*q + 1)*(d + e*x^2)^q - a*(2*q - 3)*e^q*x^(2*q - 4) - c*(4*p + 2*q + 1)*e^q*x^(2*q), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_ + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, p}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[q, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "-1/e^2* Int[(c*d - b*e - c*e*x^2)*(a + b*x^2 + c*x^4)^(p - 1), x] + (c*d^2 - b*d*e + a*e^2)/e^2* Int[(a + b*x^2 + c*x^4)^(p - 1)/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^2 + c_.*x_^4)^p_/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p + 1/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "-1/e^2*Int[(c*d - c*e*x^2)*(a + c*x^4)^(p - 1), x] + (c*d^2 + a*e^2)/e^2*Int[(a + c*x^4)^(p - 1)/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + c_.*x_^4)^p_/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p + 1/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "1/(2*d)*Int[1/Sqrt[a + b*x^2 + c*x^4], x] + 1/(2*d)*Int[(d - e*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((d_ + e_.*x_^2)*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "1/(2*d)*Int[1/Sqrt[a + c*x^4], x] + 1/(2*d)*Int[(d - e*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((d_ + e_.*x_^2)*Sqrt[a_ + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, 2*Sqrt[-c]* Int[1/((d + e*x^2)*Sqrt[b + q + 2*c*x^2]*Sqrt[-b + q - 2*c*x^2]), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/((d_ + e_.*x_^2)*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[-a*c, 2]}, Sqrt[-c]* Int[1/((d + e*x^2)*Sqrt[q + c*x^2]*Sqrt[q - c*x^2]), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/((d_ + e_.*x_^2)*Sqrt[a_ + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && GtQ[a, 0] && LtQ[c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, 2*c/(2*c*d - e*(b - q))*Int[1/Sqrt[a + b*x^2 + c*x^4], x] - e/(2*c*d - e*(b - q))* Int[(b - q + 2*c*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/((d_ + e_.*x_^2)*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0] && Not[LtQ[c, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[-a*c, 2]}, c/(c*d + e*q)*Int[1/Sqrt[a + c*x^4], x] + e/(c*d + e*q)*Int[(q - c*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/((d_ + e_.*x_^2)*Sqrt[a_ + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && GtQ[-a*c, 0] && Not[LtQ[c, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[c/a, 2]}, (c*d + a*e*q)/(c*d^2 - a*e^2)*Int[1/Sqrt[a + b*x^2 + c*x^4], x] - (a*e*(e + d*q))/(c*d^2 - a*e^2)* Int[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/((d_ + e_.*x_^2)*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[c/a, 2]}, (c*d + a*e*q)/(c*d^2 - a*e^2)*Int[1/Sqrt[a + c*x^4], x] - (a*e*(e + d*q))/(c*d^2 - a*e^2)* Int[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/((d_ + e_.*x_^2)*Sqrt[a_ + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[-c/a, 4]}, 1/(d*Sqrt[a]*q)*EllipticPi[-e/(d*q^2), ArcSin[q*x], -1]]",
     "rulenumber": 0,
            "lhs": "Int[1/((d_ + e_.*x_^2)*Sqrt[a_ + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Sqrt[1 + c*x^4/a]/Sqrt[a + c*x^4]* Int[1/((d + e*x^2)*Sqrt[1 + c*x^4/a]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((d_ + e_.*x_^2)*Sqrt[a_ + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NegQ[c/a] && Not[GtQ[a, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, Sqrt[1 + 2*c*x^2/(b - q)]* Sqrt[1 + 2*c*x^2/(b + q)]/Sqrt[a + b*x^2 + c*x^4]* Int[ 1/((d + e*x^2)*Sqrt[1 + 2*c*x^2/(b - q)]* Sqrt[1 + 2*c*x^2/(b + q)]), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/((d_ + e_.*x_^2)*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[c/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "1/(c*d^2 - b*d*e + a*e^2)* Int[(c*d - b*e - c*e*x^2)*(a + b*x^2 + c*x^4)^p, x] + e^2/(c*d^2 - b*d*e + a*e^2)* Int[(a + b*x^2 + c*x^4)^(p + 1)/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^2 + c_.*x_^4)^p_/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[p + 1/2, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "1/(c*d^2 + a*e^2)*Int[(c*d - c*e*x^2)*(a + c*x^4)^p, x] + e^2/(c*d^2 + a*e^2)*Int[(a + c*x^4)^(p + 1)/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + c_.*x_^4)^p_/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && ILtQ[p + 1/2, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "-e^2*x*(d + e*x^2)^(q + 1)* Sqrt[a + b*x^2 + c*x^4]/(2*d*(q + 1)*(c*d^2 - b*d*e + a*e^2)) + 1/(2*d*(q + 1)*(c*d^2 - b*d*e + a*e^2))* Int[(d + e*x^2)^(q + 1)/Sqrt[a + b*x^2 + c*x^4]* Simp[a*e^2*(2*q + 3) + 2*d*(c*d - b*e)*(q + 1) - 2*e*(c*d*(q + 1) - b*e*(q + 2))*x^2 + c*e^2*(2*q + 5)*x^4, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && ILtQ[q, -1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "-e^2*x*(d + e*x^2)^(q + 1)* Sqrt[a + c*x^4]/(2*d*(q + 1)*(c*d^2 + a*e^2)) + 1/(2*d*(q + 1)*(c*d^2 + a*e^2))* Int[(d + e*x^2)^(q + 1)/Sqrt[a + c*x^4]* Simp[a*e^2*(2*q + 3) + 2*c*d^2*(q + 1) - 2*e*c*d*(q + 1)*x^2 + c*e^2*(2*q + 5)*x^4, x], x]",
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            "lhs": "Int[(d_ + e_.*x_^2)^q_/Sqrt[a_ + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && ILtQ[q, -1]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[e/d, 2]}, c*(d + e*x^2)* Sqrt[(e^2*(a + b*x^2 + c*x^4))/(c*(d + e*x^2)^2)]/(2*d*e^2*q* Sqrt[a + b*x^2 + c*x^4])* EllipticE[2*ArcTan[q*x], (2*c*d - b*e)/(4*c*d)]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*x_^2 + c_.*x_^4]/(d_ + e_.*x_^2)^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0] && PosQ[e/d]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "x*Sqrt[a + b*x^2 + c*x^4]/(2*d*(d + e*x^2)) + c/(2*d*e^2)*Int[(d - e*x^2)/Sqrt[a + b*x^2 + c*x^4], x] - (c*d^2 - a*e^2)/(2*d*e^2)* Int[1/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*x_^2 + c_.*x_^4]/(d_ + e_.*x_^2)^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "x*Sqrt[a + c*x^4]/(2*d*(d + e*x^2)) + c/(2*d*e^2)*Int[(d - e*x^2)/Sqrt[a + c*x^4], x] - (c*d^2 - a*e^2)/(2*d*e^2)* Int[1/((d + e*x^2)*Sqrt[a + c*x^4]), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + c_.*x_^4]/(d_ + e_.*x_^2)^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Module[{aa, bb, cc}, Int[ReplaceAll[ ExpandIntegrand[ 1/Sqrt[aa + bb*x^2 + cc*x^4], (d + e*x^2)^ q*(aa + bb*x^2 + cc*x^4)^(p + 1/2), x], {aa -> a, bb -> b, cc -> c}], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[q, 0] && IntegerQ[p + 1/2]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Module[{aa, cc}, Int[ReplaceAll[ ExpandIntegrand[ 1/Sqrt[aa + cc*x^4], (d + e*x^2)^q*(aa + cc*x^4)^(p + 1/2), x], {aa -> a, cc -> c}], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_ + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && ILtQ[q, 0] && IntegerQ[p + 1/2]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "1/(2*Sqrt[a]*Sqrt[d]*Rt[-e/d, 2])* EllipticF[2*ArcSin[Rt[-e/d, 2]*x], b*d/(4*a*e)]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[d_ + e_.*x_^2]*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c*d - b*e, 0] && GtQ[a, 0] && GtQ[d, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Sqrt[(d + e*x^2)/d]* Sqrt[(a + b*x^2 + c*x^4)/a]/(Sqrt[d + e*x^2]* Sqrt[a + b*x^2 + c*x^4])* Int[1/(Sqrt[1 + e/d*x^2]*Sqrt[1 + b/a*x^2 + c/a*x^4]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[d_ + e_.*x_^2]*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c*d - b*e, 0] && Not[GtQ[a, 0] && GtQ[d, 0]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "x^3*Sqrt[e + d/x^2]* Sqrt[c + b/x^2 + a/x^4]/(Sqrt[d + e*x^2]*Sqrt[a + b*x^2 + c*x^4])* Int[1/(x^3*Sqrt[e + d/x^2]*Sqrt[c + b/x^2 + a/x^4]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[d_ + e_.*x_^2]*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "x^3*Sqrt[e + d/x^2]*Sqrt[c + a/x^4]/(Sqrt[d + e*x^2]*Sqrt[a + c*x^4])* Int[1/(x^3*Sqrt[e + d/x^2]*Sqrt[c + a/x^4]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[d_ + e_.*x_^2]*Sqrt[a_ + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Sqrt[a]/(2*Sqrt[d]*Rt[-e/d, 2])* EllipticE[2*ArcSin[Rt[-e/d, 2]*x], b*d/(4*a*e)]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*x_^2 + c_.*x_^4]/Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c*d - b*e, 0] && GtQ[a, 0] && GtQ[d, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Sqrt[a + b*x^2 + c*x^4]* Sqrt[(d + e*x^2)/d]/(Sqrt[d + e*x^2]*Sqrt[(a + b*x^2 + c*x^4)/a])* Int[Sqrt[1 + b/a*x^2 + c/a*x^4]/Sqrt[1 + e/d*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*x_^2 + c_.*x_^4]/Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c*d - b*e, 0] && Not[GtQ[a, 0] && GtQ[d, 0]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Sqrt[e + d/x^2]* Sqrt[a + b*x^2 + c*x^4]/(x*Sqrt[d + e*x^2]*Sqrt[c + b/x^2 + a/x^4])* Int[(x*Sqrt[c + b/x^2 + a/x^4])/Sqrt[e + d/x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*x_^2 + c_.*x_^4]/Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Sqrt[e + d/x^2]*Sqrt[a + c*x^4]/(x*Sqrt[d + e*x^2]*Sqrt[c + a/x^4])* Int[(x*Sqrt[c + a/x^4])/Sqrt[e + d/x^2], x]",
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            "lhs": "Int[Sqrt[a_ + c_.*x_^4]/Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && (IntegerQ[p] && IntegerQ[q] || IGtQ[p, 0] || IGtQ[q, 0])"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x^2)^q*(a + c*x^4)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_ + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, p, q}, x] && (IntegerQ[p] && IntegerQ[q] || IGtQ[p, 0])"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[ExpandIntegrand[(a + c*x^4)^ p, (d/(d^2 - e^2*x^4) - e*x^2/(d^2 - e^2*x^4))^(-q), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_ + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, p}, x] && NeQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]] && ILtQ[q, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Unintegrable[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p, q}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.3 (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Unintegrable[(d + e*x^2)^q*(a + c*x^4)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_.*(a_ + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, p, q}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "e^q/f^(2*q)*Int[(f*x)^(m+2*q)*(a+b*x^2+c*x^4)^p,x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(e_.*x_^2)^q_.*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_ Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,e,f,m,p},x] && IntegerQ[q] *)"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": " e^q/f^(2*q)*Int[(f*x)^(m+2*q)*(a+c*x^4)^p,x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(e_.*x_^2)^q_.*(a_+c_.*x_^4)^p_.,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,c,e,f,m,p},x] && IntegerQ[q] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "1/(2*e^((m - 1)/2))* Subst[Int[(e*x)^(q + (m - 1)/2)*(a + b*x + c*x^2)^p, x], x, x^2]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(e_.*x_^2)^q_*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, e, p, q}, x] && Not[IntegerQ[q]] && IntegerQ[(m - 1)/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "1/(2*e^((m - 1)/2))* Subst[Int[(e*x)^(q + (m - 1)/2)*(a + c*x^2)^p, x], x, x^2]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(e_.*x_^2)^q_*(a_ + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, e, p, q}, x] && Not[IntegerQ[q]] && IntegerQ[(m - 1)/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "e^IntPart[q]*(e*x^2)^ FracPart[q]/(f^(2*IntPart[q])*(f*x)^(2*FracPart[q]))* Int[(f*x)^(m + 2*q)*(a + b*x^2 + c*x^4)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(e_.*x_^2)^q_*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, e, f, m, p, q}, x] && Not[IntegerQ[q]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "e^IntPart[q]*(e*x^2)^ FracPart[q]/(f^(2*IntPart[q])*(f*x)^(2*FracPart[q]))* Int[(f*x)^(m + 2*q)*(a + c*x^4)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(e_.*x_^2)^q_*(a_ + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, e, f, m, p, q}, x] && Not[IntegerQ[q]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "1/2*Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d_ + e_.*x_^2)^q_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p, q}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "1/2*Subst[Int[(d + e*x)^q*(a + c*x^2)^p, x], x, x^2]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d_ + e_.*x_^2)^q_.*(a_ + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, p, q}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "1/c^p*Int[(f*x)^m*(d+e*x^2)^q*(b/2+c*x^2)^(2*p),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^q_.*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_ Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e,f,m,p,q},x] && EqQ[b^2-4*a*c,0] && IntegerQ[p] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "1/2*Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^2)^q_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p, q}, x] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]] && IGtQ[(m + 1)/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "(a + b*x^2 + c*x^4)^ FracPart[p]/(c^IntPart[p]*(b/2 + c*x^2)^(2*FracPart[p]))* Int[(f*x)^m*(d + e*x^2)^q*(b/2 + c*x^2)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p, q}, x] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "1/2*Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^2)^q_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p, q}, x] && IntegerQ[(m - 1)/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "1/2*Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^2]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^2)^q_.*(a_ + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, p, q}, x] && IntegerQ[(m + 1)/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[(f*x)^m*(d + e*x^2)^(q + p)*(a/d + c/e*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[(f*x)^m*(d + e*x^2)^(q + p)*(a/d + c/e*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_ + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, q, m, q}, x] && EqQ[c*d^2 + a*e^2, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "(a + b*x^2 + c*x^4)^ FracPart[ p]/((d + e*x^2)^FracPart[p]*(a/d + (c*x^2)/e)^FracPart[p])* Int[(f*x)^m*(d + e*x^2)^(q + p)*(a/d + c/e*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "(a + c*x^4)^ FracPart[ p]/((d + e*x^2)^FracPart[p]*(a/d + (c*x^2)/e)^FracPart[p])* Int[(f*x)^m*(d + e*x^2)^(q + p)*(a/d + c/e*x^2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_*(a_ + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m, p, q}, x] && EqQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "(-d)^(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p* x*(d + e*x^2)^(q + 1)/(2*e^(2*p + m/2)*(q + 1)) + 1/(2*e^(2*p + m/2)*(q + 1))*Int[(d + e*x^2)^(q + 1)* ExpandToSum[ Together[ 1/(d + e*x^2)*(2*e^(2*p + m/2)*(q + 1)* x^m*(a + b*x^2 + c*x^4)^p - (-d)^(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^ p*(d + e*(2*q + 3)*x^2))], x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && IGtQ[m/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "(-d)^(m/2 - 1)*(c*d^2 + a*e^2)^p* x*(d + e*x^2)^(q + 1)/(2*e^(2*p + m/2)*(q + 1)) + 1/(2*e^(2*p + m/2)*(q + 1))*Int[(d + e*x^2)^(q + 1)* ExpandToSum[ Together[ 1/(d + e*x^2)*(2*e^(2*p + m/2)*(q + 1)*x^m*(a + c*x^4)^p - (-d)^(m/2 - 1)*(c*d^2 + a*e^2)^ p*(d + e*(2*q + 3)*x^2))], x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_ + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && IGtQ[p, 0] && ILtQ[q, -1] && IGtQ[m/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "(-d)^(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p* x*(d + e*x^2)^(q + 1)/(2*e^(2*p + m/2)*(q + 1)) + (-d)^(m/2 - 1)/(2*e^(2*p)*(q + 1))*Int[x^m*(d + e*x^2)^(q + 1)* ExpandToSum[ Together[ 1/(d + e*x^2)*(2*(-d)^(-m/2 + 1)* e^(2*p)*(q + 1)*(a + b*x^2 + c*x^4)^p - (e^(-m/2)*(c*d^2 - b*d*e + a*e^2)^p*x^(-m))*(d + e*(2*q + 3)*x^2))], x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*(d_ + e_.*x_^2)^q_*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "(-d)^(m/2 - 1)*(c*d^2 + a*e^2)^p* x*(d + e*x^2)^(q + 1)/(2*e^(2*p + m/2)*(q + 1)) + (-d)^(m/2 - 1)/(2*e^(2*p)*(q + 1))*Int[x^m*(d + e*x^2)^(q + 1)* ExpandToSum[ Together[ 1/(d + e*x^2)*(2*(-d)^(-m/2 + 1)* e^(2*p)*(q + 1)*(a + c*x^4)^p - (e^(-m/2)*(c*d^2 + a*e^2)^p*x^(-m))*(d + e*(2*q + 3)*x^2))], x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*(d_ + e_.*x_^2)^q_*(a_ + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[ExpandIntegrand[(f*x)^m (d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[ExpandIntegrand[(f*x)^m (d + e*x^2)^q*(a + c*x^4)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_ + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m, q}, x] && IGtQ[p, 0] && IGtQ[q, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[ PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x, 0]}, -R*(f*x)^(m + 1)*(d + e*x^2)^(q + 1)/(2*d*f*(q + 1)) + f/(2*d*(q + 1))* Int[(f*x)^(m - 1)*(d + e*x^2)^(q + 1)* ExpandToSum[2*d*(q + 1)*x*Qx + R*(m + 2*q + 3)*x, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[q, -1] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{Qx = PolynomialQuotient[(a + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + c*x^4)^p, d + e*x^2, x], x, 0]}, -R*(f*x)^(m + 1)*(d + e*x^2)^(q + 1)/(2*d*f*(q + 1)) + f/(2*d*(q + 1))* Int[(f*x)^(m - 1)*(d + e*x^2)^(q + 1)* ExpandToSum[2*d*(q + 1)*x*Qx + R*(m + 2*q + 3)*x, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_*(a_ + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f}, x] && IGtQ[p, 0] && LtQ[q, -1] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, f*x, x], R = PolynomialRemainder[(a + b*x^2 + c*x^4)^p, f*x, x]}, R*(f*x)^(m + 1)*(d + e*x^2)^(q + 1)/(d*f*(m + 1)) + 1/(d*f^2*(m + 1))* Int[(f*x)^(m + 2)*(d + e*x^2)^q* ExpandToSum[d*f*(m + 1)*Qx/x - e*R*(m + 2*q + 3), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{Qx = PolynomialQuotient[(a + c*x^4)^p, f*x, x], R = PolynomialRemainder[(a + c*x^4)^p, f*x, x]}, R*(f*x)^(m + 1)*(d + e*x^2)^(q + 1)/(d*f*(m + 1)) + 1/(d*f^2*(m + 1))* Int[(f*x)^(m + 2)*(d + e*x^2)^q* ExpandToSum[d*f*(m + 1)*Qx/x - e*R*(m + 2*q + 3), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_.*(a_ + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, q}, x] && IGtQ[p, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "c^p*(f*x)^(m + 4*p - 1)*(d + e*x^2)^(q + 1)/(e* f^(4*p - 1)*(m + 4*p + 2*q + 1)) + 1/(e*(m + 4*p + 2*q + 1))*Int[(f*x)^m*(d + e*x^2)^q* ExpandToSum[ e*(m + 4*p + 2*q + 1)*((a + b*x^2 + c*x^4)^p - c^p*x^(4*p)) - d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && Not[IntegerQ[q]] && NeQ[m + 4*p + 2*q + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "c^p*(f*x)^(m + 4*p - 1)*(d + e*x^2)^(q + 1)/(e* f^(4*p - 1)*(m + 4*p + 2*q + 1)) + 1/(e*(m + 4*p + 2*q + 1))*Int[(f*x)^m*(d + e*x^2)^q* ExpandToSum[ e*(m + 4*p + 2*q + 1)*((a + c*x^4)^p - c^p*x^(4*p)) - d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_ + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m, q}, x] && IGtQ[p, 0] && Not[IntegerQ[q]] && NeQ[m + 4*p + 2*q + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{k = Denominator[m]}, k/f*Subst[ Int[x^(k*(m + 1) - 1)*(d + e*x^(2*k)/f^2)^ q*(a + b*x^(2*k)/f^k + c*x^(4*k)/f^4)^p, x], x, (f*x)^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && FractionQ[m] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{k = Denominator[m]}, k/f*Subst[ Int[x^(k*(m + 1) - 1)*(d + e*x^(2*k)/f)^q*(a + c*x^(4*k)/f)^p, x], x, (f*x)^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_.*(a_ + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, p, q}, x] && FractionQ[m] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "(f*x)^(m + 1)*(a + b*x^2 + c*x^4)^ p*(d*(m + 4*p + 3) + e*(m + 1)*x^2)/(f*(m + 1)*(m + 4*p + 3)) + 2*p/(f^2*(m + 1)*(m + 4*p + 3))* Int[(f*x)^(m + 2)*(a + b*x^2 + c*x^4)^(p - 1)* Simp[2*a*e*(m + 1) - b*d*(m + 4*p + 3) + (b*e*(m + 1) - 2*c*d*(m + 4*p + 3))*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && LtQ[m, -1] && m + 4*p + 3 != 0 && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "(f*x)^(m + 1)*(a + c*x^4)^ p*(d*(m + 4*p + 3) + e*(m + 1)*x^2)/(f*(m + 1)*(m + 4*p + 3)) + 4*p/(f^2*(m + 1)*(m + 4*p + 3))* Int[(f*x)^(m + 2)*(a + c*x^4)^(p - 1)*(a*e*(m + 1) - c*d*(m + 4*p + 3)*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)*(a_ + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1] && m + 4*p + 3 != 0 && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "(f*x)^(m + 1)*(a + b*x^2 + c*x^4)^ p*(b*e*2*p + c*d*(m + 4*p + 3) + c*e*(4*p + m + 1)*x^2)/ (c*f*(4*p + m + 1)*(m + 4*p + 3)) + 2*p/(c*(4*p + m + 1)*(m + 4*p + 3))* Int[(f*x)^m*(a + b*x^2 + c*x^4)^(p - 1)* Simp[2*a*c*d*(m + 4*p + 3) - a*b*e*(m + 1) + (2*a*c*e*(4*p + m + 1) + b*c*d*(m + 4*p + 3) - b^2*e*(m + 2*p + 1))*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && NeQ[4*p + m + 1, 0] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "(f*x)^(m + 1)*(a + c*x^4)^ p*(c*d*(m + 4*p + 3) + c*e*(4*p + m + 1)*x^2)/(c* f*(4*p + m + 1)*(m + 4*p + 3)) + 4*a*p/((4*p + m + 1)*(m + 4*p + 3))* Int[(f*x)^m*(a + c*x^4)^(p - 1)* Simp[d*(m + 4*p + 3) + e*(4*p + m + 1)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)*(a_ + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m}, x] && GtQ[p, 0] && NeQ[4*p + m + 1, 0] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "f*(f*x)^(m - 1)*(a + b*x^2 + c*x^4)^(p + 1)*(b*d - 2*a*e - (b*e - 2*c*d)*x^2)/(2*(p + 1)*(b^2 - 4*a*c)) - f^2/(2*(p + 1)*(b^2 - 4*a*c))* Int[(f*x)^(m - 2)*(a + b*x^2 + c*x^4)^(p + 1)* Simp[(m - 1)*(b*d - 2*a*e) - (4*p + 4 + m + 1)*(b*e - 2*c*d)* x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "f*(f*x)^(m - 1)*(a + c*x^4)^(p + 1)*(a*e - c*d*x^2)/(4*a*c*(p + 1)) - f^2/(4*a*c*(p + 1))* Int[(f*x)^(m - 2)*(a + c*x^4)^(p + 1)*(a*e*(m - 1) - c*d*(4*p + 4 + m + 1)*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)*(a_ + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f}, x] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "-(f*x)^(m + 1)*(a + b*x^2 + c*x^4)^(p + 1)*(d*(b^2 - 2*a*c) - a*b*e + (b*d - 2*a*e)*c*x^2)/(2*a* f*(p + 1)*(b^2 - 4*a*c)) + 1/(2*a*(p + 1)*(b^2 - 4*a*c))* Int[(f*x)^m*(a + b*x^2 + c*x^4)^(p + 1)* Simp[d*(b^2*(m + 2*(p + 1) + 1) - 2*a*c*(m + 4*(p + 1) + 1)) - a*b*e*(m + 1) + c*(m + 2*(2*p + 3) + 1)*(b*d - 2*a*e)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "-(f*x)^(m + 1)*(a + c*x^4)^(p + 1)*(d + e*x^2)/(4*a* f*(p + 1)) + 1/(4*a*(p + 1))* Int[(f*x)^m*(a + c*x^4)^(p + 1)* Simp[d*(m + 4*(p + 1) + 1) + e*(m + 2*(2*p + 3) + 1)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)*(a_ + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m}, x] && LtQ[p, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "e*f*(f*x)^(m - 1)*(a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 3)) - f^2/(c*(m + 4*p + 3))* Int[(f*x)^(m - 2)*(a + b*x^2 + c*x^4)^p* Simp[a*e*(m - 1) + (b*e*(m + 2*p + 1) - c*d*(m + 4*p + 3))*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "e*f*(f*x)^(m - 1)*(a + c*x^4)^(p + 1)/(c*(m + 4*p + 3)) - f^2/(c*(m + 4*p + 3))* Int[(f*x)^(m - 2)*(a + c*x^4)^ p*(a*e*(m - 1) - c*d*(m + 4*p + 3)*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)*(a_ + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, p}, x] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "d*(f*x)^(m + 1)*(a + b*x^2 + c*x^4)^(p + 1)/(a*f*(m + 1)) + 1/(a*f^2*(m + 1))* Int[(f*x)^(m + 2)*(a + b*x^2 + c*x^4)^p* Simp[a*e*(m + 1) - b*d*(m + 2*p + 3) - c*d*(m + 4*p + 5)*x^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "d*(f*x)^(m + 1)*(a + c*x^4)^(p + 1)/(a*f*(m + 1)) + 1/(a*f^2*(m + 1))* Int[(f*x)^(m + 2)*(a + c*x^4)^ p*(a*e*(m + 1) - c*d*(m + 4*p + 5)*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)*(a_ + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, p}, x] && LtQ[m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{r = Rt[c/e*(2*c*d - b*e), 2]}, e/2*Int[(f*x)^m/(c*d/e - r*x + c*x^2), x] + e/2*Int[(f*x)^m/(c*d/e + r*x + c*x^2), x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)/(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && GtQ[d/e, 0] && PosQ[c/e*(2*c*d - b*e)]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{r = Rt[2*c^2*d/e, 2]}, e/2*Int[(f*x)^m/(c*d/e - r*x + c*x^2), x] + e/2*Int[(f*x)^m/(c*d/e + r*x + c*x^2), x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)/(a_ + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m}, x] && EqQ[c*d^2 - a*e^2, 0] && GtQ[d/e, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, (e/2 + (2*c*d - b*e)/(2*q))* Int[(f*x)^m/(b/2 - q/2 + c*x^2), x] + (e/2 - (2*c*d - b*e)/(2*q))* Int[(f*x)^m/(b/2 + q/2 + c*x^2), x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)/(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[-a*c, 2]}, -(e/2 + c*d/(2*q))* Int[(f*x)^m/(q - c*x^2), x] + (e/2 - c*d/(2*q))* Int[(f*x)^m/(q + c*x^2), x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)/(a_ + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q/(a + b*x^2 + c*x^4), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_./(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[q] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q/(a + c*x^4), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_./(a_ + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m}, x] && IntegerQ[q] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[ExpandIntegrand[(f*x)^m, (d + e*x^2)^q/(a + b*x^2 + c*x^4), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_./(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[q] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[ExpandIntegrand[(f*x)^m, (d + e*x^2)^q/(a + c*x^4), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_./(a_ + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m}, x] && IntegerQ[q] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "f^4/c^2*Int[(f*x)^(m - 4)*(c*d - b*e + c*e*x^2)*(d + e*x^2)^(q - 1), x] - f^4/c^2* Int[(f*x)^(m - 4)*(d + e*x^2)^(q - 1)* Simp[a*(c*d - b*e) + (b*c*d - b^2*e + a*c*e)*x^2, x]/(a + b*x^2 + c*x^4), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_^2)^q_/(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && Not[IntegerQ[q]] && GtQ[q, 0] && GtQ[m, 3]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "f^4/c*Int[(f*x)^(m - 4)*(d + e*x^2)^q, x] - a*f^4/c*Int[(f*x)^(m - 4)*(d + e*x^2)^q/(a + c*x^4), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_^2)^q_/(a_ + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, q}, x] && Not[IntegerQ[q]] && GtQ[m, 3]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "e*f^2/c*Int[(f*x)^(m - 2)*(d + e*x^2)^(q - 1), x] - f^2/c* Int[(f*x)^(m - 2)*(d + e*x^2)^(q - 1)* Simp[a*e - (c*d - b*e)*x^2, x]/(a + b*x^2 + c*x^4), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_^2)^q_/(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && Not[IntegerQ[q]] && GtQ[q, 0] && GtQ[m, 1] && LeQ[m, 3]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "e*f^2/c*Int[(f*x)^(m - 2)*(d + e*x^2)^(q - 1), x] - f^2/c* Int[(f*x)^(m - 2)*(d + e*x^2)^(q - 1)* Simp[a*e - c*d*x^2, x]/(a + c*x^4), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_^2)^q_/(a_ + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f}, x] && Not[IntegerQ[q]] && GtQ[q, 0] && GtQ[m, 1] && LeQ[m, 3]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "d/a*Int[(f*x)^m*(d + e*x^2)^(q - 1), x] - 1/(a*f^2)* Int[(f*x)^(m + 2)*(d + e*x^2)^(q - 1)* Simp[b*d - a*e + c*d*x^2, x]/(a + b*x^2 + c*x^4), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_. + e_.*x_^2)^q_/(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && Not[IntegerQ[q]] && GtQ[q, 0] && LtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "d/a*Int[(f*x)^m*(d + e*x^2)^(q - 1), x] + 1/(a*f^2)* Int[(f*x)^(m + 2)*(d + e*x^2)^(q - 1)* Simp[a*e - c*d*x^2, x]/(a + c*x^4), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_. + e_.*x_^2)^q_/(a_ + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f}, x] && Not[IntegerQ[q]] && GtQ[q, 0] && LtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "d^2*f^4/(c*d^2 - b*d*e + a*e^2)*Int[(f*x)^(m - 4)*(d + e*x^2)^q, x] - f^4/(c*d^2 - b*d*e + a*e^2)* Int[(f*x)^(m - 4)*(d + e*x^2)^(q + 1)* Simp[a*d + (b*d - a*e)*x^2, x]/(a + b*x^2 + c*x^4), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_^2)^q_/(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && Not[IntegerQ[q]] && LtQ[q, -1] && GtQ[m, 3]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "d^2*f^4/(c*d^2 + a*e^2)*Int[(f*x)^(m - 4)*(d + e*x^2)^q, x] - a*f^4/(c*d^2 + a*e^2)* Int[(f*x)^(m - 4)*(d + e*x^2)^(q + 1)*(d - e*x^2)/(a + c*x^4), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_^2)^q_/(a_ + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f}, x] && Not[IntegerQ[q]] && LtQ[q, -1] && GtQ[m, 3]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "-d*e*f^2/(c*d^2 - b*d*e + a*e^2)* Int[(f*x)^(m - 2)*(d + e*x^2)^q, x] + f^2/(c*d^2 - b*d*e + a*e^2)* Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)* Simp[a*e + c*d*x^2, x]/(a + b*x^2 + c*x^4), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_^2)^q_/(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && Not[IntegerQ[q]] && LtQ[q, -1] && GtQ[m, 1] && LeQ[m, 3]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "-d*e*f^2/(c*d^2 + a*e^2)* Int[(f*x)^(m - 2)*(d + e*x^2)^q, x] + f^2/(c*d^2 + a*e^2)* Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)* Simp[a*e + c*d*x^2, x]/(a + c*x^4), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_^2)^q_/(a_ + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f}, x] && Not[IntegerQ[q]] && LtQ[q, -1] && GtQ[m, 1] && LeQ[m, 3]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "e^2/(c*d^2 - b*d*e + a*e^2)*Int[(f*x)^m*(d + e*x^2)^q, x] + 1/(c*d^2 - b*d*e + a*e^2)* Int[(f*x)^m*(d + e*x^2)^(q + 1)* Simp[c*d - b*e - c*e*x^2, x]/(a + b*x^2 + c*x^4), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_/(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4*a*c, 0] && Not[IntegerQ[q]] && LtQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "e^2/(c*d^2 + a*e^2)*Int[(f*x)^m*(d + e*x^2)^q, x] + c/(c*d^2 + a*e^2)* Int[(f*x)^m*(d + e*x^2)^(q + 1)*(d - e*x^2)/(a + c*x^4), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_/(a_ + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m}, x] && Not[IntegerQ[q]] && LtQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x^2)^q, (f*x)^m/(a + b*x^2 + c*x^4), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_/(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[b^2 - 4*a*c, 0] && Not[IntegerQ[q]] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x^2)^q, (f*x)^m/(a + c*x^4), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_/(a_ + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, q}, x] && Not[IntegerQ[q]] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q, 1/(a + b*x^2 + c*x^4), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_/(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] && Not[IntegerQ[q]] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q, 1/(a + c*x^4), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_/(a_ + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m, q}, x] && Not[IntegerQ[q]] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{r = Rt[b^2 - 4*a*c, 2]}, 2*c/r*Int[(f*x)^m*(d + e*x^2)^q/(b - r + 2*c*x^2), x] - 2*c/r*Int[(f*x)^m*(d + e*x^2)^q/(b + r + 2*c*x^2), x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_/(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{r = Rt[-a*c, 2]}, -c/(2*r)*Int[(f*x)^m*(d + e*x^2)^q/(r - c*x^2), x] - c/(2*r)*Int[(f*x)^m*(d + e*x^2)^q/(r + c*x^2), x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_/(a_ + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m, q}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "1/d^2*Int[(f*x)^ m*(a*d + (b*d - a*e)*x^2)*(a + b*x^2 + c*x^4)^(p - 1), x] + (c*d^2 - b*d*e + a*e^2)/(d^2*f^4)* Int[(f*x)^(m + 4)*(a + b*x^2 + c*x^4)^(p - 1)/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*x_^2 + c_.*x_^4)^p_./(d_. + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && LtQ[m, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "a/d^2*Int[(f*x)^m*(d - e*x^2)*(a + c*x^4)^(p - 1), x] + (c*d^2 + a*e^2)/(d^2*f^4)* Int[(f*x)^(m + 4)*(a + c*x^4)^(p - 1)/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_ + c_.*x_^4)^p_./(d_. + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "1/(d*e)*Int[(f*x)^m*(a*e + c*d*x^2)*(a + b*x^2 + c*x^4)^(p - 1), x] - (c*d^2 - b*d*e + a*e^2)/(d*e*f^2)* Int[(f*x)^(m + 2)*(a + b*x^2 + c*x^4)^(p - 1)/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*x_^2 + c_.*x_^4)^p_./(d_. + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && LtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "1/(d*e)*Int[(f*x)^m*(a*e + c*d*x^2)*(a + c*x^4)^(p - 1), x] - (c*d^2 + a*e^2)/(d*e*f^2)* Int[(f*x)^(m + 2)*(a + c*x^4)^(p - 1)/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_ + c_.*x_^4)^p_./(d_. + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "-f^4/(c*d^2 - b*d*e + a*e^2)* Int[(f*x)^(m - 4)*(a*d + (b*d - a*e)*x^2)*(a + b*x^2 + c*x^4)^p, x] + d^2*f^4/(c*d^2 - b*d*e + a*e^2)* Int[(f*x)^(m - 4)*(a + b*x^2 + c*x^4)^(p + 1)/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(a_. + b_.*x_^2 + c_.*x_^4)^p_/(d_. + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[m, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "-a*f^4/(c*d^2 + a*e^2)* Int[(f*x)^(m - 4)*(d - e*x^2)*(a + c*x^4)^p, x] + d^2*f^4/(c*d^2 + a*e^2)* Int[(f*x)^(m - 4)*(a + c*x^4)^(p + 1)/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(a_ + c_.*x_^4)^p_/(d_. + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f}, x] && LtQ[p, -1] && GtQ[m, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "f^2/(c*d^2 - b*d*e + a*e^2)* Int[(f*x)^(m - 2)*(a*e + c*d*x^2)*(a + b*x^2 + c*x^4)^p, x] - d*e*f^2/(c*d^2 - b*d*e + a*e^2)* Int[(f*x)^(m - 2)*(a + b*x^2 + c*x^4)^(p + 1)/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(a_. + b_.*x_^2 + c_.*x_^4)^p_/(d_. + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "f^2/(c*d^2 + a*e^2)* Int[(f*x)^(m - 2)*(a*e + c*d*x^2)*(a + c*x^4)^p, x] - d*e*f^2/(c*d^2 + a*e^2)* Int[(f*x)^(m - 2)*(a + c*x^4)^(p + 1)/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(a_ + c_.*x_^4)^p_/(d_. + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f}, x] && LtQ[p, -1] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "d/(2*d*e)*Int[1/Sqrt[a + b*x^2 + c*x^4], x] - d/(2*d*e)* Int[(d - e*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^2/((d_ + e_.*x_^2)*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && PosQ[c/a] && EqQ[c*d^2 - a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "d/(2*d*e)*Int[1/Sqrt[a + c*x^4], x] - d/(2*d*e)*Int[(d - e*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^2/((d_ + e_.*x_^2)*Sqrt[a_ + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && PosQ[c/a] && EqQ[c*d^2 - a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[c/a, 2]}, -a*(e + d*q)/(c*d^2 - a*e^2)* Int[1/Sqrt[a + b*x^2 + c*x^4], x] + a*d*(e + d*q)/(c*d^2 - a*e^2)* Int[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^2/((d_ + e_.*x_^2)*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && PosQ[c/a] && NeQ[c*d^2 - a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[c/a, 2]}, -a*(e + d*q)/(c*d^2 - a*e^2)*Int[1/Sqrt[a + c*x^4], x] + a*d*(e + d*q)/(c*d^2 - a*e^2)* Int[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^2/((d_ + e_.*x_^2)*Sqrt[a_ + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && PosQ[c/a] && NeQ[c*d^2 - a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "-1/e^2* Int[(d - e*x^2)/Sqrt[a + b*x^2 + c*x^4], x] + d^2/e^2*Int[1/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^4/((d_ + e_.*x_^2)*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a] && EqQ[c*d^2 - a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "-1/e^2*Int[(d - e*x^2)/Sqrt[a + c*x^4], x] + d^2/e^2*Int[1/((d + e*x^2)*Sqrt[a + c*x^4]), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^4/((d_ + e_.*x_^2)*Sqrt[a_ + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && PosQ[c/a] && EqQ[c*d^2 - a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[c/a, 2]}, -1/(e*q)*Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x] + d^2/(e*(e - d*q))* Int[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x] /; EqQ[2*c*d - a*e*q, 0]]",
     "rulenumber": 0,
            "lhs": "Int[x_^4/((d_ + e_.*x_^2)*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a] && NeQ[c*d^2 - a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[c/a, 2]}, -1/(e*q)*Int[(1 - q*x^2)/Sqrt[a + c*x^4], x] + d^2/(e*(e - d*q))* Int[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x] /; EqQ[2*c*d - a*e*q, 0]]",
     "rulenumber": 0,
            "lhs": "Int[x_^4/((d_ + e_.*x_^2)*Sqrt[a_ + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && PosQ[c/a] && NeQ[c*d^2 - a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[c/a, 2]}, -(2*c*d - a*e*q)/(c*e*(e - d*q))* Int[1/Sqrt[a + b*x^2 + c*x^4], x] - 1/(e*q)*Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x] + d^2/(e*(e - d*q))* Int[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^4/((d_ + e_.*x_^2)*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a] && NeQ[c*d^2 - a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[c/a, 2]}, -(2*c*d - a*e*q)/(c*e*(e - d*q))*Int[1/Sqrt[a + c*x^4], x] - 1/(e*q)*Int[(1 - q*x^2)/Sqrt[a + c*x^4], x] + d^2/(e*(e - d*q))* Int[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^4/((d_ + e_.*x_^2)*Sqrt[a_ + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && PosQ[c/a] && NeQ[c*d^2 - a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "x^(m - 5)*Sqrt[a + b*x^2 + c*x^4]/(c*e*(m - 3)) - 1/(c*e*(m - 3))* Int[x^(m - 6)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4])* Simp[a*d*(m - 5) + (a*e*(m - 5) + b*d*(m - 4))* x^2 + (b*e*(m - 4) + c*d*(m - 3))*x^4, x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_/((d_ + e_.*x_^2)*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m/2, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "x^(m - 5)*Sqrt[a + c*x^4]/(c*e*(m - 3)) - 1/(c*e*(m - 3))* Int[x^(m - 6)/((d + e*x^2)*Sqrt[a + c*x^4])* Simp[a*d*(m - 5) + a*e*(m - 5)*x^2 + c*d*(m - 3)*x^4, x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_/((d_ + e_.*x_^2)*Sqrt[a_ + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && IGtQ[m/2, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "x^(m + 1)*Sqrt[a + b*x^2 + c*x^4]/(a*d*(m + 1)) - 1/(a*d*(m + 1))* Int[x^(m + 2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4])* Simp[a*e*(m + 1) + b*d*(m + 2) + (b*e*(m + 2) + c*d*(m + 3))*x^2 + c*e*(m + 3)*x^4, x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_/((d_ + e_.*x_^2)*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && ILtQ[m/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "x^(m + 1)*Sqrt[a + c*x^4]/(a*d*(m + 1)) - 1/(a*d*(m + 1))* Int[x^(m + 2)/((d + e*x^2)*Sqrt[a + c*x^4])* Simp[a*e*(m + 1) + c*d*(m + 3)*x^2 + c*e*(m + 3)*x^4, x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_/((d_ + e_.*x_^2)*Sqrt[a_ + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && ILtQ[m/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "x^3*Sqrt[e + d/x^2]* Sqrt[c + b/x^2 + a/x^4]/(Sqrt[d + e*x^2]*Sqrt[a + b*x^2 + c*x^4])* Int[x^(m - 3)/(Sqrt[e + d/x^2]*Sqrt[c + b/x^2 + a/x^4]), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_/(Sqrt[d_ + e_.*x_^2]*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[m/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "x^3*Sqrt[e + d/x^2]*Sqrt[c + a/x^4]/(Sqrt[d + e*x^2]*Sqrt[a + c*x^4])* Int[x^(m - 3)/(Sqrt[e + d/x^2]*Sqrt[c + a/x^4]), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_/(Sqrt[d_ + e_.*x_^2]*Sqrt[a_ + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && IntegerQ[m/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{f = Coeff[PolynomialRemainder[x^m*(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[ PolynomialRemainder[x^m*(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 2]}, x*(a + b*x^2 + c*x^4)^(p + 1)*(a*b*g - f*(b^2 - 2*a*c) - c*(b*f - 2*a*g)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c)) + 1/(2*a*(p + 1)*(b^2 - 4*a*c))*Int[(a + b*x^2 + c*x^4)^(p + 1)* Simp[ExpandToSum[ 2*a*(p + 1)*(b^2 - 4*a*c)* PolynomialQuotient[x^m*(d + e*x^2)^q, a + b*x^2 + c*x^4, x] + b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)*x^2, x], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*(d_ + e_.*x_^2)^q_*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IGtQ[q, 1] && IGtQ[m/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{f = Coeff[PolynomialRemainder[x^m*(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[ PolynomialRemainder[x^m*(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 2]}, x*(a + b*x^2 + c*x^4)^(p + 1)*(a*b*g - f*(b^2 - 2*a*c) - c*(b*f - 2*a*g)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c)) + 1/(2*a*(p + 1)*(b^2 - 4*a*c))* Int[x^m*(a + b*x^2 + c*x^4)^(p + 1)* Simp[ExpandToSum[ 2*a*(p + 1)*(b^2 - 4*a*c)*x^(-m)* PolynomialQuotient[x^m*(d + e*x^2)^q, a + b*x^2 + c*x^4, x] + (b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5) - a*b*g)*x^(-m) + c*(4*p + 7)*(b*f - 2*a*g)*x^(2 - m), x], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*(d_ + e_.*x_^2)^q_*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IGtQ[q, 1] && ILtQ[m/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && (IGtQ[p, 0] || IGtQ[q, 0] || IntegersQ[m, q])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + c*x^4)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_ + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m, p, q}, x] && (IGtQ[p, 0] || IGtQ[q, 0] || IntegersQ[m, q])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "(f*x)^m/x^m* Int[ExpandIntegrand[ x^m*(a + c*x^4)^ p, (d/(d^2 - e^2*x^4) - e*x^2/(d^2 - e^2*x^4))^(-q), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_*(a_ + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m, p}, x] && Not[IntegerQ[p]] && ILtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Unintegrable[(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p, q}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Unintegrable[(f*x)^m*(d + e*x^2)^q*(a + c*x^4)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_ + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m, p, q}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.5 P(x) (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.5 P(x) (a+b x^2+c x^4)^p.m",
            "rhs": "Int[ExpandIntegrand[Pq*(a + b*x^2 + c*x^4)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.5 P(x) (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.5 P(x) (a+b x^2+c x^4)^p.m",
            "rhs": "Int[x*PolynomialQuotient[Pq, x, x]*(a + b*x^2 + c*x^4)^p, x] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && EqQ[Coeff[Pq, x, 0], 0] && Not[MatchQ[Pq, x^m_.*u_.",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "IntegerQ[m]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.5 P(x) (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.5 P(x) (a+b x^2+c x^4)^p.m",
            "rhs": "Module[{q = Expon[Pq, x], k}, Int[Sum[ Coeff[Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2 + c*x^4)^p, x] + Int[ x*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0, (q - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Not[PolyQ[Pq, x^2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.5 P(x) (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.5 P(x) (a+b x^2+c x^4)^p.m",
            "rhs": "With[{d = Coeff[Pq, x, 0], e = Coeff[Pq, x, 2], f = Coeff[Pq, x, 4]}, d*x*(a + b*x^2 + c*x^4)^(p + 1)/a /; EqQ[a*e - b*d*(2*p + 3), 0] && EqQ[a*f - c*d*(4*p + 5), 0]]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && EqQ[Expon[Pq, x], 4]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.5 P(x) (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.5 P(x) (a+b x^2+c x^4)^p.m",
            "rhs": "With[{d = Coeff[Pq, x, 0], e = Coeff[Pq, x, 2], f = Coeff[Pq, x, 4], g = Coeff[Pq, x, 6]}, x*(3*a*d + (a*e - b*d*(2*p + 3))* x^2)*(a + b*x^2 + c*x^4)^(p + 1)/(3*a^2) /; EqQ[3*a^2*g - c*(4*p + 7)*(a*e - b*d*(2*p + 3)), 0] && EqQ[3*a^2*f - 3*a*c*d*(4*p + 5) - b*(2*p + 5)*(a*e - b*d*(2*p + 3)), 0]]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && EqQ[Expon[Pq, x], 6]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.5 P(x) (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.5 P(x) (a+b x^2+c x^4)^p.m",
            "rhs": "Int[ExpandIntegrand[Pq/(a + b*x^2 + c*x^4), x], x]",
     "rulenumber": 0,
            "lhs": "Int[Pq_/(a_ + b_.*x_^2 + c_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.5 P(x) (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.5 P(x) (a+b x^2+c x^4)^p.m",
            "rhs": "(a + b*x^2 + c*x^4)^ FracPart[p]/((4*c)^IntPart[p]*(b + 2*c*x^2)^(2*FracPart[p]))* Int[Pq*(b + 2*c*x^2)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1 && EqQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.5 P(x) (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.5 P(x) (a+b x^2+c x^4)^p.m",
            "rhs": "With[{d = Coeff[PolynomialRemainder[Pq, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[Pq, a + b*x^2 + c*x^4, x], x, 2]}, x*(a + b*x^2 + c*x^4)^(p + 1)*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c)) + 1/(2*a*(p + 1)*(b^2 - 4*a*c))*Int[(a + b*x^2 + c*x^4)^(p + 1)* ExpandToSum[ 2*a*(p + 1)*(b^2 - 4*a*c)* PolynomialQuotient[Pq, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4*p + 7)*(b*d - 2*a*e)*x^2, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.5 P(x) (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.5 P(x) (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Expon[Pq, x^2], e = Coeff[Pq, x^2, Expon[Pq, x^2]]}, e*x^(2*q - 3)*(a + b*x^2 + c*x^4)^(p + 1)/(c*(2*q + 4*p + 1)) + 1/(c*(2*q + 4*p + 1))*Int[(a + b*x^2 + c*x^4)^p* ExpandToSum[ c*(2*q + 4*p + 1)*Pq - a*e*(2*q - 3)*x^(2*q - 4) - b*e*(2*q + 2*p - 1)*x^(2*q - 2) - c*e*(2*q + 4*p + 1)*x^(2*q), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && Not[LtQ[p, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.5 P(x) (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.5 P(x) (a+b x^2+c x^4)^p.m",
            "rhs": "With[{a = Coeff[Q4, x, 0], b = Coeff[Q4, x, 1], c = Coeff[Q4, x, 2], d = Coeff[Q4, x, 3], e = Coeff[Q4, x, 4]}, Subst[ Int[SimplifyIntegrand[ ReplaceAll[Pq, x -> -d/(4*e) + x]*(a + d^4/(256*e^3) - b*d/(8*e) + (c - 3*d^2/(8*e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2, 0] && NeQ[d, 0]]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*Q4_^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[p, x] && PolyQ[Pq, x] && PolyQ[Q4, x, 4] && Not[IGtQ[p, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.6 P(x) (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.6 P(x) (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "Module[{q = Expon[Pq, x], k}, Int[(d*x)^m* Sum[Coeff[Pq, x, 2*k]*x^(2*k), {k, 0, q/2 + 1}]*(a + b*x^2 + c*x^4)^p, x] + 1/d* Int[(d*x)^(m + 1)* Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0, (q - 1)/2 + 1}]*(a + b*x^2 + c*x^4)^p, x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*Pq_*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, p}, x] && PolyQ[Pq, x] && Not[PolyQ[Pq, x^2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.6 P(x) (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.6 P(x) (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "1/2*Subst[ Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x + c*x^2)^p, x], x, x^2]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Pq_*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.6 P(x) (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.6 P(x) (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "Int[ExpandIntegrand[(d*x)^m*Pq*(a + b*x^2 + c*x^4)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*Pq_*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && PolyQ[Pq, x^2] && IGtQ[p, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.6 P(x) (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.6 P(x) (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "1/d^2*Int[(d*x)^(m + 2)*ExpandToSum[Pq/x^2, x]*(a + b*x^2 + c*x^4)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*Pq_*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, p}, x] && PolyQ[Pq, x^2] && EqQ[Coeff[Pq, x, 0], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.6 P(x) (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.6 P(x) (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "With[{e = Coeff[Pq, x, 0], f = Coeff[Pq, x, 2], g = Coeff[Pq, x, 4]}, e*(d*x)^(m + 1)*(a + b*x^2 + c*x^4)^(p + 1)/(a* d*(m + 1)) /; EqQ[a*f*(m + 1) - b*e*(m + 2*p + 3), 0] && EqQ[a*g*(m + 1) - c*e*(m + 4*p + 5), 0] && NeQ[m, -1]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*Pq_*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, p}, x] && PolyQ[Pq, x^2] && EqQ[Expon[Pq, x], 4]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.6 P(x) (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.6 P(x) (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "(a + b*x^2 + c*x^4)^ FracPart[p]/((4*c)^IntPart[p]*(b + 2*c*x^2)^(2*FracPart[p]))* Int[(d*x)^m*Pq*(b + 2*c*x^2)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*Pq_*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, p}, x] && PolyQ[Pq, x^2] && GtQ[Expon[Pq, x^2], 1] && EqQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.6 P(x) (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.6 P(x) (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "With[{d = Coeff[PolynomialRemainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 2]}, x*(a + b*x^2 + c*x^4)^(p + 1)*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c)) + 1/(2*a*(p + 1)*(b^2 - 4*a*c))*Int[(a + b*x^2 + c*x^4)^(p + 1)* ExpandToSum[ 2*a*(p + 1)*(b^2 - 4*a*c)* PolynomialQuotient[x^m*Pq, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4*p + 7)*(b*d - 2*a*e)*x^2, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*Pq_*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && GtQ[Expon[Pq, x^2], 1] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IGtQ[m/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.6 P(x) (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.6 P(x) (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "With[{d = Coeff[PolynomialRemainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 2]}, x*(a + b*x^2 + c*x^4)^(p + 1)*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c)) + 1/(2*a*(p + 1)*(b^2 - 4*a*c))* Int[x^m*(a + b*x^2 + c*x^4)^(p + 1)* ExpandToSum[ 2*a*(p + 1)*(b^2 - 4*a*c)*x^(-m)* PolynomialQuotient[x^m*Pq, a + b*x^2 + c*x^4, x] + (b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e)*x^(-m) + c*(4*p + 7)*(b*d - 2*a*e)*x^(2 - m), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*Pq_*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && GtQ[Expon[Pq, x^2], 1] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && ILtQ[m/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.6 P(x) (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.6 P(x) (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": " With[{d=Coeff[PolynomialRemainder[x^m*Pq,a+b*x^2+c*x^4,x],x,1], e=Coeff[PolynomialRemainder[x^m*Pq,a+b*x^2+c*x^4,x],x,3]}, x^2*(a+b*x^2+c*x^4)^(p+1)*(a*b*e-d*(b^2-2*a*c)-c*(b*d-2*a*e)*x^2)/( 2*a*(p+1)*(b^2-4*a*c)) + 1/(a*(p+1)*(b^2-4*a*c))*Int[x^m*(a+b*x^2+c*x^4)^(p+1)* ExpandToSum[a*(p+1)*(b^2-4*a*c)*x^(-m)*PolynomialQuotient[x^m*Pq, a+b*x^2+c*x^4,x]+ (b^2*d*(p+2)-2*a*c*d*(2*p+3)-a*b*e)*x^(1-m)+2*c*(p+2)*(b*d-2*a* e)*x^(3-m),x],x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Pq_*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c},x] && PolyQ[Pq,x^2] &&  GtQ[Expon[Pq,x^2],1] && NeQ[b^2-4*a*c,0] && LtQ[p,-1] &&  IntegerQ[(m-1)/2] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.6 P(x) (d x)^m (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.6 P(x) (d x)^m (a+b x^2+c x^4)^p.m",
            "rhs": "Unintegrable[(d*x)^m*Pq*(a + b*x^2 + c*x^4)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*Pq_*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, p}, x] && PolyQ[Pq, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 2], C = Coeff[Px, x, 4]}, C*x^(m - 1)*Sqrt[a + b*x^2 + c*x^4]/(c*e*(m + 1)) - 1/(c*e*(m + 1))* Int[(x^(m - 2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]))* Simp[a*C*d*(m - 1) - (A*c*e*(m + 1) - C*(a*e*(m - 1) + b*d*m))* x^2 - (B*c*e*(m + 1) - C*(b*e*m + c*d*(m + 1)))*x^4, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*x_^m_/((d_ + e_.*x_^2)*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x^2, 2] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 2], C = Coeff[Px, x, 4]}, C*x^(m - 1)*Sqrt[a + c*x^4]/(c*e*(m + 1)) - 1/(c*e*(m + 1))*Int[(x^(m - 2)/((d + e*x^2)*Sqrt[a + c*x^4]))* Simp[a*C*d*(m - 1) - (A*c*e*(m + 1) - C*a*e*(m - 1))* x^2 - (B*c*e*(m + 1) - C*c*d*(m + 1))*x^4, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*x_^m_/((d_ + e_.*x_^2)*Sqrt[a_ + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && PolyQ[Px, x^2, 2] && IGtQ[m/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 2], C = Coeff[Px, x, 4]}, A*x^(m + 1)*Sqrt[a + b*x^2 + c*x^4]/(a*d*(m + 1)) + 1/(a*d*(m + 1))* Int[(x^(m + 2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]))* Simp[a*B*d*(m + 1) - A*(a*e*(m + 1) + b*d*(m + 2)) + (a*C*d*(m + 1) - A*(b*e*(m + 2) + c*d*(m + 3)))*x^2 - A*c*e*(m + 3)*x^4, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*x_^m_/((d_ + e_.*x_^2)*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x^2, 2] && NeQ[b^2 - 4*a*c, 0] && ILtQ[m/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 2], C = Coeff[Px, x, 4]}, A*x^(m + 1)*Sqrt[a + c*x^4]/(a*d*(m + 1)) + 1/(a*d*(m + 1))*Int[(x^(m + 2)/((d + e*x^2)*Sqrt[a + c*x^4]))* Simp[a*B*d*(m + 1) - A*a*e*(m + 1) + (a*C*d*(m + 1) - A*c*d*(m + 3))*x^2 - A*c*e*(m + 3)*x^4, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*x_^m_/((d_ + e_.*x_^2)*Sqrt[a_ + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && PolyQ[Px, x^2, 2] && ILtQ[m/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "1/2*Subst[ Int[ReplaceAll[Px, x -> Sqrt[x]]*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2]",
     "rulenumber": 0,
            "lhs": "Int[x_*Px_*(d_ + e_.*x_^2)^q_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p, q}, x] && PolyQ[Px, x^2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[x*PolynomialQuotient[Pr, x, x]*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^ p, x] /; FreeQ[{a, b, c, d, e, p, q}, x] && PolyQ[Pr, x] && EqQ[PolynomialRemainder[Pr, x, x], 0] && Not[MatchQ[Pr, x^m_.*u_.",
     "rulenumber": 0,
            "lhs": "Int[Pr_*(d_ + e_.*x_^2)^q_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "IntegerQ[m]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Module[{r = Expon[Pr, x], k}, Int[Sum[Coeff[Pr, x, 2*k]*x^(2*k), {k, 0, r/2}]*(d + e*x^2)^ q*(a + b*x^2 + c*x^4)^p, x] + Int[ x*Sum[Coeff[Pr, x, 2*k + 1]*x^(2*k), {k, 0, (r - 1)/2}]*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x]]",
     "rulenumber": 0,
            "lhs": "Int[Pr_*(d_ + e_.*x_^2)^q_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p, q}, x] && PolyQ[Pr, x] && Not[PolyQ[Pr, x^2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[Px*(d + e*x^2)^(p + q)*(a/d + c/e*x^2)^p, x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && (PolyQ[Px, x^2] || MatchQ[Px, (f_ + g_.*x^2)^r_.",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_ + e_.*x_^2)^q_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{f, g, r}, x]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[Px*(d + e*x^2)^(p + q)*(a/d + c/e*x^2)^p, x] /; FreeQ[{a, c, d, e, q}, x] && EqQ[c*d^2 + a*e^2, 0] && IntegerQ[p] && (PolyQ[Px, x^2] || MatchQ[Px, (f_ + g_.*x^2)^r_.",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_ + e_.*x_^2)^q_.*(a_ + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{f, g, r}, x]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "(a + b*x^2 + c*x^4)^ FracPart[p]/((d + e*x^2)^FracPart[p]*(a/d + c*x^2/e)^FracPart[p])* Int[Px*(d + e*x^2)^(p + q)*(a/d + c/e*x^2)^p, x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[p]] && (PolyQ[Px, x^2] || MatchQ[Px, (f_ + g_.*x^2)^r_.",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_ + e_.*x_^2)^q_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{f, g, r}, x]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "(a + c*x^4)^ FracPart[p]/((d + e*x^2)^FracPart[p]*(a/d + c*x^2/e)^FracPart[p])* Int[Px*(d + e*x^2)^(p + q)*(a/d + c/e*x^2)^p, x] /; FreeQ[{a, c, d, e, p, q}, x] && EqQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]] && (PolyQ[Px, x^2] || MatchQ[Px, (f_ + g_.*x^2)^r_.",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_ + e_.*x_^2)^q_.*(a_ + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{f, g, r}, x]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[ExpandIntegrand[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_ + e_.*x_^2)^q_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, q}, x] && PolyQ[Px, x^2] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[ExpandIntegrand[Px*(d + e*x^2)^q*(a + c*x^4)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_ + e_.*x_^2)^q_.*(a_ + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, q}, x] && PolyQ[Px, x^2] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, C*x*(d + e*x^2)^q*Sqrt[a + b*x^2 + c*x^4]/(c*(2*q + 3)) + 1/(c*(2*q + 3))* Int[(d + e*x^2)^(q - 1)/Sqrt[a + b*x^2 + c*x^4]* Simp[A*c*d*(2*q + 3) - a*C*d + (c*(B*d + A*e)*(2*q + 3) - C*(2*b*d + a*e + 2*a*e*q))* x^2 + (B*c*e*(2*q + 3) - 2*C*(b*e - c*d*q + b*e*q))*x^4, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*P4x_/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && PolyQ[P4x, x^2] && EqQ[Expon[P4x, x], 4] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, C*x*(d + e*x^2)^q*Sqrt[a + c*x^4]/(c*(2*q + 3)) + 1/(c*(2*q + 3))*Int[(d + e*x^2)^(q - 1)/Sqrt[a + c*x^4]* Simp[A*c*d*(2*q + 3) - a*C*d + (c*(B*d + A*e)*(2*q + 3) - a*C*e*(2*q + 1))* x^2 + (B*c*e*(2*q + 3) + 2*c*C*d*q)*x^4, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*P4x_/Sqrt[a_ + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && PolyQ[P4x, x^2] && EqQ[Expon[P4x, x], 4] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, -(C*d^2 - B*d*e + A*e^2)*x*(d + e*x^2)^(q + 1)* Sqrt[a + b*x^2 + c*x^4]/(2* d*(q + 1)*(c*d^2 - b*d*e + a*e^2)) + 1/(2*d*(q + 1)*(c*d^2 - b*d*e + a*e^2))* Int[(d + e*x^2)^(q + 1)/Sqrt[a + b*x^2 + c*x^4]* Simp[a*d*(C*d - B*e) + A*(a*e^2*(2*q + 3) + 2*d*(c*d - b*e)*(q + 1)) - 2*((B*d - A*e)*(b*e*(q + 2) - c*d*(q + 1)) - C*d*(b*d + a*e*(q + 1)))*x^2 + c*(C*d^2 - B*d*e + A*e^2)*(2*q + 5)*x^4, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*P4x_/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && PolyQ[P4x, x^2] && LeQ[Expon[P4x, x], 4] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, -(C*d^2 - B*d*e + A*e^2)*x*(d + e*x^2)^(q + 1)* Sqrt[a + c*x^4]/(2*d*(q + 1)*(c*d^2 + a*e^2)) + 1/(2*d*(q + 1)*(c*d^2 + a*e^2))* Int[(d + e*x^2)^(q + 1)/Sqrt[a + c*x^4]* Simp[a*d*(C*d - B*e) + A*(a*e^2*(2*q + 3) + 2*c*d^2*(q + 1)) + 2*d*(B*c*d - A*c*e + a*C*e)*(q + 1)*x^2 + c*(C*d^2 - B*d*e + A*e^2)*(2*q + 5)*x^4, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*P4x_/Sqrt[a_ + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && PolyQ[P4x, x^2] && LeQ[Expon[P4x, x], 4] && NeQ[c*d^2 + a*e^2, 0] && ILtQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "A*Subst[Int[1/(d - (b*d - 2*a*e)*x^2), x], x, x/Sqrt[a + b*x^2 + c*x^4]]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*x_^2)/((d_ + e_.*x_^2)*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "A*Subst[Int[1/(d + 2*a*e*x^2), x], x, x/Sqrt[a + c*x^4]]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*x_^2)/((d_ + e_.*x_^2)*Sqrt[a_ + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "(B*d + A*e)/(2*d*e)* Int[1/Sqrt[a + b*x^2 + c*x^4], x] - (B*d - A*e)/(2*d*e)* Int[(d - e*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*x_^2)/((d_ + e_.*x_^2)*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0] && NeQ[B*d + A*e, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "(B*d + A*e)/(2*d*e)*Int[1/Sqrt[a + c*x^4], x] - (B*d - A*e)/(2*d*e)* Int[(d - e*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*x_^2)/((d_ + e_.*x_^2)*Sqrt[a_ + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0] && NeQ[B*d + A*e, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Sqrt[A + B*x^2]*Sqrt[a/A + c*x^2/B]/Sqrt[a + b*x^2 + c*x^4]* Int[Sqrt[A + B*x^2]/((d + e*x^2)*Sqrt[a/A + c*x^2/B]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*x_^2)/((d_ + e_.*x_^2)*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[c*A^2 - b*A*B + a*B^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Sqrt[A + B*x^2]*Sqrt[a/A + c*x^2/B]/Sqrt[a + c*x^4]* Int[Sqrt[A + B*x^2]/((d + e*x^2)*Sqrt[a/A + c*x^2/B]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*x_^2)/((d_ + e_.*x_^2)*Sqrt[a_ + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[c*A^2 + a*B^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Sqrt[b^2 - 4*a*c]}, (2*a*B - A*(b + q))/(2*a*e - d*(b + q))* Int[1/Sqrt[a + b*x^2 + c*x^4], x] - (B*d - A*e)/(2*a*e - d*(b + q))* Int[(2*a + (b + q)*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x] /; RationalQ[q]]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*x_^2)/((d_ + e_.*x_^2)*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && GtQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*A^2 - b*A*B + a*B^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Sqrt[-a*c]}, (a*B - A*q)/(a*e - d*q)*Int[1/Sqrt[a + c*x^4], x] - (B*d - A*e)/(a*e - d*q)* Int[(a + q*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x] /; RationalQ[q]]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*x_^2)/((d_ + e_.*x_^2)*Sqrt[a_ + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, A, B}, x] && GtQ[-a*c, 0] && EqQ[c*d^2 + a*e^2, 0] && NeQ[c*A^2 + a*B^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q=Rt[B/A,2]}, -(B*d-A*e)*ArcTan[Rt[-b+c*d/e+a*e/d,2]*x/Sqrt[a+b*x^2+c*x^4]]/(2*d* e*Rt[-b+c*d/e+a*e/d,2]) + B*q*(c*d^2-a*e^2)*(A+B*x^2)*Sqrt[A^2*(a+b*x^2+c*x^4)/(a*(A+B*x^2)^2) ]/(4*c*d*e*(B*d-A*e)*Sqrt[a+b*x^2+c*x^4])* EllipticPi[-(B*d-A*e)^2/(4*d*e*A*B),2*ArcTan[q*x],1/2-b*A/(4*a*B)] ]",
     "rulenumber": 0,
            "lhs": "Int[(A_+B_.*x_^2)/((d_+e_.*x_^2)*Sqrt[a_+b_.*x_^2+c_.*x_^4]),x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,A,B},x] && NeQ[b^2-4*a*c,0] &&  NeQ[c*d^2-b*d*e+a*e^2,0] && NeQ[c*d^2-a*e^2,0] && PosQ[c/a] &&  EqQ[c*A^2-a*B^2,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": " With[{q=Rt[B/A,2]}, -(B*d-A*e)*ArcTan[Rt[c*d/e+a*e/d,2]*x/Sqrt[a+c*x^4]]/(2*d*e*Rt[c*d/ e+a*e/d,2]) + B*q*(c*d^2-a*e^2)*(A+B*x^2)*Sqrt[A^2*(a+c*x^4)/(a*(A+B*x^2)^2)]/(4* c*d*e*(B*d-A*e)*Sqrt[a+c*x^4])* EllipticPi[-(B*d-A*e)^2/(4*d*e*A*B),2*ArcTan[q*x],1/2]]",
     "rulenumber": 0,
            "lhs": "Int[(A_+B_.*x_^2)/((d_+e_.*x_^2)*Sqrt[a_+c_.*x_^4]),x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,c,d,e,A,B},x] && NeQ[c*d^2+a*e^2,0] && NeQ[c*d^2-a*e^2,0] &&  PosQ[c/a] && EqQ[c*A^2-a*B^2,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[B/A, 2]}, -(B*d - A*e)* ArcTan[Rt[-b + c*d/e + a*e/d, 2]*x/Sqrt[a + b*x^2 + c*x^4]]/(2*d* e*Rt[-b + c*d/e + a*e/d, 2]) + (B*d + A*e)*(A + B*x^2)* Sqrt[A^2*(a + b*x^2 + c*x^4)/(a*(A + B*x^2)^2)]/(4*d*e*A*q* Sqrt[a + b*x^2 + c*x^4])* EllipticPi[Cancel[-(B*d - A*e)^2/(4*d*e*A*B)], 2*ArcTan[q*x], 1/2 - b*A/(4*a*B)]]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*x_^2)/((d_ + e_.*x_^2)*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[B/A, 2]}, -(B*d - A*e)* ArcTan[Rt[c*d/e + a*e/d, 2]*x/Sqrt[a + c*x^4]]/(2*d*e* Rt[c*d/e + a*e/d, 2]) + (B*d + A*e)*(A + B*x^2)* Sqrt[A^2*(a + c*x^4)/(a*(A + B*x^2)^2)]/(4*d*e*A*q* Sqrt[a + c*x^4])* EllipticPi[Cancel[-(B*d - A*e)^2/(4*d*e*A*B)], 2*ArcTan[q*x], 1/2]]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*x_^2)/((d_ + e_.*x_^2)*Sqrt[a_ + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[c/a, 2]}, (A*(c*d + a*e*q) - a*B*(e + d*q))/(c*d^2 - a*e^2)* Int[1/Sqrt[a + b*x^2 + c*x^4], x] + a*(B*d - A*e)*(e + d*q)/(c*d^2 - a*e^2)* Int[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x]]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*x_^2)/((d_ + e_.*x_^2)*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && NeQ[c*A^2 - a*B^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[c/a, 2]}, (A*(c*d + a*e*q) - a*B*(e + d*q))/(c*d^2 - a*e^2)* Int[1/Sqrt[a + c*x^4], x] + a*(B*d - A*e)*(e + d*q)/(c*d^2 - a*e^2)* Int[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x]]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*x_^2)/((d_ + e_.*x_^2)*Sqrt[a_ + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && NeQ[c*A^2 - a*B^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "B/e*Int[1/Sqrt[a + b*x^2 + c*x^4], x] + (e*A - d*B)/e* Int[1/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*x_^2)/((d_ + e_.*x_^2)*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[c/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "B/e*Int[1/Sqrt[a + c*x^4], x] + (e*A - d*B)/e* Int[1/((d + e*x^2)*Sqrt[a + c*x^4]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*x_^2)/((d_ + e_.*x_^2)*Sqrt[a_ + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[c/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, -C/e^2*Int[(d - e*x^2)/Sqrt[a + b*x^2 + c*x^4], x] + 1/e^2* Int[(C*d^2 + A*e^2 + B*e^2*x^2)/((d + e*x^2)* Sqrt[a + b*x^2 + c*x^4]), x]]",
     "rulenumber": 0,
            "lhs": "Int[P4x_/((d_ + e_.*x_^2)*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, -C/e^2*Int[(d - e*x^2)/Sqrt[a + c*x^4], x] + 1/e^2* Int[(C*d^2 + A*e^2 + B*e^2*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x]]",
     "rulenumber": 0,
            "lhs": "Int[P4x_/((d_ + e_.*x_^2)*Sqrt[a_ + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[c*d^2 + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[c/a, 2], A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, -C/(e*q)*Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x] + 1/(c*e)* Int[(A*c*e + a*C*d*q + (B*c*e - C*(c*d - a*e*q))*x^2)/((d + e*x^2)* Sqrt[a + b*x^2 + c*x^4]), x]]",
     "rulenumber": 0,
            "lhs": "Int[P4x_/((d_ + e_.*x_^2)*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && Not[GtQ[b^2 - 4*a*c, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Rt[c/a, 2], A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, -C/(e*q)*Int[(1 - q*x^2)/Sqrt[a + c*x^4], x] + 1/(c*e)* Int[(A*c*e + a*C*d*q + (B*c*e - C*(c*d - a*e*q))*x^2)/((d + e*x^2)* Sqrt[a + c*x^4]), x]]",
     "rulenumber": 0,
            "lhs": "Int[P4x_/((d_ + e_.*x_^2)*Sqrt[a_ + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, -1/e^2*Int[(C*d - B*e - C*e*x^2)/Sqrt[a + b*x^2 + c*x^4], x] + (C*d^2 - B*d*e + A*e^2)/e^2* Int[1/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x]]",
     "rulenumber": 0,
            "lhs": "Int[P4x_/((d_ + e_.*x_^2)*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, -1/e^2*Int[(C*d - B*e - C*e*x^2)/Sqrt[a + c*x^4], x] + (C*d^2 - B*d*e + A*e^2)/e^2* Int[1/((d + e*x^2)*Sqrt[a + c*x^4]), x]]",
     "rulenumber": 0,
            "lhs": "Int[P4x_/((d_ + e_.*x_^2)*Sqrt[a_ + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Expon[Px, x]}, Coeff[Px, x, q]*x^(q - 5)*Sqrt[a + b*x^2 + c*x^4]/(c*e*(q - 3)) + 1/(c*e*(q - 3))* Int[(c*e*(q - 3)*Px - Coeff[Px, x, q]* x^(q - 6)*(d + e*x^2)*(a*(q - 5) + b*(q - 4)*x^2 + c*(q - 3)*x^4))/ ((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x] /; GtQ[q, 4]]",
     "rulenumber": 0,
            "lhs": "Int[Px_/((d_ + e_.*x_^2)*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{q = Expon[Px, x]}, Coeff[Px, x, q]*x^(q - 5)*Sqrt[a + c*x^4]/(c*e*(q - 3)) + 1/(c*e*(q - 3))* Int[(c*e*(q - 3)*Px - Coeff[Px, x, q]* x^(q - 6)*(d + e*x^2)*(a*(q - 5) + c*(q - 3)*x^4))/((d + e*x^2)*Sqrt[a + c*x^4]), x] /; GtQ[q, 4]]",
     "rulenumber": 0,
            "lhs": "Int[Px_/((d_ + e_.*x_^2)*Sqrt[a_ + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && PolyQ[Px, x] && NeQ[c*d^2 + a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": " With[{A=Coeff[PolynomialRemainder[Pq,a+b*x^2+c*x^4,x],x,0], B=Coeff[PolynomialRemainder[Pq,a+b*x^2+c*x^4,x],x,2]}, -B*x*(b^2-2*a*c+b*c*x^2)*(a+b*x^2+c*x^4)^(p+1)/(2*a*e*(p+1)*(b^2-4* a*c)) + (B*d-A*e)*x*(b^2*c*d-2*a*c^2*d-b^3*e+3*a*b*c*e+c*(b*c*d-b^2*e+2*a*c* e)*x^2)*(a+b*x^2+c*x^4)^(p+1)/ (2*a*e*(p+1)*(b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)) + Int[(a+b*x^2+c*x^4)^(p+1)/(d+e*x^2)*ExpandToSum[Pq/(a+b*x^2+c*x^4)-( d+e*x^2)/(a+b*x^2+c*x^4)^(p+1)* D[-B*x*(b^2-2*a*c+b*c*x^2)*(a+b*x^2+c*x^4)^(p+1)/(2*a*e*(p+1)*(b^ 2-4*a*c)) + (B*d-A*e)*x*(b^2*c*d-2*a*c^2*d-b^3*e+3*a*b*c*e+c*(b*c*d-b^2*e+2* a*c*e)*x^2)*(a+b*x^2+c*x^4)^(p+1)/ (2*a*e*(p+1)*(b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)),x],x],x]]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_+b_.*x_^2+c_.*x_^4)^p_/(d_+e_.*x_^2),x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e},x] && PolyQ[Pq,x^2] && Expon[Pq,x^2]>0 &&  NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && LtQ[p,-1] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[ExpandIntegrand[1/Sqrt[a + b*x^2 + c*x^4], Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^(p + 1/2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_ + e_.*x_^2)^q_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x^2] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p + 1/2] && IntegerQ[q]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[ExpandIntegrand[1/Sqrt[a + c*x^4], Px*(d + e*x^2)^q*(a + c*x^4)^(p + 1/2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_ + e_.*x_^2)^q_.*(a_ + c_.*x_^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && PolyQ[Px, x^2] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p + 1/2] && IntegerQ[q]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Unintegrable[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_ + e_.*x_^2)^q_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p, q}, x] && PolyQ[Px, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.7 P(x) (d+e x^2)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Unintegrable[Px*(d + e*x^2)^q*(a + c*x^4)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_ + e_.*x_^2)^q_.*(a_ + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, p, q}, x] && PolyQ[Px, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
            "rhs": "d*Int[1/((d^2 - e^2*x^2)*Sqrt[a + b*x^2 + c*x^4]), x] - e*Int[x/((d^2 - e^2*x^2)*Sqrt[a + b*x^2 + c*x^4]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((d_ + e_.*x_)*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
            "rhs": "d*Int[1/((d^2 - e^2*x^2)*Sqrt[a + c*x^4]), x] - e*Int[x/((d^2 - e^2*x^2)*Sqrt[a + c*x^4]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((d_ + e_.*x_)*Sqrt[a_ + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
            "rhs": "e^3*(d + e*x)^(q + 1)* Sqrt[a + b*x^2 + c*x^4]/((q + 1)*(c*d^4 + b*d^2*e^2 + a*e^4)) + 1/((q + 1)*(c*d^4 + b*d^2*e^2 + a*e^4))* Int[(d + e*x)^(q + 1)/Sqrt[a + b*x^2 + c*x^4]* Simp[d*(q + 1)*(c*d^2 + b*e^2) - e*(c*d^2*(q + 1) + b*e^2*(q + 2))*x + c*d*e^2*(q + 1)*x^2 - c*e^3*(q + 3)*x^3, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^q_/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[c*d^4 + b*d^2*e^2 + a*e^4, 0] && ILtQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
            "rhs": "e^3*(d + e*x)^(q + 1)*Sqrt[a + c*x^4]/((q + 1)*(c*d^4 + a*e^4)) + c/((q + 1)*(c*d^4 + a*e^4))* Int[(d + e*x)^(q + 1)/Sqrt[a + c*x^4]* Simp[d^3*(q + 1) - d^2*e*(q + 1)*x + d*e^2*(q + 1)*x^2 - e^3*(q + 3)*x^3, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^q_/Sqrt[a_ + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^4 + a*e^4, 0] && ILtQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
            "rhs": "d*Int[(a + b*x^2 + c*x^4)^p/(d^2 - e^2*x^2), x] - e*Int[x*(a + b*x^2 + c*x^4)^p/(d^2 - e^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^2 + c_.*x_^4)^p_./(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IntegerQ[p + 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
            "rhs": "d*Int[(a + c*x^4)^p/(d^2 - e^2*x^2), x] - e*Int[x*(a + c*x^4)^p/(d^2 - e^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + c_.*x_^4)^p_./(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && IntegerQ[p + 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[PolynomialQuotient[Px, d + e*x, x]*(d + e*x)^(q + 1)*(a + b*x^2 + c*x^4)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_ + e_.*x_)^q_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p, q}, x] && PolyQ[Px, x] && EqQ[PolynomialRemainder[Px, d + e*x, x], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[PolynomialQuotient[Px, d + e*x, x]*(d + e*x)^(q + 1)*(a + c*x^4)^ p, x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_ + e_.*x_)^q_.*(a_ + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, p, q}, x] && PolyQ[Px, x] && EqQ[PolynomialRemainder[Px, d + e*x, x], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[PolynomialQuotient[Px, a + b*x^2 + c*x^4, x]*(d + e*x)^ q*(a + b*x^2 + c*x^4)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_ + e_.*x_)^q_.*(a_ + b_.*x_^2 + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p, q}, x] && PolyQ[Px, x] && EqQ[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
            "rhs": "Int[PolynomialQuotient[Px, a + c*x^4, x]*(d + e*x)^ q*(a + c*x^4)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_ + e_.*x_)^q_.*(a_ + c_.*x_^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, p, q}, x] && PolyQ[Px, x] && EqQ[PolynomialRemainder[Px, a + c*x^4, x], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2], D = Coeff[Px, x, 3]}, Int[(d + e*x)^(q - 1)*(A*d + (B*d + A*e)*x + (C*d + B*e)*x^2 + C*e*x^3)/Sqrt[a + b*x^2 + c*x^4], x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_ + e_.*x_)^q_/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x] && LeQ[Expon[Px, x], 2] && NeQ[c*d^4 + b*d^2*e^2 + a*e^4, 0] && GtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2], D = Coeff[Px, x, 3]}, Int[(d + e*x)^(q - 1)*(A*d + (B*d + A*e)*x + (C*d + B*e)*x^2 + C*e*x^3)/Sqrt[a + c*x^4], x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_ + e_.*x_)^q_/Sqrt[a_ + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && PolyQ[Px, x] && LeQ[Expon[Px, x], 2] && NeQ[c*d^4 + a*e^4, 0] && GtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2], D = Coeff[Px, x, 3]}, D*(d + e*x)^q*Sqrt[a + b*x^2 + c*x^4]/(c*(q + 2)) - 1/(c*(q + 2))*Int[(d + e*x)^(q - 1)/Sqrt[a + b*x^2 + c*x^4]* Simp[a*D*e*q - A*c*d*(q + 2) + (b*d*D - B*c*d*(q + 2) - A*c*e*(q + 2))*x + (b*D*e*(q + 1) - c*(C*d + B*e)*(q + 2))*x^2 - c*(d*D*q + C*e*(q + 2))*x^3, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_ + e_.*x_)^q_/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x, 3] && NeQ[c*d^4 + b*d^2*e^2 + a*e^4, 0] && GtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2], D = Coeff[Px, x, 3]}, D*(d + e*x)^q*Sqrt[a + c*x^4]/(c*(q + 2)) - 1/(c*(q + 2))*Int[(d + e*x)^(q - 1)/Sqrt[a + c*x^4]* Simp[a*D*e*q - A*c*d*(q + 2) - c*(B*d*(q + 2) + A*e*(q + 2))*x - c*(C*d + B*e)*(q + 2)*x^2 - c*(d*D*q + C*e*(q + 2))*x^3, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_ + e_.*x_)^q_/Sqrt[a_ + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && PolyQ[Px, x, 3] && NeQ[c*d^4 + a*e^4, 0] && GtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2], D = Coeff[Px, x, 3]}, -(d^3*D - C*d^2*e + B*d*e^2 - A*e^3)*(d + e*x)^(q + 1)* Sqrt[a + b*x^2 + c*x^4]/((q + 1)*(c*d^4 + b*d^2*e^2 + a*e^4)) + 1/((q + 1)*(c*d^4 + b*d^2*e^2 + a*e^4))* Int[((d + e*x)^(q + 1)/Sqrt[a + b*x^2 + c*x^4])* Simp[(q + 1)*(a*e*(d^2*D - C*d*e + B*e^2) + A*d*(c*d^2 + b*e^2)) - (e*(q + 1)*(A*c*d^2 + a*e*(d*D - C*e)) - B*d*(c*d^2*(q + 1) + b*e^2*(q + 2)) - b*(d^3*D - C*d^2*e - A*e^3*(q + 2)))*x + (q + 1)*(D*e*(b*d^2 + a*e^2) + c*d*(C*d^2 - e*(B*d - A*e)))*x^2 + c*(q + 3)*(d^3*D - C*d^2*e + B*d*e^2 - A*e^3)*x^3, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_ + e_.*x_)^q_/Sqrt[a_ + b_.*x_^2 + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x] && LeQ[Expon[Px, x], 3] && NeQ[c*d^4 + b*d^2*e^2 + a*e^4, 0] && LtQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2], D = Coeff[Px, x, 3]}, -(d^3*D - C*d^2*e + B*d*e^2 - A*e^3)*(d + e*x)^(q + 1)* Sqrt[a + c*x^4]/((q + 1)*(c*d^4 + a*e^4)) + 1/((q + 1)*(c*d^4 + a*e^4))* Int[((d + e*x)^(q + 1)/Sqrt[a + c*x^4])* Simp[(q + 1)*(a*e*(d^2*D - C*d*e + B*e^2) + A*d*(c*d^2)) - (e*(q + 1)*(A*c*d^2 + a*e*(d*D - C*e)) - B*d*(c*d^2*(q + 1)))*x + (q + 1)*(D*e*(a*e^2) + c*d*(C*d^2 - e*(B*d - A*e)))* x^2 + c*(q + 3)*(d^3*D - C*d^2*e + B*d*e^2 - A*e^3)*x^3, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_ + e_.*x_)^q_/Sqrt[a_ + c_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && PolyQ[Px, x] && LeQ[Expon[Px, x], 3] && NeQ[c*d^4 + a*e^4, 0] && LtQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
            "rhs": "-A^2*(B*d + A*e)/e* Subst[Int[1/(6*A^3*B*d + 3*A^4*e - a*e*x^2), x], x, (A + B*x)^2/Sqrt[a + b*x^2 + c*x^4]]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*x_)/((d_ + e_.*x_)*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[B*d - A*e, 0] && EqQ[c^2*d^6 + a*e^4*(13*c*d^2 + b*e^2), 0] && EqQ[b^2*e^4 - 12*c*d^2*(c*d^2 - b*e^2), 0] && EqQ[4*A*c*d*e + B*(2*c*d^2 - b*e^2), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2], D = Coeff[Px, x, 3]}, Int[(x*(B*d - A*e + (d*D - C*e)*x^2))/((d^2 - e^2*x^2)* Sqrt[a + b*x^2 + c*x^4]), x] + Int[(A*d + (C*d - B*e)*x^2 - D*e*x^4)/((d^2 - e^2*x^2)* Sqrt[a + b*x^2 + c*x^4]), x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_/((d_ + e_.*x_)*Sqrt[a_ + b_.*x_^2 + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x] && LeQ[Expon[Px, x], 3] && NeQ[c*d^4 + b*d^2*e^2 + a*e^4, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
            "rhs": "With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2], D = Coeff[Px, x, 3]}, Int[(x*(B*d - A*e + (d*D - C*e)*x^2))/((d^2 - e^2*x^2)* Sqrt[a + c*x^4]), x] + Int[(A*d + (C*d - B*e)*x^2 - D*e*x^4)/((d^2 - e^2*x^2)* Sqrt[a + c*x^4]), x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_/((d_ + e_.*x_)*Sqrt[a_ + c_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && PolyQ[Px, x] && LeQ[Expon[Px, x], 3] && NeQ[c*d^4 + a*e^4, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
            "rhs": "d*Int[Px*(a + b*x^2 + c*x^4)^p/(d^2 - e^2*x^2), x] - e*Int[x*Px*(a + b*x^2 + c*x^4)^p/(d^2 - e^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_ + b_.*x_^2 + c_.*x_^4)^p_./(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x] && IntegerQ[p + 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.2 Quartic/1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
       "filename": "1.2.2.8 P(x) (d+e x)^q (a+b x^2+c x^4)^p.m",
            "rhs": "d*Int[Px*(a + c*x^4)^p/(d^2 - e^2*x^2), x] - e*Int[x*Px*(a + c*x^4)^p/(d^2 - e^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_ + c_.*x_^4)^p_./(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && PolyQ[Px, x] && IntegerQ[p + 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.1 (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.1 (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[x^(2*n*p)*(c + b*x^(-n) + a*x^(-2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && LtQ[n, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.1 (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.1 (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{k = Denominator[n]}, k*Subst[Int[x^(k - 1)*(a + b*x^(k*n) + c*x^(2*k*n))^p, x], x, x^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.1 (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.1 (a+b x^n+c x^(2 n))^p.m",
            "rhs": "-Subst[Int[(a + b*x^(-n) + c*x^(-2*n))^p/x^2, x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.1 (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.1 (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(a + b*x^n + c*x^(2*n))^p/(b + 2*c*x^n)^(2*p)* Int[(b + 2*c*x^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_. + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.1 (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.1 (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*x^n + c*x^(2*n))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.1 (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.1 (a+b x^n+c x^(2 n))^p.m",
            "rhs": "-x*(b^2 - 2*a*c + b*c*x^n)*(a + b*x^n + c*x^(2*n))^(p + 1)/(a* n*(p + 1)*(b^2 - 4*a*c)) + 1/(a*n*(p + 1)*(b^2 - 4*a*c))* Int[(b^2 - 2*a*c + n*(p + 1)*(b^2 - 4*a*c) + b*c*(n*(2*p + 3) + 1)*x^n)*(a + b*x^n + c*x^(2*n))^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && ILtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.1 (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.1 (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, 1/(2*c*q*r)*Int[(r - x^(n/2))/(q - r*x^(n/2) + x^n), x] + 1/(2*c*q*r)*Int[(r + x^(n/2))/(q + r*x^(n/2) + x^n), x]]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^n_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n/2, 0] && NegQ[b^2 - 4*a*c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.1 (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.1 (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, c/q*Int[1/(b/2 - q/2 + c*x^n), x] - c/q*Int[1/(b/2 + q/2 + c*x^n), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*x_^n_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.1 (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.1 (a+b x^n+c x^(2 n))^p.m",
            "rhs": "a^IntPart[p]*(a + b*x^n + c*x^(2*n))^FracPart[p]/ ((1 + 2*c*x^n/(b + Rt[b^2 - 4*a*c, 2]))^ FracPart[p]*(1 + 2*c*x^n/(b - Rt[b^2 - 4*a*c, 2]))^FracPart[p])* Int[(1 + 2*c*x^n/(b + Sqrt[b^2 - 4*a*c]))^ p*(1 + 2*c*x^n/(b - Sqrt[b^2 - 4*a*c]))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.1 (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.1 (a+b x^n+c x^(2 n))^p.m",
            "rhs": "1/Coefficient[u, x, 1]* Subst[Int[(a + b*x^n + c*x^(2*n))^p, x], x, u]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*u_^n_ + c_.*u_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && LinearQ[u, x] && NeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.1 (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.1 (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[(b + a*x^n + c*x^(2*n))^p/x^(n*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^mn_ + c_.*x_^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && EqQ[mn, -n] && IntegerQ[p] && PosQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.1 (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.1 (a+b x^n+c x^(2 n))^p.m",
            "rhs": "x^(n*FracPart[p])*(a + b*x^(-n) + c*x^n)^ FracPart[p]/(b + a*x^n + c*x^(2*n))^FracPart[p]* Int[(b + a*x^n + c*x^(2*n))^p/x^(n*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^mn_ + c_.*x_^n_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, p}, x] && EqQ[mn, -n] && Not[IntegerQ[p]] && PosQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": " d^m*Int[x^(m+n*p)*(b+c*x^n)^p,x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(b_.*x_^n_+c_.*x_^n2_.)^p_.,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b,c,d,m,n},x] &&  EqQ[n2,2*n] && IntegerQ[p] && (IntegerQ[m] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": " 1/d^(n*p)*Int[(d*x)^(m+n*p)*(b+c*x^n)^p,x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(b_.*x_^n_+c_.*x_^n2_.)^p_.,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b,c,d,m},x] &&  EqQ[n2,2*n] && IntegerQ[p] && IntegerQ[n] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": " (d*x)^m/x^m*Int[x^(m+n*p)*(b+c*x^n)^p,x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(b_.*x_^n_+c_.*x_^n2_.)^p_.,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b,c,d,m,n},x] &&  EqQ[n2,2*n] && IntegerQ[p] && Not[IntegerQ[m] || GtQ[d,0]] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": " (b*x^n+c*x^(2*n))^p/((d*x)^(n*p)*(b+c*x^n)^p)*Int[(d*x)^(m+n*p)*(b+c* x^2)^p,x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b,c,d,m,n,p},x] && EqQ[n2,2*n] &&  Not[IntegerQ[p]] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "1/n*Subst[Int[(a + b*x + c*x^2)^p, x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[(d*x)^m*(a + b*x^n + c*x^(2*n))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && EqQ[n2, 2*n] && IGtQ[p, 0] && Not[IntegerQ[Simplify[(m + 1)/n]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[x^(m + 2*n*p)*(c + b*x^(-n) + a*x^(-2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n}, x] && EqQ[n2, 2*n] && ILtQ[p, 0] && NegQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": " 1/c^p*Int[(d*x)^m*(b/2+c*x^n)^(2*p),x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_+b_.*x_^n_.+c_.*x_^n2_.)^p_,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,m,n,p},x] &&  EqQ[n2,2*n] && EqQ[b^2-4*a*c,0] && IntegerQ[p] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": " (d*x)^(m+1)*(a+b*x^n+c*x^(2*n))^(p+1)/(2*a*d*n*(p+1)*(2*p+1)) - (d*x)^(m+1)*(2*a+b*x^n)*(a+b*x^n+c*x^(2*n))^p/(2*a*d*n*(2*p+1))",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_+b_.*x_^n_+c_.*x_^n2_.)^p_,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,m,n,p},x] && EqQ[n2,2*n] && EqQ[b^2-4*a*c,0] &&  Not[IntegerQ[p]] && EqQ[m+2*n*(p+1)+1,0] && NeQ[2*p+1,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(a + b*x^n + c*x^(2*n))^ FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p]))* Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_ + b_.*x_^n_. + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "a^IntPart[p]*(a + b*x^n + c*x^(2*n))^ FracPart[p]/(1 + 2*c*x^n/b)^(2*FracPart[p])* Int[(d*x)^m*(1 + 2*c*x^n/b)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_ + b_.*x_^n_. + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[2*p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "1/n*Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "d^IntPart[m]*(d*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_*x_)^m_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{k = GCD[m + 1, n]}, 1/k*Subst[ Int[x^((m + 1)/k - 1)*(a + b*x^(n/k) + c*x^(2*n/k))^p, x], x, x^k] /; k != 1]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{k = Denominator[m]}, k/d*Subst[ Int[x^(k*(m + 1) - 1)*(a + b*x^(k*n)/d^n + c*x^(2*k*n)/d^(2*n))^ p, x], x, (d*x)^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && FractionQ[m] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "d^(n - 1)*(d*x)^(m - n + 1)*(a + b*x^n + c*x^(2*n))^ p*(b*n*p + c*(m + n*(2*p - 1) + 1)* x^n)/(c*(m + 2*n*p + 1)*(m + n*(2*p - 1) + 1)) - n*p*d^n/(c*(m + 2*n*p + 1)*(m + n*(2*p - 1) + 1))* Int[(d*x)^(m - n)*(a + b*x^n + c*x^(2*n))^(p - 1)* Simp[a*b*(m - n + 1) - (2*a*c*(m + n*(2*p - 1) + 1) - b^2*(m + n*(p - 1) + 1))*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IGtQ[p, 0] && GtQ[m, n - 1] && NeQ[m + 2*n*p + 1, 0] && NeQ[m + n*(2*p - 1) + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(d*x)^(m + 1)*(a + b*x^n + c*x^(2*n))^ p/(d*(m + 1)) - n*p/(d^n*(m + 1))* Int[(d*x)^(m + n)*(b + 2*c*x^n)*(a + b*x^n + c*x^(2*n))^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IGtQ[p, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(d*x)^(m + 1)*(a + b*x^n + c*x^(2*n))^ p/(d*(m + 2*n*p + 1)) + n*p/(m + 2*n*p + 1)* Int[(d*x)^m*(2*a + b*x^n)*(a + b*x^n + c*x^(2*n))^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IGtQ[p, 0] && NeQ[m + 2*n*p + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "d^(n - 1)*(d*x)^(m - n + 1)*(b + 2*c*x^n)*(a + b*x^n + c*x^(2*n))^(p + 1)/(n*(p + 1)*(b^2 - 4*a*c)) - d^n/(n*(p + 1)*(b^2 - 4*a*c))* Int[(d*x)^(m - n)*(b*(m - n + 1) + 2*c*(m + 2*n*(p + 1) + 1)*x^n)*(a + b*x^n + c*x^(2*n))^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && ILtQ[p, -1] && GtQ[m, n - 1] && LeQ[m, 2*n - 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "-d^(2*n - 1)*(d*x)^(m - 2*n + 1)*(2*a + b*x^n)*(a + b*x^n + c*x^(2*n))^(p + 1)/(n*(p + 1)*(b^2 - 4*a*c)) + d^(2*n)/(n*(p + 1)*(b^2 - 4*a*c))* Int[(d*x)^(m - 2*n)*(2*a*(m - 2*n + 1) + b*(m + n*(2*p + 1) + 1)*x^n)*(a + b*x^n + c*x^(2*n))^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && ILtQ[p, -1] && GtQ[m, 2*n - 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "-(d*x)^(m + 1)*(b^2 - 2*a*c + b*c*x^n)*(a + b*x^n + c*x^(2*n))^(p + 1)/(a*d* n*(p + 1)*(b^2 - 4*a*c)) + 1/(a*n*(p + 1)*(b^2 - 4*a*c))* Int[(d*x)^m*(a + b*x^n + c*x^(2*n))^(p + 1)* Simp[b^2*(m + n*(p + 1) + 1) - 2*a*c*(m + 2*n*(p + 1) + 1) + b*c*(m + n*(2*p + 3) + 1)*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && ILtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "d^(2*n - 1)*(d*x)^(m - 2*n + 1)*(a + b*x^n + c*x^(2*n))^(p + 1)/(c*(m + 2*n*p + 1)) - d^(2*n)/(c*(m + 2*n*p + 1))* Int[(d*x)^(m - 2*n)* Simp[a*(m - 2*n + 1) + b*(m + n*(p - 1) + 1)*x^n, x]*(a + b*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1] && NeQ[m + 2*n*p + 1, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(d*x)^(m + 1)*(a + b*x^n + c*x^(2*n))^(p + 1)/(a* d*(m + 1)) - 1/(a*d^n*(m + 1))* Int[(d*x)^(m + n)*(b*(m + n*(p + 1) + 1) + c*(m + 2*n*(p + 1) + 1)*x^n)*(a + b*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(d*x)^(m + 1)/(a*d*(m + 1)) - 1/(a*d^n)* Int[(d*x)^(m + n)*(b + c*x^n)/(a + b*x^n + c*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_/(a_ + b_.*x_^n_ + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[PolynomialDivide[x^m, (a + b*x^n + c*x^(2*n)), x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_/(a_ + b_.*x_^n_ + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IGtQ[m, 3*n - 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "d^(2*n - 1)*(d*x)^(m - 2*n + 1)/(c*(m - 2*n + 1)) - d^(2*n)/c* Int[(d*x)^(m - 2*n)*(a + b*x^n)/(a + b*x^n + c*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_/(a_ + b_.*x_^n_ + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, 1/(2*c*r)* Int[x^(m - 3*(n/2))*(q + r*x^(n/2))/(q + r*x^(n/2) + x^n), x] - 1/(2*c*r)* Int[x^(m - 3*(n/2))*(q - r*x^(n/2))/(q - r*x^(n/2) + x^n), x]]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_./(a_ + b_.*x_^n_ + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n/2, 0] && IGtQ[m, 0] && GeQ[m, 3*n/2] && LtQ[m, 2*n] && NegQ[b^2 - 4*a*c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, 1/(2*c*r)*Int[x^(m - n/2)/(q - r*x^(n/2) + x^n), x] - 1/(2*c*r)*Int[x^(m - n/2)/(q + r*x^(n/2) + x^n), x]]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_./(a_ + b_.*x_^n_ + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n/2, 0] && IGtQ[m, 0] && GeQ[m, n/2] && LtQ[m, 3*n/2] && NegQ[b^2 - 4*a*c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, d^n/2*(b/q + 1)*Int[(d*x)^(m - n)/(b/2 + q/2 + c*x^n), x] - d^n/2*(b/q - 1)*Int[(d*x)^(m - n)/(b/2 - q/2 + c*x^n), x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_/(a_ + b_.*x_^n_ + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GeQ[m, n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, c/q*Int[(d*x)^m/(b/2 - q/2 + c*x^n), x] - c/q*Int[(d*x)^m/(b/2 + q/2 + c*x^n), x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_./(a_ + b_.*x_^n_ + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "-Subst[ Int[(a + b*x^(-n) + c*x^(-2*n))^p/x^(m + 2), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && ILtQ[n, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{k = Denominator[m]}, -k/d* Subst[Int[(a + b*d^(-n)*x^(-k*n) + c*d^(-2*n)*x^(-2*k*n))^p/ x^(k*(m + 1) + 1), x], x, 1/(d*x)^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && ILtQ[n, 0] && FractionQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "-d^IntPart[m]*(d*x)^FracPart[m]*(x^(-1))^ FracPart[m]* Subst[Int[(a + b*x^(-n) + c*x^(-2*n))^p/x^(m + 2), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && ILtQ[n, 0] && Not[RationalQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{k = Denominator[n]}, k*Subst[ Int[x^(k*(m + 1) - 1)*(a + b*x^(k*n) + c*x^(2*k*n))^p, x], x, x^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "d^IntPart[m]*(d*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_*x_)^m_*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "1/(m + 1)* Subst[Int[(a + b*x^Simplify[n/(m + 1)] + c*x^Simplify[2*n/(m + 1)])^p, x], x, x^(m + 1)]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[Simplify[n/(m + 1)]] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "d^IntPart[m]*(d*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_*x_)^m_*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[Simplify[n/(m + 1)]] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, 2*c/q*Int[(d*x)^m/(b - q + 2*c*x^n), x] - 2*c/q*Int[(d*x)^m/(b + q + 2*c*x^n), x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_./(a_ + b_.*x_^n_ + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "-(d*x)^(m + 1)*(b^2 - 2*a*c + b*c*x^n)*(a + b*x^n + c*x^(2*n))^(p + 1)/(a*d* n*(p + 1)*(b^2 - 4*a*c)) + 1/(a*n*(p + 1)*(b^2 - 4*a*c))* Int[(d*x)^m*(a + b*x^n + c*x^(2*n))^(p + 1)* Simp[b^2*(n*(p + 1) + m + 1) - 2*a*c*(m + 2*n*(p + 1) + 1) + b*c*(2*n*p + 3*n + m + 1)*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && ILtQ[p + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "a^IntPart[p]*(a + b*x^n + c*x^(2*n))^FracPart[p]/ ((1 + 2*c*x^n/(b + Rt[b^2 - 4*a*c, 2]))^ FracPart[p]*(1 + 2*c*x^n/(b - Rt[b^2 - 4*a*c, 2]))^FracPart[p])* Int[(d*x)^m*(1 + 2*c*x^n/(b + Sqrt[b^2 - 4*a*c]))^ p*(1 + 2*c*x^n/(b - Sqrt[b^2 - 4*a*c]))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[x^(m - n*p)*(b + a*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^mn_ + c_.*x_^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n}, x] && EqQ[mn, -n] && IntegerQ[p] && PosQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "x^(n*FracPart[p])*(a + b/x^n + c*x^n)^ FracPart[p]/(b + a*x^n + c*x^(2*n))^FracPart[p]* Int[x^(m - n*p)*(b + a*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*x_^mn_ + c_.*x_^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n, p}, x] && EqQ[mn, -n] && Not[IntegerQ[p]] && PosQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "d^IntPart[m]*(d*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*x^(-n) + c*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_*x_)^m_.*(a_ + b_.*x_^mn_ + c_.*x_^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[mn, -n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "1/Coefficient[v, x, 1]^(m + 1)* Subst[Int[ SimplifyIntegrand[(x - Coefficient[v, x, 0])^ m*(a + b*x^n + c*x^(2*n))^p, x], x], x, v]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*v_^n_ + c_.*v_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && LinearQ[v, x] && IntegerQ[m] && NeQ[v, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "u^m/(Coefficient[v, x, 1]*v^m)* Subst[Int[x^m*(a + b*x^n + c*x^(2*n))^p, x], x, v]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*(a_. + b_.*v_^n_ + c_.*v_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && LinearPairQ[u, v, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "  1/c^p*Int[(d+e*x^n)^q*(b/2+c*x^n)^(2*p),x]",
     "rulenumber": 0,
            "lhs": "Int[(d_+e_.*x_^n_.)^q_.*(a_+b_.*x_^n_.+c_.*x_^n2_.)^p_,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e,n,p,q},x] && EqQ[n2,2*n] && EqQ[b^2-4*a*c,0] &&  IntegerQ[p] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(a + b*x^n + c*x^(2*n))^p/(d + e*x^n)^(2*p)* Int[(d + e*x^n)^(q + 2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_.)^q_.*(a_ + b_.*x_^n_. + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]] && EqQ[2*c*d - b*e, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(a + b*x^n + c*x^(2*n))^ FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p]))* Int[(d + e*x^n)^q*(b/2 + c*x^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_.)^q_.*(a_ + b_.*x_^n_. + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[x^(n*(2*p + q))*(e + d*x^(-n))^q*(c + b*x^(-n) + a*x^(-2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)^q_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && IntegersQ[p, q] && NegQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[x^(n*(2*p + q))*(e + d*x^(-n))^q*(c + a*x^(-2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)^q_.*(a_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n] && IntegersQ[p, q] && NegQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "-Subst[ Int[(d + e*x^(-n))^q*(a + b*x^(-n) + c*x^(-2*n))^p/x^2, x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)^q_.*(a_. + b_.*x_^n_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p, q}, x] && EqQ[n2, 2*n] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "-Subst[ Int[(d + e*x^(-n))^q*(a + c*x^(-2*n))^p/x^2, x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)^q_.*(a_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, p, q}, x] && EqQ[n2, 2*n] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{g = Denominator[n]}, g*Subst[ Int[x^(g - 1)*(d + e*x^(g*n))^q*(a + b*x^(g*n) + c*x^(2*g*n))^p, x], x, x^(1/g)]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)^q_.*(a_. + b_.*x_^n_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p, q}, x] && EqQ[n2, 2*n] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{g = Denominator[n]}, g*Subst[Int[x^(g - 1)*(d + e*x^(g*n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)^q_.*(a_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, p, q}, x] && EqQ[n2, 2*n] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(b*e - d*c)*(b*x^n + c*x^(2*n))^(p + 1)/(b*c*n*(p + 1)* x^(2*n*(p + 1))) + e/c*Int[x^(-n)*(b*x^n + c*x^(2*n))^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)*(b_.*x_^n_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, n, p}, x] && EqQ[n2, 2*n] && Not[IntegerQ[p]] && EqQ[n*(2*p + 1) + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "e*x^(-n + 1)*(b*x^n + c*x^(2*n))^(p + 1)/(c*(n*(2*p + 1) + 1))",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)*(b_.*x_^n_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, n, p}, x] && EqQ[n2, 2*n] && Not[IntegerQ[p]] && NeQ[n*(2*p + 1) + 1, 0] && EqQ[b*e*(n*p + 1) - c*d*(n*(2*p + 1) + 1), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "e*x^(-n + 1)*(b*x^n + c*x^(2*n))^(p + 1)/(c*(n*(2*p + 1) + 1)) - (b*e*(n*p + 1) - c*d*(n*(2*p + 1) + 1))/(c*(n*(2*p + 1) + 1))* Int[(b*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)*(b_.*x_^n_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, n, p}, x] && EqQ[n2, 2*n] && Not[IntegerQ[p]] && NeQ[n*(2*p + 1) + 1, 0] && NeQ[b*e*(n*p + 1) - c*d*(n*(2*p + 1) + 1), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(b*x^n + c*x^(2*n))^ FracPart[p]/(x^(n*FracPart[p])*(b + c*x^n)^FracPart[p])* Int[x^(n*p)*(d + e*x^n)^q*(b + c*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)^q_.*(b_.*x_^n_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[(d + e*x^n)^(p + q)*(a/d + c/e*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)^q_.*(a_ + b_.*x_^n_ + c_.*x_^n2_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[(d + e*x^n)^(p + q)*(a/d + c/e*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)^q_.*(a_ + c_.*x_^n2_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, n, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(a + b*x^n + c*x^(2*n))^ FracPart[ p]/((d + e*x^n)^FracPart[p]*(a/d + c*x^n/e)^FracPart[p])* Int[(d + e*x^n)^(p + q)*(a/d + c/e*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)^q_*(a_ + b_.*x_^n_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[p]]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(a + c*x^(2*n))^ FracPart[ p]/((d + e*x^n)^FracPart[p]*(a/d + c*x^n/e)^FracPart[p])* Int[(d + e*x^n)^(p + q)*(a/d + c/e*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)^q_*(a_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x^n)^q*(a + b*x^n + c*x^(2*n)), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)^q_.*(a_ + b_.*x_^n_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x^n)^q*(a + c*x^(2*n)), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)^q_.*(a_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "-(c*d^2 - b*d*e + a*e^2)* x*(d + e*x^n)^(q + 1)/(d*e^2*n*(q + 1)) + 1/(n*(q + 1)*d*e^2)* Int[(d + e*x^n)^(q + 1)* Simp[c*d^2 - b*d*e + a*e^2*(n*(q + 1) + 1) + c*d*e*n*(q + 1)*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)^q_*(a_ + b_.*x_^n_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "-(c*d^2 + a*e^2)* x*(d + e*x^n)^(q + 1)/(d*e^2*n*(q + 1)) + 1/(n*(q + 1)*d*e^2)* Int[(d + e*x^n)^(q + 1)* Simp[c*d^2 + a*e^2*(n*(q + 1) + 1) + c*d*e*n*(q + 1)*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)^q_*(a_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && LtQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "c*x^(n + 1)*(d + e*x^n)^(q + 1)/(e*(n*(q + 2) + 1)) + 1/(e*(n*(q + 2) + 1))* Int[(d + e*x^n)^ q*(a*e*(n*(q + 2) + 1) - (c*d*(n + 1) - b*e*(n*(q + 2) + 1))* x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)^q_*(a_ + b_.*x_^n_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "c*x^(n + 1)*(d + e*x^n)^(q + 1)/(e*(n*(q + 2) + 1)) + 1/(e*(n*(q + 2) + 1))* Int[(d + e*x^n)^q*(a*e*(n*(q + 2) + 1) - c*d*(n + 1)*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)^q_*(a_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Rt[2*d*e, 2]}, e^2/(2*c)*Int[1/(d + q*x^(n/2) + e*x^n), x] + e^2/(2*c)*Int[1/(d - q*x^(n/2) + e*x^n), x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)/(a_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 - a*e^2, 0] && IGtQ[n/2, 0] && PosQ[d*e]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Rt[-2*d*e, 2]}, d/(2*a)*Int[(d - q*x^(n/2))/(d - q*x^(n/2) - e*x^n), x] + d/(2*a)*Int[(d + q*x^(n/2))/(d + q*x^(n/2) - e*x^n), x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)/(a_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 - a*e^2, 0] && IGtQ[n/2, 0] && NegQ[d*e]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Rt[a/c, 4]}, 1/(2*Sqrt[2]*c*q^3)* Int[(Sqrt[2]*d*q - (d - e*q^2)*x^(n/2))/(q^2 - Sqrt[2]*q*x^(n/2) + x^n), x] + 1/(2*Sqrt[2]*c*q^3)* Int[(Sqrt[2]*d*q + (d - e*q^2)*x^(n/2))/(q^2 + Sqrt[2]*q*x^(n/2) + x^n), x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)/(a_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && IGtQ[n/2, 0] && PosQ[a*c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Rt[c/a, 6]}, 1/(3*a*q^2)*Int[(q^2*d - e*x)/(1 + q^2*x^2), x] + 1/(6*a*q^2)* Int[(2*q^2*d - (Sqrt[3]*q^3*d - e)*x)/(1 - Sqrt[3]*q*x + q^2*x^2), x] + 1/(6*a*q^2)* Int[(2*q^2*d + (Sqrt[3]*q^3*d + e)*x)/(1 + Sqrt[3]*q*x + q^2*x^2), x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^3)/(a_ + c_.*x_^6), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && PosQ[c/a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Rt[-a/c, 2]}, (d + e*q)/2*Int[1/(a + c*q*x^n), x] + (d - e*q)/2* Int[1/(a - c*q*x^n), x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)/(a_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && NegQ[a*c] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "d*Int[1/(a + c*x^(2*n)), x] + e*Int[x^n/(a + c*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)/(a_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && (PosQ[a*c] || Not[IntegerQ[n]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Rt[2*d/e - b/c, 2]}, e/(2*c)*Int[1/Simp[d/e + q*x^(n/2) + x^n, x], x] + e/(2*c)*Int[1/Simp[d/e - q*x^(n/2) + x^n, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)/(a_ + b_.*x_^n_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && IGtQ[n/2, 0] && (GtQ[2*d/e - b/c, 0] || Not[LtQ[2*d/e - b/c, 0]] && EqQ[d, e*Rt[a/c, 2]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, (e/2 + (2*c*d - b*e)/(2*q))* Int[1/(b/2 - q/2 + c*x^n), x] + (e/2 - (2*c*d - b*e)/(2*q))* Int[1/(b/2 + q/2 + c*x^n), x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)/(a_ + b_.*x_^n_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && IGtQ[n/2, 0] && GtQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Rt[-2*d/e - b/c, 2]}, e/(2*c*q)* Int[(q - 2*x^(n/2))/Simp[d/e + q*x^(n/2) - x^n, x], x] + e/(2*c*q)* Int[(q + 2*x^(n/2))/Simp[d/e - q*x^(n/2) - x^n, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)/(a_ + b_.*x_^n_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && IGtQ[n/2, 0] && Not[GtQ[b^2 - 4*a*c, 0]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, (e/2 + (2*c*d - b*e)/(2*q))* Int[1/(b/2 - q/2 + c*x^n), x] + (e/2 - (2*c*d - b*e)/(2*q))* Int[1/(b/2 + q/2 + c*x^n), x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)/(a_ + b_.*x_^n_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 - 4*a*c] || Not[IGtQ[n/2, 0]])"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, 1/(2*c*q*r)* Int[(d*r - (d - e*q)*x^(n/2))/(q - r*x^(n/2) + x^n), x] + 1/(2*c*q*r)* Int[(d*r + (d - e*q)*x^(n/2))/(q + r*x^(n/2) + x^n), x]]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)/(a_ + b_.*x_^n_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[n/2, 0] && NegQ[b^2 - 4*a*c]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x^n)^q/(a + b*x^n + c*x^(2*n)), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)^q_/(a_ + b_.*x_^n_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[q]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x^n)^q/(a + c*x^(2*n)), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)^q_/(a_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[q]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "e^2/(c*d^2 - b*d*e + a*e^2)*Int[(d + e*x^n)^q, x] + 1/(c*d^2 - b*d*e + a*e^2)* Int[(d + e*x^n)^(q + 1)*(c*d - b*e - c*e*x^n)/(a + b*x^n + c*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)^q_/(a_ + b_.*x_^n_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[q]] && LtQ[q, -1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "e^2/(c*d^2 + a*e^2)*Int[(d + e*x^n)^q, x] + c/(c*d^2 + a*e^2)* Int[(d + e*x^n)^(q + 1)*(d - e*x^n)/(a + c*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)^q_/(a_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[q]] && LtQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{r = Rt[b^2 - 4*a*c, 2]}, 2*c/r*Int[(d + e*x^n)^q/(b - r + 2*c*x^n), x] - 2*c/r*Int[(d + e*x^n)^q/(b + r + 2*c*x^n), x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)^q_/(a_ + b_.*x_^n_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[q]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{r = Rt[-a*c, 2]}, -c/(2*r)*Int[(d + e*x^n)^q/(r - c*x^n), x] - c/(2*r)*Int[(d + e*x^n)^q/(r + c*x^n), x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)^q_/(a_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[q]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "-x*(d*b^2 - a*b*e - 2*a*c*d + (b*d - 2*a*e)*c* x^n)*(a + b*x^n + c*x^(2*n))^(p + 1)/(a* n*(p + 1)*(b^2 - 4*a*c)) + 1/(a*n*(p + 1)*(b^2 - 4*a*c))* Int[ Simp[(n*p + n + 1)*d*b^2 - a*b*e - 2*a*c*d*(2*n*p + 2*n + 1) + (2*n*p + 3*n + 1)*(d*b - 2*a*e)*c* x^n, x]* (a + b*x^n + c*x^(2*n))^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)*(a_ + b_.*x_^n_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && ILtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "-x*(d + e*x^n)*(a + c*x^(2*n))^(p + 1)/(2*a*n*(p + 1)) + 1/(2*a*n*(p + 1))* Int[(d*(2*n*p + 2*n + 1) + e*(2*n*p + 3*n + 1)*x^n)*(a + c*x^(2*n))^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)*(a_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n] && ILtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x^n)*(a + b*x^n + c*x^(2*n))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)*(a_ + b_.*x_^n_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x^n)*(a + c*x^(2*n))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)*(a_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "c^p*x^(2*n*p - n + 1)*(d + e*x^n)^(q + 1)/(e*(2*n*p + n*q + 1)) + Int[(d + e*x^n)^q* ExpandToSum[(a + b*x^n + c*x^(2*n))^p - c^p*x^(2*n*p) - d*c^p*(2*n*p - n + 1)*x^(2*n*p - n)/(e*(2*n*p + n*q + 1)), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)^q_*(a_ + b_.*x_^n_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && NeQ[2*n*p + n*q + 1, 0] && IGtQ[n, 0] && Not[IGtQ[q, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "c^p*x^(2*n*p - n + 1)*(d + e*x^n)^(q + 1)/(e*(2*n*p + n*q + 1)) + Int[(d + e*x^n)^q* ExpandToSum[(a + c*x^(2*n))^p - c^p*x^(2*n*p) - d*c^p*(2*n*p - n + 1)*x^(2*n*p - n)/(e*(2*n*p + n*q + 1)), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)^q_*(a_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, n, q}, x] && EqQ[n2, 2*n] && IGtQ[p, 0] && NeQ[2*n*p + n*q + 1, 0] && IGtQ[n, 0] && Not[IGtQ[q, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)^q_*(a_ + b_.*x_^n_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && (IntegersQ[p, q] && Not[IntegerQ[n]] || IGtQ[p, 0] || IGtQ[q, 0] && Not[IntegerQ[n]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x^n)^q*(a + c*x^(2*n))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)^q_*(a_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && (IntegersQ[p, q] && Not[IntegerQ[n]] || IGtQ[p, 0] || IGtQ[q, 0] && Not[IntegerQ[n]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[(a + c*x^(2*n))^ p, (d/(d^2 - e^2*x^(2*n)) - e*x^n/(d^2 - e^2*x^(2*n)))^(-q), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)^q_*(a_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, n, p}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]] && ILtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Unintegrable[(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)^q_*(a_ + b_.*x_^n_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p, q}, x] && EqQ[n2, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Unintegrable[(d + e*x^n)^q*(a + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_)^q_*(a_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "1/Coefficient[u, x, 1]* Subst[Int[(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x, u]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*u_^n_)^q_.*(a_ + b_.*u_^n_ + c_.*u_^n2_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && LinearQ[u, x] && NeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "1/Coefficient[u, x, 1]* Subst[Int[(d + e*x^n)^q*(a + c*x^(2*n))^p, x], x, u]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*u_^n_)^q_.*(a_ + c_.*u_^n2_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && LinearQ[u, x] && NeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[x^(-n*q)*(e + d*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^mn_.)^q_.*(a_. + b_.*x_^n_. + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[n2, 2*n] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] || Not[IntegerQ[p]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[x^(mn*q)*(e + d*x^(-mn))^q*(a + c*x^n2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^mn_.)^q_.*(a_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, mn, p}, x] && EqQ[n2, -2*mn] && IntegerQ[q] && (PosQ[n2] || Not[IntegerQ[p]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[x^(-2*n*p)*(d + e*x^n)^q*(c + b*x^n + a*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_.)^q_.*(a_. + b_.*x_^mn_. + c_.*x_^mn2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[mn, -n] && EqQ[mn2, 2*mn] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[x^(-2*n*p)*(d + e*x^n)^q*(c + a*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_.)^q_.*(a_. + c_.*x_^mn2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, n, q}, x] && EqQ[mn2, -2*n] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "e^IntPart[q]* x^(n*FracPart[q])*(d + e*x^(-n))^FracPart[q]/(1 + d*x^n/e)^ FracPart[q]* Int[x^(-n*q)*(1 + d*x^n/e)^q*(a + b*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^mn_.)^q_*(a_. + b_.*x_^n_. + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[mn, -n] && Not[IntegerQ[p]] && Not[IntegerQ[q]] && PosQ[n]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "e^IntPart[q]* x^(-mn*FracPart[q])*(d + e*x^mn)^FracPart[q]/(1 + d*x^(-mn)/e)^ FracPart[q]*Int[x^(mn*q)*(1 + d*x^(-mn)/e)^q*(a + c*x^n2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^mn_.)^q_*(a_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, mn, p, q}, x] && EqQ[n2, -2*mn] && Not[IntegerQ[p]] && Not[IntegerQ[q]] && PosQ[n2]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "x^(n*FracPart[q])*(d+e*x^(-n))^FracPart[q]/(e+d*x^n)^FracPart[q]* Int[x^(-n*q)*(e+d*x^n)^q*(a+b*x^n+c*x^(2*n))^p,x]",
     "rulenumber": 0,
            "lhs": "Int[(d_+e_.*x_^mn_.)^q_*(a_.+b_.*x_^n_.+c_.*x_^n2_.)^p_.,x_Symbol] ",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e,n,p,q},x] && EqQ[n2,2*n] && EqQ[mn,-n] &&  Not[IntegerQ[p]] && Not[IntegerQ[q]] && PosQ[n] *)"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": " x^(-mn*FracPart[q])*(d+e*x^mn)^FracPart[q]/(e+d*x^(-mn))^FracPart[q]* Int[x^(mn*q)*(e+d*x^(-mn))^q*(a+c*x^n2)^p,x]",
     "rulenumber": 0,
            "lhs": "Int[(d_+e_.*x_^mn_.)^q_*(a_+c_.*x_^n2_.)^p_.,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,c,d,e,mn,p,q},x] && EqQ[n2,-2*mn] && Not[IntegerQ[p]] &&  Not[IntegerQ[q]] && PosQ[n2] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.3 (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "x^(2*n*FracPart[p])*(a + b*x^(-n) + c*x^(-2*n))^ FracPart[p]/(c + b*x^n + a*x^(2*n))^FracPart[p]* Int[x^(-2*n*p)*(d + e*x^n)^q*(c + b*x^n + a*x^(2*n))^p, x]",
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            "rhs": "(a + b*x^n + c*x^(2*n))^ FracPart[ p]/((d + e*x^n)^FracPart[p]*(a/d + (c*x^n)/e)^FracPart[p])* Int[(d + e*x^n)^(p + q)*(f + g*x^n)^r*(a/d + c/e*x^n)^p, x]",
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            "rhs": "(a + c*x^(2*n))^ FracPart[ p]/((d + e*x^n)^FracPart[p]*(a/d + (c*x^n)/e)^FracPart[p])* Int[(d + e*x^n)^(p + q)*(f + g*x^n)^r*(a/d + c/e*x^n)^p, x]",
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        "comment": false,
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            "lhs": "Int[x_^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p, q}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[(m + 1)/n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(a + b*x^n + c*x^(2*n))^ FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p]))* Int[(f*x)^m*(d + e*x^n)^q*(b/2 + c*x^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^ p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "f^IntPart[m]*(f*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_*x_)^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p, q}, x] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "f^IntPart[m]*(f*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(d + e*x^n)^q*(a + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_*x_)^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m, n, p, q}, x] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[(f*x)^m*(d + e*x^n)^(q + p)*(a/d + c/e*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + b_.*x_^n_ + c_.*x_^n2_)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[(f*x)^m*(d + e*x^n)^(q + p)*(a/d + c/e*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + c_.*x_^n2_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, q, m, n, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(a + b*x^n + c*x^(2*n))^ FracPart[ p]/((d + e*x^n)^FracPart[p]*(a/d + (c*x^n)/e)^FracPart[p])* Int[(f*x)^m*(d + e*x^n)^(q + p)*(a/d + c/e*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)^q_*(a_ + b_.*x_^n_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(a + c*x^(2*n))^ FracPart[ p]/((d + e*x^n)^FracPart[p]*(a/d + (c*x^n)/e)^FracPart[p])* Int[(f*x)^m*(d + e*x^n)^(q + p)*(a/d + c/e*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)^q_*(a_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(-d)^((m - Mod[m, n])/n - 1)*(c*d^2 - b*d*e + a*e^2)^p* x^(Mod[m, n] + 1)*(d + e*x^n)^(q + 1)/(n* e^(2*p + (m - Mod[m, n])/n)*(q + 1)) + 1/(n*e^(2*p + (m - Mod[m, n])/n)*(q + 1))* Int[x^Mod[m, n]*(d + e*x^n)^(q + 1)* ExpandToSum[ Together[ 1/(d + e*x^n)*(n*e^(2*p + (m - Mod[m, n])/n)*(q + 1)* x^(m - Mod[m, n])*(a + b*x^n + c*x^(2*n))^p - (-d)^((m - Mod[m, n])/n - 1)*(c*d^2 - b*d*e + a*e^2)^ p*(d*(Mod[m, n] + 1) + e*(Mod[m, n] + n*(q + 1) + 1)*x^n))], x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^n_)^q_*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[q, -1] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(-d)^((m - Mod[m, n])/n - 1)*(c*d^2 + a*e^2)^p* x^(Mod[m, n] + 1)*(d + e*x^n)^(q + 1)/(n* e^(2*p + (m - Mod[m, n])/n)*(q + 1)) + 1/(n*e^(2*p + (m - Mod[m, n])/n)*(q + 1))* Int[x^Mod[m, n]*(d + e*x^n)^(q + 1)* ExpandToSum[ Together[ 1/(d + e*x^n)*(n*e^(2*p + (m - Mod[m, n])/n)*(q + 1)* x^(m - Mod[m, n])*(a + c*x^(2*n))^p - (-d)^((m - Mod[m, n])/n - 1)*(c*d^2 + a*e^2)^ p*(d*(Mod[m, n] + 1) + e*(Mod[m, n] + n*(q + 1) + 1)*x^n))], x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^n_)^q_*(a_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[q, -1] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(-d)^((m - Mod[m, n])/n - 1)*(c*d^2 - b*d*e + a*e^2)^p* x^(Mod[m, n] + 1)*(d + e*x^n)^(q + 1)/(n* e^(2*p + (m - Mod[m, n])/n)*(q + 1)) + (-d)^((m - Mod[m, n])/n - 1)/(n*e^(2*p)*(q + 1))* Int[x^m*(d + e*x^n)^(q + 1)* ExpandToSum[ Together[ 1/(d + e*x^n)*(n*(-d)^(-(m - Mod[m, n])/n + 1)* e^(2*p)*(q + 1)*(a + b*x^n + c*x^(2*n))^p - (e^(-(m - Mod[m, n])/n)*(c*d^2 - b*d*e + a*e^2)^p* x^(-(m - Mod[m, n])))*(d*(Mod[m, n] + 1) + e*(Mod[m, n] + n*(q + 1) + 1)*x^n))], x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*(d_ + e_.*x_^n_)^q_*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(-d)^((m - Mod[m, n])/n - 1)*(c*d^2 + a*e^2)^p* x^(Mod[m, n] + 1)*(d + e*x^n)^(q + 1)/(n* e^(2*p + (m - Mod[m, n])/n)*(q + 1)) + (-d)^((m - Mod[m, n])/n - 1)/(n*e^(2*p)*(q + 1))* Int[x^m*(d + e*x^n)^(q + 1)* ExpandToSum[ Together[ 1/(d + e*x^n)*(n*(-d)^(-(m - Mod[m, n])/n + 1)* e^(2*p)*(q + 1)*(a + c*x^(2*n))^p - (e^(-(m - Mod[m, n])/n)*(c*d^2 + a*e^2)^p* x^(-(m - Mod[m, n])))*(d*(Mod[m, n] + 1) + e*(Mod[m, n] + n*(q + 1) + 1)*x^n))], x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*(d_ + e_.*x_^n_)^q_*(a_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IGtQ[p, 0] && IntegersQ[m, q] && ILtQ[q, -1] && ILtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "c^p*(f*x)^(m + 2*n*p - n + 1)*(d + e*x^n)^(q + 1)/(e* f^(2*n*p - n + 1)*(m + 2*n*p + n*q + 1)) + 1/(e*(m + 2*n*p + n*q + 1))*Int[(f*x)^m*(d + e*x^n)^q* ExpandToSum[ e*(m + 2*n*p + n*q + 1)*((a + b*x^n + c*x^(2*n))^p - c^p*x^(2*n*p)) - d*c^p*(m + 2*n*p - n + 1)*x^(2*n*p - n), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IGtQ[p, 0] && GtQ[2*n*p, n - 1] && Not[IntegerQ[q]] && NeQ[m + 2*n*p + n*q + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "c^p*(f*x)^(m + 2*n*p - n + 1)*(d + e*x^n)^(q + 1)/(e* f^(2*n*p - n + 1)*(m + 2*n*p + n*q + 1)) + 1/(e*(m + 2*n*p + n*q + 1))*Int[(f*x)^m*(d + e*x^n)^q* ExpandToSum[ e*(m + 2*n*p + n*q + 1)*((a + c*x^(2*n))^p - c^p*x^(2*n*p)) - d*c^p*(m + 2*n*p - n + 1)*x^(2*n*p - n), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IGtQ[p, 0] && GtQ[2*n*p, n - 1] && Not[IntegerQ[q]] && NeQ[m + 2*n*p + n*q + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[(f*x)^m (d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[(f*x)^m (d + e*x^n)^q*(a + c*x^(2*n))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{k = GCD[m + 1, n]}, 1/k*Subst[ Int[x^((m + 1)/k - 1)*(d + e*x^(n/k))^ q*(a + b*x^(n/k) + c*x^(2*n/k))^p, x], x, x^k] /; k != 1]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{k = GCD[m + 1, n]}, 1/k*Subst[ Int[x^((m + 1)/k - 1)*(d + e*x^(n/k))^q*(a + c*x^(2*n/k))^p, x], x, x^k] /; k != 1]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, p, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{k = Denominator[m]}, k/f*Subst[ Int[x^(k*(m + 1) - 1)*(d + e*x^(k*n)/f^n)^ q*(a + b*x^(k*n)/f^n + c*x^(2*k*n)/f^(2*n))^p, x], x, (f*x)^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^n_)^q_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^ p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && FractionQ[m] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{k = Denominator[m]}, k/f*Subst[ Int[x^(k*(m + 1) - 1)*(d + e*x^(k*n)/f)^q*(a + c*x^(2*k*n)/f)^p, x], x, (f*x)^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^n_)^q_.*(a_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, p, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && FractionQ[m] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(f*x)^(m + 1)*(a + b*x^n + c*x^(2*n))^ p*(d*(m + n*(2*p + 1) + 1) + e*(m + 1)*x^n)/(f*(m + 1)*(m + n*(2*p + 1) + 1)) + n*p/(f^n*(m + 1)*(m + n*(2*p + 1) + 1))* Int[(f*x)^(m + n)*(a + b*x^n + c*x^(2*n))^(p - 1)* Simp[2*a*e*(m + 1) - b*d*(m + n*(2*p + 1) + 1) + (b*e*(m + 1) - 2*c*d*(m + n*(2*p + 1) + 1))*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)*(a_ + b_.*x_^n_ + c_.*x_^n2_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] && NeQ[m + n*(2*p + 1) + 1, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(f*x)^(m + 1)*(a + c*x^(2*n))^ p*(d*(m + n*(2*p + 1) + 1) + e*(m + 1)*x^n)/(f*(m + 1)*(m + n*(2*p + 1) + 1)) + 2*n*p/(f^n*(m + 1)*(m + n*(2*p + 1) + 1))* Int[(f*x)^(m + n)*(a + c*x^(2*n))^(p - 1)*(a*e*(m + 1) - c*d*(m + n*(2*p + 1) + 1)*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)*(a_ + c_.*x_^n2_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] && NeQ[m + n*(2*p + 1) + 1, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(f*x)^(m + 1)*(a + b*x^n + c*x^(2*n))^ p*(b*e*n*p + c*d*(m + n*(2*p + 1) + 1) + c*e*(2*n*p + m + 1)*x^n)/ (c*f*(2*n*p + m + 1)*(m + n*(2*p + 1) + 1)) + n*p/(c*(2*n*p + m + 1)*(m + n*(2*p + 1) + 1))* Int[(f*x)^m*(a + b*x^n + c*x^(2*n))^(p - 1)* Simp[2*a*c*d*(m + n*(2*p + 1) + 1) - a*b*e*(m + 1) + (2*a*c*e*(2*n*p + m + 1) + b*c*d*(m + n*(2*p + 1) + 1) - b^2*e*(m + n*p + 1))*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)*(a_ + b_.*x_^n_ + c_.*x_^n2_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[p, 0] && NeQ[2*n*p + m + 1, 0] && NeQ[m + n*(2*p + 1) + 1, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(f*x)^(m + 1)*(a + c*x^(2*n))^ p*(c*d*(m + n*(2*p + 1) + 1) + c*e*(2*n*p + m + 1)*x^n)/(c* f*(2*n*p + m + 1)*(m + n*(2*p + 1) + 1)) + 2*a*n*p/((2*n*p + m + 1)*(m + n*(2*p + 1) + 1))* Int[(f*x)^m*(a + c*x^(2*n))^(p - 1)* Simp[d*(m + n*(2*p + 1) + 1) + e*(2*n*p + m + 1)*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)*(a_ + c_.*x_^n2_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && GtQ[p, 0] && NeQ[2*n*p + m + 1, 0] && NeQ[m + n*(2*p + 1) + 1, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "f^(n - 1)*(f*x)^(m - n + 1)*(a + b*x^n + c*x^(2*n))^(p + 1)*(b*d - 2*a*e - (b*e - 2*c*d)*x^n)/(n*(p + 1)*(b^2 - 4*a*c)) + f^n/(n*(p + 1)*(b^2 - 4*a*c))* Int[(f*x)^(m - n)*(a + b*x^n + c*x^(2*n))^(p + 1)* Simp[(n - m - 1)*(b*d - 2*a*e) + (2*n*p + 2*n + m + 1)*(b*e - 2*c*d)*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)*(a_ + b_.*x_^n_ + c_.*x_^n2_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m, n - 1] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "f^(n - 1)*(f*x)^(m - n + 1)*(a + c*x^(2*n))^(p + 1)*(a*e - c*d*x^n)/(2*a*c*n*(p + 1)) + f^n/(2*a*c*n*(p + 1))* Int[(f*x)^(m - n)*(a + c*x^(2*n))^(p + 1)*(a*e*(n - m - 1) + c*d*(2*n*p + 2*n + m + 1)*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)*(a_ + c_.*x_^n2_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m, n - 1] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "-(f*x)^(m + 1)*(a + b*x^n + c*x^(2*n))^(p + 1)*(d*(b^2 - 2*a*c) - a*b*e + (b*d - 2*a*e)*c*x^n)/(a*f* n*(p + 1)*(b^2 - 4*a*c)) + 1/(a*n*(p + 1)*(b^2 - 4*a*c))* Int[(f*x)^m*(a + b*x^n + c*x^(2*n))^(p + 1)* Simp[d*(b^2*(m + n*(p + 1) + 1) - 2*a*c*(m + 2*n*(p + 1) + 1)) - a*b*e*(m + 1) + c*(m + n*(2*p + 3) + 1)*(b*d - 2*a*e)*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)*(a_ + b_.*x_^n_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "-(f*x)^(m + 1)*(a + c*x^(2*n))^(p + 1)*(d + e*x^n)/(2*a*f*n*(p + 1)) + 1/(2*a*n*(p + 1))* Int[(f*x)^m*(a + c*x^(2*n))^(p + 1)* Simp[d*(m + 2*n*(p + 1) + 1) + e*(m + n*(2*p + 3) + 1)*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)*(a_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && LtQ[p, -1] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "e*f^(n - 1)*(f*x)^(m - n + 1)*(a + b*x^n + c*x^(2*n))^(p + 1)/(c*(m + n (2*p + 1) + 1)) - f^n/(c*(m + n (2*p + 1) + 1))* Int[(f*x)^(m - n)*(a + b*x^n + c*x^(2*n))^p* Simp[a*e*(m - n + 1) + (b*e*(m + n*p + 1) - c*d*(m + n (2*p + 1) + 1))*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)*(a_ + b_.*x_^n_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n (2*p + 1) + 1, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "e*f^(n - 1)*(f*x)^(m - n + 1)*(a + c*x^(2*n))^(p + 1)/(c*(m + n (2*p + 1) + 1)) - f^n/(c*(m + n (2*p + 1) + 1))* Int[(f*x)^(m - n)*(a + c*x^(2*n))^ p*(a*e*(m - n + 1) - c*d*(m + n (2*p + 1) + 1)*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)*(a_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, p}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n (2*p + 1) + 1, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "d*(f*x)^(m + 1)*(a + b*x^n + c*x^(2*n))^(p + 1)/(a*f*(m + 1)) + 1/(a*f^n*(m + 1))* Int[(f*x)^(m + n)*(a + b*x^n + c*x^(2*n))^p* Simp[a*e*(m + 1) - b*d*(m + n*(p + 1) + 1) - c*d*(m + 2*n (p + 1) + 1)*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)*(a_ + b_.*x_^n_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "d*(f*x)^(m + 1)*(a + c*x^(2*n))^(p + 1)/(a*f*(m + 1)) + 1/(a*f^n*(m + 1))* Int[(f*x)^(m + n)*(a + c*x^(2*n))^ p*(a*e*(m + 1) - c*d*(m + 2*n (p + 1) + 1)*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)*(a_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, p}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && LtQ[m, -1] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Rt[a*c, 2]}, With[{r = Rt[2*c*q - b*c, 2]}, c/(2*q*r)* Int[(f*x)^m* Simp[d*r - (c*d - e*q)*x^(n/2), x]/(q - r*x^(n/2) + c*x^n), x] + c/(2*q*r)* Int[(f*x)^m* Simp[d*r + (c*d - e*q)*x^(n/2), x]/(q + r*x^(n/2) + c*x^n), x]] /; Not[LtQ[2*c*q - b*c, 0]]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^n_)/(a_ + b_.*x_^n_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n] && LtQ[b^2 - 4*a*c, 0] && IntegersQ[m, n/2] && LtQ[0, m, n] && PosQ[a*c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Rt[a*c, 2]}, With[{r = Rt[2*c*q, 2]}, c/(2*q*r)* Int[(f*x)^m* Simp[d*r - (c*d - e*q)*x^(n/2), x]/(q - r*x^(n/2) + c*x^n), x] + c/(2*q*r)* Int[(f*x)^m* Simp[d*r + (c*d - e*q)*x^(n/2), x]/(q + r*x^(n/2) + c*x^n), x]] /; Not[LtQ[2*c*q, 0]]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^n_)/(a_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f}, x] && EqQ[n2, 2*n] && GtQ[a*c, 0] && IntegersQ[m, n/2] && LtQ[0, m, n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Rt[a*c, 2]}, With[{r = Rt[2*c*q - b*c, 2]}, c/(2*q*r)* Int[(f*x)^ m*(d*r - (c*d - e*q)*x^(n/2))/(q - r*x^(n/2) + c*x^n), x] + c/(2*q*r)* Int[(f*x)^ m*(d*r + (c*d - e*q)*x^(n/2))/(q + r*x^(n/2) + c*x^n), x]] /; Not[LtQ[2*c*q - b*c, 0]]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)/(a_ + b_.*x_^n_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[n2, 2*n] && LtQ[b^2 - 4*a*c, 0] && IGtQ[n/2, 1] && PosQ[a*c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Rt[a*c, 2]}, With[{r = Rt[2*c*q, 2]}, c/(2*q*r)* Int[(f*x)^ m*(d*r - (c*d - e*q)*x^(n/2))/(q - r*x^(n/2) + c*x^n), x] + c/(2*q*r)* Int[(f*x)^ m*(d*r + (c*d - e*q)*x^(n/2))/(q + r*x^(n/2) + c*x^n), x]] /; Not[LtQ[2*c*q, 0]]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)/(a_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m}, x] && EqQ[n2, 2*n] && IGtQ[n/2, 1] && GtQ[a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, (e/2 + (2*c*d - b*e)/(2*q))* Int[(f*x)^m/(b/2 - q/2 + c*x^n), x] + (e/2 - (2*c*d - b*e)/(2*q))* Int[(f*x)^m/(b/2 + q/2 + c*x^n), x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)/(a_ + b_.*x_^n_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Rt[-a*c, 2]}, -(e/2 + c*d/(2*q))* Int[(f*x)^m/(q - c*x^n), x] + (e/2 - c*d/(2*q))* Int[(f*x)^m/(q + c*x^n), x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)/(a_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[(f*x)^m*(d + e*x^n)^q/(a + b*x^n + c*x^(2*n)), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)^q_./(a_ + b_.*x_^n_ + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IntegerQ[q] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[(f*x)^m*(d + e*x^n)^q/(a + c*x^(2*n)), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)^q_./(a_ + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IntegerQ[q] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[(f*x)^m, (d + e*x^n)^q/(a + b*x^n + c*x^(2*n)), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)^q_./(a_ + b_.*x_^n_ + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IntegerQ[q] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[(f*x)^m, (d + e*x^n)^q/(a + c*x^(2*n)), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)^q_./(a_ + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IntegerQ[q] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "f^(2*n)/c^2* Int[(f*x)^(m - 2*n)*(c*d - b*e + c*e*x^n)*(d + e*x^n)^(q - 1), x] - f^(2*n)/c^2* Int[(f*x)^(m - 2*n)*(d + e*x^n)^(q - 1)* Simp[a*(c*d - b*e) + (b*c*d - b^2*e + a*c*e)*x^n, x]/(a + b*x^n + c*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_^n_)^q_/(a_ + b_.*x_^n_ + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && Not[IntegerQ[q]] && GtQ[q, 0] && GtQ[m, 2*n - 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "f^(2*n)/c*Int[(f*x)^(m - 2*n)*(d + e*x^n)^q, x] - a*f^(2*n)/c* Int[(f*x)^(m - 2*n)*(d + e*x^n)^q/(a + c*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_^n_)^q_/(a_ + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && Not[IntegerQ[q]] && GtQ[m, 2*n - 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "e*f^n/c*Int[(f*x)^(m - n)*(d + e*x^n)^(q - 1), x] - f^n/c* Int[(f*x)^(m - n)*(d + e*x^n)^(q - 1)* Simp[a*e - (c*d - b*e)*x^n, x]/(a + b*x^n + c*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_^n_)^q_/(a_ + b_.*x_^n_ + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && Not[IntegerQ[q]] && GtQ[q, 0] && GtQ[m, n - 1] && LeQ[m, 2 n - 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "e*f^n/c*Int[(f*x)^(m - n)*(d + e*x^n)^(q - 1), x] - f^n/c* Int[(f*x)^(m - n)*(d + e*x^n)^(q - 1)* Simp[a*e - c*d*x^n, x]/(a + c*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_^n_)^q_/(a_ + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && Not[IntegerQ[q]] && GtQ[q, 0] && GtQ[m, n - 1] && LeQ[m, 2 n - 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "d/a*Int[(f*x)^m*(d + e*x^n)^(q - 1), x] - 1/(a*f^n)* Int[(f*x)^(m + n)*(d + e*x^n)^(q - 1)* Simp[b*d - a*e + c*d*x^n, x]/(a + b*x^n + c*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_. + e_.*x_^n_)^q_/(a_ + b_.*x_^n_ + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && Not[IntegerQ[q]] && GtQ[q, 0] && LtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "d/a*Int[(f*x)^m*(d + e*x^n)^(q - 1), x] + 1/(a*f^n)* Int[(f*x)^(m + n)*(d + e*x^n)^(q - 1)* Simp[a*e - c*d*x^n, x]/(a + c*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_. + e_.*x_^n_)^q_/(a_ + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && Not[IntegerQ[q]] && GtQ[q, 0] && LtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "d^2*f^(2*n)/(c*d^2 - b*d*e + a*e^2)* Int[(f*x)^(m - 2*n)*(d + e*x^n)^q, x] - f^(2*n)/(c*d^2 - b*d*e + a*e^2)* Int[(f*x)^(m - 2*n)*(d + e*x^n)^(q + 1)* Simp[a*d + (b*d - a*e)*x^n, x]/(a + b*x^n + c*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_^n_)^q_/(a_ + b_.*x_^n_ + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && Not[IntegerQ[q]] && LtQ[q, -1] && GtQ[m, 2*n - 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "d^2*f^(2*n)/(c*d^2 + a*e^2)*Int[(f*x)^(m - 2*n)*(d + e*x^n)^q, x] - a*f^(2*n)/(c*d^2 + a*e^2)* Int[(f*x)^(m - 2*n)*(d + e*x^n)^(q + 1)*(d - e*x^n)/(a + c*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_^n_)^q_/(a_ + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && Not[IntegerQ[q]] && LtQ[q, -1] && GtQ[m, 2*n - 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "-d*e*f^n/(c*d^2 - b*d*e + a*e^2)* Int[(f*x)^(m - n)*(d + e*x^n)^q, x] + f^n/(c*d^2 - b*d*e + a*e^2)* Int[(f*x)^(m - n)*(d + e*x^n)^(q + 1)* Simp[a*e + c*d*x^n, x]/(a + b*x^n + c*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_^n_)^q_/(a_ + b_.*x_^n_ + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && Not[IntegerQ[q]] && LtQ[q, -1] && GtQ[m, n - 1] && LeQ[m, 2*n - 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "-d*e*f^n/(c*d^2 + a*e^2)* Int[(f*x)^(m - n)*(d + e*x^n)^q, x] + f^n/(c*d^2 + a*e^2)* Int[(f*x)^(m - n)*(d + e*x^n)^(q + 1)* Simp[a*e + c*d*x^n, x]/(a + c*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_^n_)^q_/(a_ + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && Not[IntegerQ[q]] && LtQ[q, -1] && GtQ[m, n - 1] && LeQ[m, 2*n - 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "e^2/(c*d^2 - b*d*e + a*e^2)*Int[(f*x)^m*(d + e*x^n)^q, x] + 1/(c*d^2 - b*d*e + a*e^2)* Int[(f*x)^m*(d + e*x^n)^(q + 1)* Simp[c*d - b*e - c*e*x^n, x]/(a + b*x^n + c*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)^q_/(a_ + b_.*x_^n_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && Not[IntegerQ[q]] && LtQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "e^2/(c*d^2 + a*e^2)*Int[(f*x)^m*(d + e*x^n)^q, x] + c/(c*d^2 + a*e^2)* Int[(f*x)^m*(d + e*x^n)^(q + 1)*(d - e*x^n)/(a + c*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)^q_/(a_ + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && Not[IntegerQ[q]] && LtQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x^n)^q, (f*x)^m/(a + b*x^n + c*x^(2*n)), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)^q_/(a_ + b_.*x_^n_ + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, q, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && Not[IntegerQ[q]] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x^n)^q, (f*x)^m/(a + c*x^(2*n)), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)^q_/(a_ + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, q, n}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && Not[IntegerQ[q]] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[(f*x)^m*(d + e*x^n)^q, 1/(a + b*x^n + c*x^(2*n)), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)^q_/(a_ + b_.*x_^n_ + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, q, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && Not[IntegerQ[q]] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[(f*x)^m*(d + e*x^n)^q, 1/(a + c*x^(2*n)), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)^q_/(a_ + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m, q, n}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && Not[IntegerQ[q]] && Not[IntegerQ[m]]"
  },
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "1/d^2*Int[(f*x)^ m*(a*d + (b*d - a*e)*x^n)*(a + b*x^n + c*x^(2*n))^(p - 1), x] + (c*d^2 - b*d*e + a*e^2)/(d^2*f^(2*n))* Int[(f*x)^(m + 2*n)*(a + b*x^n + c*x^(2*n))^(p - 1)/(d + e*x^n), x]",
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            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*x_^n_ + c_.*x_^n2_.)^p_./(d_. + e_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "a/d^2*Int[(f*x)^m*(d - e*x^n)*(a + c*x^(2*n))^(p - 1), x] + (c*d^2 + a*e^2)/(d^2*f^(2*n))* Int[(f*x)^(m + 2*n)*(a + c*x^(2*n))^(p - 1)/(d + e*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_ + c_.*x_^n2_.)^p_./(d_. + e_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "1/(d*e)*Int[(f*x)^m*(a*e + c*d*x^n)*(a + b*x^n + c*x^(2*n))^(p - 1), x] - (c*d^2 - b*d*e + a*e^2)/(d*e*f^n)* Int[(f*x)^(m + n)*(a + b*x^n + c*x^(2*n))^(p - 1)/(d + e*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*x_^n_ + c_.*x_^n2_.)^p_./(d_. + e_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "1/(d*e)*Int[(f*x)^m*(a*e + c*d*x^n)*(a + c*x^(2*n))^(p - 1), x] - (c*d^2 + a*e^2)/(d*e*f^n)* Int[(f*x)^(m + n)*(a + c*x^(2*n))^(p - 1)/(d + e*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_ + c_.*x_^n2_.)^p_./(d_. + e_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "-f^(2*n)/(c*d^2 - b*d*e + a*e^2)* Int[(f*x)^(m - 2*n)*(a*d + (b*d - a*e)*x^n)*(a + b*x^n + c*x^(2*n))^p, x] + d^2*f^(2*n)/(c*d^2 - b*d*e + a*e^2)* Int[(f*x)^(m - 2*n)*(a + b*x^n + c*x^(2*n))^(p + 1)/(d + e*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(a_. + b_.*x_^n_ + c_.*x_^n2_.)^p_/(d_. + e_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m, n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "-a*f^(2*n)/(c*d^2 + a*e^2)* Int[(f*x)^(m - 2*n)*(d - e*x^n)*(a + c*x^(2*n))^p, x] + d^2*f^(2*n)/(c*d^2 + a*e^2)* Int[(f*x)^(m - 2*n)*(a + c*x^(2*n))^(p + 1)/(d + e*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(a_ + c_.*x_^n2_.)^p_/(d_. + e_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m, n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "f^n/(c*d^2 - b*d*e + a*e^2)* Int[(f*x)^(m - n)*(a*e + c*d*x^n)*(a + b*x^n + c*x^(2*n))^p, x] - d*e*f^n/(c*d^2 - b*d*e + a*e^2)* Int[(f*x)^(m - n)*(a + b*x^n + c*x^(2*n))^(p + 1)/(d + e*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(a_. + b_.*x_^n_ + c_.*x_^n2_.)^p_/(d_. + e_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "f^n/(c*d^2 + a*e^2)* Int[(f*x)^(m - n)*(a*e + c*d*x^n)*(a + c*x^(2*n))^p, x] - d*e*f^n/(c*d^2 + a*e^2)* Int[(f*x)^(m - n)*(a + c*x^(2*n))^(p + 1)/(d + e*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(a_ + c_.*x_^n2_.)^p_/(d_. + e_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*x^n + c*x^(2*n))^p, (f*x)^m (d + e*x^n)^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && (IGtQ[q, 0] || IntegersQ[m, q])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[(a + c*x^(2*n))^p, (f*x)^m (d + e*x^n)^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IGtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "-Subst[ Int[(d + e*x^(-n))^q*(a + b*x^(-n) + c*x^(-2*n))^p/x^(m + 2), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && ILtQ[n, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "-Subst[ Int[(d + e*x^(-n))^q*(a + c*x^(-2*n))^p/x^(m + 2), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, p, q}, x] && EqQ[n2, 2*n] && ILtQ[n, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{g = Denominator[m]}, -g/f* Subst[Int[(d + e*f^(-n)*x^(-g*n))^ q*(a + b*f^(-n)*x^(-g*n) + c*f^(-2*n)*x^(-2*g*n))^p/ x^(g*(m + 1) + 1), x], x, 1/(f*x)^(1/g)]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^ p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && ILtQ[n, 0] && FractionQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{g = Denominator[m]}, -g/f* Subst[Int[(d + e*f^(-n)*x^(-g*n))^q*(a + c*f^(-2*n)*x^(-2*g*n))^p/ x^(g*(m + 1) + 1), x], x, 1/(f*x)^(1/g)]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, p, q}, x] && EqQ[n2, 2*n] && ILtQ[n, 0] && FractionQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "-f^IntPart[m]*(f*x)^FracPart[m]*(x^(-1))^ FracPart[m]* Subst[Int[(d + e*x^(-n))^q*(a + b*x^(-n) + c*x^(-2*n))^p/x^(m + 2), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^n_)^q_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^ p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && ILtQ[n, 0] && Not[RationalQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "-f^IntPart[m]*(f*x)^FracPart[m]*(x^(-1))^ FracPart[m]* Subst[Int[(d + e*x^(-n))^q*(a + c*x^(-2*n))^p/x^(m + 2), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^n_)^q_.*(a_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m, p, q}, x] && EqQ[n2, 2*n] && ILtQ[n, 0] && Not[RationalQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{g = Denominator[n]}, g*Subst[ Int[x^(g*(m + 1) - 1)*(d + e*x^(g*n))^ q*(a + b*x^(g*n) + c*x^(2*g*n))^p, x], x, x^(1/g)]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{g = Denominator[n]}, g*Subst[ Int[x^(g*(m + 1) - 1)*(d + e*x^(g*n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m, p, q}, x] && EqQ[n2, 2*n] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "f^IntPart[m]*(f*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_*x_)^m_*(d_ + e_.*x_^n_)^q_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "f^IntPart[m]*(f*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(d + e*x^n)^q*(a + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_*x_)^m_*(d_ + e_.*x_^n_)^q_.*(a_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m, p, q}, x] && EqQ[n2, 2*n] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "1/(m + 1)* Subst[Int[(d + e*x^Simplify[n/(m + 1)])^ q*(a + b*x^Simplify[n/(m + 1)] + c*x^Simplify[2*n/(m + 1)])^p, x], x, x^(m + 1)]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[Simplify[n/(m + 1)]] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "1/(m + 1)* Subst[Int[(d + e*x^Simplify[n/(m + 1)])^ q*(a + c*x^Simplify[2*n/(m + 1)])^p, x], x, x^(m + 1)]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && IntegerQ[Simplify[n/(m + 1)]] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "f^IntPart[m]*(f*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_*x_)^m_*(d_ + e_.*x_^n_)^q_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[Simplify[n/(m + 1)]] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "f^IntPart[m]*(f*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(d + e*x^n)^q*(a + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_*x_)^m_*(d_ + e_.*x_^n_)^q_.*(a_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m, p, q}, x] && EqQ[n2, 2*n] && IntegerQ[Simplify[n/(m + 1)]] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{r = Rt[b^2 - 4*a*c, 2]}, 2*c/r*Int[(f*x)^m*(d + e*x^n)^q/(b - r + 2*c*x^n), x] - 2*c/r*Int[(f*x)^m*(d + e*x^n)^q/(b + r + 2*c*x^n), x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)^q_/(a_ + b_.*x_^n_ + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{r = Rt[-a*c, 2]}, -c/(2*r)*Int[(f*x)^m*(d + e*x^n)^q/(r - c*x^n), x] - c/(2*r)*Int[(f*x)^m*(d + e*x^n)^q/(r + c*x^n), x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)^q_/(a_ + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m, n, q}, x] && EqQ[n2, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "-(f*x)^(m + 1)*(a + b*x^n + c*x^(2*n))^(p + 1)*(d*(b^2 - 2*a*c) - a*b*e + (b*d - 2*a*e)*c*x^n)/(a*f* n*(p + 1)*(b^2 - 4*a*c)) + 1/(a*n*(p + 1)*(b^2 - 4*a*c))* Int[(f*x)^m*(a + b*x^n + c*x^(2*n))^(p + 1)* Simp[d*(b^2*(m + n*(p + 1) + 1) - 2*a*c*(m + 2*n*(p + 1) + 1)) - a*b*e*(m + 1) + (m + n*(2*p + 3) + 1)*(b*d - 2*a*e)*c*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)*(a_ + b_.*x_^n_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && ILtQ[p + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "-(f*x)^(m + 1)*(a + c*x^(2*n))^(p + 1)*(d + e*x^n)/(2*a*f*n*(p + 1)) + 1/(2*a*n*(p + 1))* Int[(f*x)^m*(a + c*x^(2*n))^(p + 1)* Simp[d*(m + 2*n*(p + 1) + 1) + e*(m + n*(2*p + 3) + 1)*x^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)*(a_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m, n}, x] && EqQ[n2, 2*n] && ILtQ[p + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[(f*x)^m*(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && (IGtQ[p, 0] || IGtQ[q, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[(f*x)^m*(d + e*x^n)^q*(a + c*x^(2*n))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m, n, p, q}, x] && EqQ[n2, 2*n] && (IGtQ[p, 0] || IGtQ[q, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(f*x)^m/x^m* Int[ExpandIntegrand[ x^m*(a + c*x^(2*n))^ p, (d/(d^2 - e^2*x^(2*n)) - e*x^n/(d^2 - e^2*x^(2*n)))^(-q), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)^q_*(a_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m, n, p}, x] && EqQ[n2, 2*n] && Not[IntegerQ[p]] && ILtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Unintegrable[(f*x)^m*(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p, q}, x] && EqQ[n2, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Unintegrable[(f*x)^m*(d + e*x^n)^q*(a + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m, n, p, q}, x] && EqQ[n2, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "u^m/(Coefficient[v, x, 1]*v^m)* Subst[Int[x^m*(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x, v]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*(d_ + e_.*v_^n_)^q_.*(a_ + b_.*v_^n_ + c_.*v_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && LinearPairQ[u, v, x] && NeQ[v, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "u^m/(Coefficient[v, x, 1]*v^m)* Subst[Int[x^m*(d + e*x^n)^q*(a + c*x^(2*n))^p, x], x, v]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*(d_ + e_.*v_^n_)^q_.*(a_ + c_.*v_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && LinearPairQ[u, v, x] && NeQ[v, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[x^(m - n*q)*(e + d*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^mn_.)^q_.*(a_. + b_.*x_^n_. + c_.*x_^n2_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] || Not[IntegerQ[p]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[x^(m + mn*q)*(e + d*x^(-mn))^q*(a + c*x^n2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^mn_.)^q_.*(a_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m, mn, p}, x] && EqQ[n2, -2*mn] && IntegerQ[q] && (PosQ[n2] || Not[IntegerQ[p]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[x^(m - 2*n*p)*(d + e*x^n)^q*(c + b*x^n + a*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^n_.)^q_.*(a_. + b_.*x_^mn_. + c_.*x_^mn2_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, q}, x] && EqQ[mn, -n] && EqQ[mn2, 2*mn] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[x^(m - 2*n*p)*(d + e*x^n)^q*(c + a*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^n_.)^q_.*(a_. + c_.*x_^mn2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m, n, q}, x] && EqQ[mn2, -2*n] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "e^IntPart[q]* x^(n*FracPart[q])*(d + e*x^(-n))^FracPart[q]/(1 + d*x^n/e)^ FracPart[q]* Int[x^(m - n*q)*(1 + d*x^n/e)^q*(a + b*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^mn_.)^q_*(a_. + b_.*x_^n_. + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[mn, -n] && Not[IntegerQ[p]] && Not[IntegerQ[q]] && PosQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "e^IntPart[q]* x^(-mn*FracPart[q])*(d + e*x^mn)^FracPart[q]/(1 + d*x^(-mn)/e)^ FracPart[q]* Int[x^(m + mn*q)*(1 + d*x^(-mn)/e)^q*(a + c*x^n2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^mn_.)^q_*(a_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m, mn, p, q}, x] && EqQ[n2, -2*mn] && Not[IntegerQ[p]] && Not[IntegerQ[q]] && PosQ[n2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": " x^(n*FracPart[q])*(d+e*x^(-n))^FracPart[q]/(e+d*x^n)^FracPart[q]*Int[ x^(m-n*q)*(e+d*x^n)^q*(a+b*x^n+c*x^(2*n))^p,x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_+e_.*x_^mn_.)^q_*(a_.+b_.*x_^n_.+c_.*x_^n2_.)^p_.,x_ Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e,m,n,p,q},x] && EqQ[n2,2*n] && EqQ[mn,-n] &&  Not[IntegerQ[p]] && Not[IntegerQ[q]] && PosQ[n] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": " x^(-mn*FracPart[q])*(d+e*x^mn)^FracPart[q]/(e+d*x^(-mn))^FracPart[q]* Int[x^(m+mn*q)*(e+d*x^(-mn))^q*(a+c*x^n2)^p,x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_+e_.*x_^mn_.)^q_*(a_+c_.*x_^n2_.)^p_.,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,c,d,e,m,mn,p,q},x] && EqQ[n2,-2*mn] && Not[IntegerQ[p]] &&  Not[IntegerQ[q]] && PosQ[n2] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "x^(2*n*FracPart[p])*(a + b*x^(-n) + c*x^(-2*n))^ FracPart[p]/(c + b*x^n + a*x^(2*n))^FracPart[p]* Int[x^(m - 2*n*p)*(d + e*x^n)^q*(c + b*x^n + a*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^n_.)^q_.*(a_. + b_.*x_^mn_. + c_.*x_^mn2_.)^ p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p, q}, x] && EqQ[mn, -n] && EqQ[mn2, 2*mn] && Not[IntegerQ[p]] && Not[IntegerQ[q]] && PosQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "x^(2*n*FracPart[p])*(a + c*x^(-2*n))^FracPart[p]/(c + a*x^(2*n))^ FracPart[p]* Int[x^(m - 2*n*p)*(d + e*x^n)^q*(c + a*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^n_.)^q_.*(a_. + c_.*x_^mn2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[mn2, -2*n] && Not[IntegerQ[p]] && Not[IntegerQ[q]] && PosQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "f^IntPart[m]*(f*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(d + e*x^mn)^q*(a + b*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_*x_)^m_*(d_ + e_.*x_^mn_.)^ q_.*(a_. + b_.*x_^n_. + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[mn, -n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "f^IntPart[m]*(f*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(d + e*x^mn)^q*(a + c*x^n2)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_*x_)^m_*(d_ + e_.*x_^mn_.)^q_.*(a_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, m, mn, p, q}, x] && EqQ[n2, -2*mn]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[x^(m - n*p)*(d + e*x^n)^q*(b + a*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + b_.*x_^mn_ + c_.*x_^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, q}, x] && EqQ[mn, -n] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "x^(n*FracPart[p])*(a + b/x^n + c*x^n)^ FracPart[p]/(b + a*x^n + c*x^(2*n))^FracPart[p]* Int[x^(m - n*p)*(d + e*x^n)^q*(b + a*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + b_.*x_^mn_ + c_.*x_^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p, q}, x] && EqQ[mn, -n] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "f^IntPart[m]*(f*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(d + e*x^n)^q*(a + b*x^(-n) + c*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_*x_)^m_.*(d_ + e_.*x_^n_)^q_.*(a_ + b_.*x_^mn_ + c_.*x_^n_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p, q}, x] && EqQ[mn, -n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[(f*x)^m*(d1*d2 + e1*e2*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d1_ + e1_.*x_^non2_.)^q_.*(d2_ + e2_.*x_^non2_.)^ q_.*(a_. + b_.*x_^n_ + c_.*x_^n2_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[non2, n/2] && EqQ[d2*e1 + d1*e2, 0] && (IntegerQ[q] || GtQ[d1, 0] && GtQ[d2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.4 (f x)^m (d+e x^n)^q (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(d1 + e1*x^(n/2))^ FracPart[q]*(d2 + e2*x^(n/2))^FracPart[q]/(d1*d2 + e1*e2*x^n)^ FracPart[q]* Int[(f*x)^m*(d1*d2 + e1*e2*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d1_ + e1_.*x_^non2_.)^q_.*(d2_ + e2_.*x_^non2_.)^ q_.*(a_. + b_.*x_^n_ + c_.*x_^n2_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[non2, n/2] && EqQ[d2*e1 + d1*e2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[Pq*(a + b*x^n + c*x^(2*n))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^n_. + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
            "rhs": "d*x*(a + b*x^n + c*x^(2*n))^(p + 1)/a",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_. + f_.*x_^n2_.)*(a_ + b_.*x_^n_. + c_.*x_^n2_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[n2, 2*n] && EqQ[a*e - b*d*(n*(p + 1) + 1), 0] && EqQ[a*f - c*d*(2*n*(p + 1) + 1), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
            "rhs": "d*x*(a + b*x^n + c*x^(2*n))^(p + 1)/a",
     "rulenumber": 0,
            "lhs": "Int[(d_ + f_.*x_^n2_.)*(a_ + b_.*x_^n_. + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, n, p}, x] && EqQ[n2, 2*n] && EqQ[n*(p + 1) + 1, 0] && EqQ[c*d + a*f, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(a + b*x^n + c*x^(2*n))^ FracPart[p]/((4*c)^IntPart[p]*(b + 2*c*x^n)^(2*FracPart[p]))* Int[Pq*(b + 2*c*x^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^n_. + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[2*p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[x*PolynomialQuotient[Pq, x, x]*(a + b*x^n + c*x^(2*n))^p, x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && EqQ[Coeff[Pq, x, 0], 0] && Not[MatchQ[Pq, x^m_.*u_.",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^n_. + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "IntegerQ[m]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
            "rhs": "x*(a*d*(n + 1) + (a*e - b*d*(n*(p + 1) + 1))* x^n)*(a + b*x^n + c*x^(2*n))^(p + 1)/(a^2*(n + 1))",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_ + f_.*x_^n2_. + g_.*x_^n3_.)*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[n2, 2*n] && EqQ[n3, 3*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[a^2*g*(n + 1) - c*(n*(2*p + 3) + 1)*(a*e - b*d*(n*(p + 1) + 1)), 0] && EqQ[a^2*f*(n + 1) - a*c*d*(n + 1)*(2*n*(p + 1) + 1) - b*(n*(p + 2) + 1)*(a*e - b*d*(n*(p + 1) + 1)), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
            "rhs": "d*x*(a*(n + 1) - b*(n*(p + 1) + 1)* x^n)*(a + b*x^n + c*x^(2*n))^(p + 1)/(a^2*(n + 1))",
     "rulenumber": 0,
            "lhs": "Int[(d_ + f_.*x_^n2_. + g_.*x_^n3_.)*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, g, n, p}, x] && EqQ[n2, 2*n] && EqQ[n3, 3*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[a^2*g*(n + 1) + c*b*d*(n*(2*p + 3) + 1)*(n*(p + 1) + 1), 0] && EqQ[a^2*f*(n + 1) - a*c*d*(n + 1)*(2*n*(p + 1) + 1) + b^2*d*(n*(p + 2) + 1)*(n*(p + 1) + 1), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
            "rhs": "x*(a*d*(n + 1) + (a*e - b*d*(n*(p + 1) + 1))* x^n)*(a + b*x^n + c*x^(2*n))^(p + 1)/(a^2*(n + 1))",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^n_ + g_.*x_^n3_.)*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, g, n, p}, x] && EqQ[n2, 2*n] && EqQ[n3, 3*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[a^2*g*(n + 1) - c*(n*(2*p + 3) + 1)*(a*e - b*d*(n*(p + 1) + 1)), 0] && EqQ[a*c*d*(n + 1)*(2*n*(p + 1) + 1) + b*(n*(p + 2) + 1)*(a*e - b*d*(n*(p + 1) + 1)), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
            "rhs": "d*x*(a*(n + 1) - b*(n*(p + 1) + 1)* x^n)*(a + b*x^n + c*x^(2*n))^(p + 1)/(a^2*(n + 1))",
     "rulenumber": 0,
            "lhs": "Int[(d_ + g_.*x_^n3_.)*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, g, n, p}, x] && EqQ[n2, 2*n] && EqQ[n3, 3*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[a^2*g*(n + 1) + c*b*d*(n*(2*p + 3) + 1)*(n*(p + 1) + 1), 0] && EqQ[a*c*d*(n + 1)*(2*n*(p + 1) + 1) - b^2*d*(n*(p + 2) + 1)*(n*(p + 1) + 1), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Module[{q = Expon[Pq, x], i}, -x*(a + b*x^n + c*x^(2*n))^(p + 1)/(a*n*(p + 1)*(b^2 - 4*a*c))* Sum[((b^2 - 2*a*c)*Coeff[Pq, x, i] - a*b*Coeff[Pq, x, n + i])* x^i + c*(b*Coeff[Pq, x, i] - 2*a*Coeff[Pq, x, n + i])*x^(n + i), {i, 0, n - 1}] + 1/(a*n*(p + 1)*(b^2 - 4*a*c))* Int[(a + b*x^n + c*x^(2*n))^(p + 1)* Sum[((b^2*(n*(p + 1) + i + 1) - 2*a*c*(2*n*(p + 1) + i + 1))* Coeff[Pq, x, i] - a*b*(i + 1)*Coeff[Pq, x, n + i])* x^i + c*(n*(2*p + 3) + i + 1)*(b*Coeff[Pq, x, i] - 2*a*Coeff[Pq, x, n + i])*x^(n + i), {i, 0, n - 1}], x] /; LtQ[q, 2*n]]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^n_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[(b*c)^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n + c*x^(2*n), x], R = PolynomialRemainder[(b*c)^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n + c*x^(2*n), x], i}, -x*(a + b*x^n + c*x^(2*n))^(p + 1)/(a* n*(p + 1)*(b^2 - 4*a*c)*(b*c)^(Floor[(q - 1)/n] + 1))* Sum[((b^2 - 2*a*c)*Coeff[R, x, i] - a*b*Coeff[R, x, n + i])* x^i + c*(b*Coeff[R, x, i] - 2*a*Coeff[R, x, n + i])*x^(n + i), {i, 0, n - 1}] + 1/(a*n*(p + 1)*(b^2 - 4*a*c)*(b*c)^(Floor[(q - 1)/n] + 1))* Int[(a + b*x^n + c*x^(2*n))^(p + 1)* ExpandToSum[a*n*(p + 1)*(b^2 - 4*a*c)*Q + Sum[((b^2*(n*(p + 1) + i + 1) - 2*a*c*(2*n*(p + 1) + i + 1))*Coeff[R, x, i] - a*b*(i + 1)*Coeff[R, x, n + i])*x^i + c*(n*(2*p + 3) + i + 1)*(b*Coeff[R, x, i] - 2*a*Coeff[R, x, n + i])*x^(n + i), {i, 0, n - 1}], x], x]] /; GeQ[q, 2*n]]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^n_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[Pq/(a + b*x^n + c*x^(2*n)), x], x]",
     "rulenumber": 0,
            "lhs": "Int[Pq_/(a_ + b_.*x_^n_. + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x^n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && (NiceSqrtQ[b^2 - 4*a*c] || LtQ[Expon[Pq, x], n])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Expon[Pq, x]}, With[{Pqq = Coeff[Pq, x, q]}, c^p*Pqq*Log[a + b*x + c*x^2]/2 + 1/2*Int[ExpandToSum[ 2*Pq - c^p*Pqq*(b + 2*c*x)/(a + b*x + c*x^2)^(p + 1), x]*(a + b*x + c*x^2)^p, x]] /; EqQ[q + 2*p + 1, 0]]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && ILtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Expon[Pq, x]}, With[{Pqq = Coeff[Pq, x, q]}, c^p*Pqq* ArcTanh[(b + 2*c*x)/(2*Rt[c, 2]*Sqrt[a + b*x + c*x^2])] + Int[ExpandToSum[ Pq - c^(p + 1/2)*Pqq/(a + b*x + c*x^2)^(p + 1/2), x]*(a + b*x + c*x^2)^p, x]] /; EqQ[q + 2*p + 1, 0]]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && ILtQ[p + 1/2, 0] && PosQ[c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Expon[Pq, x]}, With[{Pqq = Coeff[Pq, x, q]}, -(-c)^p*Pqq* ArcTan[(b + 2*c*x)/(2*Rt[-c, 2]*Sqrt[a + b*x + c*x^2])] + Int[ExpandToSum[ Pq - (-c)^(p + 1/2)*Pqq/(a + b*x + c*x^2)^(p + 1/2), x]*(a + b*x + c*x^2)^p, x]] /; EqQ[q + 2*p + 1, 0]]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_ + c_.*x_^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && ILtQ[p + 1/2, 0] && NegQ[c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Expon[Pq, x]}, With[{Pqq = Coeff[Pq, x, q]}, Pqq* x^(q - 2*n + 1)*(a + b*x^n + c*x^(2*n))^(p + 1)/(c*(q + 2*n*p + 1)) + Int[ExpandToSum[ Pq - Pqq*x^q - Pqq*(a*(q - 2*n + 1)*x^(q - 2*n) + b*(q + n*(p - 1) + 1)*x^(q - n))/(c*(q + 2*n*p + 1)), x]*(a + b*x^n + c*x^(2*n))^p, x]] /; GeQ[q, 2*n] && NeQ[q + 2*n*p + 1, 0] && (IntegerQ[2*p] || EqQ[n, 1] && IntegerQ[4*p] || IntegerQ[p + (q + 1)/(2*n)])]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^n_. + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x^n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Module[{q = Expon[Pq, x], j, k}, Int[Sum[ x^j*Sum[Coeff[Pq, x, j + k*n]*x^(k*n), {k, 0, (q - j)/n + 1}]*(a + b*x^n + c*x^(2*n))^p, {j, 0, n - 1}], x]]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^n_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && Not[PolyQ[Pq, x^n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[RationalFunctionExpand[Pq/(a + b*x^n + c*x^(2*n)), x], x]",
     "rulenumber": 0,
            "lhs": "Int[Pq_/(a_ + b_.*x_^n_. + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{g = Denominator[n]}, g*Subst[ Int[x^(g - 1)* ReplaceAll[Pq, x -> x^g]*(a + b*x^(g*n) + c*x^(2*g*n))^p, x], x, x^(1/g)]]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, 2*c/q*Int[Pq/(b - q + 2*c*x^n), x] - 2*c/q*Int[Pq/(b + q + 2*c*x^n), x]]",
     "rulenumber": 0,
            "lhs": "Int[Pq_/(a_ + b_.*x_^n_. + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{d = Coeff[P3, x^n, 0], e = Coeff[P3, x^n, 1], f = Coeff[P3, x^n, 2], g = Coeff[P3, x^n, 3]}, -x*(b^2*c*d - 2*a*c*(c*d - a*f) - a*b*(c*e + a*g) + (b*c*(c*d + a*f) - a*b^2*g - 2*a*c*(c*e - a*g))*x^n)*(a + b*x^n + c*x^(2*n))^(p + 1)/ (a*c*n*(p + 1)*(b^2 - 4*a*c)) - 1/(a*c*n*(p + 1)*(b^2 - 4*a*c))* Int[(a + b*x^n + c*x^(2*n))^(p + 1)* Simp[a*b*(c*e + a*g) - b^2*c*d*(n + n*p + 1) - 2*a*c*(a*f - c*d*(2*n*(p + 1) + 1)) + (a*b^2*g*(n*(p + 2) + 1) - b*c*(c*d + a*f)*(n*(2*p + 3) + 1) - 2*a*c*(a*g*(n + 1) - c*e*(n*(2*p + 3) + 1)))*x^n, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[P3_*(a_ + b_.*x_^n_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && PolyQ[P3, x^n, 3] && NeQ[b^2 - 4*a*c, 0] && ILtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{d = Coeff[P2, x^n, 0], e = Coeff[P2, x^n, 1], f = Coeff[P2, x^n, 2]}, -x*(b^2*d - 2*a*(c*d - a*f) - a*b*e + (b*(c*d + a*f) - 2*a*c*e)* x^n)*(a + b*x^n + c*x^(2*n))^(p + 1)/(a* n*(p + 1)*(b^2 - 4*a*c)) - 1/(a*n*(p + 1)*(b^2 - 4*a*c))* Int[(a + b*x^n + c*x^(2*n))^(p + 1)* Simp[a*b*e - b^2*d*(n + n*p + 1) - 2*a*(a*f - c*d*(2*n*(p + 1) + 1)) - (b*(c*d + a*f)*(n*(2*p + 3) + 1) - 2*a*c*e*(n*(2*p + 3) + 1))*x^n, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[P2_*(a_ + b_.*x_^n_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && PolyQ[P2, x^n, 2] && NeQ[b^2 - 4*a*c, 0] && ILtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[Pq*(a + b*x^n + c*x^(2*n))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^n_. + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && ILtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Unintegrable[Pq*(a + b*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*x_^n_. + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && (PolyQ[Pq, x] || PolyQ[Pq, x^n])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.5 P(x) (a+b x^n+c x^(2 n))^p.m",
            "rhs": "1/Coefficient[v, x, 1]* Subst[Int[SubstFor[v, Pq, x]*(a + b*x^n + c*x^(2*n))^p, x], x, v]",
     "rulenumber": 0,
            "lhs": "Int[Pq_*(a_ + b_.*v_^n_ + c_.*v_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && LinearQ[v, x] && PolyQ[Pq, v^n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "1/n*Subst[Int[SubstFor[x^n, Pq, x]*(a + b*x + c*x^2)^p, x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Pq_*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x^n] && EqQ[Simplify[m - n + 1], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[(d*x)^m*Pq*(a + b*x^n + c*x^(2*n))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*Pq_*(a_ + b_.*x_^n_. + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "d*(g*x)^(m + 1)*(a + b*x^n + c*x^(2*n))^(p + 1)/(a*g*(m + 1))",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^ m_.*(d_ + e_.*x_^n_. + f_.*x_^n2_.)*(a_ + b_.*x_^n_. + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[a*e*(m + 1) - b*d*(m + n*(p + 1) + 1), 0] && EqQ[a*f*(m + 1) - c*d*(m + 2*n*(p + 1) + 1), 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "d*(g*x)^(m + 1)*(a + b*x^n + c*x^(2*n))^(p + 1)/(a*g*(m + 1))",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^m_.*(d_ + f_.*x_^n2_.)*(a_ + b_.*x_^n_. + c_.*x_^n2_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, g, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[m + n*(p + 1) + 1, 0] && EqQ[c*d + a*f, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(a + b*x^n + c*x^(2*n))^ FracPart[p]/((4*c)^IntPart[p]*(b + 2*c*x^n)^(2*FracPart[p]))* Int[(d*x)^m*Pq*(b + 2*c*x^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*Pq_*(a_ + b_.*x_^n_. + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[2*p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)* SubstFor[x^n, Pq, x]*(a + b*x + c*x^2)^p, x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Pq_*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x^n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(d*x)^m/x^m* Int[x^m*Pq*(a + b*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_*x_)^m_.*Pq_*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x^n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "1/d*Int[(d*x)^(m + 1)* PolynomialQuotient[Pq, x, x]*(a + b*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*Pq_*(a_ + b_.*x_^n_. + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && EqQ[Coeff[Pq, x, 0], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "-(2*c*(b*f - 2*a*g) + 2*h*(b^2 - 4*a*c)*x^(n/2) + 2*c*(2*c*f - b*g)*x^n)/(c*n*(b^2 - 4*a*c)* Sqrt[a + b*x^n + c*x^(2*n)])",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(e_ + f_.*x_^q_. + g_.*x_^r_. + h_.*x_^s_.)/(a_ + b_.*x_^n_. + c_.*x_^n2_.)^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, e, f, g, h, m, n}, x] && EqQ[n2, 2*n] && EqQ[q, n/2] && EqQ[r, 3*n/2] && EqQ[s, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*m - n + 2, 0] && EqQ[c*e + a*h, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(d*x)^m/x^m* Int[x^m*(e + f*x^(n/2) + g*x^((3*n)/2) + h*x^(2*n))/(a + b*x^n + c*x^(2*n))^(3/2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_*x_)^ m_.*(e_ + f_.*x_^q_. + g_.*x_^r_. + h_.*x_^s_.)/(a_ + b_.*x_^n_. + c_.*x_^n2_.)^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[n2, 2*n] && EqQ[q, n/2] && EqQ[r, 3*n/2] && EqQ[s, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*m - n + 2, 0] && EqQ[c*e + a*h, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Module[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[a*(b*c)^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n + c*x^(2*n), x], R = PolynomialRemainder[a*(b*c)^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n + c*x^(2*n), x], i}, -x*(a + b*x^n + c*x^(2*n))^(p + 1)/(a^2* n*(p + 1)*(b^2 - 4*a*c)*(b*c)^(Floor[(q - 1)/n] + 1))* Sum[((b^2 - 2*a*c)*Coeff[R, x, i] - a*b*Coeff[R, x, n + i])* x^i + c*(b*Coeff[R, x, i] - 2*a*Coeff[R, x, n + i])*x^(n + i), {i, 0, n - 1}] + 1/(a*n*(p + 1)*(b^2 - 4*a*c)*(b*c)^(Floor[(q - 1)/n] + 1))* Int[x^m*(a + b*x^n + c*x^(2*n))^(p + 1)* ExpandToSum[n*(p + 1)*(b^2 - 4*a*c)*x^(-m)*Q + Sum[((b^2*(n*(p + 1) + i + 1)/a - 2*c*(2*n*(p + 1) + i + 1))*Coeff[R, x, i] - b*(i + 1)*Coeff[R, x, n + i])*x^(i - m) + c*(n*(2*p + 3) + i + 1)*(b/a*Coeff[R, x, i] - 2*Coeff[R, x, n + i])*x^(n + i - m), {i, 0, n - 1}], x], x]] /; GeQ[q, 2*n]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*Pq_*(a_ + b_.*x_^n_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{g = GCD[m + 1, n]}, 1/g*Subst[ Int[x^((m + 1)/g - 1)* ReplaceAll[Pq, x -> x^(1/g)]*(a + b*x^(n/g) + c*x^(2*n/g))^p, x], x, x^g] /; NeQ[g, 1]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Pq_*(a_ + b_.*x_^n_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x^n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[(d*x)^m*Pq/(a + b*x^n + c*x^(2*n)), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*Pq_/(a_ + b_.*x_^n_. + c_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x^n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && NiceSqrtQ[b^2 - 4*a*c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Expon[Pq, x]}, With[{Pqq = Coeff[Pq, x, q]}, Pqq*(d*x)^(m + q - 2*n + 1)*(a + b*x^n + c*x^(2*n))^(p + 1)/(c* d^(q - 2*n + 1)*(m + q + 2*n*p + 1)) + Int[(d*x)^m* ExpandToSum[ Pq - Pqq*x^q - Pqq*(a*(m + q - 2*n + 1)*x^(q - 2*n) + b*(m + q + n*(p - 1) + 1)* x^(q - n))/(c*(m + q + 2*n*p + 1)), x]* (a + b*x^n + c*x^(2*n))^p, x]] /; GeQ[q, 2*n] && NeQ[m + q + 2*n*p + 1, 0] && (IntegerQ[2*p] || EqQ[n, 1] && IntegerQ[4*p] || IntegerQ[p + (q + 1)/(2*n)])]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*Pq_*(a_ + b_.*x_^n_. + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x^n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Module[{q = Expon[Pq, x], j, k}, Int[Sum[ 1/d^j*(d*x)^(m + j)* Sum[Coeff[Pq, x, j + k*n]*x^(k*n), {k, 0, (q - j)/n + 1}]*(a + b*x^n + c*x^(2*n))^p, {j, 0, n - 1}], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*Pq_*(a_ + b_.*x_^n_ + c_.*x_^n2_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && Not[PolyQ[Pq, x^n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[RationalFunctionExpand[(d*x)^m*Pq/(a + b*x^n + c*x^(2*n)), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*Pq_/(a_ + b_.*x_^n_. + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Expon[Pq, x]}, -Subst[ Int[ExpandToSum[x^q*ReplaceAll[Pq, x -> x^(-1)], x]*(a + b*x^(-n) + c*x^(-2*n))^p/x^(m + q + 2), x], x, 1/x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Pq_*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && ILtQ[n, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{g = Denominator[m], q = Expon[Pq, x]}, -g/d* Subst[Int[ ExpandToSum[x^(g*q)*ReplaceAll[Pq, x -> d^(-1)*x^(-g)], x]* (a + b*d^(-n)*x^(-g*n) + c*d^(-2*n)*x^(-2*g*n))^p/ x^(g*(m + q + 1) + 1), x], x, 1/(d*x)^(1/g)]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*Pq_*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && ILtQ[n, 0] && FractionQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Expon[Pq, x]}, -(d*x)^m*(x^(-1))^m* Subst[Int[ ExpandToSum[x^q*ReplaceAll[Pq, x -> x^(-1)], x]*(a + b*x^(-n) + c*x^(-2*n))^p/x^(m + q + 2), x], x, 1/x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_*Pq_*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && ILtQ[n, 0] && Not[RationalQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{g = Denominator[n]}, g*Subst[ Int[x^(g*(m + 1) - 1)* ReplaceAll[Pq, x -> x^g]*(a + b*x^(g*n) + c*x^(2*g*n))^p, x], x, x^(1/g)]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Pq_*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "d^(m - 1/2)*Sqrt[d*x]/Sqrt[x]* Int[x^m*Pq*(a + b*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_*x_)^m_*Pq_*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && FractionQ[n] && IGtQ[m + 1/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "d^(m + 1/2)*Sqrt[x]/Sqrt[d*x]* Int[x^m*Pq*(a + b*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_*x_)^m_*Pq_*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && FractionQ[n] && ILtQ[m - 1/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(d*x)^m/x^m* Int[x^m*Pq*(a + b*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_*x_)^m_*Pq_*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "1/(m + 1)* Subst[Int[ ReplaceAll[SubstFor[x^n, Pq, x], x -> x^Simplify[n/(m + 1)]]*(a + b*x^Simplify[n/(m + 1)] + c*x^Simplify[2*n/(m + 1)])^p, x], x, x^(m + 1)]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Pq_*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x^n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[Simplify[n/(m + 1)]] && Not[IntegerQ[n]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "(d*x)^m/x^m* Int[x^m*Pq*(a + b*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_*x_)^m_*Pq_*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x^n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[Simplify[n/(m + 1)]] && Not[IntegerQ[n]]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, 2*c/q*Int[(d*x)^m*Pq/(b - q + 2*c*x^n), x] - 2*c/q*Int[(d*x)^m*Pq/(b + q + 2*c*x^n), x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*Pq_/(a_ + b_.*x_^n_. + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Int[ExpandIntegrand[(d*x)^m*Pq*(a + b*x^n + c*x^(2*n))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*Pq_*(a_ + b_.*x_^n_. + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && ILtQ[p + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "Unintegrable[(d*x)^m*Pq*(a + b*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*Pq_*(a_ + b_.*x_^n_. + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && (PolyQ[Pq, x] || PolyQ[Pq, x^n])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.3 General/1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
       "filename": "1.2.3.6 P(x) (d x)^m (a+b x^n+c x^(2 n))^p.m",
            "rhs": "u^m/(Coefficient[v, x, 1]*v^m)* Subst[Int[x^m*SubstFor[v, Pq, x]*(a + b*x^n + c*x^(2*n))^p, x], x, v]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*Pq_*(a_ + b_.*v_^n_ + c_.*v_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && LinearPairQ[u, v, x] && PolyQ[Pq, v^n]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.1 (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.1 (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "Int[((a + b + c)*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*x_^q_. + b_.*x_^n_. + c_.*x_^r_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, p}, x] && EqQ[n, q] && EqQ[r, n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.1 (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.1 (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "Int[x^(p*q)*(a + b*x^(n - q) + c*x^(2*(n - q)))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*x_^q_. + b_.*x_^n_. + c_.*x_^r_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, q}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && IntegerQ[p]"
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            "rhs": "-x^(-q + 1)*(b^2 - 2*a*c + b*c*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1)/(a*(n - q)*(p + 1)*(b^2 - 4*a*c)) + 1/(a*(n - q)*(p + 1)*(b^2 - 4*a*c))* Int[ x^(-q)*((p*q + 1)*(b^2 - 2*a*c) + (n - q)*(p + 1)*(b^2 - 4*a*c) + b*c*(p*q + (n - q)*(2*p + 3) + 1)*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1), x]",
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            "rhs": "Unintegrable[(a*x^q + b*x^n + c*x^(2*n - q))^p, x]",
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.1 (a x^q+b x^n+c x^(2 n-q))^p.m",
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            "rhs": "1/Coefficient[u, x, 1]* Subst[Int[(a*x^q + b*x^n + c*x^(2*n - q))^p, x], x, u]",
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         "givens": "FreeQ[{a, b, c, n, p, q}, x] && EqQ[r, 2*n - q] && LinearQ[u, x] && NeQ[u, x]"
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            "rhs": "Int[x^m*((a + b + c)*x^n)^p, x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n, p}, x] && EqQ[q, n] && EqQ[r, n]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.2 (d x)^m (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.2 (d x)^m (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "Int[x^(m + p*q)*(a + b*x^(n - q) + c*x^(2*(n - q)))^p, x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n, q}, x] && EqQ[r, 2*n - q] && IntegerQ[p] && PosQ[n - q]"
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       "filename": "1.2.4.2 (d x)^m (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "-2/(n - q)* Subst[Int[1/(4*a - x^2), x], x, x^(m + 1)*(2*a + b*x^(n - q))/Sqrt[a*x^q + b*x^n + c*x^r]]",
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       "filename": "1.2.4.2 (d x)^m (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "x^(q/2)*Sqrt[a + b*x^(n - q) + c*x^(2*(n - q))]/ Sqrt[a*x^q + b*x^n + c*x^(2*n - q)]* Int[x^(m - q/2)/Sqrt[a + b*x^(n - q) + c*x^(2*(n - q))], x]",
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         "givens": "FreeQ[{a, b, c, m, n, q}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && (EqQ[m, 1] && EqQ[n, 3] && EqQ[q, 2] || (EqQ[m + 1/2] || EqQ[m, 3/2] || EqQ[m, 1/2] || EqQ[m, 5/2]) && EqQ[n, 3] && EqQ[q, 1])"
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       "filename": "1.2.4.2 (d x)^m (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "-2* x^((n - 1)/2)*(b + 2*c*x)/((b^2 - 4*a*c)* Sqrt[a*x^(n - 1) + b*x^n + c*x^(n + 1)])",
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            "lhs": "Int[x_^m_./(a_.*x_^q_. + b_.*x_^n_. + c_.*x_^r_.)^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && EqQ[m, 3*(n - 1)/2] && EqQ[q, n - 1] && EqQ[r, n + 1] && NeQ[b^2 - 4*a*c, 0]"
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       "filename": "1.2.4.2 (d x)^m (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "x^((n - 1)/2)*(4*a + 2*b*x)/((b^2 - 4*a*c)* Sqrt[a*x^(n - 1) + b*x^n + c*x^(n + 1)])",
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            "lhs": "Int[x_^m_./(a_.*x_^q_. + b_.*x_^n_. + c_.*x_^r_.)^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && EqQ[m, (3*n - 1)/2] && EqQ[q, n - 1] && EqQ[r, n + 1] && NeQ[b^2 - 4*a*c, 0]"
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       "filename": "1.2.4.2 (d x)^m (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "x^(m - n)*(a*x^(n - 1) + b*x^n + c*x^(n + 1))^(p + 1)/(2*c*(p + 1)) - b/(2*c)*Int[x^(m - 1)*(a*x^(n - 1) + b*x^n + c*x^(n + 1))^p, x]",
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         "givens": "FreeQ[{a, b, c}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && Not[IntegerQ[p]] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && RationalQ[m, p, q] && EqQ[m + p*(n - 1) - 1, 0]"
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            "rhs": "x^(m - n + q + 1)*(b + 2*c*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^ p/(2*c*(n - q)*(2*p + 1)) - p*(b^2 - 4*a*c)/(2*c*(2*p + 1))* Int[x^(m + q)*(a*x^q + b*x^n + c*x^(2*n - q))^(p - 1), x]",
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       "filename": "1.2.4.2 (d x)^m (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "x^(m - n + q + 1)*(b*(n - q)*p + c*(m + p*q + (n - q)*(2*p - 1) + 1)* x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^ p/(c*(m + p*(2*n - q) + 1)*(m + p*q + (n - q)*(2*p - 1) + 1)) + (n - q)* p/(c*(m + p*(2*n - q) + 1)*(m + p*q + (n - q)*(2*p - 1) + 1))* Int[x^(m - (n - 2*q))* Simp[-a*b*(m + p*q - n + q + 1) + (2*a*c*(m + p*q + (n - q)*(2*p - 1) + 1) - b^2*(m + p*q + (n - q)*(p - 1) + 1))*x^(n - q), x]* (a*x^q + b*x^n + c*x^(2*n - q))^(p - 1), x]",
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            "rhs": "x^(m + 1)*(a*x^q + b*x^n + c*x^(2*n - q))^p/(m + p*q + 1) - (n - q)*p/(m + p*q + 1)* Int[x^(m + n)*(b + 2*c*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p - 1), x]",
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            "rhs": "x^(m + 1)*(a*x^q + b*x^n + c*x^(2*n - q))^p/(m + p*(2*n - q) + 1) + (n - q)*p/(m + p*(2*n - q) + 1)* Int[x^(m + q)*(2*a + b*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p - 1), x]",
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            "rhs": "-x^(m - q + 1)*(b^2 - 2*a*c + b*c*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1)/(a*(n - q)*(p + 1)*(b^2 - 4*a*c)) + (2*a*c - b^2*(p + 2))/(a*(p + 1)*(b^2 - 4*a*c))* Int[x^(m - q)*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1), x]",
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.2 (d x)^m (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.2 (d x)^m (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "-x^(m - q + 1)*(b^2 - 2*a*c + b*c*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1)/(a*(n - q)*(p + 1)*(b^2 - 4*a*c)) + 1/(a*(n - q)*(p + 1)*(b^2 - 4*a*c))* Int[x^(m - q)* (b^2*(m + p*q + (n - q)*(p + 1) + 1) - 2*a*c*(m + p*q + 2*(n - q)*(p + 1) + 1) + b*c*(m + p*q + (n - q)*(2*p + 3) + 1)*x^(n - q))* (a*x^q + b*x^n + c*x^(2*n - q))^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_.*x_^q_. + b_.*x_^n_. + c_.*x_^r_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && Not[IntegerQ[p]] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && RationalQ[m, q] && LtQ[m + p*q + 1, n - q]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.2 (d x)^m (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.2 (d x)^m (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "x^(m - n + 1)*(b + 2*c*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1)/((n - q)*(p + 1)*(b^2 - 4*a*c)) - 1/((n - q)*(p + 1)*(b^2 - 4*a*c))* Int[ x^(m - n)*(b*(m + p*q - n + q + 1) + 2*c*(m + p*q + 2*(n - q)*(p + 1) + 1)*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_.*x_^q_. + b_.*x_^n_. + c_.*x_^r_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && Not[IntegerQ[p]] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && RationalQ[m, q] && LtQ[n - q, m + p*q + 1, 2*(n - q)]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.2 (d x)^m (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.2 (d x)^m (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "x^(m - 2*n + q + 1)*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1)/(2* c*(n - q)*(p + 1)) - b/(2*c)*Int[x^(m - n + q)*(a*x^q + b*x^n + c*x^(2*n - q))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_.*x_^q_. + b_.*x_^n_. + c_.*x_^r_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && Not[IntegerQ[p]] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GeQ[p, -1] && LtQ[p, 0] && RationalQ[m, q] && EqQ[m + p*q + 1, 2*(n - q)]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.2 (d x)^m (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.2 (d x)^m (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "-x^(m - q + 1)*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1)/(2*a*(n - q)*(p + 1)) - b/(2*a)*Int[x^(m + n - q)*(a*x^q + b*x^n + c*x^(2*n - q))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_.*x_^q_. + b_.*x_^n_. + c_.*x_^r_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && Not[IntegerQ[p]] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GeQ[p, -1] && LtQ[p, 0] && RationalQ[m, q] && EqQ[m + p*q + 1, -2*(n - q)*(p + 1)]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.2 (d x)^m (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.2 (d x)^m (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "x^(m - 2*n + q + 1)*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1)/(c*(m + p*q + 2*(n - q)*p + 1)) - 1/(c*(m + p*q + 2*(n - q)*p + 1))* Int[ x^(m - 2*(n - q))*(a*(m + p*q - 2*(n - q) + 1) + b*(m + p*q + (n - q)*(p - 1) + 1)*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_.*x_^q_. + b_.*x_^n_. + c_.*x_^r_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && Not[IntegerQ[p]] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GeQ[p, -1] && LtQ[p, 0] && RationalQ[m, q] && GtQ[m + p*q + 1, 2*(n - q)]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.2 (d x)^m (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.2 (d x)^m (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "x^(m - q + 1)*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1)/(a*(m + p*q + 1)) - 1/(a*(m + p*q + 1))* Int[ x^(m + n - q)*(b*(m + p*q + (n - q)*(p + 1) + 1) + c*(m + p*q + 2*(n - q)*(p + 1) + 1)*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_.*x_^q_. + b_.*x_^n_. + c_.*x_^r_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && Not[IntegerQ[p]] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GeQ[p, -1] && LtQ[p, 0] && RationalQ[m, q] && LtQ[m + p*q + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.2 (d x)^m (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.2 (d x)^m (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "(a*x^q + b*x^n + c*x^(2*n - q))^ p/(x^(p*q)*(a + b*x^(n - q) + c*x^(2*(n - q)))^p)* Int[x^(m + p*q)*(a + b*x^(n - q) + c*x^(2*(n - q)))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_.*x_^q_. + b_.*x_^n_. + c_.*x_^r_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n, p, q}, x] && EqQ[r, 2*n - q] && Not[IntegerQ[p]] && PosQ[n - q]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.2 (d x)^m (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.2 (d x)^m (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "1/Coefficient[u, x, 1]* Subst[Int[x^m*(a*x^q + b*x^n + c*x^(2*n - q))^p, x], x, u]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*(a_.*u_^q_. + b_.*u_^n_. + c_.*u_^r_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n, p, q}, x] && EqQ[r, 2*n - q] && LinearQ[u, x] && NeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.3 (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.3 (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "Int[x^(p*q)*(A + B*x^(n - q))*(a + b*x^(n - q) + c*x^(2*(n - q)))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*x_^r_.)*(a_.*x_^q_. + b_.*x_^n_. + c_.*x_^j_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, B, n, q}, x] && EqQ[r, n - q] && EqQ[j, 2*n - q] && IntegerQ[p] && PosQ[n - q]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.3 (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.3 (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "Sqrt[a*x^q+b*x^n+c*x^(2*n-q)]/(x^(q/2)*Sqrt[a+b*x^(n-q)+c*x^(2*( n-q))])* Int[x^(q*p)*(A+B*x^(n-q))*(a+b*x^(n-q)+c*x^(2*(n-q)))^p,x]",
     "rulenumber": 0,
            "lhs": "Int[(A_+B_.*x_^j_.)*(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.)^p_,x_Symbol] ",
        "comment": false,
         "givens": " FreeQ[{a,b,c,A,B,n,p,q},x] && EqQ[j,n-q] && EqQ[r,2*n-q] && PosQ[n-q]  && IGtQ[p+1/2,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.3 (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.3 (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "x^(q/2)*Sqrt[a+b*x^(n-q)+c*x^(2*(n-q))]/Sqrt[a*x^q+b*x^n+c*x^(2* n-q)]* Int[x^(q*p)*(A+B*x^(n-q))*(a+b*x^(n-q)+c*x^(2*(n-q)))^p,x]",
     "rulenumber": 0,
            "lhs": "Int[(A_+B_.*x_^j_.)*(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.)^p_,x_Symbol] ",
        "comment": false,
         "givens": " FreeQ[{a,b,c,A,B,n,p,q},x] && EqQ[j,n-q] && EqQ[r,2*n-q] && PosQ[n-q]  && ILtQ[p-1/2,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.3 (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.3 (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": " Sqrt[a*x^q+b*x^n+c*x^(2*n-q)]/(x^(q/2)*Sqrt[a+b*x^(n-q)+c*x^(2*(n-q))] )* Int[x^(q/2)*(A+B*x^(n-q))*Sqrt[a+b*x^(n-q)+c*x^(2*(n-q))],x]",
     "rulenumber": 0,
            "lhs": "Int[(A_+B_.*x_^j_.)*Sqrt[a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.],x_ Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,A,B,n,q},x] && EqQ[j,n-q] && EqQ[r,2*n-q] && PosQ[n-q] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.3 (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.3 (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "x^(q/2)*Sqrt[a + b*x^(n - q) + c*x^(2*(n - q))]/ Sqrt[a*x^q + b*x^n + c*x^(2*n - q)]* Int[(A + B*x^(n - q))/(x^(q/2)* Sqrt[a + b*x^(n - q) + c*x^(2*(n - q))]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*x_^j_.)/Sqrt[a_.*x_^q_. + b_.*x_^n_. + c_.*x_^r_.], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, B, n, q}, x] && EqQ[j, n - q] && EqQ[r, 2*n - q] && PosQ[n - q] && EqQ[n, 3] && EqQ[q, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.3 (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.3 (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "x*(b*B*(n - q)*p + A*c*(p*q + (n - q)*(2*p + 1) + 1) + B*c*(p*(2*n - q) + 1)* x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^p/ (c*(p*(2*n - q) + 1)*(p*q + (n - q)*(2*p + 1) + 1)) + (n - q)*p/(c*(p*(2*n - q) + 1)*(p*q + (n - q)*(2*p + 1) + 1))* Int[x^q* (2*a*A*c*(p*q + (n - q)*(2*p + 1) + 1) - a*b*B*(p*q + 1) + (2*a*B*c*(p*(2*n - q) + 1) + A*b*c*(p*q + (n - q)*(2*p + 1) + 1) - b^2*B*(p*q + (n - q)*p + 1))*x^(n - q))* (a*x^q + b*x^n + c*x^(2*n - q))^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*x_^r_.)*(a_.*x_^q_. + b_.*x_^n_. + c_.*x_^j_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, B, n, q}, x] && EqQ[r, n - q] && EqQ[j, 2*n - q] && Not[IntegerQ[p]] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && NeQ[p*(2*n - q) + 1, 0] && NeQ[p*q + (n - q)*(2*p + 1) + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.3 (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.3 (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "With[{n = q + r}, x*(A*(p*q + (n - q)*(2*p + 1) + 1) + B*(p*(2*n - q) + 1)*x^(n - q))*(a*x^q + c*x^(2*n - q))^ p/((p*(2*n - q) + 1)*(p*q + (n - q)*(2*p + 1) + 1)) + (n - q)*p/((p*(2*n - q) + 1)*(p*q + (n - q)*(2*p + 1) + 1))* Int[x^q*(2*a* A*(p*q + (n - q)*(2*p + 1) + 1) + (2*a* B*(p*(2*n - q) + 1))*x^(n - q))*(a*x^q + c*x^(2*n - q))^(p - 1), x] /; EqQ[j, 2*n - q] && NeQ[p*(2*n - q) + 1, 0] && NeQ[p*q + (n - q)*(2*p + 1) + 1, 0]]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*x_^r_.)*(a_.*x_^q_. + c_.*x_^j_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, A, B, q}, x] && Not[IntegerQ[p]] && GtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.3 (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.3 (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "-x^(-q + 1)*(A*b^2 - a*b*B - 2*a*A*c + (A*b - 2*a*B)*c* x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1)/(a*(n - q)*(p + 1)*(b^2 - 4*a*c)) + 1/(a*(n - q)*(p + 1)*(b^2 - 4*a*c))* Int[x^(-q)* ((A*b^2*(p*q + (n - q)*(p + 1) + 1) - a*b*B*(p*q + 1) - 2*a*A*c*(p*q + 2*(n - q)*(p + 1) + 1) + (p*q + (n - q)*(2*p + 3) + 1)*(A*b - 2*a*B)*c* x^(n - q))* (a*x^q + b*x^n + c*x^(2*n - q))^(p + 1)), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*x_^r_.)*(a_.*x_^q_. + b_.*x_^n_. + c_.*x_^j_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, B, n, q}, x] && EqQ[r, n - q] && EqQ[j, 2*n - q] && Not[IntegerQ[p]] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.3 (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.3 (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "With[{n = q + r}, -x^(-q + 1)*(a*A*c + a*B*c*x^(n - q))*(a*x^q + c*x^(2*n - q))^(p + 1)/(a*(n - q)*(p + 1)*(2*a*c)) + 1/(a*(n - q)*(p + 1)*(2*a*c))* Int[x^(-q)*((a*A*c*(p*q + 2*(n - q)*(p + 1) + 1) + a*B*c*(p*q + (n - q)*(2*p + 3) + 1)*x^(n - q))*(a*x^q + c*x^(2*n - q))^(p + 1)), x] /; EqQ[j, 2*n - q]]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*x_^r_.)*(a_.*x_^q_. + c_.*x_^j_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, A, B, q}, x] && Not[IntegerQ[p]] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.3 (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.3 (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "(a*x^j+b*x^k+c*x^n)^p/(x^(j*p)*(a+b*x^(k-j)+c*x^(2*(k-j)))^p)* Int[x^(j*p)*(A+B*x^(k-j))*(a+b*x^(k-j)+c*x^(2*(k-j)))^p,x]",
     "rulenumber": 0,
            "lhs": "Int[(A_+B_.*x_^q_)*(a_.*x_^j_.+b_.*x_^k_.+c_.*x_^n_.)^p_,x_Symbol] ",
        "comment": false,
         "givens": " FreeQ[{a,b,c,A,B,j,k,p},x] && EqQ[q,k-j] && EqQ[n,2*k-j] && PosQ[k-j]  && Not[IntegerQ[p]] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.3 (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.3 (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "Unintegrable[(A + B*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*x_^j_.)*(a_.*x_^q_. + b_.*x_^n_. + c_.*x_^r_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, B, n, p, q}, x] && EqQ[j, n - q] && EqQ[r, 2*n - q]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.3 (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.3 (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "1/Coefficient[u, x, 1]* Subst[Int[(A + B*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^p, x], x, u]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*u_^j_.)*(a_.*u_^q_. + b_.*u_^n_. + c_.*u_^r_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, B, n, p, q}, x] && EqQ[j, n - q] && EqQ[r, 2*n - q] && LinearQ[u, x] && NeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "Int[x^(m + p*q)*(A + B*x^(n - q))*(a + b*x^(n - q) + c*x^(2*(n - q)))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(A_ + B_.*x_^r_.)*(a_.*x_^q_. + b_.*x_^n_. + c_.*x_^j_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, B, m, n, q}, x] && EqQ[r, n - q] && EqQ[j, 2*n - q] && IntegerQ[p] && PosQ[n - q]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "x^(m + 1)*(A*(m + p*q + (n - q)*(2*p + 1) + 1) + B*(m + p*q + 1)*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^ p/((m + p*q + 1)*(m + p*q + (n - q)*(2*p + 1) + 1)) + (n - q)*p/((m + p*q + 1)*(m + p*q + (n - q)*(2*p + 1) + 1))* Int[x^(n + m)* Simp[2*a*B*(m + p*q + 1) - A*b*(m + p*q + (n - q)*(2*p + 1) + 1) + (b*B*(m + p*q + 1) - 2*A*c*(m + p*q + (n - q)*(2*p + 1) + 1))*x^(n - q), x]* (a*x^q + b*x^n + c*x^(2*n - q))^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(A_ + B_.*x_^r_.)*(a_.*x_^q_. + b_.*x_^n_. + c_.*x_^j_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, B}, x] && EqQ[r, n - q] && EqQ[j, 2*n - q] && Not[IntegerQ[p]] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[p, 0] && RationalQ[m, q] && LeQ[m + p*q, -(n - q)] && NeQ[m + p*q + 1, 0] && NeQ[m + p*q + (n - q)*(2*p + 1) + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "With[{n = q + r}, x^(m + 1)*(A*(m + p*q + (n - q)*(2*p + 1) + 1) + B*(m + p*q + 1)*x^(n - q))*(a*x^q + c*x^(2*n - q))^ p/((m + p*q + 1)*(m + p*q + (n - q)*(2*p + 1) + 1)) + 2*(n - q)*p/((m + p*q + 1)*(m + p*q + (n - q)*(2*p + 1) + 1))* Int[x^(n + m)* Simp[a*B*(m + p*q + 1) - A*c*(m + p*q + (n - q)*(2*p + 1) + 1)*x^(n - q), x]*(a*x^q + c*x^(2*n - q))^(p - 1), x] /; EqQ[j, 2*n - q] && IGtQ[n, 0] && LeQ[m + p*q, -(n - q)] && NeQ[m + p*q + 1, 0] && NeQ[m + p*q + (n - q)*(2*p + 1) + 1, 0]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(A_ + B_.*x_^r_.)*(a_.*x_^q_. + c_.*x_^j_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, A, B}, x] && Not[IntegerQ[p]] && RationalQ[m, p, q] && GtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "x^(m - n + 1)*(A*b - 2*a*B - (b*B - 2*A*c)* x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1)/((n - q)*(p + 1)*(b^2 - 4*a*c)) + 1/((n - q)*(p + 1)*(b^2 - 4*a*c))* Int[x^(m - n)* Simp[(m + p*q - n + q + 1)*(2*a*B - A*b) + (m + p*q + 2*(n - q)*(p + 1) + 1)*(b*B - 2*A*c)*x^(n - q), x]* (a*x^q + b*x^n + c*x^(2*n - q))^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(A_ + B_.*x_^r_.)*(a_.*x_^q_. + b_.*x_^n_. + c_.*x_^j_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, B}, x] && EqQ[r, n - q] && EqQ[j, 2*n - q] && Not[IntegerQ[p]] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && RationalQ[m, q] && GtQ[m + p*q, n - q - 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "With[{n = q + r}, x^(m - n + 1)*(a*B - A*c*x^(n - q))*(a*x^q + c*x^(2*n - q))^(p + 1)/(2*a* c*(n - q)*(p + 1)) - 1/(2*a*c*(n - q)*(p + 1))* Int[x^(m - n)* Simp[a*B*(m + p*q - n + q + 1) - A*c*(m + p*q + (n - q)*2*(p + 1) + 1)*x^(n - q), x]*(a*x^q + c*x^(2*n - q))^(p + 1), x] /; EqQ[j, 2*n - q] && IGtQ[n, 0] && m + p*q > n - q - 1]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(A_ + B_.*x_^r_.)*(a_.*x_^q_. + c_.*x_^j_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, A, B}, x] && Not[IntegerQ[p]] && RationalQ[m, q] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "x^(m + 1)*(b*B*(n - q)*p + A*c*(m + p*q + (n - q)*(2*p + 1) + 1) + B*c*(m + p*q + 2*(n - q)*p + 1)* x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^p/ (c*(m + p*(2*n - q) + 1)*(m + p*q + (n - q)*(2*p + 1) + 1)) + (n - q)* p/(c*(m + p*(2*n - q) + 1)*(m + p*q + (n - q)*(2*p + 1) + 1))* Int[x^(m + q)* Simp[2*a*A*c*(m + p*q + (n - q)*(2*p + 1) + 1) - a*b*B*(m + p*q + 1) + (2*a*B*c*(m + p*q + 2*(n - q)*p + 1) + A*b*c*(m + p*q + (n - q)*(2*p + 1) + 1) - b^2*B*(m + p*q + (n - q)*p + 1))*x^(n - q), x]* (a*x^q + b*x^n + c*x^(2*n - q))^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(A_ + B_.*x_^r_.)*(a_.*x_^q_. + b_.*x_^n_. + c_.*x_^j_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, B}, x] && EqQ[r, n - q] && EqQ[j, 2*n - q] && Not[IntegerQ[p]] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[p, 0] && RationalQ[m, q] && GtQ[m + p*q, -(n - q) - 1] && NeQ[m + p*(2*n - q) + 1, 0] && NeQ[m + p*q + (n - q)*(2*p + 1) + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "With[{n = q + r}, x^(m + 1)*(A*(m + p*q + (n - q)*(2*p + 1) + 1) + B*(m + p*q + 2*(n - q)*p + 1)* x^(n - q))*(a*x^q + c*x^(2*n - q))^p/ ((m + p*(2*n - q) + 1)*(m + p*q + (n - q)*(2*p + 1) + 1)) + (n - q)* p/((m + p*(2*n - q) + 1)*(m + p*q + (n - q)*(2*p + 1) + 1))* Int[x^(m + q)* Simp[2*a*A*(m + p*q + (n - q)*(2*p + 1) + 1) + 2*a*B*(m + p*q + 2*(n - q)*p + 1)*x^(n - q), x]*(a*x^q + c*x^(2*n - q))^(p - 1), x] /; EqQ[j, 2*n - q] && IGtQ[n, 0] && GtQ[m + p*q, -(n - q)] && NeQ[m + p*q + 2*(n - q)*p + 1, 0] && NeQ[m + p*q + (n - q)*(2*p + 1) + 1, 0] && NeQ[m + 1, n]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(A_ + B_.*x_^r_.)*(a_.*x_^q_. + c_.*x_^j_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, A, B}, x] && Not[IntegerQ[p]] && RationalQ[m, q] && GtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "-x^(m - q + 1)*(A*b^2 - a*b*B - 2*a*A*c + (A*b - 2*a*B)*c* x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1)/(a*(n - q)*(p + 1)*(b^2 - 4*a*c)) + 1/(a*(n - q)*(p + 1)*(b^2 - 4*a*c))* Int[x^(m - q)* Simp[A*b^2*(m + p*q + (n - q)*(p + 1) + 1) - a*b*B*(m + p*q + 1) - 2*a*A*c*(m + p*q + 2*(n - q)*(p + 1) + 1) + (m + p*q + (n - q)*(2*p + 3) + 1)*(A*b - 2*a*B)*c* x^(n - q), x]* (a*x^q + b*x^n + c*x^(2*n - q))^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(A_ + B_.*x_^r_.)*(a_.*x_^q_. + b_.*x_^n_. + c_.*x_^j_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, B}, x] && EqQ[r, n - q] && EqQ[j, 2*n - q] && Not[IntegerQ[p]] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && RationalQ[m, q] && m + p*q < n - q - 1"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "With[{n = q + r}, -x^(m - q + 1)*(A*c + B*c*x^(n - q))*(a*x^q + c*x^(2*n - q))^(p + 1)/(2*a* c*(n - q)*(p + 1)) + 1/(2*a*c*(n - q)*(p + 1))* Int[x^(m - q)* Simp[A*c*(m + p*q + 2*(n - q)*(p + 1) + 1) + B*(m + p*q + (n - q)*(2*p + 3) + 1)*c*x^(n - q), x]*(a*x^q + c*x^(2*n - q))^(p + 1), x] /; EqQ[j, 2*n - q] && IGtQ[n, 0] && LtQ[m + p*q, n - q - 1]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(A_ + B_.*x_^r_.)*(a_.*x_^q_. + c_.*x_^j_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, A, B}, x] && Not[IntegerQ[p]] && RationalQ[m, q] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "B*x^(m - n + 1)*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1)/(c*(m + p*q + (n - q)*(2*p + 1) + 1)) - 1/(c*(m + p*q + (n - q)*(2*p + 1) + 1))* Int[x^(m - n + q)* Simp[a*B*(m + p*q - n + q + 1) + (b*B*(m + p*q + (n - q)*p + 1) - A*c*(m + p*q + (n - q)*(2*p + 1) + 1))*x^(n - q), x]* (a*x^q + b*x^n + c*x^(2*n - q))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(A_ + B_.*x_^r_.)*(a_.*x_^q_. + b_.*x_^n_. + c_.*x_^j_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, B}, x] && EqQ[r, n - q] && EqQ[j, 2*n - q] && Not[IntegerQ[p]] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GeQ[p, -1] && LtQ[p, 0] && RationalQ[m, q] && GeQ[m + p*q, n - q - 1] && NeQ[m + p*q + (n - q)*(2*p + 1) + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "With[{n = q + r}, B*x^(m - n + 1)*(a*x^q + c*x^(2*n - q))^(p + 1)/(c*(m + p*q + (n - q)*(2*p + 1) + 1)) - 1/(c*(m + p*q + (n - q)*(2*p + 1) + 1))* Int[x^(m - n + q)* Simp[a*B*(m + p*q - n + q + 1) - A*c*(m + p*q + (n - q)*(2*p + 1) + 1)*x^(n - q), x]*(a*x^q + c*x^(2*n - q))^p, x] /; EqQ[j, 2*n - q] && IGtQ[n, 0] && GeQ[m + p*q, n - q - 1] && NeQ[m + p*q + (n - q)*(2*p + 1) + 1, 0]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(A_ + B_.*x_^r_.)*(a_.*x_^q_. + c_.*x_^j_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, A, B}, x] && Not[IntegerQ[p]] && RationalQ[m, p, q] && GeQ[p, -1] && LtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "A*x^(m - q + 1)*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1)/(a*(m + p*q + 1)) + 1/(a*(m + p*q + 1))* Int[x^(m + n - q)* Simp[a*B*(m + p*q + 1) - A*b*(m + p*q + (n - q)*(p + 1) + 1) - A*c*(m + p*q + 2*(n - q)*(p + 1) + 1)*x^(n - q), x]* (a*x^q + b*x^n + c*x^(2*n - q))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(A_ + B_.*x_^r_.)*(a_.*x_^q_. + b_.*x_^n_. + c_.*x_^j_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, B}, x] && EqQ[r, n - q] && EqQ[j, 2*n - q] && Not[IntegerQ[p]] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && RationalQ[m, p, q] && (GeQ[p, -1] && LtQ[p, 0] || EqQ[m + p*q + (n - q)*(2*p + 1) + 1, 0]) && LeQ[m + p*q, -(n - q)] && NeQ[m + p*q + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "With[{n = q + r}, A*x^(m - q + 1)*(a*x^q + c*x^(2*n - q))^(p + 1)/(a*(m + p*q + 1)) + 1/(a*(m + p*q + 1))* Int[x^(m + n - q)* Simp[a*B*(m + p*q + 1) - A*c*(m + p*q + 2*(n - q)*(p + 1) + 1)*x^(n - q), x]*(a*x^q + c*x^(2*n - q))^p, x] /; EqQ[j, 2*n - q] && IGtQ[n, 0] && (GeQ[p, -1] && LtQ[p, 0] || EqQ[m + p*q + (n - q)*(2*p + 1) + 1, 0]) && LeQ[m + p*q, -(n - q)] && NeQ[m + p*q + 1, 0]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(A_ + B_.*x_^r_.)*(a_.*x_^q_. + c_.*x_^j_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, A, B}, x] && Not[IntegerQ[p]] && RationalQ[m, p, q]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "x^(q/2)*Sqrt[a + b*x^(n - q) + c*x^(2*(n - q))]/ Sqrt[a*x^q + b*x^n + c*x^(2*n - q)]* Int[ x^(m - q/2)*(A + B*x^(n - q))/ Sqrt[a + b*x^(n - q) + c*x^(2*(n - q))], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(A_ + B_.*x_^j_.)/ Sqrt[a_.*x_^q_. + b_.*x_^n_. + c_.*x_^r_.], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, B, m, n, q}, x] && EqQ[j, n - q] && EqQ[r, 2*n - q] && PosQ[n - q] && (EqQ[m, 1/2] || EqQ[m, -1/2]) && EqQ[n, 3] && EqQ[q, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": " Sqrt[a*x^q+b*x^n+c*x^(2*n-q)]/(x^(q/2)*Sqrt[a+b*x^(n-q)+c*x^(2*(n-q))] )* Int[x^(m+q*p)*(A+B*x^(n-q))*(a+b*x^(n-q)+c*x^(2*(n-q)))^p,x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(A_+B_.*x_^j_.)*(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.)^p_,x_ Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,A,B,m,n,p,q},x] && EqQ[j,n-q] && EqQ[r,2*n-q] &&  IGtQ[p+1/2,0] && PosQ[n-q] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": " x^(q/2)*Sqrt[a+b*x^(n-q)+c*x^(2*(n-q))]/Sqrt[a*x^q+b*x^n+c*x^(2*n-q)]*  Int[x^(m+q*p)*(A+B*x^(n-q))*(a+b*x^(n-q)+c*x^(2*(n-q)))^p,x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(A_+B_.*x_^j_.)*(a_.*x_^q_.+b_.*x_^n_.+c_.*x_^r_.)^p_,x_ Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,A,B,m,n,p,q},x] && EqQ[j,n-q] && EqQ[r,2*n-q] &&  ILtQ[p-1/2,0] && PosQ[n-q] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "(a*x^j + b*x^k + c*x^n)^ p/(x^(j*p)*(a + b*x^(k - j) + c*x^(2*(k - j)))^p)* Int[ x^(m + j*p)*(A + B*x^(k - j))*(a + b*x^(k - j) + c*x^(2*(k - j)))^ p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(A_ + B_.*x_^q_)*(a_.*x_^j_. + b_.*x_^k_. + c_.*x_^n_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, B, j, k, m, p}, x] && EqQ[q, k - j] && EqQ[n, 2*k - j] && Not[IntegerQ[p]] && PosQ[k - j]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.2 Trinomial products/1.2.4 Improper/1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
       "filename": "1.2.4.4 (f x)^m (d+e x^(n-q)) (a x^q+b x^n+c x^(2 n-q))^p.m",
            "rhs": "1/Coefficient[u, x, 1]* Subst[Int[x^m*(A + B*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^p, x], x, u]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*(A_ + B_.*u_^j_.)*(a_.*u_^q_. + b_.*u_^n_. + c_.*u_^r_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, B, m, n, p, q}, x] && EqQ[j, n - q] && EqQ[r, 2*n - q] && LinearQ[u, x] && NeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.1 P(x)^p.m",
       "filename": "1.3.1 P(x)^p.m",
            "rhs": "Module[{gcd = PolyGCD[P, Q, x]}, Int[u*gcd^(p + q)*PolynomialQuotient[P, gcd, x]^p* PolynomialQuotient[Q, gcd, x]^q, x] /; NeQ[gcd, 1]]",
     "rulenumber": 0,
            "lhs": "Int[u_.*P_^p_*Q_^q_, x_Symbol]",
        "comment": false,
         "givens": "IGtQ[p, 0] && ILtQ[q, 0] && PolyQ[P, x] && PolyQ[Q, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.1 P(x)^p.m",
       "filename": "1.3.1 P(x)^p.m",
            "rhs": "Module[{gcd = PolyGCD[P, Q, x]}, Int[u*gcd^(q + 1)*PolynomialQuotient[P, gcd, x]* PolynomialQuotient[Q, gcd, x]^q, x] /; NeQ[gcd, 1]]",
     "rulenumber": 0,
            "lhs": "Int[u_.*P_*Q_^q_, x_Symbol]",
        "comment": false,
         "givens": "ILtQ[q, 0] && PolyQ[P, x] && PolyQ[Q, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.1 P(x)^p.m",
       "filename": "1.3.1 P(x)^p.m",
            "rhs": "With[{m = MinimumMonomialExponent[P, x]}, P^FracPart[p]/(x^(m*FracPart[p])*Distrib[1/x^m, P]^FracPart[p])* Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_.*P_^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[p, x] && Not[IntegerQ[p]] && SumQ[P] && EveryQ[Function[BinomialQ[#, x]], P] && Not[PolyQ[P, x, 2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.1 P(x)^p.m",
       "filename": "1.3.1 P(x)^p.m",
            "rhs": "With[{u = Factor[ReplaceAll[P, x -> Sqrt[x]]]}, Int[ExpandIntegrand[ReplaceAll[u, x -> x^2]^p, x], x] /; Not[SumQ[NonfreeFactors[u, x]]]]",
     "rulenumber": 0,
            "lhs": "Int[P_^p_, x_Symbol]",
        "comment": false,
         "givens": "PolyQ[P, x^2] && ILtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.1 P(x)^p.m",
       "filename": "1.3.1 P(x)^p.m",
            "rhs": "With[{u = Factor[P]}, Int[ExpandIntegrand[u^p, x], x] /; Not[SumQ[NonfreeFactors[u, x]]]]",
     "rulenumber": 0,
            "lhs": "Int[P_^p_, x_Symbol]",
        "comment": false,
         "givens": "PolyQ[P, x] && ILtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.1 P(x)^p.m",
       "filename": "1.3.1 P(x)^p.m",
            "rhs": "With[{u = Factor[P]}, Int[u^p, x] /; Not[SumQ[NonfreeFactors[u, x]]]]",
     "rulenumber": 0,
            "lhs": "Int[P_^p_, x_Symbol]",
        "comment": false,
         "givens": "PolyQ[P, x] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.1 P(x)^p.m",
       "filename": "1.3.1 P(x)^p.m",
            "rhs": "With[{u=Factor[Pn]}, Pn^p/DistributeDegree[u,p]*Int[DistributeDegree[u,p],x] /;  Not[SumQ[u]]]",
     "rulenumber": 0,
            "lhs": "Int[Pn_^p_,x_Symbol]",
        "comment": false,
         "givens": "PolyQ[Pn,x] && Not[IntegerQ[p]] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.1 P(x)^p.m",
       "filename": "1.3.1 P(x)^p.m",
            "rhs": "1/3^p*Subst[Int[Simp[(3*a*c - b^2)/c + c^2*x^3/b, x]^p, x], x, c/(3*d) + x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_ + c_.*x_^2 + d_.*x_^3)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IGtQ[p, 0] && EqQ[c^2 - 3*b*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.1 P(x)^p.m",
       "filename": "1.3.1 P(x)^p.m",
            "rhs": "Int[ExpandToSum[P^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[P_^p_, x_Symbol]",
        "comment": false,
         "givens": "PolyQ[P, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.1 P(x)^p.m",
       "filename": "1.3.1 P(x)^p.m",
            "rhs": "Int[ExpandIntegrand[P^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[P_^p_, x_Symbol]",
        "comment": false,
         "givens": "PolyQ[P, x] && IntegerQ[p] && QuadraticProductQ[Factor[P], x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.1 P(x)^p.m",
       "filename": "1.3.1 P(x)^p.m",
            "rhs": "1/(3^(3*p)*a^(2*p))*Int[(3*a - b*x)^p*(3*a + 2*b*x)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_ + d_.*x_^3)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x] && EqQ[4*b^3 + 27*a^2*d, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.1 P(x)^p.m",
       "filename": "1.3.1 P(x)^p.m",
            "rhs": "(a + b*x + d*x^3)^ p/((3*a - b*x)^p*(3*a + 2*b*x)^(2*p))* Int[(3*a - b*x)^p*(3*a + 2*b*x)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_ + d_.*x_^3)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, p}, x] && EqQ[4*b^3 + 27*a^2*d, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.1 P(x)^p.m",
       "filename": "1.3.1 P(x)^p.m",
            "rhs": "With[{r = Rt[-9*a*d^2 + Sqrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]}, 1/d^(2*p)*Int[Simp[18^(1/3)*b*d/(3*r) - r/18^(1/3) + d*x, x]^p* Simp[b*d/3 + 12^(1/3)*b^2*d^2/(3*r^2) + r^2/(3*12^(1/3)) - d*(2^(1/3)*b*d/(3^(1/3)*r) - r/18^(1/3))*x + d^2*x^2, x]^p, x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_ + d_.*x_^3)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.1 P(x)^p.m",
       "filename": "1.3.1 P(x)^p.m",
            "rhs": "With[{r = Rt[-9*a*d^2 + Sqrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]}, (a + b*x + d*x^3)^p/ (Simp[18^(1/3)*b*d/(3*r) - r/18^(1/3) + d*x, x]^p* Simp[b*d/3 + 12^(1/3)*b^2*d^2/(3*r^2) + r^2/(3*12^(1/3)) - d*(2^(1/3)*b*d/(3^(1/3)*r) - r/18^(1/3))*x + d^2*x^2, x]^ p)* Int[Simp[18^(1/3)*b*d/(3*r) - r/18^(1/3) + d*x, x]^p* Simp[b*d/3 + 12^(1/3)*b^2*d^2/(3*r^2) + r^2/(3*12^(1/3)) - d*(2^(1/3)*b*d/(3^(1/3)*r) - r/18^(1/3))*x + d^2*x^2, x]^p, x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_ + d_.*x_^3)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, p}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.1 P(x)^p.m",
       "filename": "1.3.1 P(x)^p.m",
            "rhs": "With[{a = Coeff[P3, x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3, x, 3]}, Subst[ Int[Simp[(2*c^3 - 9*b*c*d + 27*a*d^2)/(27*d^2) - (c^2 - 3*b*d)* x/(3*d) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c, 0]]",
     "rulenumber": 0,
            "lhs": "Int[P3_^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[p, x] && PolyQ[P3, x, 3]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.1 P(x)^p.m",
       "filename": "1.3.1 P(x)^p.m",
            "rhs": "With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1], c = Coeff[P4, x, 2], d = Coeff[P4, x, 3], e = Coeff[P4, x, 4]}, 1/a^(3*p)* Int[ExpandIntegrand[(a - b*x)^(-p)/(a^5 - b^5*x^5)^(-p), x], x] /; NeQ[a, 0] && EqQ[c, b^2/a] && EqQ[d, b^3/a^2] && EqQ[e, b^4/a^3]]",
     "rulenumber": 0,
            "lhs": "Int[P4_^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[p, x] && PolyQ[P4, x, 4] && ILtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.1 P(x)^p.m",
       "filename": "1.3.1 P(x)^p.m",
            "rhs": "With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1], c = Coeff[P4, x, 2], d = Coeff[P4, x, 3], e = Coeff[P4, x, 4]}, -16*a^2*Subst[ Int[1/(b - 4*a*x)^2*(a*(-3*b^4 + 16*a*b^2*c - 64*a^2*b*d + 256*a^3*e - 32*a^2*(3*b^2 - 8*a*c)*x^2 + 256*a^4*x^4)/(b - 4*a*x)^4)^p, x], x, b/(4*a) + 1/x] /; NeQ[a, 0] && NeQ[b, 0] && EqQ[b^3 - 4*a*b*c + 8*a^2*d, 0]]",
     "rulenumber": 0,
            "lhs": "Int[P4_^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[p, x] && PolyQ[P4, x, 4] && IntegerQ[2*p] && Not[IGtQ[p, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.1 P(x)^p.m",
       "filename": "1.3.1 P(x)^p.m",
            "rhs": "With[{a = Coeff[Q6, x, 0], b = Coeff[Q6, x, 2], c = Coeff[Q6, x, 3], d = Coeff[Q6, x, 4], e = Coeff[Q6, x, 6]}, 1/(3^(3*p)*a^(2*p))*Int[ExpandIntegrand[ (3*a + 3*Rt[a, 3]^2*Rt[c, 3]*x + b*x^2)^p* (3*a - 3*(-1)^(1/3)*Rt[a, 3]^2*Rt[c, 3]*x + b*x^2)^p* (3*a + 3*(-1)^(2/3)*Rt[a, 3]^2*Rt[c, 3]*x + b*x^2)^p, x], x] /; EqQ[b^2 - 3*a*d, 0] && EqQ[b^3 - 27*a^2*e, 0]]",
     "rulenumber": 0,
            "lhs": "Int[Q6_^p_, x_Symbol]",
        "comment": false,
         "givens": "ILtQ[p, 0] && PolyQ[Q6, x, 6] && EqQ[Coeff[Q6, x, 1], 0] && EqQ[Coeff[Q6, x, 5], 0] && RationalFunctionQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "With[{a = Coeff[v, x, 0], b = Coeff[v, x, 2], c = Coeff[v, x, 4]}, a/d*Subst[Int[1/(1 - 2*b*x^2 + (b^2 - 4*a*c)*x^4), x], x, x/Sqrt[v]] /; EqQ[c*d + a*e, 0] && PosQ[a*c]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[v_]/(d_ + e_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{d, e}, x] && PolyQ[v, x^2, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "With[{q = Sqrt[b^2 - 4*a*c]}, -a*Sqrt[b + q]/(2*Sqrt[2]*Rt[-a*c, 2]*d)* ArcTan[Sqrt[b + q]* x*(b - q + 2*c*x^2)/(2*Sqrt[2]*Rt[-a*c, 2]* Sqrt[a + b*x^2 + c*x^4])] + a*Sqrt[-b + q]/(2*Sqrt[2]*Rt[-a*c, 2]*d)* ArcTanh[Sqrt[-b + q]* x*(b + q + 2*c*x^2)/(2*Sqrt[2]*Rt[-a*c, 2]* Sqrt[a + b*x^2 + c*x^4])]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*x_^2 + c_.*x_^4]/(d_ + e_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c*d + a*e, 0] && NegQ[a*c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "With[{PP = Factor[ReplaceAll[P, x -> Sqrt[x]]]}, Int[ExpandIntegrand[ReplaceAll[PP, x -> x^2]^p*Q^q, x], x] /; Not[SumQ[NonfreeFactors[PP, x]]]]",
     "rulenumber": 0,
            "lhs": "Int[P_^p_*Q_^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[q, x] && PolyQ[P, x^2] && PolyQ[Q, x] && ILtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /; Not[SumQ[NonfreeFactors[PP, x]]]]",
     "rulenumber": 0,
            "lhs": "Int[P_^p_*Q_^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Qm, x], x] /; QuadraticProductQ[PP, x]]",
     "rulenumber": 0,
            "lhs": "Int[P_^p_*Qm_, x_Symbol]",
        "comment": false,
         "givens": "PolyQ[Qm, x] && PolyQ[P, x] && ILtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "1/(3^(3*p)*a^(2*p))* Int[(e + f*x)^m*(3*a - b*x)^p*(3*a + 2*b*x)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_ + b_.*x_ + d_.*x_^3)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m}, x] && EqQ[4*b^3 + 27*a^2*d, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "(a + b*x + d*x^3)^ p/((3*a - b*x)^p*(3*a + 2*b*x)^(2*p))* Int[(e + f*x)^m*(3*a - b*x)^p*(3*a + 2*b*x)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_ + b_.*x_ + d_.*x_^3)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, p}, x] && EqQ[4*b^3 + 27*a^2*d, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "Int[ExpandIntegrand[(e + f*x)^m*(a + b*x + d*x^3)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_ + b_.*x_ + d_.*x_^3)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "With[{r = Rt[-9*a*d^2 + Sqrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]}, 1/d^(2*p)* Int[(e + f*x)^m*Simp[18^(1/3)*b*d/(3*r) - r/18^(1/3) + d*x, x]^p* Simp[b*d/3 + 12^(1/3)*b^2*d^2/(3*r^2) + r^2/(3*12^(1/3)) - d*(2^(1/3)*b*d/(3^(1/3)*r) - r/18^(1/3))*x + d^2*x^2, x]^p, x]]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_ + b_.*x_ + d_.*x_^3)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && ILtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "With[{r = Rt[-9*a*d^2 + Sqrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]}, (a + b*x + d*x^3)^p/ (Simp[18^(1/3)*b*d/(3*r) - r/18^(1/3) + d*x, x]^p* Simp[b*d/3 + 12^(1/3)*b^2*d^2/(3*r^2) + r^2/(3*12^(1/3)) - d*(2^(1/3)*b*d/(3^(1/3)*r) - r/18^(1/3))*x + d^2*x^2, x]^ p)* Int[(e + f*x)^m* Simp[18^(1/3)*b*d/(3*r) - r/18^(1/3) + d*x, x]^p* Simp[b*d/3 + 12^(1/3)*b^2*d^2/(3*r^2) + r^2/(3*12^(1/3)) - d*(2^(1/3)*b*d/(3^(1/3)*r) - r/18^(1/3))*x + d^2*x^2, x]^p, x]]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_ + b_.*x_ + d_.*x_^3)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, p}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "With[{a = Coeff[P3, x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3, x, 3]}, Subst[ Int[((3*d*e - c*f)/(3*d) + f*x)^m* Simp[(2*c^3 - 9*b*c*d + 27*a*d^2)/(27*d^2) - (c^2 - 3*b*d)* x/(3*d) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c, 0]]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*P3_^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{e, f, m, p}, x] && PolyQ[P3, x, 3]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "With[{Px = 1/320*(33*b^2*c + 6*a*c^2 + 40*a^2*e) - 22/5*a*c*e*x^2 + 22/15*b*c*e*x^3 + 1/4*e*(5*c^2 + 4*a*e)*x^4 + 4/3*b*e^2*x^5 + 2*c*e^2*x^6 + e^3*x^8}, 1/(8*Rt[e, 2])* Log[Px + Dist[1/(8*Rt[e, 2]*x), D[Px, x], x]* Sqrt[a + b*x + c*x^2 + e*x^4]]]",
     "rulenumber": 0,
            "lhs": "Int[x_/Sqrt[a_ + b_.*x_ + c_.*x_^2 + e_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, e}, x] && EqQ[71*c^2 + 100*a*e, 0] && EqQ[1152*c^3 - 125*b^2*e, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "B*Subst[Int[ x/Sqrt[(-3*d^4 + 16*c*d^2*e - 64*b*d*e^2 + 256*a*e^3)/(256* e^3) + (d^3 - 4*c*d*e + 8*b*e^2)*x/(8*e^2) - (3*d^2 - 8*c*e)*x^2/(8*e) + e*x^4], x], x, d/(4*e) + x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*x_)/Sqrt[a_ + b_.*x_ + c_.*x_^2 + d_.*x_^3 + e_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[B*d - 4*A*e, 0] && EqQ[d*(141*d^3 - 752*c*d*e - 400*b*e^2) + 16*e^2*(71*c^2 + 100*a*e), 0] && EqQ[144*(3*d^2 - 8*c*e)^3 + 125*(d^3 - 4*c*d*e + 8*b*e^2)^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "a*f/(d*Rt[a^2*(2*a - c), 2])* ArcTan[(a*b + (4*a^2 + b^2 - 2*a*c)*x + a*b*x^2)/(2* Rt[a^2*(2*a - c), 2]*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_^2)/((d_ + e_.*x_ + d_.*x_^2)* Sqrt[a_ + b_.*x_ + c_.*x_^2 + b_.*x_^3 + a_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[b*d - a*e, 0] && EqQ[f + g, 0] && PosQ[a^2*(2*a - c)]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "-a*f/(d*Rt[-a^2*(2*a - c), 2])* ArcTanh[(a*b + (4*a^2 + b^2 - 2*a*c)*x + a*b*x^2)/(2* Rt[-a^2*(2*a - c), 2]*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_^2)/((d_ + e_.*x_ + d_.*x_^2)* Sqrt[a_ + b_.*x_ + c_.*x_^2 + b_.*x_^3 + a_.*x_^4]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[b*d - a*e, 0] && EqQ[f + g, 0] && NegQ[a^2*(2*a - c)]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "With[{q = Sqrt[8*a^2 + b^2 - 4*a*c], A = Coeff[P3, x, 0], B = Coeff[P3, x, 1], C = Coeff[P3, x, 2], D = Coeff[P3, x, 3]}, 1/q*Int[(b*A - 2*a*B + 2*a*D + A*q + (2*a*A - 2*a*C + b*D + D*q)*x)/(2*a + (b + q)*x + 2*a*x^2), x] - 1/q* Int[(b*A - 2*a*B + 2*a*D - A*q + (2*a*A - 2*a*C + b*D - D*q)*x)/(2*a + (b - q)*x + 2*a*x^2), x]]",
     "rulenumber": 0,
            "lhs": "Int[P3_/(a_ + b_.*x_ + c_.*x_^2 + d_.*x_^3 + e_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && PolyQ[P3, x, 3] && EqQ[a, e] && EqQ[b, d]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "With[{q = Sqrt[8*a^2 + b^2 - 4*a*c], A = Coeff[P3, x, 0], B = Coeff[P3, x, 1], C = Coeff[P3, x, 2], D = Coeff[P3, x, 3]}, 1/q*Int[ x^m*(b*A - 2*a*B + 2*a*D + A*q + (2*a*A - 2*a*C + b*D + D*q)*x)/(2*a + (b + q)*x + 2*a*x^2), x] - 1/q* Int[x^m*(b*A - 2*a*B + 2*a*D - A*q + (2*a*A - 2*a*C + b*D - D*q)*x)/(2*a + (b - q)*x + 2*a*x^2), x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*P3_/(a_ + b_.*x_ + c_.*x_^2 + d_.*x_^3 + e_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m}, x] && PolyQ[P3, x, 3] && EqQ[a, e] && EqQ[b, d]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "With[{q = Rt[C*(2*e*(B*d - 4*A*e) + C*(d^2 - 4*c*e)), 2]}, -2*C^2/q*ArcTanh[(C*d - B*e + 2*C*e*x)/q] + 2*C^2/q* ArcTanh[C*(4*B*c*C - 3*B^2*d - 4*A*C*d + 12*A*B*e + 4*C*(2*c*C - B*d + 2*A*e)*x + 4*C*(2*C*d - B*e)*x^2 + 8*C^2*e*x^3)/(q*(B^2 - 4*A*C))]]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*x_ + C_.*x_^2)/(a_ + b_.*x_ + c_.*x_^2 + d_.*x_^3 + e_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B, C}, x] && EqQ[B^2*d + 2*C*(b*C + A*d) - 2*B*(c*C + 2*A*e), 0] && EqQ[2*B^2*c*C - 8*a*C^3 - B^3*d - 4*A*B*C*d + 4*A*(B^2 + 2*A*C)*e, 0] && PosQ[C*(2*e*(B*d - 4*A*e) + C*(d^2 - 4*c*e))]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "With[{q = Rt[C*(-8*A*e^2 + C*(d^2 - 4*c*e)), 2]}, -2*C^2/q*ArcTanh[C*(d + 2*e*x)/q] + 2*C^2/q*ArcTanh[ C*(A*d - 2*(c*C + A*e)*x - 2*C*d*x^2 - 2*C*e*x^3)/(A*q)]]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + C_.*x_^2)/(a_ + b_.*x_ + c_.*x_^2 + d_.*x_^3 + e_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, C}, x] && EqQ[b*C + A*d, 0] && EqQ[a*C^2 - A^2*e, 0] && PosQ[C*(-8*A*e^2 + C*(d^2 - 4*c*e))]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "With[{q = Rt[-C*(2*e*(B*d - 4*A*e) + C*(d^2 - 4*c*e)), 2]}, 2*C^2/q*ArcTan[(C*d - B*e + 2*C*e*x)/q] - 2*C^2/q* ArcTan[C*(4*B*c*C - 3*B^2*d - 4*A*C*d + 12*A*B*e + 4*C*(2*c*C - B*d + 2*A*e)*x + 4*C*(2*C*d - B*e)*x^2 + 8*C^2*e*x^3)/(q*(B^2 - 4*A*C))]]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*x_ + C_.*x_^2)/(a_ + b_.*x_ + c_.*x_^2 + d_.*x_^3 + e_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B, C}, x] && EqQ[B^2*d + 2*C*(b*C + A*d) - 2*B*(c*C + 2*A*e), 0] && EqQ[2*B^2*c*C - 8*a*C^3 - B^3*d - 4*A*B*C*d + 4*A*(B^2 + 2*A*C)*e, 0] && NegQ[C*(2*e*(B*d - 4*A*e) + C*(d^2 - 4*c*e))]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "With[{q = Rt[-C*(-8*A*e^2 + C*(d^2 - 4*c*e)), 2]}, 2*C^2/q*ArcTan[(C*d + 2*C*e*x)/q] - 2*C^2/q*ArcTan[-C*(-A*d + 2*(c*C + A*e)*x + 2*C*d*x^2 + 2*C*e*x^3)/(A*q)]]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + C_.*x_^2)/(a_ + b_.*x_ + c_.*x_^2 + d_.*x_^3 + e_.*x_^4), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, C}, x] && EqQ[b*C + A*d, 0] && EqQ[a*C^2 - A^2*e, 0] && NegQ[C*(-8*A*e^2 + C*(d^2 - 4*c*e))]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1], c = Coeff[P4, x, 2], d = Coeff[P4, x, 3], e = Coeff[P4, x, 4]}, 1/a^(3*p)* Int[ExpandIntegrand[Px*(a - b*x)^(-p)/(a^5 - b^5*x^5)^(-p), x], x] /; NeQ[a, 0] && EqQ[c, b^2/a] && EqQ[d, b^3/a^2] && EqQ[e, b^4/a^3]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*P4_^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[p, x] && PolyQ[P4, x, 4] && PolyQ[Px, x] && ILtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "A^2*(n - 1)* Subst[Int[1/(a + A^2*b*(n - 1)^2*x^2), x], x, x/(A*(n - 1) - B*x^n)]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*x_^n_)/(a_ + b_.*x_^2 + c_.*x_^n_ + d_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, A, B, n}, x] && EqQ[n2, 2*n] && NeQ[n, 2] && EqQ[a*B^2 - A^2*d*(n - 1)^2, 0] && EqQ[B*c + 2*A*d*(n - 1), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "A^2*(m - n + 1)/(m + 1)* Subst[Int[1/(a + A^2*b*(m - n + 1)^2*x^2), x], x, x^(m + 1)/(A*(m - n + 1) + B*(m + 1)*x^n)]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(A_ + B_.*x_^n_.)/(a_ + b_.*x_^k_. + c_.*x_^n_. + d_.*x_^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, A, B, m, n}, x] && EqQ[n2, 2*n] && EqQ[k, 2*(m + 1)] && EqQ[a*B^2*(m + 1)^2 - A^2*d*(m - n + 1)^2, 0] && EqQ[B*c*(m + 1) - 2*A*d*(m - n + 1), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "With[{q = Rt[(-a*c*f^2 + 12*a^2*g^2 + f*(3*c^2*d - 2*a*b*g))/(c* g*(3*c*d - a*f)), 2], r = Rt[(a*c*f^2 + 4*g*(b*c*d + a^2*g) - f*(3*c^2*d + 2*a*b*g))/(c* g*(3*c*d - a*f)), 2]}, c/(g*q)*ArcTan[(r + 2*x)/q] - c/(g*q)*ArcTan[(r - 2*x)/q] - c/(g*q)* ArcTan[(3*c*d - a*f)* x/(g*q*(b*c*d - 2*a^2*g)*(b*c*d - a*b*f + 4*a^2*g))* (b*c^2*d*f - a*b^2*f*g - 2*a^2*c*f*g + 6*a^2*b*g^2 + c*(3*c^2*d*f - a*c*f^2 - b*c*d*g + 2*a^2*g^2)*x^2 + c^2*g*(3*c*d - a*f)*x^4)]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^2 + c_.*x_^4)/(d_ + e_.*x_^2 + f_.*x_^4 + g_.*x_^6), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[9*c^3*d^2 - c*(b^2 + 6*a*c)*d*f + a^2*c*f^2 + 2*a*b*(3*c*d + a*f)*g - 12*a^3*g^2, 0] && EqQ[3*c^4*d^2*e - 3*a^2*c^2*d*f*g + a^3*c*f^2*g + 2*a^3*g^2*(b*f - 6*a*g) - c^3*d*(2*b*d*f + a*e*f - 12*a*d*g), 0] && NeQ[3*c*d - a*f, 0] && NeQ[b*c*d - 2*a^2*g, 0] && NeQ[b*c*d - a*b*f + 4*a^2*g, 0] && PosQ[(-a*c*f^2 + 12*a^2*g^2 + f*(3*c^2*d - 2*a*b*g))/(c* g*(3*c*d - a*f))]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "With[{q = Rt[(-a*c*f^2 + 12*a^2*g^2 + 3*f*c^2*d)/(c*g*(3*c*d - a*f)), 2], r = Rt[(a*c*f^2 + 4*a^2*g^2 - 3*c^2*d*f)/(c*g*(3*c*d - a*f)), 2]}, c/(g*q)*ArcTan[(r + 2*x)/q] - c/(g*q)*ArcTan[(r - 2*x)/q] - c/(g*q)* ArcTan[(c*(3*c*d - a*f)* x*(2*a^2*f*g - (3*c^2*d*f - a*c*f^2 + 2*a^2*g^2)*x^2 - c*(3*c*d - a*f)*g*x^4))/(8*a^4*g^3*q)]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + c_.*x_^4)/(d_ + e_.*x_^2 + f_.*x_^4 + g_.*x_^6), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g}, x] && EqQ[9*c^3*d^2 - 6*a*c^2*d*f + a^2*c*f^2 - 12*a^3*g^2, 0] && EqQ[3*c^4*d^2*e - 3*a^2*c^2*d*f*g + a^3*c*f^2*g - 12*a^4*g^3 - a*c^3*d*(e*f - 12*d*g), 0] && NeQ[3*c*d - a*f, 0] && PosQ[(-a*c*f^2 + 12*a^2*g^2 + 3*c^2*d*f)/(c*g*(3*c*d - a*f))]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "With[{a = Coeff[Q6, x, 0], b = Coeff[Q6, x, 2], c = Coeff[Q6, x, 3], d = Coeff[Q6, x, 4], e = Coeff[Q6, x, 6]}, 1/(3^(3*p)*a^(2*p))*Int[ExpandIntegrand[u* (3*a + 3*Rt[a, 3]^2*Rt[c, 3]*x + b*x^2)^p* (3*a - 3*(-1)^(1/3)*Rt[a, 3]^2*Rt[c, 3]*x + b*x^2)^p* (3*a + 3*(-1)^(2/3)*Rt[a, 3]^2*Rt[c, 3]*x + b*x^2)^p, x], x] /; EqQ[b^2 - 3*a*d, 0] && EqQ[b^3 - 27*a^2*e, 0]]",
     "rulenumber": 0,
            "lhs": "Int[u_*Q6_^p_, x_Symbol]",
        "comment": false,
         "givens": "ILtQ[p, 0] && PolyQ[Q6, x, 6] && EqQ[Coeff[Q6, x, 1], 0] && EqQ[Coeff[Q6, x, 5], 0] && RationalFunctionQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Coeff[Pm, x, m]*Log[Qn]/(n*Coeff[Qn, x, n]) + Simplify[Pm - Coeff[Pm, x, m]*D[Qn, x]/(n*Coeff[Qn, x, n])]* Int[1/Qn, x] /; EqQ[m, n - 1] && EqQ[D[Simplify[ Pm - Coeff[Pm, x, m]/(n*Coeff[Qn, x, n])*D[Qn, x]], x], 0]]",
     "rulenumber": 0,
            "lhs": "Int[Pm_/Qn_, x_Symbol]",
        "comment": false,
         "givens": "PolyQ[Pm, x] && PolyQ[Qn, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Coeff[Pm, x, m]*Qn^(p + 1)/(n*(p + 1)*Coeff[Qn, x, n]) + Simplify[Pm - Coeff[Pm, x, m]*D[Qn, x]/(n*Coeff[Qn, x, n])]* Int[Qn^p, x] /; EqQ[m, n - 1] && EqQ[D[Simplify[ Pm - Coeff[Pm, x, m]/(n*Coeff[Qn, x, n])*D[Qn, x]], x], 0]]",
     "rulenumber": 0,
            "lhs": "Int[Pm_*Qn_^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[p, x] && PolyQ[Pm, x] && PolyQ[Qn, x] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Coeff[Pm, x, m]*Log[Qn]/(n*Coeff[Qn, x, n]) + 1/(n*Coeff[Qn, x, n]) Int[ ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x, m]*D[Qn, x], x]/Qn, x] /; EqQ[m, n - 1]]",
     "rulenumber": 0,
            "lhs": "Int[Pm_/Qn_, x_Symbol]",
        "comment": false,
         "givens": "PolyQ[Pm, x] && PolyQ[Qn, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Coeff[Pm, x, m]*Qn^(p + 1)/(n*(p + 1)*Coeff[Qn, x, n]) + 1/(n*Coeff[Qn, x, n])* Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x, m]*D[Qn, x], x]*Qn^p, x] /; EqQ[m, n - 1]]",
     "rulenumber": 0,
            "lhs": "Int[Pm_*Qn_^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[p, x] && PolyQ[Pm, x] && PolyQ[Qn, x] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.2 P(x) Q(x)^p.m",
       "filename": "1.3.2 P(x) Q(x)^p.m",
            "rhs": "With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Coeff[Pm, x, m]*x^(m - n + 1)* Qn^(p + 1)/((m + n*p + 1)*Coeff[Qn, x, n]) + 1/((m + n*p + 1)*Coeff[Qn, x, n])* Int[ExpandToSum[(m + n*p + 1)*Coeff[Qn, x, n]*Pm - Coeff[Pm, x, m]* x^(m - n)*((m - n + 1)*Qn + (p + 1)*x*D[Qn, x]), x]*Qn^p, x] /; LtQ[1, n, m + 1] && m + n*p + 1 < 0]",
     "rulenumber": 0,
            "lhs": "Int[Pm_*Qn_^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[p, x] && PolyQ[Pm, x] && PolyQ[Qn, x] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "c/(e*(b*c - a*d))*Int[(u*Sqrt[a + b*x])/x, x] - a/(f*(b*c - a*d))*Int[(u*Sqrt[c + d*x])/x, x]",
     "rulenumber": 0,
            "lhs": "Int[u_/(e_.*Sqrt[a_. + b_.*x_] + f_.*Sqrt[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a*e^2 - c*f^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "-d/(e*(b*c - a*d))*Int[u*Sqrt[a + b*x], x] + b/(f*(b*c - a*d))*Int[u*Sqrt[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[u_/(e_.*Sqrt[a_. + b_.*x_] + f_.*Sqrt[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[b*e^2 - d*f^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "e*Int[(u*Sqrt[a + b*x])/(a*e^2 - c*f^2 + (b*e^2 - d*f^2)*x), x] - f*Int[(u*Sqrt[c + d*x])/(a*e^2 - c*f^2 + (b*e^2 - d*f^2)*x), x]",
     "rulenumber": 0,
            "lhs": "Int[u_/(e_.*Sqrt[a_. + b_.*x_] + f_.*Sqrt[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[a*e^2 - c*f^2, 0] && NeQ[b*e^2 - d*f^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "-b/(a*d)*Int[u*x^n, x] + 1/(a*c)*Int[u*Sqrt[a + b*x^(2*n)], x]",
     "rulenumber": 0,
            "lhs": "Int[u_./(d_.*x_^n_. + c_.*Sqrt[a_. + b_.*x_^p_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && EqQ[p, 2*n] && EqQ[b*c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "-d* Int[x^(m + n)/(a*c^2 + (b*c^2 - d^2)*x^(2*n)), x] + c*Int[(x^m*Sqrt[a + b*x^(2*n)])/(a*c^2 + (b*c^2 - d^2)*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_./(d_.*x_^n_. + c_.*Sqrt[a_. + b_.*x_^p_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && EqQ[p, 2*n] && NeQ[b*c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "With[{r = Numerator[Rt[a/b, 3]], s = Denominator[Rt[a/b, 3]]}, r/(3*a)*Int[1/((r + s*x)*Sqrt[d + e*x + f*x^2]), x] + r/(3*a)* Int[(2*r - s*x)/((r^2 - r*s*x + s^2*x^2)*Sqrt[d + e*x + f*x^2]), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_ + b_.*x_^3)*Sqrt[d_. + e_.*x_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && PosQ[a/b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "With[{r = Numerator[Rt[a/b, 3]], s = Denominator[Rt[a/b, 3]]}, r/(3*a)*Int[1/((r + s*x)*Sqrt[d + f*x^2]), x] + r/(3*a)* Int[(2*r - s*x)/((r^2 - r*s*x + s^2*x^2)*Sqrt[d + f*x^2]), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_ + b_.*x_^3)*Sqrt[d_. + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, f}, x] && PosQ[a/b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "With[{r = Numerator[Rt[-a/b, 3]], s = Denominator[Rt[-a/b, 3]]}, r/(3*a)*Int[1/((r - s*x)*Sqrt[d + e*x + f*x^2]), x] + r/(3*a)* Int[(2*r + s*x)/((r^2 + r*s*x + s^2*x^2)*Sqrt[d + e*x + f*x^2]), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_ + b_.*x_^3)*Sqrt[d_. + e_.*x_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NegQ[a/b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "With[{r = Numerator[Rt[-a/b, 3]], s = Denominator[Rt[-a/b, 3]]}, r/(3*a)*Int[1/((r - s*x)*Sqrt[d + f*x^2]), x] + r/(3*a)* Int[(2*r + s*x)/((r^2 + r*s*x + s^2*x^2)*Sqrt[d + f*x^2]), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_ + b_.*x_^3)*Sqrt[d_. + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, f}, x] && NegQ[a/b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "With[{a = Coeff[v, x, 0], b = Coeff[v, x, 2], c = Coeff[v, x, 4], d = Coeff[1/u, x, 0], e = Coeff[1/u, x, 2], f = Coeff[1/u, x, 4]}, A*Subst[Int[1/(d - (b*d - a*e)*x^2), x], x, x/Sqrt[v]] /; EqQ[a*B + A*c, 0] && EqQ[c*d - a*f, 0]]",
     "rulenumber": 0,
            "lhs": "Int[u_*(A_ + B_.*x_^4)/Sqrt[v_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{A, B}, x] && PolyQ[v, x^2, 2] && PolyQ[1/u, x^2, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "a*Int[1/((a^2 - b^2*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x] - b*Int[x/((a^2 - b^2*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_ + b_.*x_)*Sqrt[c_ + d_.*x_^2]*Sqrt[e_ + f_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "2*(f*(5*b*c*g^2 - 2*b^2*g*h - 3*a*c*g*h + 2*a*b*h^2) + c*f*(10*c*g^2 - b*g*h + a*h^2)*x + 9*c^2*f*g*h*x^2 + 3*c^2*f*h^2*x^3 - (e*g - d*h)*(5*c*g - 2*b*h + c*h*x)*Sqrt[a + b*x + c*x^2])/ (15*c^2*f*(g + h*x))*Sqrt[d + e*x + f*Sqrt[a + b*x + c*x^2]]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)* Sqrt[d_. + e_.*x_ + f_.*Sqrt[a_. + b_.*x_ + c_.*x_^2]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h}, x] && EqQ[(e*g - d*h)^2 - f^2*(c*g^2 - b*g*h + a*h^2), 0] && EqQ[2*e^2*g - 2*d*e*h - f^2*(2*c*g - b*h), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "Int[(g + h*x)^ m*(ExpandToSum[u + f*j, x] + f*k*Sqrt[ExpandToSum[v, x]])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)^m_.*(u_ + f_.*(j_. + k_.*Sqrt[v_]))^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{f, g, h, j, k, m, n}, x] && LinearQ[u, x] && QuadraticQ[v, x] && Not[LinearMatchQ[u, x] && QuadraticMatchQ[v, x] && (EqQ[j, 0] || EqQ[f, 1])] && EqQ[(Coefficient[u, x, 1]*g - h*(Coefficient[u, x, 0] + f*j))^2 - f^2*k^2*(Coefficient[v, x, 2]*g^2 - Coefficient[v, x, 1]*g*h + Coefficient[v, x, 0]*h^2), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": " Int[(d+e*x)/(d^2-a*f^2+(2*d*e-b*f^2)*x),x] - f*Int[Sqrt[a+b*x+c*x^2]/(d^2-a*f^2+(2*d*e-b*f^2)*x),x]",
     "rulenumber": 0,
            "lhs": "Int[1/(d_.+e_.*x_+f_.*Sqrt[a_.+b_.*x_+c_.*x_^2]),x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e,f},x] && EqQ[e^2-c*f^2,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": " Int[(d+e*x)/(d^2-a*f^2+2*d*e*x),x] - f*Int[Sqrt[a+c*x^2]/(d^2-a*f^2+2*d*e*x),x]",
     "rulenumber": 0,
            "lhs": "Int[1/(d_.+e_.*x_+f_.*Sqrt[a_.+c_.*x_^2]),x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,c,d,e,f},x] &&  EqQ[e^2-c*f^2,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "2*Subst[Int[(g + h*x^n)^ p*(d^2*e - (b*d - a*e)*f^2 - (2*d*e - b*f^2)*x + e*x^2)/(-2*d*e + b*f^2 + 2*e*x)^2, x], x, d + e*x + f*Sqrt[a + b*x + c*x^2]]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*(d_. + e_.*x_ + f_.*Sqrt[a_. + b_.*x_ + c_.*x_^2])^n_)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "1/(2*e)*Subst[ Int[(g + h*x^n)^p*(d^2 + a*f^2 - 2*d*x + x^2)/(d - x)^2, x], x, d + e*x + f*Sqrt[a + c*x^2]]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*(d_. + e_.*x_ + f_.*Sqrt[a_ + c_.*x_^2])^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "Int[(g + h*(ExpandToSum[u, x] + f*Sqrt[ExpandToSum[v, x]])^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*(u_ + f_. Sqrt[v_])^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{f, g, h, n}, x] && LinearQ[u, x] && QuadraticQ[v, x] && Not[LinearMatchQ[u, x] && QuadraticMatchQ[v, x]] && EqQ[Coefficient[u, x, 1]^2 - Coefficient[v, x, 2]*f^2, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "1/(2^(m + 1)*e^(m + 1))* Subst[Int[ x^(n - m - 2)*(a*f^2 + x^2)*(-a*f^2*h + 2*e*g*x + h*x^2)^m, x], x, e*x + f*Sqrt[a + c*x^2]]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)^m_.*(e_.*x_ + f_.*Sqrt[a_. + c_.*x_^2])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, e, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "1/(2^(2*m + p + 1)*e^(p + 1)*f^(2*m))*(i/c)^m* Subst[Int[ x^(n - 2*m - p - 2)*(-a*f^2 + x^2)^p*(a*f^2 + x^2)^(2*m + 1), x], x, e*x + f*Sqrt[a + c*x^2]]",
     "rulenumber": 0,
            "lhs": "Int[x_^p_.*(g_ + i_.*x_^2)^m_.*(e_.*x_ + f_.*Sqrt[a_ + c_.*x_^2])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, e, f, g, i, n}, x] && EqQ[e^2 - c*f^2, 0] && EqQ[c*g - a*i, 0] && IntegersQ[p, 2*m] && (IntegerQ[m] || GtQ[i/c, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "2/f^(2*m)*(i/c)^m* Subst[ Int[x^n*(d^2*e - (b*d - a*e)*f^2 - (2*d*e - b*f^2)*x + e*x^2)^(2*m + 1)/(-2*d*e + b*f^2 + 2*e*x)^(2*(m + 1)), x], x, d + e*x + f*Sqrt[a + b*x + c*x^2]]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_ + i_.*x_^2)^ m_.*(d_. + e_.*x_ + f_.*Sqrt[a_. + b_.*x_ + c_.*x_^2])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, i, n}, x] && EqQ[e^2 - c*f^2, 0] && EqQ[c*g - a*i, 0] && EqQ[c*h - b*i, 0] && IntegerQ[2*m] && (IntegerQ[m] || GtQ[i/c, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "1/(2^(2*m + 1)*e*f^(2*m))*(i/c)^m* Subst[ Int[x^n*(d^2 + a*f^2 - 2*d*x + x^2)^(2*m + 1)/(-d + x)^(2*(m + 1)), x], x, d + e*x + f*Sqrt[a + c*x^2]]",
     "rulenumber": 0,
            "lhs": "Int[(g_ + i_.*x_^2)^m_.*(d_. + e_.*x_ + f_.*Sqrt[a_ + c_.*x_^2])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, i, n}, x] && EqQ[e^2 - c*f^2, 0] && EqQ[c*g - a*i, 0] && IntegerQ[2*m] && (IntegerQ[m] || GtQ[i/c, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "(i/c)^(m - 1/2)* Sqrt[g + h*x + i*x^2]/Sqrt[a + b*x + c*x^2]* Int[(a + b*x + c*x^2)^m*(d + e*x + f*Sqrt[a + b*x + c*x^2])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_ + i_.*x_^2)^ m_.*(d_. + e_.*x_ + f_.*Sqrt[a_. + b_.*x_ + c_.*x_^2])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, i, n}, x] && EqQ[e^2 - c*f^2, 0] && EqQ[c*g - a*i, 0] && EqQ[c*h - b*i, 0] && IGtQ[m + 1/2, 0] && Not[GtQ[i/c, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "(i/c)^(m - 1/2)*Sqrt[g + i*x^2]/Sqrt[a + c*x^2]* Int[(a + c*x^2)^m*(d + e*x + f*Sqrt[a + c*x^2])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_ + i_.*x_^2)^m_.*(d_. + e_.*x_ + f_.*Sqrt[a_ + c_.*x_^2])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, i, n}, x] && EqQ[e^2 - c*f^2, 0] && EqQ[c*g - a*i, 0] && IGtQ[m + 1/2, 0] && Not[GtQ[i/c, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "(i/c)^(m + 1/2)* Sqrt[a + b*x + c*x^2]/Sqrt[g + h*x + i*x^2]* Int[(a + b*x + c*x^2)^m*(d + e*x + f*Sqrt[a + b*x + c*x^2])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_ + i_.*x_^2)^ m_.*(d_. + e_.*x_ + f_.*Sqrt[a_. + b_.*x_ + c_.*x_^2])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, i, n}, x] && EqQ[e^2 - c*f^2, 0] && EqQ[c*g - a*i, 0] && EqQ[c*h - b*i, 0] && ILtQ[m - 1/2, 0] && Not[GtQ[i/c, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "(i/c)^(m + 1/2)*Sqrt[a + c*x^2]/Sqrt[g + i*x^2]* Int[(a + c*x^2)^m*(d + e*x + f*Sqrt[a + c*x^2])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_ + i_.*x_^2)^m_.*(d_. + e_.*x_ + f_.*Sqrt[a_ + c_.*x_^2])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, i, n}, x] && EqQ[e^2 - c*f^2, 0] && EqQ[c*g - a*i, 0] && ILtQ[m - 1/2, 0] && Not[GtQ[i/c, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "Int[ExpandToSum[w, x]^ m*(ExpandToSum[u + f*j, x] + f*k*Sqrt[ExpandToSum[v, x]])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[w_^m_.*(u_ + f_.*(j_. + k_.*Sqrt[v_]))^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{f, j, k, m, n}, x] && LinearQ[u, x] && QuadraticQ[{v, w}, x] && Not[LinearMatchQ[u, x] && QuadraticMatchQ[{v, w}, x] && (EqQ[j, 0] || EqQ[f, 1])] && EqQ[Coefficient[u, x, 1]^2 - Coefficient[v, x, 2]*f^2*k^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "1/a*Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[c*x^2 + d*(a + b*x^n)^(2/n)]]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_ + b_.*x_^n_.)*Sqrt[c_.*x_^2 + d_.*(a_ + b_.*x_^n_.)^p_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && EqQ[p, 2/n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "2*b^2*d*x^3/(3*(a + b*Sqrt[c + d*x^2])^(3/2)) + 2*a*x/Sqrt[a + b*Sqrt[c + d*x^2]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*Sqrt[c_ + d_.*x_^2]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "Sqrt[2]*b/a* Subst[Int[1/Sqrt[1 + x^2/a], x], x, a*x + b*Sqrt[c + d*x^2]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_.*x_^2 + b_.*x_*Sqrt[c_ + d_.*x_^2]]/(x_* Sqrt[c_ + d_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2*d, 0] && EqQ[b^2*c + a, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "Int[Sqrt[a*e*x^2 + b*e*x*Sqrt[c + d*x^2]]/(x*Sqrt[c + d*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[e_.*x_*(a_.*x_ + b_.*Sqrt[c_ + d_.*x_^2])]/(x_* Sqrt[c_ + d_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2*d, 0] && EqQ[b^2*c*e + a, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "d*Subst[Int[1/(1 - 2*c*x^2), x], x, x/Sqrt[c*x^2 + d*Sqrt[a + b*x^4]]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[c_.*x_^2 + d_.*Sqrt[a_ + b_.*x_^4]]/Sqrt[a_ + b_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[c^2 - b*d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "(1 - I)/2* Int[(c + d*x)^m/Sqrt[Sqrt[a] - I*b*x^2], x] + (1 + I)/2*Int[(c + d*x)^m/Sqrt[Sqrt[a] + I*b*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.* Sqrt[b_.*x_^2 + Sqrt[a_ + e_.*x_^4]]/Sqrt[a_ + e_.*x_^4], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && EqQ[e, b^2] && GtQ[a, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "2/(3*c)*Int[1/Sqrt[a + b*x^3], x] + 1/(3*c)*Int[(c - 2*d*x)/((c + d*x)*Sqrt[a + b*x^3]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((c_ + d_.*x_)*Sqrt[a_ + b_.*x_^3]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[b*c^3 - 4*a*d^3, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "-6*a*d^3/(c*(b*c^3 - 28*a*d^3))* Int[1/Sqrt[a + b*x^3], x] + 1/(c*(b*c^3 - 28*a*d^3))* Int[Simp[c*(b*c^3 - 22*a*d^3) + 6*a*d^4*x, x]/((c + d*x)*Sqrt[a + b*x^3]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((c_ + d_.*x_)*Sqrt[a_ + b_.*x_^3]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[b^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "With[{q = Rt[b/a, 3]}, -q/((1 + Sqrt[3])*d - c*q)*Int[1/Sqrt[a + b*x^3], x] + d/((1 + Sqrt[3])*d - c*q)* Int[(1 + Sqrt[3] + q*x)/((c + d*x)*Sqrt[a + b*x^3]), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/((c_ + d_.*x_)*Sqrt[a_ + b_.*x_^3]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "2*e/d*Subst[Int[1/(1 + 3*a*x^2), x], x, (1 + 2*d*x/c)/Sqrt[a + b*x^3]]",
     "rulenumber": 0,
            "lhs": "Int[(e_ + f_.*x_)/((c_ + d_.*x_)*Sqrt[a_ + b_.*x_^3]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b*c^3 - 4*a*d^3, 0] && EqQ[2*d*e + c*f, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "-2*e/d* Subst[Int[1/(9 - a*x^2), x], x, (1 + f*x/e)^2/Sqrt[a + b*x^3]]",
     "rulenumber": 0,
            "lhs": "Int[(e_ + f_.*x_)/((c_ + d_.*x_)*Sqrt[a_ + b_.*x_^3]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b*c^3 + 8*a*d^3, 0] && EqQ[2*d*e + c*f, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "(2*d*e + c*f)/(3*c*d)*Int[1/Sqrt[a + b*x^3], x] + (d*e - c*f)/(3*c*d)* Int[(c - 2*d*x)/((c + d*x)*Sqrt[a + b*x^3]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)/((c_ + d_.*x_)*Sqrt[a_ + b_.*x_^3]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && (EqQ[b*c^3 - 4*a*d^3, 0] || EqQ[b*c^3 + 8*a*d^3, 0]) && NeQ[2*d*e + c*f, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "With[{k = Simplify[(d*e + 2*c*f)/(c*f)]}, (1 + k)*e/d* Subst[Int[1/(1 + (3 + 2*k)*a*x^2), x], x, (1 + (1 + k)*d*x/c)/Sqrt[a + b*x^3]]]",
     "rulenumber": 0,
            "lhs": "Int[(e_ + f_.*x_)/((c_ + d_.*x_)*Sqrt[a_ + b_.*x_^3]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6, 0] && EqQ[6*a*d^4*e - c*f*(b*c^3 - 22*a*d^3), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "-(6*a*d^4*e - c*f*(b*c^3 - 22*a*d^3))/(c* d*(b*c^3 - 28*a*d^3))*Int[1/Sqrt[a + b*x^3], x] + (d*e - c*f)/(c*d*(b*c^3 - 28*a*d^3))* Int[(c*(b*c^3 - 22*a*d^3) + 6*a*d^4*x)/((c + d*x)* Sqrt[a + b*x^3]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)/((c_ + d_.*x_)*Sqrt[a_ + b_.*x_^3]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6, 0] && NeQ[6*a*d^4*e - c*f*(b*c^3 - 22*a*d^3), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": " With[{q=(1+Sqrt[3])*f/e}, 4*3^(1/4)*Sqrt[2-Sqrt[3]]*f*(1+Sqrt[3]+q*x)^2*Sqrt[(1+q^3*x^3)/(1+ Sqrt[3]+q*x)^4]/(q*Sqrt[a+b*x^3])* Subst[Int[1/(((1-Sqrt[3])*d-c*q+((1+Sqrt[3])*d-c*q)*x)* Sqrt[7-4*Sqrt[3]-2*(3-2*Sqrt[3])*x^2-x^4]),x],x,(-1+Sqrt[3]-q*x) /(1+Sqrt[3]+q*x)]]",
     "rulenumber": 0,
            "lhs": "Int[(e_+f_.*x_)/((c_+d_.*x_)*Sqrt[a_+b_.*x_^3]),x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f},x] && NeQ[d*e-c*f,0] &&  EqQ[b*e^3-2*(5+3*Sqrt[3])*a*f^3,0] &&  NeQ[b*c^3-2*(5-3*Sqrt[3])*a*d^3,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "With[{q = Simplify[(1 + Sqrt[3])*f/e]}, 4*3^(1/4)*Sqrt[2 - Sqrt[3]]*f*(1 + q*x)* Sqrt[(1 - q*x + q^2*x^2)/(1 + Sqrt[3] + q*x)^2]/ (q*Sqrt[a + b*x^3]*Sqrt[(1 + q*x)/(1 + Sqrt[3] + q*x)^2])* Subst[ Int[1/(((1 - Sqrt[3])*d - c*q + ((1 + Sqrt[3])*d - c*q)*x)* Sqrt[1 - x^2]*Sqrt[7 - 4*Sqrt[3] + x^2]), x], x, (-1 + Sqrt[3] - q*x)/(1 + Sqrt[3] + q*x)]]",
     "rulenumber": 0,
            "lhs": "Int[(e_ + f_.*x_)/((c_ + d_.*x_)*Sqrt[a_ + b_.*x_^3]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b*e^3 - 2*(5 + 3*Sqrt[3])*a*f^3, 0] && NeQ[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "With[{q = Simplify[(-1 + Sqrt[3])*f/e]}, 4*3^(1/4)*Sqrt[2 + Sqrt[3]]*f*(1 - q*x)* Sqrt[(1 + q*x + q^2*x^2)/(1 - Sqrt[3] - q*x)^2]/ (q*Sqrt[a + b*x^3]*Sqrt[-(1 - q*x)/(1 - Sqrt[3] - q*x)^2])* Subst[ Int[1/(((1 + Sqrt[3])*d + c*q + ((1 - Sqrt[3])*d + c*q)*x)* Sqrt[1 - x^2]*Sqrt[7 + 4*Sqrt[3] + x^2]), x], x, (1 + Sqrt[3] - q*x)/(-1 + Sqrt[3] + q*x)]]",
     "rulenumber": 0,
            "lhs": "Int[(e_ + f_.*x_)/((c_ + d_.*x_)*Sqrt[a_ + b_.*x_^3]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b*e^3 - 2*(5 - 3*Sqrt[3])*a*f^3, 0] && NeQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "With[{q = Rt[b/a, 3]}, ((1 + Sqrt[3])*f - e*q)/((1 + Sqrt[3])*d - c*q)* Int[1/Sqrt[a + b*x^3], x] + (d*e - c*f)/((1 + Sqrt[3])*d - c*q)* Int[(1 + Sqrt[3] + q*x)/((c + d*x)*Sqrt[a + b*x^3]), x]]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)/((c_ + d_.*x_)*Sqrt[a_ + b_.*x_^3]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && NeQ[b^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6, 0] && NeQ[b^2*e^6 - 20*a*b*e^3*f^3 - 8*a^2*f^6, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "-2*g*h* Subst[Int[1/(2*e*h - (b*d*f - 2*a*e*h)*x^2), x], x, (1 + 2*h*x/g)/Sqrt[a + b*x^3]]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_ + h_.*x_^2)/((c_ + d_.*x_ + e_.*x_^2)* Sqrt[a_ + b_.*x_^3]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b*d*f - 2*a*e*h, 0] && EqQ[b*g^3 - 8*a*h^3, 0] && EqQ[g^2 + 2*f*h, 0] && EqQ[b*d*f + b*c*g - 4*a*e*h, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "-g/e* Subst[Int[1/(1 + a*x^2), x], x, (1 + 2*h*x/g)/Sqrt[a + b*x^3]]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_ + h_.*x_^2)/((c_ + e_.*x_^2)*Sqrt[a_ + b_.*x_^3]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, e, f, g, h}, x] && EqQ[b*g^3 - 8*a*h^3, 0] && EqQ[g^2 + 2*f*h, 0] && EqQ[b*c*g - 4*a*e*h, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "b/d*Int[x^2/Sqrt[a + b*x^3], x] - (b*c^3 - a*d^3)/d^3*Int[1/((c + d*x)*Sqrt[a + b*x^3]), x] + b*c/d^3*Int[(c - d*x)/Sqrt[a + b*x^3], x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*x_^3]/(c_ + d_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "Sqrt[3]*ArcTan[(1 - 2^(1/3)*Rt[b, 3]*(c - d*x)/(d*(a + b*x^3)^(1/3)))/ Sqrt[3]]/(2^(4/3)*Rt[b, 3]*c) + Log[(c + d*x)^2*(c - d*x)]/(2^(7/3)*Rt[b, 3]*c) - (3*Log[ Rt[b, 3]*(c - d*x) + 2^(2/3)*d*(a + b*x^3)^(1/3)])/(2^(7/3)* Rt[b, 3]*c)",
     "rulenumber": 0,
            "lhs": "Int[1/((c_ + d_.*x_)*(a_ + b_.*x_^3)^(1/3)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[b*c^3 + a*d^3, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "1/(2*c)*Int[1/(a + b*x^3)^(1/3), x] + 1/(2*c)*Int[(c - d*x)/((c + d*x)*(a + b*x^3)^(1/3)), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((c_ + d_.*x_)*(a_ + b_.*x_^3)^(1/3)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[2*b*c^3 - a*d^3, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "Unintegrable[1/((c + d*x)*(a + b*x^3)^(1/3)), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((c_ + d_.*x_)*(a_ + b_.*x_^3)^(1/3)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "Sqrt[3]*f* ArcTan[(1 + 2*Rt[b, 3]*(2*c + d*x)/(d*(a + b*x^3)^(1/3)))/ Sqrt[3]]/(Rt[b, 3]*d) + (f*Log[c + d*x])/(Rt[b, 3]*d) - (3*f*Log[Rt[b, 3]*(2*c + d*x) - d*(a + b*x^3)^(1/3)])/(2* Rt[b, 3]*d)",
     "rulenumber": 0,
            "lhs": "Int[(e_ + f_.*x_)/((c_ + d_.*x_)*(a_ + b_.*x_^3)^(1/3)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[d*e + c*f, 0] && EqQ[2*b*c^3 - a*d^3, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "f/d*Int[1/(a + b*x^3)^(1/3), x] + (d*e - c*f)/d* Int[1/((c + d*x)*(a + b*x^3)^(1/3)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)/((c_. + d_.*x_)*(a_ + b_.*x_^3)^(1/3)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "Int[ExpandIntegrand[(a + b*x^nn)^ p, (c/(c^2 - d^2*x^(2*n)) - d*x^n/(c^2 - d^2*x^(2*n)))^(-q), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_ + d_.*x_^n_.)^q_*(a_ + b_.*x_^nn_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, nn, p}, x] && Not[IntegerQ[p]] && ILtQ[q, 0] && IGtQ[Log[2, nn/n], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "(e*x)^m/x^m* Int[ExpandIntegrand[ x^m*(a + b*x^nn)^ p, (c/(c^2 - d^2*x^(2*n)) - d*x^n/(c^2 - d^2*x^(2*n)))^(-q), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(c_ + d_.*x_^n_.)^q_*(a_ + b_.*x_^nn_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, nn, p}, x] && Not[IntegerQ[p]] && ILtQ[q, 0] && IGtQ[Log[2, nn/n], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "1/n*Subst[Int[x^((m + 1)/n - 1)/(c + d*x + e*Sqrt[a + b*x]), x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_./(c_ + d_.*x_^n_ + e_.*Sqrt[a_ + b_.*x_^n_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[(m + 1)/n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.3 Miscellaneous algebraic functions.m",
       "filename": "1.3.3 Miscellaneous algebraic functions.m",
            "rhs": "c*Int[u/(c^2 - a*e^2 + c*d*x^n), x] - a*e*Int[u/((c^2 - a*e^2 + c*d*x^n)*Sqrt[a + b*x^n]), x]",
     "rulenumber": 0,
            "lhs": "Int[u_./(c_ + d_.*x_^n_ + e_.*Sqrt[a_ + b_.*x_^n_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.4 Normalizing algebraic functions.m",
       "filename": "1.3.4 Normalizing algebraic functions.m",
            "rhs": "(e*(a + b*x^n)^r)^ p*(f*(c + d*x^n)^s)^q/((a + b*x^n)^(p*r)*(c + d*x^n)^(q*s))* Int[x^m*(a + b*x^n)^(p*r)*(c + d*x^n)^(q*s), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(e_.*(a_ + b_.*x_^n_.)^r_.)^p_*(f_.*(c_ + d_.*x_^n_.)^s_)^ q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p, q, r, s}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.4 Normalizing algebraic functions.m",
       "filename": "1.3.4 Normalizing algebraic functions.m",
            "rhs": "(b*e/d)^p*Int[u, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(e_.*(a_. + b_.*x_^n_.)/(c_ + d_.*x_^n_.))^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.4 Normalizing algebraic functions.m",
       "filename": "1.3.4 Normalizing algebraic functions.m",
            "rhs": "Int[u*(e*(a + b*x^n))^p/(c + d*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(e_.*(a_. + b_.*x_^n_.)/(c_ + d_.*x_^n_.))^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p}, x] && GtQ[b*d*e, 0] && GtQ[c - a*d/b, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.4 Normalizing algebraic functions.m",
       "filename": "1.3.4 Normalizing algebraic functions.m",
            "rhs": "With[{q = Denominator[p]}, q*e*(b*c - a*d)/n*Subst[ Int[ x^(q*(p + 1) - 1)*(-a*e + c*x^q)^(1/n - 1)/(b*e - d*x^q)^(1/n + 1), x], x, (e*(a + b*x^n)/(c + d*x^n))^(1/q)]]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*(a_. + b_.*x_^n_.)/(c_ + d_.*x_^n_.))^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && FractionQ[p] && IntegerQ[1/n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.4 Normalizing algebraic functions.m",
       "filename": "1.3.4 Normalizing algebraic functions.m",
            "rhs": "With[{q = Denominator[p]}, q*e*(b*c - a*d)/n*Subst[ Int[ x^(q*(p + 1) - 1)*(-a*e + c*x^q)^(Simplify[(m + 1)/n] - 1)/(b*e - d*x^q)^(Simplify[(m + 1)/n] + 1), x], x, (e*(a + b*x^n)/(c + d*x^n))^(1/q)]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(e_.*(a_. + b_.*x_^n_.)/(c_ + d_.*x_^n_.))^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && FractionQ[p] && IntegerQ[Simplify[(m + 1)/n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.4 Normalizing algebraic functions.m",
       "filename": "1.3.4 Normalizing algebraic functions.m",
            "rhs": "With[{q = Denominator[p]}, q*e*(b*c - a*d)/n* Subst[Int[ SimplifyIntegrand[ x^(q*(p + 1) - 1)*(-a*e + c*x^q)^(1/n - 1)/(b*e - d*x^q)^(1/n + 1)* ReplaceAll[u, x -> (-a*e + c*x^q)^(1/n)/(b*e - d*x^q)^(1/n)]^r, x], x], x, (e*(a + b*x^n)/(c + d*x^n))^(1/q)]]",
     "rulenumber": 0,
            "lhs": "Int[u_^r_.*(e_.*(a_. + b_.*x_^n_.)/(c_ + d_.*x_^n_.))^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && PolynomialQ[u, x] && FractionQ[p] && IntegerQ[1/n] && IntegerQ[r]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.4 Normalizing algebraic functions.m",
       "filename": "1.3.4 Normalizing algebraic functions.m",
            "rhs": "With[{q = Denominator[p]}, q*e*(b*c - a*d)/n* Subst[Int[ SimplifyIntegrand[ x^(q*(p + 1) - 1)*(-a*e + c*x^q)^((m + 1)/n - 1)/(b*e - d*x^q)^((m + 1)/n + 1)* ReplaceAll[u, x -> (-a*e + c*x^q)^(1/n)/(b*e - d*x^q)^(1/n)]^r, x], x], x, (e*(a + b*x^n)/(c + d*x^n))^(1/q)]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*u_^r_.*(e_.*(a_. + b_.*x_^n_.)/(c_ + d_.*x_^n_.))^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && PolynomialQ[u, x] && FractionQ[p] && IntegerQ[1/n] && IntegersQ[m, r]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.4 Normalizing algebraic functions.m",
       "filename": "1.3.4 Normalizing algebraic functions.m",
            "rhs": "-c*Subst[Int[(a + b*x^n)^p/x^2, x], x, c/x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*(c_./x_)^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.4 Normalizing algebraic functions.m",
       "filename": "1.3.4 Normalizing algebraic functions.m",
            "rhs": "-c^(m + 1)* Subst[Int[(a + b*x^n)^p/x^(m + 2), x], x, c/x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*(c_./x_)^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, p}, x] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.4 Normalizing algebraic functions.m",
       "filename": "1.3.4 Normalizing algebraic functions.m",
            "rhs": "-c*(d*x)^m*(c/x)^m* Subst[Int[(a + b*x^n)^p/x^(m + 2), x], x, c/x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_*(a_. + b_.*(c_./x_)^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.4 Normalizing algebraic functions.m",
       "filename": "1.3.4 Normalizing algebraic functions.m",
            "rhs": "-d* Subst[Int[(a + b*x^n + c*x^(2*n))^p/x^2, x], x, d/x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*(d_./x_)^n_ + c_.*(d_./x_)^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x] && EqQ[n2, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.4 Normalizing algebraic functions.m",
       "filename": "1.3.4 Normalizing algebraic functions.m",
            "rhs": "-d^(m + 1)* Subst[Int[(a + b*x^n + c*x^(2*n))^p/x^(m + 2), x], x, d/x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*(d_./x_)^n_ + c_.*(d_./x_)^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x] && EqQ[n2, 2*n] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.4 Normalizing algebraic functions.m",
       "filename": "1.3.4 Normalizing algebraic functions.m",
            "rhs": "-d*(e*x)^m*(d/x)^m* Subst[Int[(a + b*x^n + c*x^(2*n))^p/x^(m + 2), x], x, d/x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_*(a_ + b_.*(d_./x_)^n_ + c_.*(d_./x_)^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.4 Normalizing algebraic functions.m",
       "filename": "1.3.4 Normalizing algebraic functions.m",
            "rhs": "-d* Subst[Int[(a + b*x^n + c/d^(2*n)*x^(2*n))^p/x^2, x], x, d/x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*(d_./x_)^n_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x] && EqQ[n2, -2*n] && IntegerQ[2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.4 Normalizing algebraic functions.m",
       "filename": "1.3.4 Normalizing algebraic functions.m",
            "rhs": "-d^(m + 1)* Subst[Int[(a + b*x^n + c/d^(2*n)*x^(2*n))^p/x^(m + 2), x], x, d/x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*(d_./x_)^n_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x] && EqQ[n2, -2*n] && IntegerQ[2*n] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/1 Algebraic functions/1.3 Miscellaneous/1.3.4 Normalizing algebraic functions.m",
       "filename": "1.3.4 Normalizing algebraic functions.m",
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            "rhs": "(-d)^m* F^(g*(e - c*f/d))/(f^(m + 1)*g^(m + 1)*Log[F]^(m + 1))* Gamma[m + 1, -f*g*Log[F]/d*(c + d*x)]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*F_^(g_.*(e_. + f_.*x_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, c, d, e, f, g}, x] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.1 (c+d x)^m (a+b (F^(g (e+f x)))^n)^p.m",
       "filename": "2.1 (c+d x)^m (a+b (F^(g (e+f x)))^n)^p.m",
            "rhs": "2/d*Subst[Int[F^(g*(e - c*f/d) + f*g*x^2/d), x], x, Sqrt[c + d*x]]",
     "rulenumber": 0,
            "lhs": "Int[F_^(g_.*(e_. + f_.*x_))/Sqrt[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, c, d, e, f, g}, x] && Not[$UseGamma === True]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.1 (c+d x)^m (a+b (F^(g (e+f x)))^n)^p.m",
       "filename": "2.1 (c+d x)^m (a+b (F^(g (e+f x)))^n)^p.m",
            "rhs": "-F^(g*(e - c*f/d))*(c + d*x)^ FracPart[ m]/(d*(-f*g*Log[F]/d)^(IntPart[m] + 1)*(-f*g* Log[F]*(c + d*x)/d)^FracPart[m])* Gamma[m + 1, (-f*g*Log[F]/d)*(c + d*x)]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_*F_^(g_.*(e_. + f_.*x_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, c, d, e, f, g, m}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.1 (c+d x)^m (a+b (F^(g (e+f x)))^n)^p.m",
       "filename": "2.1 (c+d x)^m (a+b (F^(g (e+f x)))^n)^p.m",
            "rhs": "(b*F^(g*(e + f*x)))^n/F^(g*n*(e + f*x))* Int[(c + d*x)^m*F^(g*n*(e + f*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(b_.*F_^(g_.*(e_. + f_.*x_)))^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, b, c, d, e, f, g, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.1 (c+d x)^m (a+b (F^(g (e+f x)))^n)^p.m",
       "filename": "2.1 (c+d x)^m (a+b (F^(g (e+f x)))^n)^p.m",
            "rhs": "Int[ExpandIntegrand[(c + d*x)^m, (a + b*(F^(g*(e + f*x)))^n)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(a_ + b_.*(F_^(g_.*(e_. + f_.*x_)))^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g, m, n}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.1 (c+d x)^m (a+b (F^(g (e+f x)))^n)^p.m",
       "filename": "2.1 (c+d x)^m (a+b (F^(g (e+f x)))^n)^p.m",
            "rhs": "(c + d*x)^(m + 1)/(a*d*(m + 1)) - b/a*Int[(c + d*x)^ m*(F^(g*(e + f*x)))^n/(a + b*(F^(g*(e + f*x)))^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_./(a_ + b_.*(F_^(g_.*(e_. + f_.*x_)))^n_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.1 (c+d x)^m (a+b (F^(g (e+f x)))^n)^p.m",
       "filename": "2.1 (c+d x)^m (a+b (F^(g (e+f x)))^n)^p.m",
            "rhs": "-(c+d*x)^m/(a*f*g*n*Log[F])*Log[1+a/(b*(F^(g*(e+f*x)))^n)] + d*m/(a*f*g*n*Log[F])*Int[(c+d*x)^(m-1)*Log[1+a/(b*(F^(g*(e+f*x)))^n) ],x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.+d_.*x_)^m_./(a_+b_.*(F_^(g_.*(e_.+f_.*x_)))^n_.),x_Symbol] ",
        "comment": false,
         "givens": "FreeQ[{F,a,b,c,d,e,f,g,n},x] && IGtQ[m,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.1 (c+d x)^m (a+b (F^(g (e+f x)))^n)^p.m",
       "filename": "2.1 (c+d x)^m (a+b (F^(g (e+f x)))^n)^p.m",
            "rhs": "1/a*Int[(c + d*x)^m*(a + b*(F^(g*(e + f*x)))^n)^(p + 1), x] - b/a*Int[(c + d*x)^m*(F^(g*(e + f*x)))^ n*(a + b*(F^(g*(e + f*x)))^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(a_ + b_.*(F_^(g_.*(e_. + f_.*x_)))^n_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g, n}, x] && ILtQ[p, 0] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.1 (c+d x)^m (a+b (F^(g (e+f x)))^n)^p.m",
       "filename": "2.1 (c+d x)^m (a+b (F^(g (e+f x)))^n)^p.m",
            "rhs": "With[{u = IntHide[(a + b*(F^(g*(e + f*x)))^n)^p, x]}, Dist[(c + d*x)^m, u, x] - d*m*Int[(c + d*x)^(m - 1)*u, x]]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(a_ + b_.*(F_^(g_.*(e_. + f_.*x_)))^n_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.1 (c+d x)^m (a+b (F^(g (e+f x)))^n)^p.m",
       "filename": "2.1 (c+d x)^m (a+b (F^(g (e+f x)))^n)^p.m",
            "rhs": "Int[NormalizePowerOfLinear[u, x]^ m*(a + b*(F^(g*ExpandToSum[v, x]))^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*(a_. + b_.*(F_^(g_.*v_))^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, g, n, p}, x] && LinearQ[v, x] && PowerOfLinearQ[u, x] && Not[LinearMatchQ[v, x] && PowerOfLinearMatchQ[u, x]] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.1 (c+d x)^m (a+b (F^(g (e+f x)))^n)^p.m",
       "filename": "2.1 (c+d x)^m (a+b (F^(g (e+f x)))^n)^p.m",
            "rhs": "Module[{uu = NormalizePowerOfLinear[u, x], z}, z = If[PowerQ[uu] && FreeQ[uu[[2]], x], uu[[1]]^(m*uu[[2]]), uu^m]; uu^m/z*Int[z*(a + b*(F^(g*ExpandToSum[v, x]))^n)^p, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*(a_. + b_.*(F_^(g_.*v_))^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, g, m, n, p}, x] && LinearQ[v, x] && PowerOfLinearQ[u, x] && Not[LinearMatchQ[v, x] && PowerOfLinearMatchQ[u, x]] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.1 (c+d x)^m (a+b (F^(g (e+f x)))^n)^p.m",
       "filename": "2.1 (c+d x)^m (a+b (F^(g (e+f x)))^n)^p.m",
            "rhs": "Unintegrable[(c + d*x)^m*(a + b*(F^(g*(e + f*x)))^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(a_ + b_.*(F_^(g_.*(e_. + f_.*x_)))^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.2 (c+d x)^m (F^(g (e+f x)))^n (a+b (F^(g (e+f x)))^n)^p.m",
       "filename": "2.2 (c+d x)^m (F^(g (e+f x)))^n (a+b (F^(g (e+f x)))^n)^p.m",
            "rhs": "(c + d*x)^m/(b*f*g*n*Log[F])* Log[1 + b*(F^(g*(e + f*x)))^n/a] - d*m/(b*f*g*n*Log[F])* Int[(c + d*x)^(m - 1)*Log[1 + b*(F^(g*(e + f*x)))^n/a], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^ m_.*(F_^(g_.*(e_. + f_.*x_)))^ n_./(a_ + b_.*(F_^(g_.*(e_. + f_.*x_)))^n_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.2 (c+d x)^m (F^(g (e+f x)))^n (a+b (F^(g (e+f x)))^n)^p.m",
       "filename": "2.2 (c+d x)^m (F^(g (e+f x)))^n (a+b (F^(g (e+f x)))^n)^p.m",
            "rhs": "(c + d*x)^ m*(a + b*(F^(g*(e + f*x)))^n)^(p + 1)/(b*f*g*n*(p + 1)* Log[F]) - d*m/(b*f*g*n*(p + 1)*Log[F])* Int[(c + d*x)^(m - 1)*(a + b*(F^(g*(e + f*x)))^n)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(F_^(g_.*(e_. + f_.*x_)))^ n_.*(a_. + b_.*(F_^(g_.*(e_. + f_.*x_)))^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g, m, n, p}, x] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.2 (c+d x)^m (F^(g (e+f x)))^n (a+b (F^(g (e+f x)))^n)^p.m",
       "filename": "2.2 (c+d x)^m (F^(g (e+f x)))^n (a+b (F^(g (e+f x)))^n)^p.m",
            "rhs": "Unintegrable[(c + d*x)^m*(F^(g*(e + f*x)))^ n*(a + b*(F^(g*(e + f*x)))^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(F_^(g_.*(e_. + f_.*x_)))^ n_.*(a_. + b_.*(F_^(g_.*(e_. + f_.*x_)))^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g, m, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.2 (c+d x)^m (F^(g (e+f x)))^n (a+b (F^(g (e+f x)))^n)^p.m",
       "filename": "2.2 (c+d x)^m (F^(g (e+f x)))^n (a+b (F^(g (e+f x)))^n)^p.m",
            "rhs": "(k*G^(j*(h + i*x)))^q/(F^(g*(e + f*x)))^n* Int[(c + d*x)^m*(F^(g*(e + f*x)))^n*(a + b*(F^(g*(e + f*x)))^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(k_.*G_^(j_.*(h_. + i_.*x_)))^ q_.*(a_. + b_.*(F_^(g_.*(e_. + f_.*x_)))^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g, h, i, j, k, m, n, p, q}, x] && EqQ[f*g*n*Log[F] - i*j*q*Log[G], 0] && NeQ[(k*G^(j*(h + i*x)))^q - (F^(g*(e + f*x)))^n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "(F^(c*(a + b*x)))^n/(b*c*n*Log[F])",
     "rulenumber": 0,
            "lhs": "Int[(F_^(c_.*(a_. + b_.*x_)))^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Int[ExpandIntegrand[u*F^(c*ExpandToSum[v, x]), x], x]",
     "rulenumber": 0,
            "lhs": "Int[u_*F_^(c_.*v_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, c}, x] && PolynomialQ[u, x] && LinearQ[v, x] && $UseGamma === True"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x]",
     "rulenumber": 0,
            "lhs": "Int[u_*F_^(c_.*v_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, c}, x] && PolynomialQ[u, x] && LinearQ[v, x] && Not[$UseGamma === True]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0], e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, g*u^(m + 1)*F^(c*v)/(b*c*e*Log[F]) /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*F_^(c_.*v_)*w_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Int[ExpandIntegrand[ w*NormalizePowerOfLinear[u, x]^m*F^(c*ExpandToSum[v, x]), x], x]",
     "rulenumber": 0,
            "lhs": "Int[w_*u_^m_.*F_^(c_.*v_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x] && IntegerQ[m] && $UseGamma === True"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePowerOfLinear[u, x]^m, x], x]",
     "rulenumber": 0,
            "lhs": "Int[w_*u_^m_.*F_^(c_.*v_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x] && IntegerQ[m] && Not[$UseGamma === True]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Module[{uu = NormalizePowerOfLinear[u, x], z}, z = If[PowerQ[uu] && FreeQ[uu[[2]], x], uu[[1]]^(m*uu[[2]]), uu^m]; uu^m/z*Int[ExpandIntegrand[w*z*F^(c*ExpandToSum[v, x]), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[w_*u_^m_.*F_^(c_.*v_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, c, m}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "e*x*F^(c*(a + b*x))*Log[d*x]^(n + 1)/(n + 1)",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))* Log[d_.*x_]^n_.*(e_ + h_.*(f_. + g_.*x_)*Log[d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g, h, n}, x] && EqQ[e - f*h*(n + 1), 0] && EqQ[g*h*(n + 1) - b*c*e*Log[F], 0] && NeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "e*x^(m + 1)*F^(c*(a + b*x))*Log[d*x]^(n + 1)/(n + 1)",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*F_^(c_.*(a_. + b_.*x_))* Log[d_.*x_]^n_.*(e_ + h_.*(f_. + g_.*x_)*Log[d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*(m + 1) - f*h*(n + 1), 0] && EqQ[g*h*(n + 1) - b*c*e*Log[F], 0] && NeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "F^(a + b*(c + d*x))/(b*d*Log[F])",
     "rulenumber": 0,
            "lhs": "Int[F_^(a_. + b_.*(c_. + d_.*x_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])",
     "rulenumber": 0,
            "lhs": "Int[F_^(a_. + b_.*(c_. + d_.*x_)^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d}, x] && PosQ[b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-b*Log[F], 2]]/(2*d*Rt[-b*Log[F], 2])",
     "rulenumber": 0,
            "lhs": "Int[F_^(a_. + b_.*(c_. + d_.*x_)^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d}, x] && NegQ[b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "(c + d*x)*F^(a + b*(c + d*x)^n)/d - b*n*Log[F]*Int[(c + d*x)^n*F^(a + b*(c + d*x)^n), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(a_. + b_.*(c_. + d_.*x_)^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d}, x] && IntegerQ[2/n] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "With[{k = Denominator[n]}, k/d*Subst[Int[x^(k - 1)*F^(a + b*x^(k*n)), x], x, (c + d*x)^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[F_^(a_. + b_.*(c_. + d_.*x_)^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d}, x] && IntegerQ[2/n] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "-F^a*(c + d*x)* Gamma[1/n, -b*(c + d*x)^n*Log[F]]/(d* n*(-b*(c + d*x)^n*Log[F])^(1/n))",
     "rulenumber": 0,
            "lhs": "Int[F_^(a_. + b_.*(c_. + d_.*x_)^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, n}, x] && Not[IntegerQ[2/n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "(e + f*x)^n* F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n*Log[F])",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*F_^(a_. + b_.*(c_. + d_.*x_)^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] && EqQ[d*e - c*f, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "F^a*ExpIntegralEi[b*(c + d*x)^n*Log[F]]/(f*n)",
     "rulenumber": 0,
            "lhs": "Int[F_^(a_. + b_.*(c_. + d_.*x_)^n_)/(e_. + f_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "1/(d*(m + 1))*Subst[Int[F^(a + b*x^2), x], x, (c + d*x)^(m + 1)]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*F_^(a_. + b_.*(c_. + d_.*x_)^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, m, n}, x] && EqQ[n, 2*(m + 1)]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "(c + d*x)^(m - n + 1)* F^(a + b*(c + d*x)^n)/(b*d*n*Log[F]) - (m - n + 1)/(b*n*Log[F])* Int[(c + d*x)^(m - n)*F^(a + b*(c + d*x)^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*F_^(a_. + b_.*(c_. + d_.*x_)^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d}, x] && IntegerQ[2*(m + 1)/n] && LtQ[0, (m + 1)/n, 5] && IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "(c + d*x)^(m - n + 1)* F^(a + b*(c + d*x)^n)/(b*d*n*Log[F]) - (m - n + 1)/(b*n*Log[F])* Int[(c + d*x)^Simplify[m - n]*F^(a + b*(c + d*x)^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*F_^(a_. + b_.*(c_. + d_.*x_)^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, m, n}, x] && IntegerQ[2*Simplify[(m + 1)/n]] && LtQ[0, Simplify[(m + 1)/n], 5] && Not[RationalQ[m]] && SumSimplerQ[m, -n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "(c + d*x)^(m + 1)*F^(a + b*(c + d*x)^n)/(d*(m + 1)) - b*n*Log[F]/(m + 1)* Int[(c + d*x)^(m + n)*F^(a + b*(c + d*x)^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*F_^(a_. + b_.*(c_. + d_.*x_)^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d}, x] && IntegerQ[2*(m + 1)/n] && LtQ[-4, (m + 1)/n, 5] && IntegerQ[ n] && (GtQ[n, 0] && LtQ[m, -1] || GtQ[-n, 0] && LeQ[-n, m + 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "(c + d*x)^(m + 1)*F^(a + b*(c + d*x)^n)/(d*(m + 1)) - b*n*Log[F]/(m + 1)* Int[(c + d*x)^Simplify[m + n]*F^(a + b*(c + d*x)^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*F_^(a_. + b_.*(c_. + d_.*x_)^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, m, n}, x] && IntegerQ[2*Simplify[(m + 1)/n]] && LtQ[-4, Simplify[(m + 1)/n], 5] && Not[RationalQ[m]] && SumSimplerQ[m, n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "With[{k = Denominator[n]}, k/d*Subst[Int[x^(k*(m + 1) - 1)*F^(a + b*x^(k*n)), x], x, (c + d*x)^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*F_^(a_. + b_.*(c_. + d_.*x_)^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, m, n}, x] && IntegerQ[2*(m + 1)/n] && LtQ[0, (m + 1)/n, 5] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "(e + f*x)^m/(c + d*x)^m* Int[(c + d*x)^m*F^(a + b*(c + d*x)^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*F_^(a_. + b_.*(c_. + d_.*x_)^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0] && IntegerQ[2*Simplify[(m + 1)/n]] && NeQ[f, d] && Not[IntegerQ[m]] && NeQ[c*e, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": " (*-F^a*(e+f*x)^(m+1)/(f*n)*ExpIntegralE[1-(m+1)/n,-b*(c+d*x)^n*Log[F]]  *) -F^ a*(e + f*x)^(m + 1)/(f*n*(-b*(c + d*x)^n*Log[F])^((m + 1)/n))* Gamma[(m + 1)/n, -b*(c + d*x)^n*Log[F]]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*F_^(a_. + b_.*(c_. + d_.*x_)^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "f*(e + f*x)^(m - 1)*F^(a + b*(c + d*x)^2)/(2*b*d^2*Log[F]) + (d*e - c*f)/d*Int[(e + f*x)^(m - 1)*F^(a + b*(c + d*x)^2), x] - (m - 1)*f^2/(2*b*d^2*Log[F])* Int[(e + f*x)^(m - 2)*F^(a + b*(c + d*x)^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_*F_^(a_. + b_.*(c_. + d_.*x_)^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && FractionQ[m] && GtQ[m, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "f*(e + f*x)^(m + 1)*F^(a + b*(c + d*x)^2)/((m + 1)*f^2) + 2*b*d*(d*e - c*f)*Log[F]/(f^2*(m + 1))* Int[(e + f*x)^(m + 1)*F^(a + b*(c + d*x)^2), x] - 2*b*d^2*Log[F]/(f^2*(m + 1))* Int[(e + f*x)^(m + 2)*F^(a + b*(c + d*x)^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_*F_^(a_. + b_.*(c_. + d_.*x_)^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "(e + f*x)^(m + 1)*F^(a + b*(c + d*x)^n)/(f*(m + 1)) - b*d*n*Log[F]/(f*(m + 1))* Int[(e + f*x)^(m + 1)*(c + d*x)^(n - 1)*F^(a + b*(c + d*x)^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_*F_^(a_. + b_.*(c_. + d_.*x_)^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && IGtQ[n, 2] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "d/f*Int[F^(a + b/(c + d*x))/(c + d*x), x] - (d*e - c*f)/f*Int[F^(a + b/(c + d*x))/((c + d*x)*(e + f*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(a_. + b_./(c_. + d_.*x_))/(e_. + f_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "(e + f*x)^(m + 1)* F^(a + b/(c + d*x))/(f*(m + 1)) + b*d*Log[F]/(f*(m + 1))* Int[(e + f*x)^(m + 1)*F^(a + b/(c + d*x))/(c + d*x)^2, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_*F_^(a_. + b_./(c_. + d_.*x_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && ILtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Unintegrable[F^(a + b*(c + d*x)^n)/(e + f*x), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(a_. + b_.*(c_. + d_.*x_)^n_)/(e_. + f_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, n}, x] && NeQ[d*e - c*f, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Int[ExpandToSum[u, x]^m*F^ExpandToSum[v, x], x]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*F_^v_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, m}, x] && LinearQ[u, x] && BinomialQ[v, x] && Not[LinearMatchQ[u, x] && BinomialMatchQ[v, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Int[ExpandLinearProduct[F^(a + b*(c + d*x)^n), u, c, d, x], x]",
     "rulenumber": 0,
            "lhs": "Int[u_*F_^(a_. + b_.*(c_. + d_.*x_)^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Int[u*F^(a + b*NormalizePowerOfLinear[v, x]), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*F_^(a_. + b_.*v_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b}, x] && PolynomialQ[u, x] && PowerOfLinearQ[v, x] && Not[PowerOfLinearMatchQ[v, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": " Int[u*F^(a+b*ExpandToSum[v,x]^n),x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*F_^(a_.+b_.*v_^n_),x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F,a,b,n},x] &&  PolynomialQ[u,x] && LinearQ[v,x] && Not[LinearMatchQ[v,x]] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Int[u*F^ExpandToSum[u,x],x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*F_^u_,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[F,x] && PolynomialQ[u,x] && BinomialQ[u,x] &&  Not[BinomialMatchQ[u,x]] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "-d/(f*(d*g - c*h))* Subst[Int[F^(a - b*h/(d*g - c*h) + d*b*x/(d*g - c*h))/x, x], x, (g + h*x)/(c + d*x)]",
     "rulenumber": 0,
            "lhs": "Int[F_^(a_. + b_./(c_. + d_.*x_))/((e_. + f_.*x_)*(g_. + h_.*x_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f}, x] && EqQ[d*e - c*f, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "F^(e + f*b/d)*Int[(g + h*x)^m, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)^m_.*F_^(e_. + f_.*(a_. + b_.*x_)/(c_. + d_.*x_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g, h, m}, x] && EqQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Int[(g + h*x)^m*F^((d*e + b*f)/d - f*(b*c - a*d)/(d*(c + d*x))), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)^m_.*F_^(e_. + f_.*(a_. + b_.*x_)/(c_. + d_.*x_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g, h, m}, x] && NeQ[b*c - a*d, 0] && EqQ[d*g - c*h, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "d/h*Int[F^(e + f*(a + b*x)/(c + d*x))/(c + d*x), x] - (d*g - c*h)/h* Int[F^(e + f*(a + b*x)/(c + d*x))/((c + d*x)*(g + h*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(e_. + f_.*(a_. + b_.*x_)/(c_. + d_.*x_))/(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g, h}, x] && NeQ[b*c - a*d, 0] && NeQ[d*g - c*h, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "(g + h*x)^(m + 1)* F^(e + f*(a + b*x)/(c + d*x))/(h*(m + 1)) - f*(b*c - a*d)*Log[F]/(h*(m + 1))* Int[(g + h*x)^(m + 1)*F^(e + f*(a + b*x)/(c + d*x))/(c + d*x)^2, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)^m_*F_^(e_. + f_.*(a_. + b_.*x_)/(c_. + d_.*x_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g, h}, x] && NeQ[b*c - a*d, 0] && NeQ[d*g - c*h, 0] && ILtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "-d/(h*(d*i - c*j))* Subst[Int[ F^(e + f*(b*i - a*j)/(d*i - c*j) - (b*c - a*d)*f*x/(d*i - c*j))/ x, x], x, (i + j*x)/(c + d*x)]",
     "rulenumber": 0,
            "lhs": "Int[F_^(e_. + f_.*(a_. + b_.*x_)/(c_. + d_.*x_))/((g_. + h_.*x_)*(i_. + j_.*x_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g, h}, x] && EqQ[d*g - c*h, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "F^(a - b^2/(4*c))*Int[F^((b + 2*c*x)^2/(4*c)), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(a_. + b_.*x_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Int[F^ExpandToSum[v, x], x]",
     "rulenumber": 0,
            "lhs": "Int[F_^v_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[F, x] && QuadraticQ[v, x] && Not[QuadraticMatchQ[v, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "e*F^(a + b*x + c*x^2)/(2*c*Log[F])",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)*F_^(a_. + b_.*x_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "e*(d + e*x)^(m - 1)*F^(a + b*x + c*x^2)/(2*c*Log[F]) - (m - 1)*e^2/(2*c*Log[F])* Int[(d + e*x)^(m - 2)*F^(a + b*x + c*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*F_^(a_. + b_.*x_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0] && GtQ[m, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "1/(2*e)*F^(a - b^2/(4*c))*ExpIntegralEi[(b + 2*c*x)^2*Log[F]/(4*c)]",
     "rulenumber": 0,
            "lhs": "Int[F_^(a_. + b_.*x_ + c_.*x_^2)/(d_. + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "(d + e*x)^(m + 1)* F^(a + b*x + c*x^2)/(e*(m + 1)) - 2*c*Log[F]/(e^2*(m + 1))* Int[(d + e*x)^(m + 2)*F^(a + b*x + c*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*F_^(a_. + b_.*x_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "e*F^(a + b*x + c*x^2)/(2*c*Log[F]) - (b*e - 2*c*d)/(2*c)*Int[F^(a + b*x + c*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)*F_^(a_. + b_.*x_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "e*(d + e*x)^(m - 1)*F^(a + b*x + c*x^2)/(2*c*Log[F]) - (b*e - 2*c*d)/(2*c)* Int[(d + e*x)^(m - 1)*F^(a + b*x + c*x^2), x] - (m - 1)*e^2/(2*c*Log[F])* Int[(d + e*x)^(m - 2)*F^(a + b*x + c*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*F_^(a_. + b_.*x_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && GtQ[m, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "(d + e*x)^(m + 1)*F^(a + b*x + c*x^2)/(e*(m + 1)) - (b*e - 2*c*d)*Log[F]/(e^2*(m + 1))* Int[(d + e*x)^(m + 1)*F^(a + b*x + c*x^2), x] - 2*c*Log[F]/(e^2*(m + 1))* Int[(d + e*x)^(m + 2)*F^(a + b*x + c*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*F_^(a_. + b_.*x_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Unintegrable[(d + e*x)^m*F^(a + b*x + c*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*F_^(a_. + b_.*x_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, m}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Int[ExpandToSum[u, x]^m*F^ExpandToSum[v, x], x]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*F_^v_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, m}, x] && LinearQ[u, x] && QuadraticQ[v, x] && Not[LinearMatchQ[u, x] && QuadraticMatchQ[v, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "With[{u = IntHide[F^(e*(c + d*x))*(a + b*F^v)^p, x]}, Dist[x^m, u, x] - m*Int[x^(m - 1)*u, x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*F_^(e_.*(c_. + d_.*x_))*(a_. + b_.*F_^v_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && EqQ[v, 2*e*(c + d*x)] && GtQ[m, 0] && ILtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "1/(d*e*n*Log[F])*Subst[Int[(a + b*x)^p, x], x, (F^(e*(c + d*x)))^n]",
     "rulenumber": 0,
            "lhs": "Int[(F_^(e_.*(c_. + d_.*x_)))^ n_.*(a_ + b_.*(F_^(e_.*(c_. + d_.*x_)))^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "(G^(h*(f + g*x)))^m/(F^(e*(c + d*x)))^n* Int[(F^(e*(c + d*x)))^n*(a + b*(F^(e*(c + d*x)))^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(G_^(h_. (f_. + g_.*x_)))^ m_.*(a_ + b_.*(F_^(e_.*(c_. + d_.*x_)))^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, G, a, b, c, d, e, f, g, h, m, n, p}, x] && EqQ[d*e*n*Log[F], g*h*m*Log[G]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "With[{m = FullSimplify[g*h*Log[G]/(d*e*Log[F])]}, Denominator[m]*G^(f*h - c*g*h/d)/(d*e*Log[F])* Subst[Int[x^(Numerator[m] - 1)*(a + b*x^Denominator[m])^p, x], x, F^(e*(c + d*x)/Denominator[m])] /; LeQ[m, -1] || GeQ[m, 1]]",
     "rulenumber": 0,
            "lhs": "Int[G_^(h_. (f_. + g_.*x_))*(a_ + b_.*F_^(e_.*(c_. + d_.*x_)))^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "With[{m = FullSimplify[d*e*Log[F]/(g*h*Log[G])]}, Denominator[m]/(g*h*Log[G])* Subst[Int[ x^(Denominator[m] - 1)*(a + b*F^(c*e - d*e*f/g)*x^Numerator[m])^p, x], x, G^(h*(f + g*x)/Denominator[m])] /; LtQ[m, -1] || GtQ[m, 1]]",
     "rulenumber": 0,
            "lhs": "Int[G_^(h_. (f_. + g_.*x_))*(a_ + b_.*F_^(e_.*(c_. + d_.*x_)))^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Int[Expand[G^(h*(f + g*x))*(a + b*F^(e*(c + d*x)))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[G_^(h_. (f_. + g_.*x_))*(a_ + b_.*F_^(e_.*(c_. + d_.*x_)))^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, G, a, b, c, d, e, f, g, h}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "a^p*G^(h*(f + g*x))/(g*h*Log[G])* Hypergeometric2F1[-p, g*h*Log[G]/(d*e*Log[F]), g*h*Log[G]/(d*e*Log[F]) + 1, Simplify[-b/a*F^(e*(c + d*x))]]",
     "rulenumber": 0,
            "lhs": "Int[G_^(h_. (f_. + g_.*x_))*(a_ + b_.*F_^(e_.*(c_. + d_.*x_)))^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x] && (ILtQ[p, 0] || GtQ[a, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "(a + b*F^(e*(c + d*x)))^ p/(1 + (b/a)*F^(e*(c + d*x)))^p* Int[G^(h*(f + g*x))*(1 + b/a*F^(e*(c + d*x)))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[G_^(h_. (f_. + g_.*x_))*(a_ + b_.*F_^(e_.*(c_. + d_.*x_)))^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x] && Not[ILtQ[p, 0] || GtQ[a, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Int[G^(h*ExpandToSum[u, x])*(a + b*F^(e*ExpandToSum[v, x]))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[G_^(h_. u_)*(a_ + b_.*F_^(e_.*v_))^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, G, a, b, e, h, p}, x] && LinearQ[{u, v}, x] && Not[LinearMatchQ[{u, v}, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "1/b*Int[(c+d*x)^m*F^((g-h)*(e+f*x)),x] - a/b*Int[(c+d*x)^m*F^((g-h)*(e+f*x))/(a+b*F^(h*(e+f*x))),x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.+d_.*x_)^m_.*F_^(g_.*(e_.+f_.*x_))/(a_+b_.*F_^(h_.*(e_.+f_. *x_))),x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{F,a,b,c,d,e,f,g,h,m},x] && LeQ[0,g/h-1,g/h] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "1/a*Int[(c+d*x)^m*F^(g*(e+f*x)),x] - b/a*Int[(c+d*x)^m*F^((g+h)*(e+f*x))/(a+b*F^(h*(e+f*x))),x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.+d_.*x_)^m_.*F_^(g_.*(e_.+f_.*x_))/(a_+b_.*F_^(h_.*(e_.+f_. *x_))),x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{F,a,b,c,d,e,f,g,h,m},x] && LeQ[g/h,g/h+1,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "With[{w = ExpandIntegrand[(e + f*x)^m, (a + b*F^u)^p*(c + d*F^v)^q, x]}, Int[w, x] /; SumQ[w]]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_. + b_.*F_^u_)^p_.*(c_. + d_.*F_^v_)^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, m}, x] && IntegersQ[p, q] && LinearQ[{u, v}, x] && RationalQ[Simplify[u/v]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "With[{m = FullSimplify[(g*h*Log[G] + s*t*Log[H])/(d*e*Log[F])]}, Denominator[m]*G^(f*h - c*g*h/d)*H^(r*t - c*s*t/d)/(d*e*Log[F])* Subst[ Int[x^(Numerator[m] - 1)*(a + b*x^Denominator[m])^p, x], x, F^(e*(c + d*x)/Denominator[m])] /; RationalQ[m]]",
     "rulenumber": 0,
            "lhs": "Int[G_^(h_. (f_. + g_.*x_))* H_^(t_. (r_. + s_.*x_))*(a_ + b_.*F_^(e_.*(c_. + d_.*x_)))^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, G, H, a, b, c, d, e, f, g, h, r, s, t, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "G^((f - c*g/d)*h)* Int[H^(t*(r + s*x))*(b + a*F^(-e*(c + d*x)))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[G_^(h_. (f_. + g_.*x_))* H_^(t_. (r_. + s_.*x_))*(a_ + b_.*F_^(e_.*(c_. + d_.*x_)))^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, G, H, a, b, c, d, e, f, g, h, r, s, t}, x] && EqQ[d*e*p*Log[F] + g*h*Log[G], 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Int[Expand[G^(h*(f + g*x))*H^(t*(r + s*x))*(a + b*F^(e*(c + d*x)))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[G_^(h_. (f_. + g_.*x_))* H_^(t_. (r_. + s_.*x_))*(a_ + b_.*F_^(e_.*(c_. + d_.*x_)))^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, G, H, a, b, c, d, e, f, g, h, r, s, t}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "a^p*G^(h*(f + g*x))*H^(t*(r + s*x))/(g*h*Log[G] + s*t*Log[H])* Hypergeometric2F1[-p, (g*h*Log[G] + s*t*Log[H])/(d*e* Log[F]), (g*h*Log[G] + s*t*Log[H])/(d*e*Log[F]) + 1, Simplify[-b/a*F^(e*(c + d*x))]]",
     "rulenumber": 0,
            "lhs": "Int[G_^(h_. (f_. + g_.*x_))* H_^(t_. (r_. + s_.*x_))*(a_ + b_.*F_^(e_.*(c_. + d_.*x_)))^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, G, H, a, b, c, d, e, f, g, h, r, s, t}, x] && ILtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "G^(h*(f + g*x))* H^(t*(r + s*x))*(a + b*F^(e*(c + d*x)))^ p/((g*h*Log[G] + s*t*Log[H])*((a + b*F^(e*(c + d*x)))/a)^p)* Hypergeometric2F1[-p, (g*h*Log[G] + s*t*Log[H])/(d*e* Log[F]), (g*h*Log[G] + s*t*Log[H])/(d*e*Log[F]) + 1, Simplify[-b/a*F^(e*(c + d*x))]]",
     "rulenumber": 0,
            "lhs": "Int[G_^(h_. (f_. + g_.*x_))* H_^(t_. (r_. + s_.*x_))*(a_ + b_.*F_^(e_.*(c_. + d_.*x_)))^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, G, H, a, b, c, d, e, f, g, h, r, s, t, p}, x] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Int[G^(h*ExpandToSum[u, x])* H^(t*ExpandToSum[w, x])*(a + b*F^(e*ExpandToSum[v, x]))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[G_^(h_. u_)*H_^(t_. w_)*(a_ + b_.*F_^(e_.*v_))^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, G, H, a, b, e, h, t, p}, x] && LinearQ[{u, v, w}, x] && Not[LinearMatchQ[{u, v, w}, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "(a*x^n + b*F^(e*(c + d*x)))^(p + 1)/(b*d*e*(p + 1)* Log[F]) - a*n/(b*d*e*Log[F])* Int[x^(n - 1)*(a*x^n + b*F^(e*(c + d*x)))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(e_.*(c_. + d_.*x_))*(a_.*x_^n_. + b_.*F_^(e_.*(c_. + d_.*x_)))^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, n, p}, x] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "x^m*(a*x^n + b*F^(e*(c + d*x)))^(p + 1)/(b*d*e*(p + 1)*Log[F]) - a*n/(b*d*e*Log[F])* Int[x^(m + n - 1)*(a*x^n + b*F^(e*(c + d*x)))^p, x] - m/(b*d*e*(p + 1)*Log[F])* Int[x^(m - 1)*(a*x^n + b*F^(e*(c + d*x)))^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.* F_^(e_.*(c_. + d_.*x_))*(a_.*x_^n_. + b_.*F_^(e_.*(c_. + d_.*x_)))^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, m, n, p}, x] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, 2*c/q*Int[(f + g*x)^m/(b - q + 2*c*F^u), x] - 2*c/q*Int[(f + g*x)^m/(b + q + 2*c*F^u), x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^m_./(a_. + b_.*F_^u_ + c_.*F_^v_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, 2*c/q*Int[(f + g*x)^m*F^u/(b - q + 2*c*F^u), x] - 2*c/q*Int[(f + g*x)^m*F^u/(b + q + 2*c*F^u), x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^m_.*F_^u_/(a_. + b_.*F_^u_ + c_.*F_^v_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "With[{q = Rt[b^2 - 4*a*c, 2]}, (Simplify[(2*c*h - b*i)/q] + i)* Int[(f + g*x)^m/(b - q + 2*c*F^u), x] - (Simplify[(2*c*h - b*i)/q] - i)* Int[(f + g*x)^m/(b + q + 2*c*F^u), x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^m_.*(h_ + i_.*F_^u_)/(a_. + b_.*F_^u_ + c_.*F_^v_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, f, g, h, i}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "With[{u = IntHide[1/(a*F^(c + d*x) + b*F^v), x]}, x^m*u - m*Int[x^(m - 1)*u, x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_./(a_.*F_^(c_. + d_.*x_) + b_.*F_^v_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d}, x] && EqQ[v, -(c + d*x)] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Int[u*F^v/(c + a*F^v + b*F^(2*v)), x]",
     "rulenumber": 0,
            "lhs": "Int[u_/(a_ + b_.*F_^v_ + c_.*F_^w_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c}, x] && EqQ[w, -v] && LinearQ[v, x] && If[RationalQ[Coefficient[v, x, 1]], GtQ[Coefficient[v, x, 1], 0], LtQ[LeafCount[v], LeafCount[w]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Int[ExpandIntegrand[F^(g*(d + e*x)^n), 1/(a + b*x + c*x^2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(g_.*(d_. + e_.*x_)^n_.)/(a_. + b_.*x_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, g, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Int[ExpandIntegrand[F^(g*(d + e*x)^n), 1/(a + c*x^2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(g_.*(d_. + e_.*x_)^n_.)/(a_ + c_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, c, d, e, g, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Int[ExpandIntegrand[F^(g*(d + e*x)^n), u^m/(a + b*x + c*x^2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*F_^(g_.*(d_. + e_.*x_)^n_.)/(a_. + b_.*x_ + c_*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, g, n}, x] && PolynomialQ[u, x] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Int[ExpandIntegrand[F^(g*(d + e*x)^n), u^m/(a + c*x^2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*F_^(g_.*(d_. + e_.*x_)^n_.)/(a_ + c_*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, c, d, e, g, n}, x] && PolynomialQ[u, x] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Sqrt[Pi]*Exp[2*Sqrt[-a*Log[F]]*Sqrt[-b*Log[F]]]* Erf[(Sqrt[-a*Log[F]] + Sqrt[-b*Log[F]]*x^2)/x]/ (4*Sqrt[-b*Log[F]]) - Sqrt[Pi]*Exp[-2*Sqrt[-a*Log[F]]*Sqrt[-b*Log[F]]]* Erf[(Sqrt[-a*Log[F]] - Sqrt[-b*Log[F]]*x^2)/x]/ (4*Sqrt[-b*Log[F]])",
     "rulenumber": 0,
            "lhs": "Int[F_^((a_. + b_.*x_^4)/x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "-(E^x + x^m)^(n + 1)/(n + 1) + Int[(E^x + x^m)^(n + 1), x] + m*Int[x^(m - 1)*(E^x + x^m)^n, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(E^x_ + x_^m_.)^n_, x_Symbol]",
        "comment": false,
         "givens": "RationalQ[m, n] && GtQ[m, 0] && LtQ[n, 0] && NeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Int[u*F^(a*v)*z^(a*b*Log[F]), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*F_^(a_.*(v_. + b_.*Log[z_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "x/(n*(c*x^n)^(1/n))* Subst[Int[E^(a*d*Log[F] + x/n + b*d*Log[F]*x^2), x], x, Log[c*x^n]]",
     "rulenumber": 0,
            "lhs": "Int[F_^(d_.*(a_. + b_.*Log[c_.*x_^n_.]^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "(e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n))* Subst[Int[E^(a*d*Log[F] + (m + 1)*x/n + b*d*Log[F]*x^2), x], x, Log[c*x^n]]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*F_^(d_.*(a_. + b_.*Log[c_.*x_^n_.]^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Int[F^(a^2*d + 2*a*b*d*Log[c*x^n] + b^2*d*Log[c*x^n]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(d_.*(a_. + b_.*Log[c_.*x_^n_.])^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Int[(e*x)^m*F^(a^2*d + 2*a*b*d*Log[c*x^n] + b^2*d*Log[c*x^n]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*F_^(d_.*(a_. + b_.*Log[c_.*x_^n_.])^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "1/(d*e*n*Log[F])* Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n]",
     "rulenumber": 0,
            "lhs": "Int[Log[a_ + b_.*(F_^(e_.*(c_. + d_.*x_)))^n_.], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "x*Log[a + b*(F^(e*(c + d*x)))^n] - b*d*e*n*Log[F]* Int[x*(F^(e*(c + d*x)))^n/(a + b*(F^(e*(c + d*x)))^n), x]",
     "rulenumber": 0,
            "lhs": "Int[Log[a_ + b_.*(F_^(e_.*(c_. + d_.*x_)))^n_.], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, n}, x] && Not[GtQ[a, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "a^n*Int[u*F^(n*v),x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_.*F_^v_)^n_,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{F,a},x] && IntegerQ[n] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "(a*F^v)^n/F^(n*v)*Int[u*F^(n*v), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_.*F_^v_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, n}, x] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "With[{v = FunctionOfExponential[u, x]}, v/D[v, x]* Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v]] /; FunctionOfExponentialQ[u, x] && Not[MatchQ[u, w_*(a_.*v_^n_)^m_ /; FreeQ[{a, m, n}, x] && IntegerQ[m*n]]] && Not[MatchQ[u, E^(c_.*(a_. + b_.*x))*F_[v_]",
     "rulenumber": 0,
            "lhs": "Int[u_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Int[u*F^(n*v)*(a + b*F^ExpandToSum[w - v, x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_.*F_^v_ + b_.*F_^w_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, n}, x] && ILtQ[n, 0] && LinearQ[{v, w}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Int[u*F^(n*v)*(a + b*E^ExpandToSum[Log[G]*w - Log[F]*v, x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_.*F_^v_ + b_.*G_^w_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, G, a, b, n}, x] && ILtQ[n, 0] && LinearQ[{v, w}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "(a*F^v + b*F^w)^ n/(F^(n*v)*(a + b*F^ExpandToSum[w - v, x])^n)* Int[u*F^(n*v)*(a + b*F^ExpandToSum[w - v, x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_.*F_^v_ + b_.*F_^w_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, n}, x] && Not[IntegerQ[n]] && LinearQ[{v, w}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "(a*F^v + b*G^w)^ n/(F^(n*v)*(a + b*E^ExpandToSum[Log[G]*w - Log[F]*v, x])^n)* Int[u*F^(n*v)*(a + b*E^ExpandToSum[Log[G]*w - Log[F]*v, x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_.*F_^v_ + b_.*G_^w_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, G, a, b, n}, x] && Not[IntegerQ[n]] && LinearQ[{v, w}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "With[{z = v*Log[F] + w*Log[G]}, Int[u*NormalizeIntegrand[E^z, x], x] /; BinomialQ[z, x] || PolynomialQ[z, x] && LeQ[Exponent[z, x], 2]]",
     "rulenumber": 0,
            "lhs": "Int[u_.*F_^v_*G_^w_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, G}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "With[{z = v*y/(Log[F]*D[u, x])}, F^u*z /; EqQ[D[z, x], w*y]]",
     "rulenumber": 0,
            "lhs": "Int[F_^u_*(v_ + w_)*y_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[F, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "With[{z = Log[F]*v*D[u, x] + (n + 1)*D[v, x]}, Coefficient[w, x, Exponent[w, x]]/ Coefficient[z, x, Exponent[z, x]]*F^u*v^(n + 1) /; EqQ[Exponent[w, x], Exponent[z, x]] && EqQ[w*Coefficient[z, x, Exponent[z, x]], z*Coefficient[w, x, Exponent[w, x]]]]",
     "rulenumber": 0,
            "lhs": "Int[F_^u_*v_^n_.*w_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, n}, x] && PolynomialQ[u, x] && PolynomialQ[v, x] && PolynomialQ[w, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "2*e*g/(C*(e*f - d*g))* Subst[Int[(a + b*F^(c*x))^n/x, x], x, Sqrt[d + e*x]/Sqrt[f + g*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*F_^(c_.*Sqrt[d_. + e_.*x_]/Sqrt[f_. + g_.*x_]))^ n_./(A_. + B_.*x_ + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, A, B, C, F}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[B*e*g - C*(e*f + d*g), 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "2*e*g/(C*(e*f - d*g))* Subst[Int[(a + b*F^(c*x))^n/x, x], x, Sqrt[d + e*x]/Sqrt[f + g*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*F_^(c_.*Sqrt[d_. + e_.*x_]/Sqrt[f_. + g_.*x_]))^ n_./(A_ + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, A, C, F}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[e*f + d*g, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Unintegrable[(a + b*F^(c*Sqrt[d + e*x]/Sqrt[f + g*x]))^ n/(A + B*x + C*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*F_^(c_.*Sqrt[d_. + e_.*x_]/Sqrt[f_. + g_.*x_]))^ n_/(A_. + B_.*x_ + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, A, B, C, F, n}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[B*e*g - C*(e*f + d*g), 0] && Not[IGtQ[n, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/2 Exponentials/2.3 Miscellaneous exponentials.m",
       "filename": "2.3 Miscellaneous exponentials.m",
            "rhs": "Unintegrable[(a + b*F^(c*Sqrt[d + e*x]/Sqrt[f + g*x]))^n/(A + C*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*F_^(c_.*Sqrt[d_. + e_.*x_]/Sqrt[f_. + g_.*x_]))^ n_/(A_ + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, A, C, F, n}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[e*f + d*g, 0] && Not[IGtQ[n, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "B*(d + e*x)*Sqrt[a + b*Log[c*(d + e*x)^n]]/(b*e) + (2*A*b - B*(2*a + b*n))/(2*b)* Int[1/Sqrt[a + b*Log[c*(d + e*x)^n]], x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*Log[c_.*(d_. + e_.*x_)^n_.])/ Sqrt[a_ + b_.*Log[c_.*(d_. + e_.*x_)^n_.]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "x*Log[c*x^n] - n*x",
     "rulenumber": 0,
            "lhs": "Int[Log[c_.*x_^n_.], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "x*(a + b*Log[c*x^n])^p - b*n*p*Int[(a + b*Log[c*x^n])^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "x*(a + b*Log[c*x^n])^(p + 1)/(b*n*(p + 1)) - 1/(b*n*(p + 1))*Int[(a + b*Log[c*x^n])^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*x_^n_.])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && LtQ[p, -1] && IntegerQ[2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "LogIntegral[c*x]/c",
     "rulenumber": 0,
            "lhs": "Int[1/Log[c_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[c, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "1/(n*c^(1/n))*Subst[Int[E^(x/n)*(a + b*x)^p, x], x, Log[c*x^n]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*x_^n_.])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && IntegerQ[1/n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "x/(n*(c*x^n)^(1/n))* Subst[Int[E^(x/n)*(a + b*x)^p, x], x, Log[c*x^n]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*x_^n_.])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "(a + b*Log[c*x^n])^2/(2*b*n)",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*x_^n_.])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "1/(b*n)*Subst[Int[x^p, x], x, a + b*Log[c*x^n]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*x_^n_.])^p_./x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "b*(d*x)^(m + 1)*Log[c*x^n]/(d*(m + 1))",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && EqQ[a*(m + 1) - b*n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "(d*x)^(m + 1)*(a + b*Log[c*x^n])/(d*(m + 1)) - b*n*(d*x)^(m + 1)/(d*(m + 1)^2)",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "(d*x)^(m + 1)*(a + b*Log[c*x^n])^p/(d*(m + 1)) - b*n*p/(m + 1)*Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "(d*x)^(m + 1)*(a + b*Log[c*x^n])^(p + 1)/(b*d* n*(p + 1)) - (m + 1)/(b*n*(p + 1))* Int[(d*x)^m*(a + b*Log[c*x^n])^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_. + b_.*Log[c_.*x_^n_.])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "1/n*Subst[Int[1/Log[c*x], x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_./Log[c_.*x_^n_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, m, n}, x] && EqQ[m, n - 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "(d*x)^m/x^m*Int[x^m/Log[c*x^n], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_*x_)^m_./Log[c_.*x_^n_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, m, n}, x] && EqQ[m, n - 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "1/c^(m + 1)*Subst[Int[E^((m + 1)*x)*(a + b*x)^p, x], x, Log[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*Log[c_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "(d*x)^(m + 1)/(d*n*(c*x^n)^((m + 1)/n))* Subst[Int[E^((m + 1)/n*x)*(a + b*x)^p, x], x, Log[c*x^n]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_. + b_.*Log[c_.*x_^n_.])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "(d*x^q)^m/x^(m*q)* Int[x^(m*q)*(a + b*Log[c*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_^q_)^m_*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p, q}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "(d1*x^q1)^m1*(d2*x^q2)^m2/x^(m1*q1 + m2*q2)* Int[x^(m1*q1 + m2*q2)*(a + b*Log[c*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d1_.*x_^q1_)^m1_*(d2_.*x_^q2_)^m2_*(a_. + b_.*Log[c_.*x_^n_.])^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, d2, m1, m2, n, p, q1, q2}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "With[{u = IntHide[(d + e*x^r)^q, x]}, u*(a + b*Log[c*x^n]) - b*n*Int[SimplifyIntegrand[u/x, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^r_.)^q_.*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "x*(d + e*x^r)^(q + 1)*(a + b*Log[c*x^n])/d - b*n/d*Int[(d + e*x^r)^(q + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^r_.)^q_*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, q, r}, x] && EqQ[r*(q + 1) + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "-1/e*PolyLog[2, 1 - c*x]",
     "rulenumber": 0,
            "lhs": "Int[Log[c_.*x_]/(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "(a + b*Log[-c*d/e])*Log[d + e*x]/e + b*Int[Log[-e*x/d]/(d + e*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*x_])/(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && GtQ[-c*d/e, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "Log[1 + e*x/d]*(a + b*Log[c*x^n])^p/e - b*n*p/e*Int[Log[1 + e*x/d]*(a + b*Log[c*x^n])^(p - 1)/x, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*x_^n_.])^p_./(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "x*(a + b*Log[c*x^n])^p/(d*(d + e*x)) - b*n*p/d*Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*x_^n_.])^p_./(d_ + e_.*x_)^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p}, x] && GtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^ p/(e*(q + 1)) - b*n*p/(e*(q + 1))* Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1))/x, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^q_.*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || IntegersQ[2*p, 2*q] && Not[IGtQ[q, 0]] || EqQ[p, 2] && NeQ[q, 1])"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "x*(d + e*x)^q*(a + b*Log[c*x^n])^(p + 1)/(b*n*(p + 1)) + d*q/(b*n*(p + 1))* Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^(p + 1), x] - (q + 1)/(b*n*(p + 1))* Int[(d + e*x)^q*(a + b*Log[c*x^n])^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^q_.*(a_. + b_.*Log[c_.*x_^n_.])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && LtQ[p, -1] && GtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "x*(d + e*x^2)^q*(a + b*Log[c*x^n])/(2*q + 1) - b*n/(2*q + 1)*Int[(d + e*x^2)^q, x] + 2*d*q/(2*q + 1)*Int[(d + e*x^2)^(q - 1)*(a + b*Log[c*x^n]), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_.*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && GtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "x*(a + b*Log[c*x^n])/(d*Sqrt[d + e*x^2]) - b*n/d*Int[1/Sqrt[d + e*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*x_^n_.])/(d_ + e_.*x_^2)^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "-x*(d + e*x^2)^(q + 1)*(a + b*Log[c*x^n])/(2* d*(q + 1)) + b*n/(2*d*(q + 1))*Int[(d + e*x^2)^(q + 1), x] + (2*q + 3)/(2*d*(q + 1))* Int[(d + e*x^2)^(q + 1)*(a + b*Log[c*x^n]), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && LtQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "With[{u = IntHide[1/(d + e*x^2), x]}, u*(a + b*Log[c*x^n]) - b*n*Int[u/x, x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*x_^n_.])/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "ArcSinh[Rt[e, 2]*x/Sqrt[d]]*(a + b*Log[c*x^n])/Rt[e, 2] - b*n/Rt[e, 2]*Int[ArcSinh[Rt[e, 2]*x/Sqrt[d]]/x, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*x_^n_.])/Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && GtQ[d, 0] && PosQ[e]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "ArcSin[Rt[-e, 2]*x/Sqrt[d]]*(a + b*Log[c*x^n])/Rt[-e, 2] - b*n/Rt[-e, 2]*Int[ArcSin[Rt[-e, 2]*x/Sqrt[d]]/x, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*x_^n_.])/Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && GtQ[d, 0] && NegQ[e]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "Sqrt[1 + e/d*x^2]/Sqrt[d + e*x^2]* Int[(a + b*Log[c*x^n])/Sqrt[1 + e/d*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*x_^n_.])/Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && Not[GtQ[d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "Sqrt[1 + e1*e2/(d1*d2)*x^2]/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])* Int[(a + b*Log[c*x^n])/Sqrt[1 + e1*e2/(d1*d2)*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*x_^n_.])/(Sqrt[d1_ + e1_.*x_]* Sqrt[d2_ + e2_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2*e1 + d1*e2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "With[{u = IntHide[(d + e*x^r)^q, x]}, Dist[(a + b*Log[c*x^n]), u, x] - b*n*Int[SimplifyIntegrand[u/x, x], x] /; EqQ[r, 1] && IntegerQ[q - 1/2] || EqQ[r, 2] && EqQ[q, -1] || InverseFunctionFreeQ[u, x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^r_.)^q_.*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, q, r}, x] && IntegerQ[2*q] && IntegerQ[r]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^r_.)^q_.*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || IGtQ[p, 0] && IntegerQ[r])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "Unintegrable[(d + e*x^r)^q*(a + b*Log[c*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^r_.)^q_.*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p, q, r}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "Int[ExpandToSum[u, x]^q*(a + b*Log[c*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[u_^q_.*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, p, q}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "Int[(e + d*x)^q*(a + b*Log[c*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_./x_)^q_.*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[m, q] && IntegerQ[q]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "With[{u = IntHide[x^m*(d + e*x^r)^q, x]}, u*(a + b*Log[c*x^n]) - b*n*Int[SimplifyIntegrand[u/x, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^r_.)^q_.*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] && Not[EqQ[q, 1] && EqQ[m, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "(f*x)^(m + 1)*(d + e*x^r)^(q + 1)*(a + b*Log[c*x^n])/(d*f*(m + 1)) - b*n/(d*(m + 1))*Int[(f*x)^m*(d + e*x^r)^(q + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^r_.)^q_*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[m + r*(q + 1) + 1, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "f^m/n*Subst[Int[(d + e*x)^q*(a + b*Log[c*x])^p, x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^r_)^q_.*(a_. + b_.*Log[c_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || GtQ[f, 0]) && EqQ[r, n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "f^m*Log[1 + e*x^r/d]*(a + b*Log[c*x^n])^p/(e*r) - b*f^m*n*p/(e*r)* Int[Log[1 + e*x^r/d]*(a + b*Log[c*x^n])^(p - 1)/x, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(a_. + b_.*Log[c_.*x_^n_.])^p_./(d_ + e_.*x_^r_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "f^m*(d + e*x^r)^(q + 1)*(a + b*Log[c*x^n])^p/(e*r*(q + 1)) - b*f^m*n*p/(e*r*(q + 1))* Int[(d + e*x^r)^(q + 1)*(a + b*Log[c*x^n])^(p - 1)/x, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^r_)^q_.*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n] && NeQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "(f*x)^m/x^m* Int[x^m*(d + e*x^r)^q*(a + b*Log[c*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_*x_)^m_.*(d_ + e_.*x_^r_)^q_.*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] && Not[(IntegerQ[m] || GtQ[f, 0])]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "-(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*(a + b*Log[c*x^n])/(2*d*f*(q + 1)) + 1/(2*d*(q + 1))* Int[(f*x)^ m*(d + e*x^2)^(q + 1)*(a*(m + 2*q + 3) + b*n + b*(m + 2*q + 3)*Log[c*x^n]), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && ILtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "d^IntPart[q]*(d + e*x^2)^FracPart[q]/(1 + e/d*x^2)^FracPart[q]* Int[x^m*(1 + e/d*x^2)^q*(a + b*Log[c*x^n]), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IntegerQ[m/2] && IntegerQ[q - 1/2] && Not[LtQ[m + 2*q, -2] || GtQ[d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "(d1 + e1*x)^ q*(d2 + e2*x)^q/(1 + e1*e2/(d1*d2)*x^2)^q* Int[x^m*(1 + e1*e2/(d1*d2)*x^2)^q*(a + b*Log[c*x^n]), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d1_ + e1_.*x_)^q_*(d2_ + e2_.*x_)^ q_*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[m] && IntegerQ[q - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "1/n*Subst[Int[(a + b*Log[c*x])/(x*(d + e*x^(r/n))), x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*x_^n_])/(x_*(d_ + e_.*x_^r_.)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, r}, x] && IntegerQ[r/n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "1/d*Int[(a + b*Log[c*x^n])^p/x, x] - e/d*Int[(a + b*Log[c*x^n])^p/(d + e*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*x_^n_.])^p_./(x_*(d_ + e_.*x_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "  (r*Log[x]-Log[1+(e*x^r)/d])*(a+b*Log[c*x^n])^p/(d*r) - b*n*p/d*Int[Log[x]*(a+b*Log[c*x^n])^(p-1)/x,x] + b*n*p/(d*r)*Int[Log[1+(e*x^r)/d]*(a+b*Log[c*x^n])^(p-1)/x,x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.+b_.*Log[c_.*x_^n_.])^p_./(x_*(d_+e_.*x_^r_.)),x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e,n,r},x] && IGtQ[p,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "-Log[1 + d/(e*x^r)]*(a + b*Log[c*x^n])^p/(d*r) + b*n*p/(d*r)* Int[Log[1 + d/(e*x^r)]*(a + b*Log[c*x^n])^(p - 1)/x, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*x_^n_.])^p_./(x_*(d_ + e_.*x_^r_.)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "d*Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p/x, x] + e*Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^q_.*(a_. + b_.*Log[c_.*x_^n_.])^p_./x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && IntegerQ[2*q]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "1/d*Int[(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p/x, x] - e/d*Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^q_*(a_. + b_.*Log[c_.*x_^n_.])^p_./x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "With[{u = IntHide[(d + e*x^r)^q/x, x]}, u*(a + b*Log[c*x^n]) - b*n*Int[Dist[1/x, u, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^r_.)^q_.*(a_. + b_.*Log[c_.*x_^n_.])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, r}, x] && IntegerQ[q - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "1/d*Int[(d + e*x^r)^(q + 1)*(a + b*Log[c*x^n])^p/x, x] - e/d*Int[x^(r - 1)*(d + e*x^r)^q*(a + b*Log[c*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^r_.)^q_*(a_. + b_.*Log[c_.*x_^n_.])^p_./x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0] && ILtQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "With[{u = IntHide[(f*x)^m*(d + e*x^r)^q, x]}, Dist[(a + b*Log[c*x^n]), u, x] - b*n*Int[SimplifyIntegrand[u/x, x], x] /; (EqQ[r, 1] || EqQ[r, 2]) && IntegerQ[m] && IntegerQ[q - 1/2] || InverseFunctionFreeQ[u, x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^r_.)^q_.*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[2*q] && (IntegerQ[m] && IntegerQ[r] || IGtQ[q, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "With[{u = ExpandIntegrand[(a + b*Log[c*x^n]), (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^r_.)^q_.*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || IntegerQ[m] && IntegerQ[r])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(d + e*x^(r/n))^ q*(a + b*Log[c*x])^p, x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^r_.)^q_.*(a_. + b_.*Log[c_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p, q, r}, x] && IntegerQ[q] && IntegerQ[r/n] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[p, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^r_.)^q_.*(a_. + b_.*Log[c_.*x_^n_.])^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[ q] && (GtQ[q, 0] || IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "Unintegrable[(f*x)^m*(d + e*x^r)^q*(a + b*Log[c*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^r_.)^q_.*(a_. + b_.*Log[c_.*x_^n_.])^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p, q, r}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "Int[(f*x)^m*ExpandToSum[u, x]^q*(a + b*Log[c*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*u_^q_.*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, f, m, n, p, q}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "Int[ExpandIntegrand[Polyx*(a + b*Log[c*x^n])^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Polyx_*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, p}, x] && PolynomialQ[Polyx, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[RFx_*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "With[{u = ExpandIntegrand[RFx*(a + b*Log[c*x^n])^p, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[RFx_*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "Unintegrable[AFx*(a + b*Log[c*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[AFx_*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, p}, x] && AlgebraicFunctionQ[AFx, x, True]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*Log[c*x^n])^p*(d + e*Log[c*x^n])^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*x_^n_.])^p_.*(d_ + e_.*Log[c_.*x_^n_.])^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IntegerQ[p] && IntegerQ[q]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "With[{u = IntHide[(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - e*r*Int[SimplifyIntegrand[u/x, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*x_^n_.])^p_.*(d_. + e_.*Log[f_.*x_^r_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, p, r}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "x*(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r])^q - e*q*r*Int[(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r])^(q - 1), x] - b*n*p*Int[(a + b*Log[c*x^n])^(p - 1)*(d + e*Log[f*x^r])^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*x_^n_.])^p_.*(d_. + e_.*Log[f_.*x_^r_.])^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, r}, x] && IGtQ[p, 0] && IGtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "Unintegrable[(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r])^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*x_^n_.])^p_.*(d_. + e_.*Log[f_.*x_^r_.])^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, p, q, r}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "1/Coeff[v, x, 1]* Subst[Int[(a + b*Log[x])^p*(c + d*Log[x])^q, x], x, v]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[v_])^p_.*(c_. + d_.*Log[v_])^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, p, q}, x] && LinearQ[v, x] && NeQ[Coeff[v, x, 0], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "1/n*Subst[Int[(a + b*x)^p*(d + e*x)^q, x], x, Log[c*x^n]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*x_^n_.])^p_.*(d_. + e_.*Log[c_.*x_^n_.])^q_./ x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p, q}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Dist[(d + e*Log[f*x^r]), u, x] - e*r*Int[SimplifyIntegrand[u/x, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^m_.*(a_. + b_.*Log[c_.*x_^n_.])^ p_.*(d_. + e_.*Log[f_.*x_^r_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, x] && Not[EqQ[p, 1] && EqQ[a, 0] && NeQ[d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "(g*x)^(m + 1)*(a + b*Log[c*x^n])^ p*(d + e*Log[f*x^r])^q/(g*(m + 1)) - e*q*r/(m + 1)* Int[(g*x)^m*(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r])^(q - 1), x] - b*n*p/(m + 1)* Int[(g*x)^m*(a + b*Log[c*x^n])^(p - 1)*(d + e*Log[f*x^r])^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^m_.*(a_. + b_.*Log[c_.*x_^n_.])^ p_.*(d_. + e_.*Log[f_.*x_^r_.])^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, r}, x] && IGtQ[p, 0] && IGtQ[q, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "Unintegrable[(g*x)^m*(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r])^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^m_.*(a_. + b_.*Log[c_.*x_^n_.])^ p_.*(d_. + e_.*Log[f_.*x_^r_.])^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "With[{e = Coeff[u, x, 0], f = Coeff[u, x, 1], g = Coeff[v, x, 0], h = Coeff[v, x, 1]}, 1/h*Subst[Int[(f*x/h)^m*(a + b*Log[x])^p*(c + d*Log[x])^q, x], x, v] /; EqQ[f*g - e*h, 0] && NeQ[g, 0]]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*(a_. + b_.*Log[v_])^p_.*(c_. + d_.*Log[v_])^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, p, q}, x] && LinearQ[{u, v}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "With[{u = IntHide[Log[d*(e + f*x^m)^r], x]}, Dist[(a + b*Log[c*x^n])^p, u, x] - b*n*p*Int[Dist[(a + b*Log[c*x^n])^(p - 1)/x, u, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Log[d_.*(e_ + f_.*x_^m_.)^r_.]*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p, 0] && RationalQ[ m] && (EqQ[p, 1] || FractionQ[m] && IntegerQ[1/m] || EqQ[r, 1] && EqQ[m, 1] && EqQ[d*e, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "With[{u = IntHide[(a + b*Log[c*x^n])^p, x]}, Dist[Log[d*(e + f*x^m)^r], u, x] - f*m*r*Int[Dist[x^(m - 1)/(e + f*x^m), u, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Log[d_.*(e_ + f_.*x_^m_.)^r_.]*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "Unintegrable[Log[d*(e + f*x^m)^r]*(a + b*Log[c*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[Log[d_.*(e_ + f_.*x_^m_.)^r_.]*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, r, m, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "Int[Log[d*ExpandToSum[u, x]^r]*(a + b*Log[c*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[Log[d_.*u_^r_.]*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, r, n, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "-PolyLog[2, -d*f*x^m]*(a + b*Log[c*x^n])^p/m + b*n*p/m* Int[PolyLog[2, -d*f*x^m]*(a + b*Log[c*x^n])^(p - 1)/x, x]",
     "rulenumber": 0,
            "lhs": "Int[Log[d_.*(e_ + f_.*x_^m_.)]*(a_. + b_.*Log[c_.*x_^n_.])^p_./x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "Log[d*(e + f*x^m)^r]*(a + b*Log[c*x^n])^(p + 1)/(b*n*(p + 1)) - f*m*r/(b*n*(p + 1))* Int[x^(m - 1)*(a + b*Log[c*x^n])^(p + 1)/(e + f*x^m), x]",
     "rulenumber": 0,
            "lhs": "Int[Log[d_.*(e_ + f_.*x_^m_.)^r_.]*(a_. + b_.*Log[c_.*x_^n_.])^p_./x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p, 0] && NeQ[d*e, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)^r], x]}, Dist[(a + b*Log[c*x^n]), u, x] - b*n*Int[Dist[1/x, u, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^q_.* Log[d_.*(e_ + f_.*x_^m_.)^r_.]*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || RationalQ[m] && RationalQ[q]) && NeQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)], x]}, Dist[(a + b*Log[c*x^n])^p, u, x] - b*n*p*Int[Dist[(a + b*Log[c*x^n])^(p - 1)/x, u, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^q_.* Log[d_.*(e_ + f_.*x_^m_.)]*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && IGtQ[p, 0] && RationalQ[m] && RationalQ[q] && NeQ[q, -1] && (EqQ[p, 1] || FractionQ[m] && IntegerQ[(q + 1)/m] || IGtQ[q, 0] && IntegerQ[(q + 1)/m] && EqQ[d*e, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "With[{u = IntHide[(g*x)^q*(a + b*Log[c*x^n])^p, x]}, Dist[Log[d*(e + f*x^m)^r], u, x] - f*m*r*Int[Dist[x^(m - 1)/(e + f*x^m), u, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^q_.* Log[d_.*(e_ + f_.*x_^m_.)^r_.]*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && IGtQ[p, 0] && RationalQ[m] && RationalQ[q]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "Unintegrable[(g*x)^q*Log[d*(e + f*x^m)^r]*(a + b*Log[c*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^q_.* Log[d_.*(e_ + f_.*x_^m_.)^r_.]*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, r, m, n, p, q}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "Int[(g*x)^q*Log[d*ExpandToSum[u, x]^r]*(a + b*Log[c*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^q_.*Log[d_.*u_^r_.]*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, g, r, n, p, q}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "-b*n*x*PolyLog[k, e*x^q] + x*PolyLog[k, e*x^q]*(a + b*Log[c*x^n]) + b*n*q*Int[PolyLog[k - 1, e*x^q], x] - q*Int[PolyLog[k - 1, e*x^q]*(a + b*Log[c*x^n]), x]",
     "rulenumber": 0,
            "lhs": "Int[PolyLog[k_, e_.*x_^q_.]*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, e, n, q}, x] && IGtQ[k, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "Unintegrable[PolyLog[k, e*x^q]*(a + b*Log[c*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[PolyLog[k_, e_.*x_^q_.]*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, e, n, p, q}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "PolyLog[k + 1, e*x^q]*(a + b*Log[c*x^n])^p/q - b*n*p/q*Int[PolyLog[k + 1, e*x^q]*(a + b*Log[c*x^n])^(p - 1)/x, x]",
     "rulenumber": 0,
            "lhs": "Int[PolyLog[k_, e_.*x_^q_.]*(a_. + b_.*Log[c_.*x_^n_.])^p_./x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "PolyLog[k, e*x^q]*(a + b*Log[c*x^n])^(p + 1)/(b*n*(p + 1)) - q/(b*n*(p + 1))* Int[PolyLog[k - 1, e*x^q]*(a + b*Log[c*x^n])^(p + 1)/x, x]",
     "rulenumber": 0,
            "lhs": "Int[PolyLog[k_, e_.*x_^q_.]*(a_. + b_.*Log[c_.*x_^n_.])^p_./x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, e, k, n, q}, x] && LtQ[p, -1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "-b*n*(d*x)^(m + 1)*PolyLog[k, e*x^q]/(d*(m + 1)^2) + (d*x)^(m + 1)* PolyLog[k, e*x^q]*(a + b*Log[c*x^n])/(d*(m + 1)) + b*n*q/(m + 1)^2*Int[(d*x)^m*PolyLog[k - 1, e*x^q], x] - q/(m + 1)* Int[(d*x)^m*PolyLog[k - 1, e*x^q]*(a + b*Log[c*x^n]), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*PolyLog[k_, e_.*x_^q_.]*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, q}, x] && IGtQ[k, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "Unintegrable[(d*x)^m*PolyLog[k, e*x^q]*(a + b*Log[c*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.* PolyLog[k_, e_.*x_^q_.]*(a_. + b_.*Log[c_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p, q}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "With[{u = IntHide[Px*F[d*(e + f*x)]^m, x]}, Dist[(a + b*Log[c*x^n]), u, x] - b*n*Int[Dist[1/x, u, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_.*F_[d_.*(e_. + f_.*x_)]^m_.*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && PolynomialQ[Px, x] && IGtQ[m, 0] && MemberQ[{ArcSin, ArcCos, ArcSinh, ArcCosh}, F]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.1 u (a+b log(c x^n))^p.m",
       "filename": "3.1 u (a+b log(c x^n))^p.m",
            "rhs": "With[{u = IntHide[Px*F[d*(e + f*x)], x]}, Dist[(a + b*Log[c*x^n]), u, x] - b*n*Int[Dist[1/x, u, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_.*F_[d_.*(e_. + f_.*x_)]*(a_. + b_.*Log[c_.*x_^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && PolynomialQ[Px, x] && MemberQ[{ArcTan, ArcCot, ArcTanh, ArcCoth}, F]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "1/e*Subst[Int[(a + b*Log[c*x^n])^p, x], x, d + e*x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "1/e*Subst[Int[(f*x/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_. x_)^q_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p, q}, x] && EqQ[e*f - d*g, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "-PolyLog[2, -c*e*x^n]/n",
     "rulenumber": 0,
            "lhs": "Int[Log[c_.*(d_ + e_.*x_^n_.)]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "(a + b*Log[c*d])*Log[x] + b*Int[Log[1 + e*x/d]/x, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_)])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && GtQ[c*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "1/g*Subst[Int[(a + b*Log[1 + c*e*x/g])/x, x], x, f + g*x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_)])/(f_. + g_. x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g + c*(e*f - d*g), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "Log[e*(f + g*x)/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n])/g - b*e*n/g*Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d + e*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])/(f_. + g_. x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "(f + g*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n])/(g*(q + 1)) - b*e*n/(g*(q + 1))*Int[(f + g*x)^(q + 1)/(d + e*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^q_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "Log[e*(f + g*x)/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n])^p/g - b*e*n*p/g* Int[Log[(e*(f + g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_/(f_. + g_. x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && IGtQ[p, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "(d + e*x)*(a + b*Log[c*(d + e*x)^n])^ p/((e*f - d*g)*(f + g*x)) - b*e*n*p/(e*f - d*g)* Int[(a + b*Log[c*(d + e*x)^n])^(p - 1)/(f + g*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_/(f_. + g_.*x_)^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "(f + g*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n])^ p/(g*(q + 1)) - b*e*n*p/(g*(q + 1))* Int[(f + g*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^q_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && (Not[IGtQ[q, 0]] || EqQ[p, 2] && NeQ[q, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(f + g*x)^q/(a + b*Log[c*(d + e*x)^n]), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^q_./(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && IGtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "(d + e*x)*(f + g*x)^ q*(a + b*Log[c*(d + e*x)^n])^(p + 1)/(b*e*n*(p + 1)) + q*(e*f - d*g)/(b*e*n*(p + 1))* Int[(f + g*x)^(q - 1)*(a + b*Log[c*(d + e*x)^n])^(p + 1), x] - (q + 1)/(b*n*(p + 1))* Int[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^q_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && LtQ[p, -1] && GtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^q_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && IGtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "-e/g* Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)]",
     "rulenumber": 0,
            "lhs": "Int[Log[c_./(d_ + e_.*x_)]/(f_ + g_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "(a + b*Log[c/(2*d)])*Int[1/(f + g*x^2), x] + b*Int[Log[2*d/(d + e*x)]/(f + g*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_./(d_ + e_.*x_)])/(f_ + g_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e^2*f + d^2*g, 0] && GtQ[c/(2*d), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "With[{u = IntHide[1/Sqrt[f + g*x^2], x]}, u*(a + b*Log[c*(d + e*x)^n]) - b*e*n*Int[SimplifyIntegrand[u/(d + e*x), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])/Sqrt[f_ + g_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n}, x] && GtQ[f, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "With[{u = IntHide[1/Sqrt[f1*f2 + g1*g2*x^2], x]}, u*(a + b*Log[c*(d + e*x)^n]) - b*e*n*Int[SimplifyIntegrand[u/(d + e*x), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])/(Sqrt[f1_ + g1_.*x_]* Sqrt[f2_ + g2_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f1, g1, f2, g2, n}, x] && EqQ[f2*g1 + f1*g2, 0] && GtQ[f1, 0] && GtQ[f2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "Sqrt[1 + g/f*x^2]/Sqrt[f + g*x^2]* Int[(a + b*Log[c*(d + e*x)^n])/Sqrt[1 + g/f*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])/Sqrt[f_ + g_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n}, x] && Not[GtQ[f, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "Sqrt[1 + g1*g2/(f1*f2)*x^2]/(Sqrt[f1 + g1*x]*Sqrt[f2 + g2*x])* Int[(a + b*Log[c*(d + e*x)^n])/Sqrt[1 + g1*g2/(f1*f2)*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])/(Sqrt[f1_ + g1_.*x_]* Sqrt[f2_ + g2_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f1, g1, f2, g2, n}, x] && EqQ[f2*g1 + f1*g2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "With[{k = Denominator[r]}, k*Subst[ Int[x^(k - 1)*(f + g*x^(k*r))^q*(a + b*Log[c*(d + e*x^k)^n])^p, x], x, x^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_^r_)^q_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p, q}, x] && FractionQ[r] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_^r_)^q_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, r}, x] && IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || IntegerQ[r] && NeQ[r, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "Int[ExpandIntegrand[Log[c*(d + e*x)], x^m/(f + g*x), x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Log[c_.*(d_ + e_.*x_)]/(f_ + g_. x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e, f, g}, x] && EqQ[e*f - d*g, 0] && EqQ[c*d, 1] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "1/e*Subst[ Int[(g*x/e)^q*((e*h - d*i)/e + i*x/e)^r*(a + b*Log[c*x^n])^p, x], x, d + e*x]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_. x_)^q_.*(h_. + i_. x_)^ r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "Int[(g + f*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(f_ + g_./x_)^q_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p, q}, x] && EqQ[m, q] && IntegerQ[q]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "(f + g*x^r)^(q + 1)*(a + b*Log[c*(d + e*x)^n])^ p/(g*r*(q + 1)) - b*e*n*p/(g*r*(q + 1))* Int[(f + g*x^r)^(q + 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(f_. + g_.*x_^r_.)^ q_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, q, r}, x] && EqQ[m, r - 1] && NeQ[q, -1] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "With[{u = IntHide[x^m*(f + g*x^r)^q, x]}, Dist[(a + b*Log[c*(d + e*x)^n]), u, x] - b*e*n*Int[SimplifyIntegrand[u/(d + e*x), x], x] /; InverseFunctionFreeQ[u, x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(f_ + g_.*x_^r_.)^ q_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, q, r}, x] && IntegerQ[m] && IntegerQ[q] && IntegerQ[r]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "With[{k = Denominator[r]}, k*Subst[ Int[x^(k*(m + 1) - 1)*(f + g*x^(k*r))^ q*(a + b*Log[c*(d + e*x^k)^n])^p, x], x, x^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(f_. + g_.*x_^r_)^ q_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p, q}, x] && FractionQ[r] && IGtQ[p, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^ p, (h*x)^m*(f + g*x^r)^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(h_.*x_)^m_.*(f_ + g_.*x_^r_.)^ q_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "Int[ExpandIntegrand[Polyx*(a + b*Log[c*(d + e*x)^n])^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Polyx_*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p}, x] && PolynomialQ[Polyx, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "With[{u = ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[RFx_*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && RationalFunctionQ[RFx, x] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "With[{u = ExpandIntegrand[RFx*(a + b*Log[c*(d + e*x)^n])^p, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[RFx_*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && RationalFunctionQ[RFx, x] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "Unintegrable[AFx*(a + b*Log[c*(d + e*x)^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[AFx_*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p}, x] && AlgebraicFunctionQ[AFx, x, True]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "Int[ExpandToSum[u, x]^q*(a + b*Log[c*ExpandToSum[v, x]^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[u_^q_.*(a_. + b_.*Log[c_.*v_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, p, q}, x] && BinomialQ[u, x] && LinearQ[v, x] && Not[BinomialMatchQ[u, x] && LinearMatchQ[v, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "-x*(m - Log[f*x^m])*(a + b*Log[c*(d + e*x)^n]) + b*e*m*n*Int[x/(d + e*x), x] - b*e*n*Int[(x*Log[f*x^m])/(d + e*x), x]",
     "rulenumber": 0,
            "lhs": "Int[Log[f_.*x_^m_.]*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "With[{u = IntHide[(a + b*Log[c*(d + e*x)^n])^p, x]}, Dist[Log[f*x^m], u, x] - m*Int[Dist[1/x, u, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Log[f_.*x_^m_.]*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "Unintegrable[Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[Log[f_.*x_^m_.]*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "Log[f*x^m]^2*(a + b*Log[c*(d + e*x)^n])/(2*m) - b*e*n/(2*m)*Int[Log[f*x^m]^2/(d + e*x), x]",
     "rulenumber": 0,
            "lhs": "Int[Log[f_.*x_^m_.]*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "-1/(g*(q + 1))*(m*(g*x)^(q + 1)/(q + 1) - (g*x)^(q + 1)*Log[f*x^m])*(a + b*Log[c*(d + e*x)^n]) + b*e*m*n/(g*(q + 1)^2)*Int[(g*x)^(q + 1)/(d + e*x), x] - b*e*n/(g*(q + 1))*Int[(g*x)^(q + 1)*Log[f*x^m]/(d + e*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^q_.* Log[f_.*x_^m_.]*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && NeQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "Log[f*x^m]^2*(a + b*Log[c*(d + e*x)^n])^p/(2*m) - b*e*n*p/(2*m)* Int[Log[f*x^m]^2*(a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x), x]",
     "rulenumber": 0,
            "lhs": "Int[Log[f_.*x_^m_.]*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_./x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "With[{u = IntHide[(g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x]}, Dist[Log[f*x^m], u, x] - m*Int[Dist[1/x, u, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^q_.* Log[f_.*x_^m_.]*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && IGtQ[p, 1] && IGtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "With[{u=IntHide[(a+b*Log[c*(d+e*x)^n])^p,x]}, Dist[(g*x)^q*Log[f*x^m],u,x] - g*m*Int[Dist[(g*x)^(q-1),u,x],x] -  g*q*Int[Dist[(g*x)^(q-1)*Log[f*x^m],u,x],x]]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^q_.*Log[f_.*x_^m_.]*(a_.+b_.*Log[c_.*(d_+e_.*x_)^n_.]) ^p_,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e,f,g,m,n,q},x] && IGtQ[p,1] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "Unintegrable[(g*x)^q*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^q_.* Log[f_.*x_^m_.]*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p, q}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "x*(a + b*Log[c*(d + e*x)^n])^p*(f + g*Log[h*(i + j*x)^m]) - g*j*m*Int[x*(a + b*Log[c*(d + e*x)^n])^p/(i + j*x), x] - b*e*n*p* Int[x*(a + b*Log[c*(d + e*x)^n])^(p - 1)*(f + g*Log[h*(i + j*x)^m])/(d + e*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^ p_.*(f_. + g_.*Log[h_.*(i_. + j_.*x_)^m_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "1/e*Subst[Int[Log[f*(g*x/d)^m]*(a + b*Log[c*x^n])^p, x], x, d + e*x]",
     "rulenumber": 0,
            "lhs": "Int[Log[f_.*(g_. + h_.*x_)^m_.]*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n, p}, x] && EqQ[e*f - d*g, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "Unintegrable[(a + b*Log[c*(d + e*x)^n])^p*(f + g*Log[h*(i + j*x)^m])^ q, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^ p_.*(f_. + g_.*Log[h_.*(i_. + j_.*x_)^m_.])^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "1/e*Subst[ Int[(k*x/d)^r*(a + b*Log[c*x^n])^ p*(f + g*Log[h*((e*i - d*j)/e + j*x/e)^m]), x], x, d + e*x]",
     "rulenumber": 0,
            "lhs": "Int[(k_. + l_.*x_)^r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^ p_.*(f_. + g_.*Log[h_.*(i_. + j_.*x_)^m_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r}, x] && EqQ[e*k - d*l, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "Log[x]*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]) - e*g*m*Int[Log[x]*(a + b*Log[c*(d + e*x)^n])/(d + e*x), x] - b*j*n*Int[Log[x]*(f + g*Log[h*(i + j*x)^m])/(i + j*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])*(f_. + g_.*Log[h_.*(i_. + j_.*x_)^m_.])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && EqQ[e*i - d*j, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "Log[-b*x/a]*Log[a + b*x]*Log[c + d*x] - 1/2*(Log[-b*x/a] - Log[-d*x/c])*(Log[a + b*x] + Log[a*(c + d*x)/(c*(a + b*x))])^2 + 1/2*(Log[-b*x/a] - Log[-(b*c - a*d)*x/(a*(c + d*x))] + Log[(b*c - a*d)/(b*(c + d*x))])* Log[a*(c + d*x)/(c*(a + b*x))]^2 + (Log[c + d*x] - Log[a*(c + d*x)/(c*(a + b*x))])* PolyLog[2, 1 + b*x/a] + (Log[a + b*x] + Log[a*(c + d*x)/(c*(a + b*x))])* PolyLog[2, 1 + d*x/c] - Log[a*(c + d*x)/(c*(a + b*x))]* PolyLog[2, d*(a + b*x)/(b*(c + d*x))] + Log[a*(c + d*x)/(c*(a + b*x))]* PolyLog[2, c*(a + b*x)/(a*(c + d*x))] - PolyLog[3, 1 + b*x/a] - PolyLog[3, 1 + d*x/c] - PolyLog[3, d*(a + b*x)/(b*(c + d*x))] + PolyLog[3, c*(a + b*x)/(a*(c + d*x))]",
     "rulenumber": 0,
            "lhs": "Int[Log[a_ + b_.*x_]*Log[c_ + d_.*x_]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "Int[Log[ExpandToSum[v, x]]*Log[ExpandToSum[w, x]]/x, x]",
     "rulenumber": 0,
            "lhs": "Int[Log[v_]*Log[w_]/x_, x_Symbol]",
        "comment": false,
         "givens": "LinearQ[{v, w}, x] && Not[LinearMatchQ[{v, w}, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "m*Int[Log[i + j*x]*Log[c*(d + e*x)^n]/x, x] - (m*Log[i + j*x] - Log[h*(i + j*x)^m])* Int[Log[c*(d + e*x)^n]/x, x]",
     "rulenumber": 0,
            "lhs": "Int[Log[c_.*(d_ + e_.*x_)^n_.]*Log[h_.*(i_. + j_.*x_)^m_.]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e, h, i, j, m, n}, x] && NeQ[e*i - d*j, 0] && NeQ[i + j*x, h*(i + j*x)^m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "f*Int[(a + b*Log[c*(d + e*x)^n])/x, x] + g*Int[Log[h*(i + j*x)^m]*(a + b*Log[c*(d + e*x)^n])/x, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])*(f_ + g_.*Log[h_.*(i_. + j_.*x_)^m_.])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && NeQ[e*i - d*j, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "x^(r + 1)*(a + b*Log[c*(d + e*x)^n])^ p*(f + g*Log[h*(i + j*x)^m])/(r + 1) - g*j*m/(r + 1)* Int[x^(r + 1)*(a + b*Log[c*(d + e*x)^n])^p/(i + j*x), x] - b*e*n*p/(r + 1)* Int[x^(r + 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1)*(f + g*Log[h*(i + j*x)^m])/(d + e*x), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^ p_.*(f_. + g_.*Log[h_.*(i_. + j_.*x_)^m_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && IGtQ[p, 0] && IntegerQ[r] && (EqQ[p, 1] || GtQ[r, 0]) && NeQ[r, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "1/l*Subst[ Int[x^r*(a + b*Log[c*(-(e*k - d*l)/l + e*x/l)^n])*(f + g*Log[h*(-(j*k - i*l)/l + j*x/l)^m]), x], x, k + l*x]",
     "rulenumber": 0,
            "lhs": "Int[(k_ + l_.*x_)^ r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])*(f_. + g_.*Log[h_.*(i_. + j_.*x_)^m_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, m, n}, x] && IntegerQ[r]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "Unintegrable[(k + l*x)^r*(a + b*Log[c*(d + e*x)^n])^ p*(f + g*Log[h*(i + j*x)^m])^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(k_. + l_.*x_)^r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^ p_.*(f_. + g_.*Log[h_.*(i_. + j_.*x_)^m_.])^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, m, n, p, q, r}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "1/g*Subst[Int[PolyLog[k, h*x/d]*(a + b*Log[c*x^n])^p/x, x], x, d + e*x]",
     "rulenumber": 0,
            "lhs": "Int[PolyLog[k_, h_ + i_.*x_]*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])^ p_./(f_ + g_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, i, k, n}, x] && EqQ[e*f - d*g, 0] && EqQ[g*h - f*i, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "With[{u = IntHide[Px*F[f*(g + h*x)], x]}, Dist[(a + b*Log[c*(d + e*x)^n]), u, x] - b*e*n*Int[SimplifyIntegrand[u/(d + e*x), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_.*F_[ f_.*(g_. + h_.*x_)]*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, n}, x] && PolynomialQ[Px, x] && MemberQ[{ArcSin, ArcCos, ArcTan, ArcCot, ArcSinh, ArcCosh, ArcTanh, ArcCoth}, F]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "Int[u*(a + b*Log[c*ExpandToSum[v, x]^n])^p, x] /; FreeQ[{a, b, c, n, p}, x] && LinearQ[v, x] && Not[LinearMatchQ[v, x]] && Not[EqQ[n, 1] && MatchQ[c*v, e_.*(f_ + g_.*x)",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_. + b_.*Log[c_.*v_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{e, f, g}, x]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "Subst[Int[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_. + b_.*Log[c_.*(d_.*(e_. + f_. x_)^m_.)^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[n]] && Not[EqQ[d, 1] && EqQ[m, 1]] && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.2 u (a+b log(c (d+e x)^n))^p.m",
       "filename": "3.2 u (a+b log(c (d+e x)^n))^p.m",
            "rhs": "Unintegrable[AFx*(a + b*Log[c*(d*(e + f*x)^m)^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[AFx_*(a_. + b_.*Log[c_.*(d_.*(e_. + f_. x_)^m_.)^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && AlgebraicFunctionQ[AFx, x, True]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "With[{C = FullSimplify[Pq^m*(1 - u)/D[u, x]]}, C*PolyLog[2, 1 - u] /; FreeQ[C, x]]",
     "rulenumber": 0,
            "lhs": "Int[Pq_^m_.*Log[u_], x_Symbol]",
        "comment": false,
         "givens": "IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, x][[2]], Expon[Pq, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "x*Log[c*(d + e*x^n)^p] - e*n*p*Int[x^n/(d + e*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[Log[c_.*(d_ + e_.*x_^n_)^p_.], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "(e + d*x)*(a + b*Log[c*(d + e/x)^p])^q/d + b*e*p*q/d*Int[(a + b*Log[c*(d + e/x)^p])^(q - 1)/x, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*(d_ + e_./x_)^p_.])^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && IGtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "x*(a + b*Log[c*(d + e*x^n)^p])^q - b*e*n*p*q* Int[x^n*(a + b*Log[c*(d + e*x^n)^p])^(q - 1)/(d + e*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p}, x] && IGtQ[q, 0] && (EqQ[q, 1] || IntegerQ[n])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": " With[{k=Denominator[n]}, k*Subst[Int[x^(k-1)*(a+b*Log[c*(d+e*x^(k*n))^p])^q,x],x,x^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(a_.+b_.*Log[c_.*(d_+e_.*x_^n_)^p_.])^q_,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e,p,q},x] && LtQ[-1,n,1] && (GtQ[n,0] || IGtQ[q,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "With[{k = Denominator[n]}, k*Subst[Int[x^(k - 1)*(a + b*Log[c*(d + e*x^(k*n))^p])^q, x], x, x^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p, q}, x] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "Unintegrable[(a + b*Log[c*(d + e*x^n)^p])^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p, q}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "Int[(a + b*Log[c*ExpandToSum[v, x]^p])^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*v_^p_.])^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p, q}, x] && BinomialQ[v, x] && Not[BinomialMatchQ[v, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p, q}, x] && IntegerQ[ Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) && Not[EqQ[q, 1] && ILtQ[n, 0] && IGtQ[m, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "(f*x)^(m + 1)*(a + b*Log[c*(d + e*x^n)^p])/(f*(m + 1)) - b*e*n*p/(f*(m + 1))*Int[x^(n - 1)*(f*x)^(m + 1)/(d + e*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "(f*x)^m/x^m* Int[x^m*(a + b*Log[c*(d + e*x^n)^p])^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_*x_)^m_*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p, q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "(f*x)^(m + 1)*(a + b*Log[c*(d + e*x^n)^p])^ q/(f*(m + 1)) - b*e*n*p*q/(f^n*(m + 1))* Int[(f*x)^(m + n)*(a + b*Log[c*(d + e*x^n)^p])^(q - 1)/(d + e*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p}, x] && IGtQ[q, 1] && IntegerQ[n] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "With[{k = Denominator[n]}, k*Subst[ Int[x^(k*(m + 1) - 1)*(a + b*Log[c*(d + e*x^(k*n))^p])^q, x], x, x^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, p, q}, x] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "(f*x)^m/x^m* Int[x^m*(a + b*Log[c*(d + e*x^n)^p])^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_*x_)^m_*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p, q}, x] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "Unintegrable[(f*x)^m*(a + b*Log[c*(d + e*x^n)^p])^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p, q}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "Int[(f*x)^m*(a + b*Log[c*ExpandToSum[v, x]^p])^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(a_. + b_.*Log[c_.*v_^p_.])^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, f, m, p, q}, x] && BinomialQ[v, x] && Not[BinomialMatchQ[v, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "Log[f + g*x]*(a + b*Log[c*(d + e*x^n)^p])/g - b*e*n*p/g*Int[x^(n - 1)*Log[f + g*x]/(d + e*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])/(f_. + g_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p}, x] && RationalQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "(f + g*x)^(r + 1)*(a + b*Log[c*(d + e*x^n)^p])/(g*(r + 1)) - b*e*n*p/(g*(r + 1))* Int[x^(n - 1)*(f + g*x)^(r + 1)/(d + e*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p, r}, x] && (IGtQ[r, 0] || RationalQ[n]) && NeQ[r, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "Unintegrable[(f + g*x)^r*(a + b*Log[c*(d + e*x^n)^p])^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p, q, r}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "Int[ExpandToSum[u, x]^r*(a + b*Log[c*ExpandToSum[v, x]^p])^q, x]",
     "rulenumber": 0,
            "lhs": "Int[u_^r_.*(a_. + b_.*Log[c_.*v_^p_.])^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p, q, r}, x] && LinearQ[u, x] && BinomialQ[v, x] && Not[LinearMatchQ[u, x] && BinomialMatchQ[v, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x)^r, x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(f_. + g_.*x_)^ r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p, q}, x] && IntegerQ[m] && IntegerQ[r]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "With[{k = Denominator[m]}, k/h*Subst[ Int[x^(k*(m + 1) - 1)*(f + g*x^k/h)^ r*(a + b*Log[c*(d + e*x^(k*n)/h^n)^p])^q, x], x, (h*x)^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(h_.*x_)^m_*(f_. + g_.*x_)^ r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_.)^p_.])^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, p, r}, x] && FractionQ[m] && IntegerQ[n] && IntegerQ[r]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "Unintegrable[(h*x)^m*(f + g*x)^r*(a + b*Log[c*(d + e*x^n)^p])^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(h_.*x_)^m_.*(f_. + g_.*x_)^ r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n, p, q, r}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "Int[(h*x)^m*ExpandToSum[u, x]^r*(a + b*Log[c*ExpandToSum[v, x]^p])^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(h_.*x_)^m_.*u_^r_.*(a_. + b_.*Log[c_.*v_^p_.])^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, h, m, p, q, r}, x] && LinearQ[u, x] && BinomialQ[v, x] && Not[LinearMatchQ[u, x] && BinomialMatchQ[v, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "With[{u = IntHide[1/(f + g*x^2), x]}, u*(a + b*Log[c*(d + e*x^n)^p]) - b*e*n*p*Int[u*x^(n - 1)/(d + e*x^n), x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])/(f_ + g_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "With[{t = ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, (f + g*x^s)^r, x]}, Int[t, x] /; SumQ[t]]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_^s_)^r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^ q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p, q, r, s}, x] && IntegerQ[n] && IGtQ[q, 0] && IntegerQ[r] && IntegerQ[s] && (EqQ[q, 1] || GtQ[r, 0] && GtQ[s, 1] || LtQ[s, 0] && LtQ[r, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "With[{k = Denominator[n]}, k*Subst[ Int[x^(k - 1)*(f + g*x^(k*s))^ r*(a + b*Log[c*(d + e*x^(k*n))^p])^q, x], x, x^(1/k)] /; IntegerQ[k*s]]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_^s_)^r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^ q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p, q, r, s}, x] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "Unintegrable[(f + g*x^s)^r*(a + b*Log[c*(d + e*x^n)^p])^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_^s_)^r_. (a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^ q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p, q, r, s}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "Int[ExpandToSum[u, x]^r*(a + b*Log[c*ExpandToSum[v, x]^p])^q, x]",
     "rulenumber": 0,
            "lhs": "Int[u_^r_.*(a_. + b_.*Log[c_.*v_^p_.])^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p, q, r}, x] && BinomialQ[{u, v}, x] && Not[BinomialMatchQ[{u, v}, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^ r*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(f_ + g_.*x_^s_)^ r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && IntegerQ[r] && IntegerQ[s/n] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(f_ + g_.*x_^s_)^ r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] && IntegerQ[s]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "With[{k = Denominator[n]}, k*Subst[ Int[x^(k - 1)*(f + g*x^(k*s))^ r*(a + b*Log[c*(d + e*x^(k*n))^p])^q, x], x, x^(1/k)] /; IntegerQ[k*s]]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_^s_)^r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^ q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p, q, r, s}, x] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "1/n*Subst[ Int[x^(m + 1/n - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^ q, x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(f_ + g_.*x_^s_)^ r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && FractionQ[n] && IntegerQ[1/n] && IntegerQ[s/n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "With[{k = Denominator[m]}, k/h*Subst[ Int[x^(k*(m + 1) - 1)*(f + g*x^(k*s)/h^s)^ r*(a + b*Log[c*(d + e*x^(k*n)/h^n)^p])^q, x], x, (h*x)^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(h_.*x_)^m_*(f_. + g_.*x_^s_.)^ r_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_.)^p_.])^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, p, r}, x] && FractionQ[m] && IntegerQ[n] && IntegerQ[s]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "Unintegrable[(h*x)^m*(f + g*x^s)^r*(a + b*Log[c*(d + e*x^n)^p])^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(h_.*x_)^m_.*(f_ + g_.*x_^s_)^ r_. (a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n, p, q, r, s}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "Int[(h*x)^m*ExpandToSum[u, x]^r*(a + b*Log[c*ExpandToSum[v, x]^p])^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(h_.*x_)^m_.*u_^r_.*(a_. + b_.*Log[c_.*v_^p_.])^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, h, m, p, q, r}, x] && BinomialQ[{u, v}, x] && Not[BinomialMatchQ[{u, v}, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "Log[f*x^q]^(m + 1)*(a + b*Log[c*(d + e*x^n)^p])/(q*(m + 1)) - b*e*n*p/(q*(m + 1))* Int[x^(n - 1)*Log[f*x^q]^(m + 1)/(d + e*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[Log[f_.*x_^q_.]^m_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p, q}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "With[{u = IntHide[F[f*x]^m, x]}, Dist[a + b*Log[c*(d + e*x^n)^p], u, x] - b*e*n*p*Int[SimplifyIntegrand[u*x^(n - 1)/(d + e*x^n), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[F_[f_.*x_]^m_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_^n_)^p_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p}, x] && MemberQ[{ArcSin, ArcCos, ArcSinh, ArcCosh}, F] && IGtQ[m, 0] && IGtQ[n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "1/g*Subst[Int[(a + b*Log[c*(d + e*x^n)^p])^q, x], x, f + g*x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*(d_ + e_.*(f_. + g_.*x_)^n_)^p_.])^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IGtQ[q, 0] && (EqQ[q, 1] || IntegerQ[n])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.3 u (a+b log(c (d+e x^m)^n))^p.m",
       "filename": "3.3 u (a+b log(c (d+e x^m)^n))^p.m",
            "rhs": "Unintegrable[(a + b*Log[c*(d + e*(f + g*x)^n)^p])^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*(d_ + e_.*(f_. + g_.*x_)^n_)^p_.])^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "Int[u*Log[e*(b^p*f/d^p*(c + d*x)^(p + q))^r]^s, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^s_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && EqQ[b*c - a*d, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "(a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/b + q*r*s*(b*c - a*d)/b* Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/(c + d*x), x]",
     "rulenumber": 0,
            "lhs": "Int[Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^s_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && IGtQ[s, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "(a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/b - r*s*(p + q)* Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1), x] + q*r*s*(b*c - a*d)/b* Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/(c + d*x), x]",
     "rulenumber": 0,
            "lhs": "Int[Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^s_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && NeQ[p + q, 0] && IGtQ[s, 0] && LtQ[s, 4]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "-Log[-(b*c - a*d)/(d*(a + b*x))]* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/h + p*r*s (b*c - a*d)/h* Int[Log[-(b*c - a*d)/(d*(a + b*x))]* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/((a + b*x)*(c + d*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^ s_./(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && EqQ[b*g - a*h, 0] && IGtQ[s, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "d/h*Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/(c + d*x), x] - (d*g - c*h)/h* Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/((c + d*x)*(g + h*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^ s_/(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && NeQ[b*g - a*h, 0] && NeQ[d*g - c*h, 0] && IGtQ[s, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "(a + b*x)* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/((b*g - a*h)*(g + h*x)) - p*r*s*(b*c - a*d)/(b*g - a*h)* Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/((c + d*x)*(g + h*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^ s_./(g_. + h_.*x_)^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && NeQ[b*g - a*h, 0] && IGtQ[s, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "d/(d*g - c*h)* Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/(g + h*x)^2, x] - h/(d*g - c*h)* Int[(c + d*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/(g + h*x)^3, x]",
     "rulenumber": 0,
            "lhs": "Int[Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^ s_/(g_. + h_.*x_)^3, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && EqQ[b*g - a*h, 0] && NeQ[d*g - c*h, 0] && IGtQ[s, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "(g + h*x)^(m + 1)* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/(h*(m + 1)) - p*r*s*(b*c - a*d)/(h*(m + 1))* Int[(g + h*x)^(m + 1)* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/((a + b*x)*(c + d*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)^m_.* Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^s_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && IGtQ[s, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "b*(c + d*x)*(e*(f*(a + b*x)^p*(c + d*x)^q)^r)^(1/(p*r))/(h*p* r*(b*c - a*d)*(g + h*x))* ExpIntegralEi[-1/(p*r)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]]",
     "rulenumber": 0,
            "lhs": "Int[1/((g_. + h_.*x_)^2* Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, p, q, r}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && EqQ[b*g - a*h, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "Log[g + h*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/h - b*p*r/h*Int[Log[g + h*x]/(a + b*x), x] - d*q*r/h*Int[Log[g + h*x]/(c + d*x), x]",
     "rulenumber": 0,
            "lhs": "Int[Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]/(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, p, q, r}, x] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "(g + h*x)^(m + 1)* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(h*(m + 1)) - b*p*r/(h*(m + 1))*Int[(g + h*x)^(m + 1)/(a + b*x), x] - d*q*r/(h*(m + 1))*Int[(g + h*x)^(m + 1)/(c + d*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)^m_.* Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, p, q, r}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "Int[(Log[(a + b*x)^(p*r)] + Log[(c + d*x)^(q*r)])^2/(g + h*x), x] + (Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - Log[(a + b*x)^(p*r)] - Log[(c + d*x)^(q*r)])* (2*Int[Log[(c + d*x)^(q*r)]/(g + h*x), x] + Int[(Log[(a + b*x)^(p*r)] - Log[(c + d*x)^(q*r)] + Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(g + h*x), x])",
     "rulenumber": 0,
            "lhs": "Int[Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^2/(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, p, q, r}, x] && NeQ[b*c - a*d, 0] && EqQ[b*g - a*h, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": " Int[(Log[(a+b*x)^(p*r)]+Log[(c+d*x)^(q*r)])^2/(g+h*x),x] + (Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r]-Log[(a+b*x)^(p*r)]-Log[(c+d*x)^(q* r)])* Int[(Log[(a+b*x)^(p*r)]+Log[(c+d*x)^(q*r)]+Log[e*(f*(a+b*x)^p*(c+ d*x)^q)^r])/(g+h*x),x]",
     "rulenumber": 0,
            "lhs": "Int[Log[e_.*(f_.*(a_.+b_.*x_)^p_.*(c_.+d_.*x_)^q_.)^r_.]^2/(g_.+h_. *x_),x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,g,h,p,q,r},x] &&  NeQ[b*c-a*d,0] && EqQ[b*g-a*h,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "Log[g + h*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/h - 2*b*p*r/h* Int[Log[g + h*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(a + b*x), x] - 2*d*q*r/h* Int[Log[g + h*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(c + d*x), x]",
     "rulenumber": 0,
            "lhs": "Int[Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^2/(g_. + h_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, p, q, r}, x] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "(g + h*x)^(m + 1)* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/(h*(m + 1)) - b*p*r*s/(h*(m + 1))* Int[(g + h*x)^(m + 1)* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/(a + b*x), x] - d*q*r*s/(h*(m + 1))* Int[(g + h*x)^(m + 1)* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/(c + d*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)^m_.* Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^s_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "(s + t*Log[i*(g + h*x)^n])^(m + 1)* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(k*n*t*(m + 1)) - b*p*r/(k*n*t*(m + 1))* Int[(s + t*Log[i*(g + h*x)^n])^(m + 1)/(a + b*x), x] - d*q*r/(k*n*t*(m + 1))* Int[(s + t*Log[i*(g + h*x)^n])^(m + 1)/(c + d*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(s_. + t_.*Log[i_.*(g_. + h_.*x_)^n_.])^m_.* Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]/(j_. + k_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, i, j, k, s, t, m, n, p, q, r}, x] && NeQ[b*c - a*d, 0] && EqQ[h*j - g*k, 0] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - Log[(a + b*x)^(p*r)] - Log[(c + d*x)^(q*r)])* Int[(s + t*Log[i*(g + h*x)^n])/(j + k*x), x] + Int[(Log[(a + b*x)^(p*r)]*(s + t*Log[i*(g + h*x)^n]))/(j + k*x), x] + Int[(Log[(c + d*x)^(q*r)]*(s + t*Log[i*(g + h*x)^n]))/(j + k*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(s_. + t_.*Log[i_.*(g_. + h_.*x_)^n_.])* Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]/(j_. + k_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, i, j, k, s, t, n, p, q, r}, x] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "Unintegrable[(s + t*Log[i*(g + h*x)^n])^m* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^u/(j + k*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(s_. + t_.*Log[i_.*(g_. + h_.*x_)^n_.])^m_.* Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^ u_./(j_. + k_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, i, j, k, s, t, m, n, p, q, r, u}, x] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "With[{g = Coeff[Simplify[1/(u*(a + b*x))], x, 0], h = Coeff[Simplify[1/(u*(a + b*x))], x, 1]}, -(b - d*e)/(h*(b*c - a*d))* Subst[Int[Log[e*x]/(1 - e*x), x], x, (c + d*x)/(a + b*x)] /; EqQ[g*(b - d*e) - h*(a - c*e), 0]]",
     "rulenumber": 0,
            "lhs": "Int[u_*Log[e_.*(c_. + d_.*x_)/(a_. + b_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && LinearQ[Simplify[1/(u*(a + b*x))], x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "With[{g = Coeff[Simplify[1/(u*(a + b*x))], x, 0], h = Coeff[Simplify[1/(u*(a + b*x))], x, 1]}, -Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/(b*g - a*h)* Log[-(b*c - a*d)*(g + h*x)/((d*g - c*h)*(a + b*x))] + p*r*s*(b*c - a*d)/(b*g - a*h)* Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/((a + b*x)*(c + d*x))* Log[-(b*c - a*d)*(g + h*x)/((d*g - c*h)*(a + b*x))], x] /; NeQ[b*g - a*h, 0] && NeQ[d*g - c*h, 0]]",
     "rulenumber": 0,
            "lhs": "Int[u_*Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^s_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && EqQ[p + q, 0] && LinearQ[Simplify[1/(u*(a + b*x))], x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "With[{h = Simplify[u*(a + b*x)*(c + d*x)]}, h*Log[ Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]]/(p* r*(b*c - a*d)) /; FreeQ[h, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_/Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p, q, r}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "With[{h = Simplify[u*(a + b*x)*(c + d*x)]}, h*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s + 1)/(p* r*(s + 1)*(b*c - a*d)) /; FreeQ[h, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^s_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && NeQ[s, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "With[{g = Simplify[(v - 1)*(c + d*x)/(a + b*x)], h = Simplify[u*(a + b*x)*(c + d*x)]}, -h*PolyLog[2, 1 - v]* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/(b*c - a*d) + h*p*r*s* Int[PolyLog[2, 1 - v]* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/((a + b*x)*(c + d*x)), x] /; FreeQ[{g, h}, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*Log[v_]* Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^s_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && EqQ[p + q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "With[{k = Simplify[v*(a + b*x)*(c + d*x)]}, k*Log[i*(j*(g + h*x)^t)^u]* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s + 1)/(p* r*(s + 1)*(b*c - a*d)) - k*h*t*u/(p*r*(s + 1)*(b*c - a*d))* Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s + 1)/(g + h*x), x] /; FreeQ[k, x]]",
     "rulenumber": 0,
            "lhs": "Int[v_*Log[i_.*(j_.*(g_. + h_.*x_)^t_.)^u_.]* Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^s_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, i, j, p, q, r, s, t, u}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && NeQ[s, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "With[{g = Simplify[v*(c + d*x)/(a + b*x)], h = Simplify[u*(a + b*x)*(c + d*x)]}, h*PolyLog[n + 1, v]* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/(b*c - a*d) - h*p*r*s* Int[PolyLog[n + 1, v]* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/((a + b*x)*(c + d*x)), x] /; FreeQ[{g, h}, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*PolyLog[n_, v_]* Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^s_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && EqQ[p + q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "(a + b*x)^(m + 1)*(c + d*x)^(n + 1)* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/((m + 1)*(b*c - a*d)) - p*r*s*(b*c - a*d)/((m + 1)*(b*c - a*d))* Int[(a + b*x)^m*(c + d*x)^n* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_.*(c_. + d_.*x_)^n_.* Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^s_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && EqQ[m + n + 2, 0] && NeQ[m, -1] && IGtQ[s, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "(a + b*x)^(m + 1)*(c + d*x)^(n + 1)/(p* r*(b*c - a*d)*(e*(f*(a + b*x)^p*(c + d*x)^q)^r)^((m + 1)/(p*r)))* ExpIntegralEi[(m + 1)/(p*r)* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_.*(c_. + d_.*x_)^n_./ Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p, q, r}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && EqQ[m + n + 2, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "2*e*g/(C*(e*f - d*g))* Subst[Int[(a + b*Log[c*x])^n/x, x], x, Sqrt[d + e*x]/Sqrt[f + g*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*Sqrt[d_. + e_.*x_]/Sqrt[f_. + g_.*x_]])^ n_./(A_. + B_.*x_ + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, A, B, C, n}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[B*e*g - C*(e*f + d*g), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "g/(C*f)*Subst[Int[(a + b*Log[c*x])^n/x, x], x, Sqrt[d + e*x]/Sqrt[f + g*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*Sqrt[d_. + e_.*x_]/Sqrt[f_. + g_.*x_]])^ n_./(A_. + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, A, C, n}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[e*f + d*g, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "p*r*Int[RFx*Log[a + b*x], x] + q*r*Int[RFx*Log[c + d*x], x] - (p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])*Int[RFx, x] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] && RationalFunctionQ[RFx, x] && NeQ[b*c - a*d, 0] && Not[MatchQ[RFx, u_.*(a + b*x)^m_.*(c + d*x)^n_.",
     "rulenumber": 0,
            "lhs": "Int[RFx_.*Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.], x_Symbol]",
        "comment": false,
         "givens": "IntegersQ[m, n]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "With[{u=IntHide[RFx,x]}, u*Log[e*(f*(a+b*x)^p*(c+d*x)^q)^r] - b*p*r*Int[u/(a+b*x),x] -  d*q*r*Int[u/(c+d*x),x] /; NonsumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[RFx_*Log[e_.*(f_.*(a_.+b_.*x_)^p_.*(c_.+d_.*x_)^q_.)^r_.],x_ Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e,f,p,q,r},x] && RationalFunctionQ[RFx,x] &&  NeQ[b*c-a*d,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "With[{u = ExpandIntegrand[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s, RFx, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[RFx_*Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^s_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && RationalFunctionQ[RFx, x] && IGtQ[s, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "Unintegrable[RFx*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s, x]",
     "rulenumber": 0,
            "lhs": "Int[RFx_*Log[e_.*(f_.*(a_. + b_.*x_)^p_.*(c_. + d_.*x_)^q_.)^r_.]^s_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && RationalFunctionQ[RFx, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "Int[u*Log[e*(f*ExpandToSum[v, x]^p*ExpandToSum[w, x]^q)^r]^s, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*Log[e_.*(f_.*v_^p_.*w_^q_.)^r_.]^s_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{e, f, p, q, r, s}, x] && LinearQ[{v, w}, x] && Not[LinearMatchQ[{v, w}, x]] && AlgebraicFunctionQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": "Int[u*Log[e*(f*ExpandToSum[v + g*w, x]/ExpandToSum[w, x])^r]^s, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*Log[e_.*(f_.*(g_ + v_./w_))^r_.]^s_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{e, f, g, r, s}, x] && LinearQ[w, x] && (FreeQ[v, x] || LinearQ[v, x]) && AlgebraicFunctionQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
       "filename": "3.4 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m",
            "rhs": " 1/f*Subst[Int[Log[g*(h*Simp[-(b*e-a*f)/f+b*x/f,x]^p)^q]*Log[i*(j*Simp[ -(d*e-c*f)/f+d*x/f,x]^r)^s]/x,x],x,e+f*x]",
     "rulenumber": 0,
            "lhs": "Int[Log[g_.*(h_.*(a_.+b_.*x_)^p_.)^q_.]*Log[i_.*(j_.*(c_.+d_.*x_)^ r_.)^s_.]/(e_+f_.*x_),x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e,f,g,h,i,j,p,q,r,s},x] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "With[{w = DerivativeDivides[v, u*(1 - v), x]}, w*PolyLog[2, 1 - v]",
     "rulenumber": 0,
            "lhs": "Int[u_*Log[v_], x_Symbol]",
        "comment": false,
         "givens": "Not[FalseQ[w]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "With[{z = DerivativeDivides[v, w*(1 - v), x]}, z*(a + b*Log[u])*PolyLog[2, 1 - v] - b*Int[SimplifyIntegrand[z*PolyLog[2, 1 - v]*D[u, x]/u, x], x] /; Not[FalseQ[z]]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[u_])*Log[v_]*w_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "x*Log[c*Log[d*x^n]^p] - n*p*Int[1/Log[d*x^n], x]",
     "rulenumber": 0,
            "lhs": "Int[Log[c_.*Log[d_.*x_^n_.]^p_.], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "Log[d*x^n]*(a + b*Log[c*Log[d*x^n]^p])/n - b*p*Log[x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*Log[d_.*x_^n_.]^p_.])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "(e*x)^(m + 1)*(a + b*Log[c*Log[d*x^n]^p])/(e*(m + 1)) - b*n*p/(m + 1)*Int[(e*x)^m/Log[d*x^n], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_. + b_.*Log[c_.*Log[d_.*x_^n_.]^p_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "x*(a + b*Log[c*RFx^p])^n - b*n*p* Int[SimplifyIntegrand[ x*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x]/RFx, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*RFx_^p_.])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "Log[d + e*x]*(a + b*Log[c*RFx^p])^n/e - b*n*p/e* Int[Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x]/RFx, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Log[c_.*RFx_^p_.])^n_./(d_. + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "(d + e*x)^(m + 1)*(a + b*Log[c*RFx^p])^ n/(e*(m + 1)) - b*n*p/(e*(m + 1))* Int[SimplifyIntegrand[(d + e*x)^(m + 1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x]/RFx, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*(a_. + b_.*Log[c_.*RFx_^p_.])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "With[{u = IntHide[1/(d + e*x^2), x]}, u*Log[c*RFx^n] - n*Int[SimplifyIntegrand[u*D[RFx, x]/RFx, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Log[c_.*RFx_^n_.]/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e, n}, x] && RationalFunctionQ[RFx, x] && Not[PolynomialQ[RFx, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "With[{u = IntHide[1/Qx, x]}, u*Log[c*Px^n] - n*Int[SimplifyIntegrand[u*D[Px, x]/Px, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Log[c_.*Px_^n_.]/Qx_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, n}, x] && QuadraticQ[{Qx, Px}, x] && EqQ[D[Px/Qx, x], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "With[{u = ExpandIntegrand[(a + b*Log[c*RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[RGx_*(a_. + b_.*Log[c_.*RFx_^p_.])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalFunctionQ[RGx, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "With[{u = ExpandIntegrand[RGx*(a + b*Log[c*RFx^p])^n, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[RGx_*(a_. + b_.*Log[c_.*RFx_^p_.])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalFunctionQ[RGx, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "With[{lst = SubstForFractionalPowerOfLinear[RFx*(a + b*Log[u]), x]}, lst[[2]]*lst[[4]]* Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])] /; Not[FalseQ[lst]]]",
     "rulenumber": 0,
            "lhs": "Int[RFx_*(a_. + b_.*Log[u_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && RationalFunctionQ[RFx, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "-(f + g*x)^m* PolyLog[2, -e*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F]) + g*m/(b*c*n*Log[F])* Int[(f + g*x)^(m - 1)*PolyLog[2, -e*(F^(c*(a + b*x)))^n], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^m_.*Log[1 + e_.*(F_^(c_.*(a_. + b_.*x_)))^n_.], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "(f + g*x)^(m + 1)* Log[d + e*(F^(c*(a + b*x)))^n]/(g*(m + 1)) - (f + g*x)^(m + 1)* Log[1 + e/d*(F^(c*(a + b*x)))^n]/(g*(m + 1)) + Int[(f + g*x)^m*Log[1 + e/d*(F^(c*(a + b*x)))^n], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^m_.*Log[d_ + e_.*(F_^(c_.*(a_. + b_.*x_)))^n_.], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && NeQ[d, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "x*Log[d + e*x + f*Sqrt[a + b*x + c*x^2]] + f^2*(b^2 - 4*a*c)/2* Int[x/((2*d*e - b*f^2)*(a + b*x + c*x^2) - f*(b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[Log[d_. + e_.*x_ + f_.*Sqrt[a_. + b_.*x_ + c_.*x_^2]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[e^2 - c*f^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "x*Log[d + e*x + f*Sqrt[a + c*x^2]] - a*c*f^2* Int[x/(d*e*(a + c*x^2) + f*(a*e - c*d*x)*Sqrt[a + c*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[Log[d_. + e_.*x_ + f_.*Sqrt[a_. + c_.*x_^2]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f}, x] && EqQ[e^2 - c*f^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "(g*x)^(m + 1)* Log[d + e*x + f*Sqrt[a + b*x + c*x^2]]/(g*(m + 1)) + f^2*(b^2 - 4*a*c)/(2*g*(m + 1))* Int[(g*x)^(m + 1)/((2*d*e - b*f^2)*(a + b*x + c*x^2) - f*(b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^m_.* Log[d_. + e_.*x_ + f_.*Sqrt[a_. + b_.*x_ + c_.*x_^2]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[e^2 - c*f^2, 0] && NeQ[m, -1] && IntegerQ[2*m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "(g*x)^(m + 1)* Log[d + e*x + f*Sqrt[a + c*x^2]]/(g*(m + 1)) - a*c*f^2/(g*(m + 1))* Int[(g*x)^(m + 1)/(d*e*(a + c*x^2) + f*(a*e - c*d*x)*Sqrt[a + c*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*x_)^m_.*Log[d_. + e_.*x_ + f_.*Sqrt[a_. + c_.*x_^2]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, e, f, g, m}, x] && EqQ[e^2 - c*f^2, 0] && NeQ[m, -1] && IntegerQ[2*m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "Int[v*Log[d + e*x + f*Sqrt[ExpandToSum[u, x]]], x] /; FreeQ[{d, e, f}, x] && QuadraticQ[u, x] && Not[QuadraticMatchQ[u, x]] && (EqQ[v, 1] || MatchQ[v, (g_.*x)^m_.",
     "rulenumber": 0,
            "lhs": "Int[v_.*Log[d_. + e_.*x_ + f_.*Sqrt[u_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{g, m}, x]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "Log[a*x^m + b*Log[c*x^n]^q]/(b*n*q) - a*m/(b*n*q)*Int[x^(m - 1)/(a*x^m + b*Log[c*x^n]^q), x]",
     "rulenumber": 0,
            "lhs": "Int[Log[c_.*x_^n_.]^r_./(x_*(a_.*x_^m_. + b_.*Log[c_.*x_^n_.]^q_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n, q, r}, x] && EqQ[r, q - 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "Int[ExpandIntegrand[Log[c*x^n]^r/x, (a*x^m + b*Log[c*x^n]^q)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Log[c_.*x_^n_.]^r_.*(a_.*x_^m_. + b_.*Log[c_.*x_^n_.]^q_)^p_./x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n, p, q, r}, x] && EqQ[r, q - 1] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "(a*x^m + b*Log[c*x^n]^q)^(p + 1)/(b*n*q*(p + 1)) - a*m/(b*n*q)*Int[x^(m - 1)*(a*x^m + b*Log[c*x^n]^q)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[Log[c_.*x_^n_.]^r_.*(a_.*x_^m_. + b_.*Log[c_.*x_^n_.]^q_)^p_./x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n, p, q, r}, x] && EqQ[r, q - 1] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "e*Log[a*x^m + b*Log[c*x^n]^q]/(b*n*q)",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_^m_. + e_.*Log[c_.*x_^n_.]^r_.)/(x_*(a_.*x_^m_. + b_.*Log[c_.*x_^n_.]^q_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, q, r}, x] && EqQ[r, q - 1] && EqQ[a*e*m - b*d*n*q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "e*Log[a*x^m + b*Log[c*x^n]^q]/(b*n*q) + Int[u/(x*(a*x^m + b*Log[c*x^n]^q)), x]",
     "rulenumber": 0,
            "lhs": "Int[(u_ + d_.*x_^m_. + e_.*Log[c_.*x_^n_.]^r_.)/(x_*(a_.*x_^m_. + b_.*Log[c_.*x_^n_.]^q_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, q, r}, x] && EqQ[r, q - 1] && EqQ[a*e*m - b*d*n*q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "e*Log[a*x^m + b*Log[c*x^n]^q]/(b*n*q) - (a*e*m - b*d*n*q)/(b*n*q)* Int[x^(m - 1)/(a*x^m + b*Log[c*x^n]^q), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_^m_. + e_.*Log[c_.*x_^n_.]^r_.)/(x_*(a_.*x_^m_. + b_.*Log[c_.*x_^n_.]^q_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, q, r}, x] && EqQ[r, q - 1] && NeQ[a*e*m - b*d*n*q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "e*(a*x^m + b*Log[c*x^n]^q)^(p + 1)/(b*n*q*(p + 1))",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_^m_. + e_.*Log[c_.*x_^n_.]^r_.)*(a_.*x_^m_. + b_.*Log[c_.*x_^n_.]^q_)^ p_./x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p, q, r}, x] && EqQ[r, q - 1] && NeQ[p, -1] && EqQ[a*e*m - b*d*n*q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "e*(a*x^m + b*Log[c*x^n]^q)^(p + 1)/(b*n*q*(p + 1)) - (a*e*m - b*d*n*q)/(b*n*q)* Int[x^(m - 1)*(a*x^m + b*Log[c*x^n]^q)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_^m_. + e_.*Log[c_.*x_^n_.]^r_.)*(a_.*x_^m_. + b_.*Log[c_.*x_^n_.]^q_)^ p_./x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p, q, r}, x] && EqQ[r, q - 1] && NeQ[p, -1] && NeQ[a*e*m - b*d*n*q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "d*Log[c*x^n]/(a*n*(a*x^m + b*Log[c*x^n]^q))",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_^m_. + e_.*x_^m_.*Log[c_.*x_^n_.] + f_.*Log[c_.*x_^n_.]^ q_.)/(x_*(a_.*x_^m_. + b_.*Log[c_.*x_^n_.]^q_)^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, q}, x] && EqQ[e*n + d*m, 0] && EqQ[a*f + b*d*(q - 1), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "-e* Log[c*x^n]/(a*(a*x + b*Log[c*x^n]^q)) + (d + e*n)/a* Int[1/(x*(a*x + b*Log[c*x^n]^q)), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*Log[c_.*x_^n_.])/(a_.*x_ + b_.*Log[c_.*x_^n_.]^q_)^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[d + e*n*q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "x*Log[u] - Int[SimplifyIntegrand[x*D[u, x]/u, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Log[u_], x_Symbol]",
        "comment": false,
         "givens": "InverseFunctionFreeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "x*Log[u] - Int[SimplifyIntegrand[x*Simplify[D[u, x]/u], x], x]",
     "rulenumber": 0,
            "lhs": "Int[Log[u_], x_Symbol]",
        "comment": false,
         "givens": "ProductQ[u]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "Log[a + b*x]*Log[u]/b - 1/b*Int[SimplifyIntegrand[Log[a + b*x]*D[u, x]/u, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Log[u_]/(a_. + b_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && RationalFunctionQ[D[u, x]/u, x] && (NeQ[a, 0] || Not[BinomialQ[u, x] && EqQ[BinomialDegree[u, x]^2, 1]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "(a + b*x)^(m + 1)*Log[u]/(b*(m + 1)) - 1/(b*(m + 1))* Int[SimplifyIntegrand[(a + b*x)^(m + 1)*D[u, x]/u, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*x_)^m_.*Log[u_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m}, x] && InverseFunctionFreeQ[u, x] && NeQ[m, -1] (* && Not[FunctionOfQ[x^(m+1),u,x]] && FalseQ[PowerVariableExpn[u,m+1, x]] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "With[{v = IntHide[1/Qx, x]}, v*Log[u] - Int[SimplifyIntegrand[v*D[u, x]/u, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Log[u_]/Qx_, x_Symbol]",
        "comment": false,
         "givens": "QuadraticQ[Qx, x] && InverseFunctionFreeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "u^(a*x)/a - Int[SimplifyIntegrand[x*u^(a*x - 1)*D[u, x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[u_^(a_.*x_)*Log[u_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[a, x] && InverseFunctionFreeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[w*D[u, x]/u, x], x] /; InverseFunctionFreeQ[w, x]]",
     "rulenumber": 0,
            "lhs": "Int[v_*Log[u_], x_Symbol]",
        "comment": false,
         "givens": "InverseFunctionFreeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[w*Simplify[D[u, x]/u], x], x] /; InverseFunctionFreeQ[w, x]]",
     "rulenumber": 0,
            "lhs": "Int[v_*Log[u_], x_Symbol]",
        "comment": false,
         "givens": "ProductQ[u]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "x*Log[v]*Log[w] - Int[SimplifyIntegrand[x*Log[w]*D[v, x]/v, x], x] - Int[SimplifyIntegrand[x*Log[v]*D[w, x]/w, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Log[v_]*Log[w_], x_Symbol]",
        "comment": false,
         "givens": "InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[w, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "With[{z = IntHide[u, x]}, Dist[Log[v]*Log[w], z, x] - Int[SimplifyIntegrand[z*Log[w]*D[v, x]/v, x], x] - Int[SimplifyIntegrand[z*Log[v]*D[w, x]/w, x], x] /; InverseFunctionFreeQ[z, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*Log[v_]*Log[w_], x_Symbol]",
        "comment": false,
         "givens": "InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[w, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "Int[u^(a*Log[f]), x]",
     "rulenumber": 0,
            "lhs": "Int[f_^(a_.*Log[u_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, f}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "With[{lst=FunctionOfLog[u,x]}, 1/lst[[3]]*Subst[Int[lst[[1]],x],x,Log[lst[[2]]]] /;  Not[FalseQ[lst]]]",
     "rulenumber": 0,
            "lhs": "Int[u_/x_,x_Symbol] := With[{lst=FunctionOfLog[u,x]}, ShowStep[\"\",\"Int[F[Log[a*x^n]]/x,x]\",\"Subst[Int[F[x],x],x,Log[a*x^n] ]/n\",Hold[ 1/lst[[3]]*Subst[Int[lst[[1]],x],x,Log[lst[[2]]]]]] /;  Not[FalseQ[lst]]] /; SimplifyFlag && NonsumQ[u], Int[u_/x_,x_Symbol]",
        "comment": false,
         "givens": "NonsumQ[u]] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "With[{lst = FunctionOfLog[Cancel[x*u], x]}, 1/lst[[3]]*Subst[Int[lst[[1]], x], x, Log[lst[[2]]]] /; Not[FalseQ[lst]]]",
     "rulenumber": 0,
            "lhs": "Int[u_, x_Symbol] := With[{lst = FunctionOfLog[Cancel[x*u], x]}, ShowStep[\"\", \"Int[F[Log[a*x^n]]/x,x]\", \"Subst[Int[F[x],x],x,Log[a*x^n]]/n\", Hold[ 1/lst[[3]]* Subst[Int[lst[[1]], x], x, Log[lst[[2]]]]]] /; Not[FalseQ[lst]]] /; SimplifyFlag && NonsumQ[u], Int[u_, x_Symbol]",
        "comment": false,
         "givens": "NonsumQ[u]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/3 Logarithms/3.5 Miscellaneous logarithms.m",
       "filename": "3.5 Miscellaneous logarithms.m",
            "rhs": "Int[u*x^(p*r)*(a*x^(m - r) + b*Log[c*x^n]^q)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_.*x_^m_. + b_.*x_^r_.*Log[c_.*x_^n_.]^q_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n, p, q, r}, x] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m",
       "filename": "4.1.0.1 (a sin)^m (b trg)^n.m",
            "rhs": "Int[DeactivateTrig[u, x], x]",
     "rulenumber": 0,
            "lhs": "Int[u_, x_Symbol] := Int[DeactivateTrig[u, x], x] /; SimplifyFlag && FunctionOfTrigOfLinearQ[u, x], Int[u_, x_Symbol]",
        "comment": false,
         "givens": "FunctionOfTrigOfLinearQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m",
       "filename": "4.1.0.1 (a sin)^m (b trg)^n.m",
            "rhs": "(a*Sin[e + f*x])^(m + 1)*(b*Cos[e + f*x])^(n + 1)/(a*b*f*(m + 1))",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_. + f_.*x_])^m_.*(b_.*cos[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m",
       "filename": "4.1.0.1 (a sin)^m (b trg)^n.m",
            "rhs": "1/(a*f)*Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_. + f_.*x_])^m_.*cos[e_. + f_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && Not[IntegerQ[(m - 1)/2] && LtQ[0, m, n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m",
       "filename": "4.1.0.1 (a sin)^m (b trg)^n.m",
            "rhs": "-1/(a*f)* Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*cos[e_. + f_.*x_])^m_.*sin[e_. + f_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && Not[IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m",
       "filename": "4.1.0.1 (a sin)^m (b trg)^n.m",
            "rhs": "-a*(a*Sin[e + f*x])^(m - 1)*(b*Cos[e + f*x])^(n + 1)/(b*f*(n + 1)) + a^2*(m - 1)/(b^2*(n + 1))* Int[(a*Sin[e + f*x])^(m - 2)*(b*Cos[e + f*x])^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*cos[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && GtQ[m, 1] && LtQ[n, -1] && (IntegersQ[2*m, 2*n] || EqQ[m + n, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m",
       "filename": "4.1.0.1 (a sin)^m (b trg)^n.m",
            "rhs": "a*(a*Cos[e + f*x])^(m - 1)*(b*Sin[e + f*x])^(n + 1)/(b*f*(n + 1)) + a^2*(m - 1)/(b^2*(n + 1))* Int[(a*Cos[e + f*x])^(m - 2)*(b*Sin[e + f*x])^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*cos[e_. + f_.*x_])^m_*(b_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && GtQ[m, 1] && LtQ[n, -1] && (IntegersQ[2*m, 2*n] || EqQ[m + n, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m",
       "filename": "4.1.0.1 (a sin)^m (b trg)^n.m",
            "rhs": "-a*(b*Cos[e + f*x])^(n + 1)*(a*Sin[e + f*x])^(m - 1)/(b*f*(m + n)) + a^2*(m - 1)/(m + n)* Int[(b*Cos[e + f*x])^n*(a*Sin[e + f*x])^(m - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*cos[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m",
       "filename": "4.1.0.1 (a sin)^m (b trg)^n.m",
            "rhs": "a*(b*Sin[e + f*x])^(n + 1)*(a*Cos[e + f*x])^(m - 1)/(b*f*(m + n)) + a^2*(m - 1)/(m + n)* Int[(b*Sin[e + f*x])^n*(a*Cos[e + f*x])^(m - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*cos[e_. + f_.*x_])^m_*(b_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m",
       "filename": "4.1.0.1 (a sin)^m (b trg)^n.m",
            "rhs": "(b*Cos[e + f*x])^(n + 1)*(a*Sin[e + f*x])^(m + 1)/(a*b*f*(m + 1)) + (m + n + 2)/(a^2*(m + 1))* Int[(b*Cos[e + f*x])^n*(a*Sin[e + f*x])^(m + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*cos[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, n}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m",
       "filename": "4.1.0.1 (a sin)^m (b trg)^n.m",
            "rhs": "-(b*Sin[e + f*x])^(n + 1)*(a*Cos[e + f*x])^(m + 1)/(a*b*f*(m + 1)) + (m + n + 2)/(a^2*(m + 1))* Int[(b*Sin[e + f*x])^n*(a*Cos[e + f*x])^(m + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*cos[e_. + f_.*x_])^m_*(b_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, n}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m",
       "filename": "4.1.0.1 (a sin)^m (b trg)^n.m",
            "rhs": "Sqrt[a*Sin[e + f*x]]*Sqrt[b*Cos[e + f*x]]/Sqrt[Sin[2*e + 2*f*x]]* Int[Sqrt[Sin[2*e + 2*f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_.*sin[e_. + f_.*x_]]*Sqrt[b_.*cos[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m",
       "filename": "4.1.0.1 (a sin)^m (b trg)^n.m",
            "rhs": "Sqrt[Sin[2*e + 2*f*x]]/(Sqrt[a*Sin[e + f*x]]*Sqrt[b*Cos[e + f*x]])* Int[1/Sqrt[Sin[2*e + 2*f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_.*sin[e_. + f_.*x_]]*Sqrt[b_.*cos[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m",
       "filename": "4.1.0.1 (a sin)^m (b trg)^n.m",
            "rhs": "  (a*Sin[e+f*x])^m*(b*Cos[e+f*x])^n/(a*Tan[e+f*x])^m*Int[(a*Tan[e+f*x] )^m,x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_.+f_.*x_])^m_*(b_.*cos[e_.+f_.*x_])^n_,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,e,f,m,n},x] && EqQ[m+n,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m",
       "filename": "4.1.0.1 (a sin)^m (b trg)^n.m",
            "rhs": "With[{k = Denominator[m]}, k*a*b/f* Subst[Int[x^(k*(m + 1) - 1)/(a^2 + b^2*x^(2*k)), x], x, (a*Sin[e + f*x])^(1/k)/(b*Cos[e + f*x])^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*cos[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] && LtQ[m, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m",
       "filename": "4.1.0.1 (a sin)^m (b trg)^n.m",
            "rhs": "With[{k = Denominator[m]}, -k*a*b/f* Subst[Int[x^(k*(m + 1) - 1)/(a^2 + b^2*x^(2*k)), x], x, (a*Cos[e + f*x])^(1/k)/(b*Sin[e + f*x])^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*cos[e_. + f_.*x_])^m_*(b_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] && LtQ[m, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m",
       "filename": "4.1.0.1 (a sin)^m (b trg)^n.m",
            "rhs": "  b^(2*IntPart[(n-1)/2]+1)*(b*Cos[e+f*x])^(2*FracPart[(n-1)/2])/(a*f*( Cos[e+f*x]^2)^FracPart[(n-1)/2])* Subst[Int[x^m*(1-x^2/a^2)^((n-1)/2),x],x,a*Sin[e+f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_.+f_.*x_])^m_*(b_.*cos[e_.+f_.*x_])^n_,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,e,f,m,n},x] && (RationalQ[n] || Not[RationalQ[m]] &&  (EqQ[b,1] || NeQ[a,1])) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m",
       "filename": "4.1.0.1 (a sin)^m (b trg)^n.m",
            "rhs": "  -b^(2*IntPart[(n-1)/2]+1)*(b*Sin[e+f*x])^(2*FracPart[(n-1)/2])/(a*f* (Sin[e+f*x]^2)^FracPart[(n-1)/2])* Subst[Int[x^m*(1-x^2/a^2)^((n-1)/2),x],x,a*Cos[e+f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*cos[e_.+f_.*x_])^m_*(b_.*sin[e_.+f_.*x_])^n_,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,e,f,m,n},x] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m",
       "filename": "4.1.0.1 (a sin)^m (b trg)^n.m",
            "rhs": "-b^(2*IntPart[(n - 1)/2] + 1)*(b*Sin[e + f*x])^(2* FracPart[(n - 1)/2])*(a*Cos[e + f*x])^(m + 1)/(a* f*(m + 1)*(Sin[e + f*x]^2)^FracPart[(n - 1)/2])* Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, Cos[e + f*x]^2]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*cos[e_. + f_.*x_])^m_*(b_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m, n}, x] && SimplerQ[n, m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m",
       "filename": "4.1.0.1 (a sin)^m (b trg)^n.m",
            "rhs": "b^(2*IntPart[(n - 1)/2] + 1)*(b*Cos[e + f*x])^(2* FracPart[(n - 1)/2])*(a*Sin[e + f*x])^(m + 1)/(a* f*(m + 1)*(Cos[e + f*x]^2)^FracPart[(n - 1)/2])* Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, Sin[e + f*x]^2]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*cos[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m",
       "filename": "4.1.0.1 (a sin)^m (b trg)^n.m",
            "rhs": "b*(a*Sin[e + f*x])^(m + 1)*(b*Sec[e + f*x])^(n - 1)/(a*f*(m + 1))",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_. + f_.*x_])^m_.*(b_.*sec[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m, n}, x] && EqQ[m - n + 2, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m",
       "filename": "4.1.0.1 (a sin)^m (b trg)^n.m",
            "rhs": "a*b*(a*Sin[e + f*x])^(m - 1)*(b*Sec[e + f*x])^(n - 1)/(f*(n - 1)) - a^2*b^2*(m - 1)/(n - 1)* Int[(a*Sin[e + f*x])^(m - 2)*(b*Sec[e + f*x])^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && GtQ[n, 1] && GtQ[m, 1] && IntegersQ[2*m, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m",
       "filename": "4.1.0.1 (a sin)^m (b trg)^n.m",
            "rhs": "(a*Sin[e + f*x])^(m + 1)*(b*Sec[e + f*x])^(n + 1)/(a*b*f*(m - n)) - (n + 1)/(b^2*(m - n))* Int[(a*Sin[e + f*x])^m*(b*Sec[e + f*x])^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && IntegersQ[2*m, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m",
       "filename": "4.1.0.1 (a sin)^m (b trg)^n.m",
            "rhs": "(a*Sin[e + f*x])^(m + 1)*(b*Sec[e + f*x])^(n + 1)/(a*b*f*(m + 1)) - (n + 1)/(a^2*b^2*(m + 1))* Int[(a*Sin[e + f*x])^(m + 2)*(b*Sec[e + f*x])^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && LtQ[n, -1] && LtQ[m, -1] && IntegersQ[2*m, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m",
       "filename": "4.1.0.1 (a sin)^m (b trg)^n.m",
            "rhs": "(a*Sin[e + f*x])^(m + 1)*(b*Sec[e + f*x])^(n + 1)/(a*b*f*(m - n)) - (n + 1)/(b^2*(m - n))* Int[(a*Sin[e + f*x])^m*(b*Sec[e + f*x])^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m}, x] && LtQ[n, -1] && NeQ[m - n, 0] && IntegersQ[2*m, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m",
       "filename": "4.1.0.1 (a sin)^m (b trg)^n.m",
            "rhs": "-a* b*(a*Sin[e + f*x])^(m - 1)*(b*Sec[e + f*x])^(n - 1)/(f*(m - n)) + a^2*(m - 1)/(m - n)* Int[(a*Sin[e + f*x])^(m - 2)*(b*Sec[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m - n, 0] && IntegersQ[2*m, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m",
       "filename": "4.1.0.1 (a sin)^m (b trg)^n.m",
            "rhs": "b*(a*Sin[e + f*x])^(m + 1)*(b*Sec[e + f*x])^(n - 1)/(a*f*(m + 1)) + (m - n + 2)/(a^2*(m + 1))* Int[(a*Sin[e + f*x])^(m + 2)*(b*Sec[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, n}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m",
       "filename": "4.1.0.1 (a sin)^m (b trg)^n.m",
            "rhs": "(b*Cos[e + f*x])^n*(b*Sec[e + f*x])^n* Int[(a*Sin[e + f*x])^m/(b*Cos[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m, n}, x] && IntegerQ[m - 1/2] && IntegerQ[n - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m",
       "filename": "4.1.0.1 (a sin)^m (b trg)^n.m",
            "rhs": "1/b^2*(b*Cos[e + f*x])^(n + 1)*(b*Sec[e + f*x])^(n + 1)* Int[(a*Sin[e + f*x])^m/(b*Cos[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m, n}, x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && LtQ[n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m",
       "filename": "4.1.0.1 (a sin)^m (b trg)^n.m",
            "rhs": "b^2*(b*Cos[e + f*x])^(n - 1)*(b*Sec[e + f*x])^(n - 1)* Int[(a*Sin[e + f*x])^m/(b*Cos[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m, n}, x] && Not[IntegerQ[m]] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.1 (a sin)^m (b trg)^n.m",
       "filename": "4.1.0.1 (a sin)^m (b trg)^n.m",
            "rhs": "(a*b)^IntPart[n]*(a*Sin[e + f*x])^ FracPart[n]*(b*Csc[e + f*x])^FracPart[n]* Int[(a*Sin[e + f*x])^(m - n), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_. + f_.*x_])^m_.*(b_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m, n}, x] && Not[IntegerQ[m]] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m",
       "filename": "4.1.0.2 (a trg)^m (b tan)^n.m",
            "rhs": "-b*(a*Sin[e + f*x])^ m*(b*Tan[e + f*x])^(n - 1)/(f*m)",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n - 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m",
       "filename": "4.1.0.2 (a trg)^m (b tan)^n.m",
            "rhs": "-1/f* Subst[Int[(1 - x^2)^((m + n - 1)/2)/x^n, x], x, Cos[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[sin[e_. + f_.*x_]^m_.*tan[e_. + f_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n - 1)/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m",
       "filename": "4.1.0.2 (a trg)^m (b tan)^n.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, b*ff/f* Subst[Int[(ff*x)^(m + n)/(b^2 + ff^2*x^2)^(m/2 + 1), x], x, b*Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[sin[e_. + f_.*x_]^m_*(b_.*tan[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, e, f, n}, x] && IntegerQ[m/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m",
       "filename": "4.1.0.2 (a trg)^m (b tan)^n.m",
            "rhs": "With[{ff = FreeFactors[Sin[e + f*x], x]}, ff/f* Subst[Int[(ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, a*Sin[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_. + f_.*x_])^m_.*tan[e_. + f_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m",
       "filename": "4.1.0.2 (a trg)^m (b tan)^n.m",
            "rhs": "b*(a*Sin[e + f*x])^(m + 2)*(b*Tan[e + f*x])^(n - 1)/(a^2*f*(n - 1)) - b^2*(m + 2)/(a^2*(n - 1))* Int[(a*Sin[e + f*x])^(m + 2)*(b*Tan[e + f*x])^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && GtQ[n, 1] && (LtQ[m, -1] || EqQ[m, -1] && EqQ[n, 3/2]) && IntegersQ[2*m, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m",
       "filename": "4.1.0.2 (a trg)^m (b tan)^n.m",
            "rhs": "b*(a*Sin[e + f*x])^m*(b*Tan[e + f*x])^(n - 1)/(f*(n - 1)) - b^2*(m + n - 1)/(n - 1)* Int[(a*Sin[e + f*x])^m*(b*Tan[e + f*x])^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_. + f_.*x_])^m_.*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && IntegersQ[2*m, 2*n] && Not[GtQ[m, 1] && Not[IntegerQ[(m - 1)/2]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m",
       "filename": "4.1.0.2 (a trg)^m (b tan)^n.m",
            "rhs": "2*Sqrt[a*Sin[e + f*x]]/(b*f*Sqrt[b*Tan[e + f*x]]) + a^2/b^2*Int[Sqrt[b*Tan[e + f*x]]/(a*Sin[e + f*x])^(3/2), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_.*sin[e_. + f_.*x_]]/(b_.*tan[e_. + f_.*x_])^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m",
       "filename": "4.1.0.2 (a trg)^m (b tan)^n.m",
            "rhs": "(a*Sin[e + f*x])^ m*(b*Tan[e + f*x])^(n + 1)/(b*f*m) - a^2*(n + 1)/(b^2*m)* Int[(a*Sin[e + f*x])^(m - 2)*(b*Tan[e + f*x])^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && LtQ[n, -1] && GtQ[m, 1] && IntegersQ[2*m, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m",
       "filename": "4.1.0.2 (a trg)^m (b tan)^n.m",
            "rhs": "(a*Sin[e + f*x])^ m*(b*Tan[e + f*x])^(n + 1)/(b*f*(m + n + 1)) - (n + 1)/(b^2*(m + n + 1))* Int[(a*Sin[e + f*x])^m*(b*Tan[e + f*x])^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_. + f_.*x_])^m_.*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m}, x] && LtQ[n, -1] && NeQ[m + n + 1, 0] && IntegersQ[2*m, 2*n] && Not[EqQ[n, -3/2] && EqQ[m, 1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m",
       "filename": "4.1.0.2 (a trg)^m (b tan)^n.m",
            "rhs": "-b*(a*Sin[e + f*x])^ m*(b*Tan[e + f*x])^(n - 1)/(f*m) + a^2*(m + n - 1)/m* Int[(a*Sin[e + f*x])^(m - 2)*(b*Tan[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_. + f_.*x_])^m_.*(b_.*tan[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, n}, x] && (GtQ[m, 1] || EqQ[m, 1] && EqQ[n, 1/2]) && IntegersQ[2*m, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m",
       "filename": "4.1.0.2 (a trg)^m (b tan)^n.m",
            "rhs": "b*(a*Sin[e + f*x])^(m + 2)*(b*Tan[e + f*x])^(n - 1)/(a^2* f*(m + n + 1)) + (m + 2)/(a^2*(m + n + 1))* Int[(a*Sin[e + f*x])^(m + 2)*(b*Tan[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_. + f_.*x_])^m_*(b_.*tan[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, n}, x] && LtQ[m, -1] && NeQ[m + n + 1, 0] && IntegersQ[2*m, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m",
       "filename": "4.1.0.2 (a trg)^m (b tan)^n.m",
            "rhs": "1/a^n*Int[(a*Sin[e + f*x])^(m + n)/Cos[e + f*x]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_. + f_.*x_])^m_*tan[e_. + f_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, e, f, m}, x] && IntegerQ[n] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m",
       "filename": "4.1.0.2 (a trg)^m (b tan)^n.m",
            "rhs": "Cos[e + f*x]^n*(b*Tan[e + f*x])^n/(a*Sin[e + f*x])^n* Int[(a*Sin[e + f*x])^(m + n)/Cos[e + f*x]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_. + f_.*x_])^m_.*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m, n}, x] && Not[IntegerQ[n]] && (ILtQ[m, 0] || EqQ[m, 1] && EqQ[n, -1/2] || IntegersQ[m - 1/2, n - 1/2])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m",
       "filename": "4.1.0.2 (a trg)^m (b tan)^n.m",
            "rhs": "a*Cos[e + f*x]^(n + 1)*(b*Tan[e + f*x])^(n + 1)/(b*(a*Sin[e + f*x])^(n + 1))* Int[(a*Sin[e + f*x])^(m + n)/Cos[e + f*x]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sin[e_. + f_.*x_])^m_.*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m, n}, x] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m",
       "filename": "4.1.0.2 (a trg)^m (b tan)^n.m",
            "rhs": "(a*Cos[e + f*x])^FracPart[m]*(Sec[e + f*x]/a)^ FracPart[m]*Int[(b*Tan[e + f*x])^n/(Sec[e + f*x]/a)^m, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*cos[e_. + f_.*x_])^m_*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m, n}, x] && Not[IntegerQ[m]] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m",
       "filename": "4.1.0.2 (a trg)^m (b tan)^n.m",
            "rhs": "(a*Cot[e + f*x])^m*(b*Tan[e + f*x])^m* Int[(b*Tan[e + f*x])^(n - m), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*cot[e_. + f_.*x_])^m_*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m, n}, x] && Not[IntegerQ[m]] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m",
       "filename": "4.1.0.2 (a trg)^m (b tan)^n.m",
            "rhs": "-(a*Sec[e + f*x])^ m*(b*Tan[e + f*x])^(n + 1)/(b*f*m)",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sec[e_. + f_.*x_])^m_.*(b_.*tan[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m",
       "filename": "4.1.0.2 (a trg)^m (b tan)^n.m",
            "rhs": "a/f*Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sec[e_. + f_.*x_])^m_.*(b_.*tan[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && Not[IntegerQ[m/2] && LtQ[0, m, n + 1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m",
       "filename": "4.1.0.2 (a trg)^m (b tan)^n.m",
            "rhs": "1/f*Subst[Int[(b*x)^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[sec[e_. + f_.*x_]^m_*(b_.*tan[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] && Not[IntegerQ[(n - 1)/2] && LtQ[0, n, m - 1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m",
       "filename": "4.1.0.2 (a trg)^m (b tan)^n.m",
            "rhs": "a^2*(a*Sec[e + f*x])^(m - 2)*(b*Tan[e + f*x])^(n + 1)/(b*f*(n + 1)) - a^2*(m - 2)/(b^2*(n + 1))* Int[(a*Sec[e + f*x])^(m - 2)*(b*Tan[e + f*x])^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sec[e_. + f_.*x_])^m_.*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && LtQ[n, -1] && (GtQ[m, 1] || EqQ[m, 1] && EqQ[n, -3/2]) && IntegersQ[2*m, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m",
       "filename": "4.1.0.2 (a trg)^m (b tan)^n.m",
            "rhs": "(a*Sec[e + f*x])^ m*(b*Tan[e + f*x])^(n + 1)/(b*f*(n + 1)) - (m + n + 1)/(b^2*(n + 1))* Int[(a*Sec[e + f*x])^m*(b*Tan[e + f*x])^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sec[e_. + f_.*x_])^m_.*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m}, x] && LtQ[n, -1] && IntegersQ[2*m, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m",
       "filename": "4.1.0.2 (a trg)^m (b tan)^n.m",
            "rhs": "b*(a*Sec[e + f*x])^m*(b*Tan[e + f*x])^(n - 1)/(f*m) - b^2*(n - 1)/(a^2*m)* Int[(a*Sec[e + f*x])^(m + 2)*(b*Tan[e + f*x])^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sec[e_. + f_.*x_])^m_*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && GtQ[n, 1] && (LtQ[m, -1] || EqQ[m, -1] && EqQ[n, 3/2]) && IntegersQ[2*m, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m",
       "filename": "4.1.0.2 (a trg)^m (b tan)^n.m",
            "rhs": "b*(a*Sec[e + f*x])^m*(b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1)) - b^2*(n - 1)/(m + n - 1)* Int[(a*Sec[e + f*x])^m*(b*Tan[e + f*x])^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sec[e_. + f_.*x_])^m_.*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m",
       "filename": "4.1.0.2 (a trg)^m (b tan)^n.m",
            "rhs": "-(a*Sec[e + f*x])^ m*(b*Tan[e + f*x])^(n + 1)/(b*f*m) + (m + n + 1)/(a^2*m)* Int[(a*Sec[e + f*x])^(m + 2)*(b*Tan[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sec[e_. + f_.*x_])^m_*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, n}, x] && (LtQ[m, -1] || EqQ[m, -1] && EqQ[n, -1/2]) && IntegersQ[2*m, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m",
       "filename": "4.1.0.2 (a trg)^m (b tan)^n.m",
            "rhs": "a^2*(a*Sec[e + f*x])^(m - 2)*(b*Tan[e + f*x])^(n + 1)/(b* f*(m + n - 1)) + a^2*(m - 2)/(m + n - 1)* Int[(a*Sec[e + f*x])^(m - 2)*(b*Tan[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sec[e_. + f_.*x_])^m_.*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, n}, x] && (GtQ[m, 1] || EqQ[m, 1] && EqQ[n, 1/2]) && NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m",
       "filename": "4.1.0.2 (a trg)^m (b tan)^n.m",
            "rhs": "Sqrt[Sin[e + f*x]]/(Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]])* Int[1/(Sqrt[Cos[e + f*x]]*Sqrt[Sin[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[sec[e_. + f_.*x_]/Sqrt[b_.*tan[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, e, f}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m",
       "filename": "4.1.0.2 (a trg)^m (b tan)^n.m",
            "rhs": "Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]]/Sqrt[Sin[e + f*x]]* Int[Sqrt[Cos[e + f*x]]*Sqrt[Sin[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[b_.*tan[e_. + f_.*x_]]/sec[e_. + f_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, e, f}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m",
       "filename": "4.1.0.2 (a trg)^m (b tan)^n.m",
            "rhs": "a^(m + n)*(b*Tan[e + f*x])^n/((a*Sec[e + f*x])^n*(b*Sin[e + f*x])^n)* Int[(b*Sin[e + f*x])^n/Cos[e + f*x]^(m + n), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sec[e_. + f_.*x_])^m_*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m, n}, x] && IntegerQ[n + 1/2] && IntegerQ[m + 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m",
       "filename": "4.1.0.2 (a trg)^m (b tan)^n.m",
            "rhs": "(a*Sec[e + f*x])^ m*(b*Tan[e + f*x])^(n + 1)*(Cos[e + f*x]^2)^((m + n + 1)/2)/(b* f*(n + 1))* Hypergeometric2F1[(n + 1)/2, (m + n + 1)/2, (n + 3)/2, Sin[e + f*x]^2]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sec[e_. + f_.*x_])^m_.*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m, n}, x] && Not[IntegerQ[(n - 1)/2]] && Not[IntegerQ[m/2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.2 (a trg)^m (b tan)^n.m",
       "filename": "4.1.0.2 (a trg)^m (b tan)^n.m",
            "rhs": "(a*Csc[e + f*x])^FracPart[m]*(Sin[e + f*x]/a)^ FracPart[m]*Int[(b*Tan[e + f*x])^n/(Sin[e + f*x]/a)^m, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*csc[e_. + f_.*x_])^m_*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m, n}, x] && Not[IntegerQ[m]] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.3 (a csc)^m (b sec)^n.m",
       "filename": "4.1.0.3 (a csc)^m (b sec)^n.m",
            "rhs": "a*b*(a*Csc[e + f*x])^(m - 1)*(b*Sec[e + f*x])^(n - 1)/(f*(n - 1))",
     "rulenumber": 0,
            "lhs": "Int[(a_.*csc[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n - 2, 0] && NeQ[n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.3 (a csc)^m (b sec)^n.m",
       "filename": "4.1.0.3 (a csc)^m (b sec)^n.m",
            "rhs": "1/f*Subst[Int[(1 + x^2)^((m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]^m_.*sec[e_. + f_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.3 (a csc)^m (b sec)^n.m",
       "filename": "4.1.0.3 (a csc)^m (b sec)^n.m",
            "rhs": "-1/(f*a^n)* Subst[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*csc[e_. + f_.*x_])^m_*sec[e_. + f_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2] && Not[IntegerQ[(m + 1)/2] && LtQ[0, m, n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.3 (a csc)^m (b sec)^n.m",
       "filename": "4.1.0.3 (a csc)^m (b sec)^n.m",
            "rhs": "1/(f*a^n)* Subst[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sec[e_. + f_.*x_])^m_*csc[e_. + f_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2] && Not[IntegerQ[(m + 1)/2] && LtQ[0, m, n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.3 (a csc)^m (b sec)^n.m",
       "filename": "4.1.0.3 (a csc)^m (b sec)^n.m",
            "rhs": "-a*(a*Csc[e + f*x])^(m - 1)*(b*Sec[e + f*x])^(n + 1)/(f*b*(m - 1)) + a^2*(n + 1)/(b^2*(m - 1))* Int[(a*Csc[e + f*x])^(m - 2)*(b*Sec[e + f*x])^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*csc[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && GtQ[m, 1] && LtQ[n, -1] && IntegersQ[2*m, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.3 (a csc)^m (b sec)^n.m",
       "filename": "4.1.0.3 (a csc)^m (b sec)^n.m",
            "rhs": "b*(a*Csc[e + f*x])^(m + 1)*(b*Sec[e + f*x])^(n - 1)/(f*a*(n - 1)) + b^2*(m + 1)/(a^2*(n - 1))* Int[(a*Csc[e + f*x])^(m + 2)*(b*Sec[e + f*x])^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*csc[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && GtQ[n, 1] && LtQ[m, -1] && IntegersQ[2*m, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.3 (a csc)^m (b sec)^n.m",
       "filename": "4.1.0.3 (a csc)^m (b sec)^n.m",
            "rhs": "-a* b*(a*Csc[e + f*x])^(m - 1)*(b*Sec[e + f*x])^(n - 1)/(f*(m - 1)) + a^2*(m + n - 2)/(m - 1)* Int[(a*Csc[e + f*x])^(m - 2)*(b*Sec[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*csc[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && IntegersQ[2*m, 2*n] && Not[GtQ[n, m]]"
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       "filename": "4.1.0.3 (a csc)^m (b sec)^n.m",
            "rhs": "a*b*(a*Csc[e + f*x])^(m - 1)*(b*Sec[e + f*x])^(n - 1)/(f*(n - 1)) + b^2*(m + n - 2)/(n - 1)* Int[(a*Csc[e + f*x])^m*(b*Sec[e + f*x])^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*csc[e_. + f_.*x_])^m_.*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && IntegersQ[2*m, 2*n]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.3 (a csc)^m (b sec)^n.m",
       "filename": "4.1.0.3 (a csc)^m (b sec)^n.m",
            "rhs": "b*(a*Csc[e + f*x])^(m + 1)*(b*Sec[e + f*x])^(n - 1)/(a*f*(m + n)) + (m + 1)/(a^2*(m + n))* Int[(a*Csc[e + f*x])^(m + 2)*(b*Sec[e + f*x])^n, x]",
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            "lhs": "Int[(a_.*csc[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, n}, x] && LtQ[m, -1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.3 (a csc)^m (b sec)^n.m",
       "filename": "4.1.0.3 (a csc)^m (b sec)^n.m",
            "rhs": "-a*(a*Csc[e + f*x])^(m - 1)*(b*Sec[e + f*x])^(n + 1)/(b*f*(m + n)) + (n + 1)/(b^2*(m + n))* Int[(a*Csc[e + f*x])^m*(b*Sec[e + f*x])^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*csc[e_. + f_.*x_])^m_.*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m}, x] && LtQ[n, -1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]"
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       "filename": "4.1.0.3 (a csc)^m (b sec)^n.m",
            "rhs": "(a*Csc[e + f*x])^m*(b*Sec[e + f*x])^n/ Tan[e + f*x]^n*Int[Tan[e + f*x]^n, x]",
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            "lhs": "Int[(a_.*csc[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m, n}, x] && Not[IntegerQ[n]] && EqQ[m + n, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.3 (a csc)^m (b sec)^n.m",
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            "rhs": "(a*Csc[e + f*x])^m*(b*Sec[e + f*x])^ n*(a*Sin[e + f*x])^m*(b*Cos[e + f*x])^n* Int[(a*Sin[e + f*x])^(-m)*(b*Cos[e + f*x])^(-n), x]",
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            "lhs": "Int[(a_.*csc[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m, n}, x] && IntegerQ[m - 1/2] && IntegerQ[n - 1/2]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.3 (a csc)^m (b sec)^n.m",
       "filename": "4.1.0.3 (a csc)^m (b sec)^n.m",
            "rhs": "a^2/b^2*(a*Csc[e + f*x])^(m - 1)*(b*Sec[e + f*x])^(n + 1)*(a* Sin[e + f*x])^(m - 1)*(b*Cos[e + f*x])^(n + 1)* Int[(a*Sin[e + f*x])^(-m)*(b*Cos[e + f*x])^(-n), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*csc[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m, n}, x] && Not[SimplerQ[-m, -n]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.0.3 (a csc)^m (b sec)^n.m",
       "filename": "4.1.0.3 (a csc)^m (b sec)^n.m",
            "rhs": "a^2/b^2*(a*Sec[e + f*x])^(m - 1)*(b*Csc[e + f*x])^(n + 1)*(a* Cos[e + f*x])^(m - 1)*(b*Sin[e + f*x])^(n + 1)* Int[(a*Cos[e + f*x])^(-m)*(b*Sin[e + f*x])^(-n), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*sec[e_. + f_.*x_])^m_*(b_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m, n}, x]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
       "filename": "4.1.1.1 (a+b sin)^n.m",
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     "rulenumber": 0,
            "lhs": "Int[sin[c_. + d_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
       "filename": "4.1.1.1 (a+b sin)^n.m",
            "rhs": "x/2 - Sin[2*c + d*x]/(2*d)",
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            "lhs": "Int[sin[c_. + d_.*x_/2]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d}, x]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
       "filename": "4.1.1.1 (a+b sin)^n.m",
            "rhs": "(* -Cot[c+d*x]*(c*Sin[c+d*x])^n/(d*n) +  b^2*(n-1)/n*Int[(b*Sin[c+d*x])^(n-2),x] *) -b* Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1)/(d*n) + b^2*(n - 1)/n*Int[(b*Sin[c + d*x])^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
       "filename": "4.1.1.1 (a+b sin)^n.m",
            "rhs": "Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1)) + (n + 2)/(b^2*(n + 1))*Int[(b*Sin[c + d*x])^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
       "filename": "4.1.1.1 (a+b sin)^n.m",
            "rhs": "Sin[c + d*x]/d",
     "rulenumber": 0,
            "lhs": "Int[sin[c_. + Pi/2 + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d}, x]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
       "filename": "4.1.1.1 (a+b sin)^n.m",
            "rhs": "-Cos[c + d*x]/d",
     "rulenumber": 0,
            "lhs": "Int[sin[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d}, x]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
       "filename": "4.1.1.1 (a+b sin)^n.m",
            "rhs": "Int[Csc[c+d*x],x]",
     "rulenumber": 0,
            "lhs": "Int[1/sin[c_.+d_.*x_],x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{c,d},x] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
       "filename": "4.1.1.1 (a+b sin)^n.m",
            "rhs": "2/d*EllipticE[1/2*(c - Pi/2 + d*x), 2]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[sin[c_. + d_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d}, x]"
  },
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
       "filename": "4.1.1.1 (a+b sin)^n.m",
            "rhs": "Sqrt[b*Sin[c + d*x]]/Sqrt[Sin[c + d*x]]*Int[Sqrt[Sin[c + d*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[b_*sin[c_. + d_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
       "filename": "4.1.1.1 (a+b sin)^n.m",
            "rhs": "2/d*EllipticF[1/2*(c - Pi/2 + d*x), 2]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[sin[c_. + d_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
       "filename": "4.1.1.1 (a+b sin)^n.m",
            "rhs": "Sqrt[Sin[c + d*x]]/Sqrt[b*Sin[c + d*x]]* Int[1/Sqrt[Sin[c + d*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[b_*sin[c_. + d_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d}, x]"
  },
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
       "filename": "4.1.1.1 (a+b sin)^n.m",
            "rhs": " Cos[c+d*x]/(b*d*Sqrt[Cos[c+d*x]^2])*Subst[Int[x^n/Sqrt[1-x^2/b^2],x], x,b*Sin[c+d*x]]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*sin[c_.+d_.*x_])^n_,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b,c,d,n},x] && Not[IntegerQ[2*n]] *)"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
       "filename": "4.1.1.1 (a+b sin)^n.m",
            "rhs": "Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1)* Sqrt[Cos[c + d*x]^2])* Hypergeometric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, n}, x] && Not[IntegerQ[2*n]]"
  },
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
       "filename": "4.1.1.1 (a+b sin)^n.m",
            "rhs": "(2*a^2 + b^2)*x/2 - 2*a*b*Cos[c + d*x]/d - b^2*Cos[c + d*x]*Sin[c + d*x]/(2*d)",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[c_. + d_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x]"
  },
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
       "filename": "4.1.1.1 (a+b sin)^n.m",
            "rhs": "Int[ExpandTrig[(a + b*sin[c + d*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
       "filename": "4.1.1.1 (a+b sin)^n.m",
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            "lhs": "Int[Sqrt[a_ + b_.*sin[c_. + d_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
       "filename": "4.1.1.1 (a+b sin)^n.m",
            "rhs": "-b* Cos[c + d*x]*(a + b*Sin[c + d*x])^(n - 1)/(d*n) + a*(2*n - 1)/n*Int[(a + b*Sin[c + d*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && IGtQ[n - 1/2, 0]"
  },
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       "filename": "4.1.1.1 (a+b sin)^n.m",
            "rhs": "-Cos[c + d*x]/(d*(b + a*Sin[c + d*x]))",
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            "lhs": "Int[1/(a_ + b_.*sin[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
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            "rhs": "-2/d* Subst[Int[1/(2*a - x^2), x], x, b*Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]]]",
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            "lhs": "Int[1/Sqrt[a_ + b_.*sin[c_. + d_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
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            "rhs": "b*Cos[c + d*x]*(a + b*Sin[c + d*x])^n/(a*d*(2*n + 1)) + (n + 1)/(a*(2*n + 1))*Int[(a + b*Sin[c + d*x])^(n + 1), x]",
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            "lhs": "Int[(a_ + b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]"
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       "filename": "4.1.1.1 (a+b sin)^n.m",
            "rhs": " a^2*Cos[c+d*x]/(d*Sqrt[a+b*Sin[c+d*x]]*Sqrt[a-b*Sin[c+d*x]])*Subst[ Int[(a+b*x)^(n-1/2)/Sqrt[a-b*x],x],x,Sin[c+d*x]]",
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            "lhs": "Int[(a_+b_.*sin[c_.+d_.*x_])^n_,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,n},x] && EqQ[a^2-b^2,0] && Not[IntegerQ[2*n]] *)"
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            "rhs": "-2^(n + 1/2)*a^(n - 1/2)*b* Cos[c + d*x]/(d*Sqrt[a + b*Sin[c + d*x]])* Hypergeometric2F1[1/2, 1/2 - n, 3/2, 1/2*(1 - b*Sin[c + d*x]/a)]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[2*n]] && GtQ[a, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
       "filename": "4.1.1.1 (a+b sin)^n.m",
            "rhs": "a^IntPart[n]*(a + b*Sin[c + d*x])^FracPart[n]/(1 + b/a*Sin[c + d*x])^ FracPart[n]*Int[(1 + b/a*Sin[c + d*x])^n, x]",
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            "lhs": "Int[(a_ + b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[2*n]] && Not[GtQ[a, 0]]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
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            "lhs": "Int[Sqrt[a_ + b_.*sin[c_. + d_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
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            "lhs": "Int[Sqrt[a_ + b_.*sin[c_. + d_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a - b, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
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            "lhs": "Int[Sqrt[a_ + b_.*sin[c_. + d_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && Not[GtQ[a + b, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
       "filename": "4.1.1.1 (a+b sin)^n.m",
            "rhs": "-b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(n - 1)/(d*n) + 1/n*Int[(a + b*Sin[c + d*x])^(n - 2)* Simp[a^2*n + b^2*(n - 1) + a*b*(2*n - 1)*Sin[c + d*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 1] && IntegerQ[2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
       "filename": "4.1.1.1 (a+b sin)^n.m",
            "rhs": "With[{q = Rt[a^2 - b^2, 2]}, x/q + 2/(d*q)*ArcTan[b*Cos[c + d*x]/(a + q + b*Sin[c + d*x])]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*sin[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && GtQ[a^2 - b^2, 0] && PosQ[a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
       "filename": "4.1.1.1 (a+b sin)^n.m",
            "rhs": "With[{q = Rt[a^2 - b^2, 2]}, -x/q - 2/(d*q)*ArcTan[b*Cos[c + d*x]/(a - q + b*Sin[c + d*x])]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*sin[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && GtQ[a^2 - b^2, 0] && NegQ[a]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
       "filename": "4.1.1.1 (a+b sin)^n.m",
            "rhs": "With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, 2*e/d* Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*sin[c_. + Pi/2 + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
       "filename": "4.1.1.1 (a+b sin)^n.m",
            "rhs": "With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, 2*e/d* Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*sin[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
       "filename": "4.1.1.1 (a+b sin)^n.m",
            "rhs": "2/(d*Sqrt[a + b])*EllipticF[1/2*(c - Pi/2 + d*x), 2*b/(a + b)]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*sin[c_. + d_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
       "filename": "4.1.1.1 (a+b sin)^n.m",
            "rhs": "2/(d*Sqrt[a - b])*EllipticF[1/2*(c + Pi/2 + d*x), -2*b/(a - b)]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*sin[c_. + d_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a - b, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
       "filename": "4.1.1.1 (a+b sin)^n.m",
            "rhs": "Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]]* Int[1/Sqrt[a/(a + b) + b/(a + b)*Sin[c + d*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*sin[c_. + d_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && Not[GtQ[a + b, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
       "filename": "4.1.1.1 (a+b sin)^n.m",
            "rhs": "-b* Cos[c + d* x]*(a + b*Sin[c + d*x])^(n + 1)/(d*(n + 1)*(a^2 - b^2)) + 1/((n + 1)*(a^2 - b^2))* Int[(a + b*Sin[c + d*x])^(n + 1)* Simp[a*(n + 1) - b*(n + 2)*Sin[c + d*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
       "filename": "4.1.1.1 (a+b sin)^n.m",
            "rhs": "Cos[c + d*x]/(d*Sqrt[1 + Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]])* Subst[Int[(a + b*x)^n/(Sqrt[1 + x]*Sqrt[1 - x]), x], x, Sin[c + d*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 - b^2, 0] && Not[IntegerQ[2*n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m",
       "filename": "4.1.1.1 (a+b sin)^n.m",
            "rhs": "Int[(a + b*Sin[2*c + 2*d*x]/2)^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[c_. + d_.*x_]*cos[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "1/(b^p*f)* Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^p_.*(a_ + b_.*sin[e_. + f_.*x_])^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] || Not[IntegerQ[m + 1/2]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "1/(b^p*f)* Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^p_.*(a_ + b_.*sin[e_. + f_.*x_])^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "-b*(g*Cos[e + f*x])^(p + 1)/(f*g*(p + 1)) + a*Int[(g*Cos[e + f*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, p}, x] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "(a/g)^(2*m)* Int[(g*Cos[e + f*x])^(2*m + p)/(a - b*Sin[e + f*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && LtQ[p, -1] && GeQ[2*m + p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m/(a*f*g*m)",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[Simplify[m + p + 1], 0] && Not[ILtQ[p, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ m/(a*f*g*Simplify[2*m + p + 1]) + Simplify[m + p + 1]/(a*Simplify[2*m + p + 1])* Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + p + 1], 0] && NeQ[2*m + p + 1, 0] && Not[IGtQ[m, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)/(f* g*(m - 1))",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "-b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p)) + a*(2*m + p - 1)/(m + p)* Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[Simplify[(2*m + p - 1)/2], 0] && NeQ[m + p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "-b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ m/(a*f*g*(p + 1)) + a*(m + p + 1)/(g^2*(p + 1))* Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[p, -2*m] && IntegersQ[m + 1/2, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "-2* b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)/(f* g*(p + 1)) + b^2*(2*m + p - 1)/(g^2*(p + 1))* Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 1] && LtQ[p, -1] && IntegersQ[2*m, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "a*Sqrt[1 + Cos[e + f*x]]* Sqrt[a + b*Sin[e + f*x]]/(a + a*Cos[e + f*x] + b*Sin[e + f*x])* Int[Sqrt[1 + Cos[e + f*x]]/Sqrt[g*Cos[e + f*x]], x] + b*Sqrt[1 + Cos[e + f*x]]* Sqrt[a + b*Sin[e + f*x]]/(a + a*Cos[e + f*x] + b*Sin[e + f*x])* Int[Sin[e + f*x]/(Sqrt[g*Cos[e + f*x]]*Sqrt[1 + Cos[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]/Sqrt[g_.*cos[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "-b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p)) + a*(2*m + p - 1)/(m + p)* Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && NeQ[m + p, 0] && IntegersQ[2*m, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)/(b* f*(m + p)) + g^2*(p - 1)/(a*(m + p))* Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && (GtQ[m, -2] || EqQ[2*m + p + 1, 0] || EqQ[m, -2] && IntegerQ[p]) && NeQ[m + p, 0] && IntegersQ[2*m, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "2*g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)/(b* f*(2*m + p + 1)) + g^2*(p - 1)/(b^2*(2*m + p + 1))* Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] && NeQ[2*m + p + 1, 0] && Not[ILtQ[m + p + 1, 0]] && IntegersQ[2*m, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ m/(a*f*g*(2*m + p + 1)) + (m + p + 1)/(a*(2*m + p + 1))* Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && NeQ[2*m + p + 1, 0] && IntegersQ[2*m, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "g*(g*Cos[e + f*x])^(p - 1)/(b*f*(p - 1)) + g^2/a*Int[(g*Cos[e + f*x])^(p - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && GtQ[p, 1] && IntegerQ[2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "b*(g*Cos[e + f*x])^(p + 1)/(a*f*g*(p - 1)*(a + b*Sin[e + f*x])) + p/(a*(p - 1))*Int[(g*Cos[e + f*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && Not[GeQ[p, 1]] && IntegerQ[2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "g*Sqrt[1 + Cos[e + f*x]]* Sqrt[a + b*Sin[e + f*x]]/(a + a*Cos[e + f*x] + b*Sin[e + f*x])* Int[Sqrt[1 + Cos[e + f*x]]/Sqrt[g*Cos[e + f*x]], x] - g*Sqrt[1 + Cos[e + f*x]]* Sqrt[a + b*Sin[e + f*x]]/(b + b*Cos[e + f*x] + a*Sin[e + f*x])* Int[Sin[e + f*x]/(Sqrt[g*Cos[e + f*x]]*Sqrt[1 + Cos[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[g_.*cos[e_. + f_.*x_]]/Sqrt[a_ + b_.*sin[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "g*Sqrt[g*Cos[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]/(b*f) + g^2/(2*a)*Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[g*Cos[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^(3/2)/Sqrt[a_ + b_.*sin[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "-2* b*(g*Cos[e + f*x])^(p + 1)/(f* g*(2*p - 1)*(a + b*Sin[e + f*x])^(3/2)) + 2*a*(p - 2)/(2*p - 1)* Int[(g*Cos[e + f*x])^p/(a + b*Sin[e + f*x])^(3/2), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_/Sqrt[a_ + b_.*sin[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && GtQ[p, 2] && IntegerQ[2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "-b*(g*Cos[e + f*x])^(p + 1)/(a*f*g*(p + 1)* Sqrt[a + b*Sin[e + f*x]]) + a*(2*p + 1)/(2*g^2*(p + 1))* Int[(g*Cos[e + f*x])^(p + 2)/(a + b*Sin[e + f*x])^(3/2), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_/Sqrt[a_ + b_.*sin[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && LtQ[p, -1] && IntegerQ[2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "a^m*(g*Cos[e + f*x])^(p + 1)/(f* g*(1 + Sin[e + f*x])^((p + 1)/2)*(1 - Sin[e + f*x])^((p + 1)/2))* Subst[ Int[(1 + b/a*x)^(m + (p - 1)/2)*(1 - b/a*x)^((p - 1)/2), x], x, Sin[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "a^2*(g*Cos[e + f*x])^(p + 1)/(f* g*(a + b*Sin[e + f*x])^((p + 1)/2)*(a - b*Sin[e + f*x])^((p + 1)/2))* Subst[Int[(a + b*x)^(m + (p - 1)/2)*(a - b*x)^((p - 1)/2), x], x, Sin[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "-(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m* Sin[e + f*x]/(f*g*(p + 1)) + 1/(g^2*(p + 1))* Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1)*(a*(p + 2) + b*(m + p + 2)*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && LtQ[0, m, 1] && LtQ[p, -1] && (IntegersQ[2*m, 2*p] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "-(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*(b + a*Sin[e + f*x])/(f*g*(p + 1)) + 1/(g^2*(p + 1))* Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(p + 2) + a*b*(m + p + 1)*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && LtQ[p, -1] && (IntegersQ[2*m, 2*p] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "-b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p)) + 1/(m + p)* Int[(g*Cos[e + f*x])^ p*(a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(m + p) + a*b*(2*m + p - 1)*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && NeQ[m + p, 0] && (IntegersQ[2*m, 2*p] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)/(b* f*(m + 1)) + g^2*(p - 1)/(b*(m + 1))* Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1)* Sin[e + f*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && IntegersQ[2*m, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "-b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)/(f*g*(a^2 - b^2)*(m + 1)) + 1/((a^2 - b^2)*(m + 1))* Int[(g*Cos[e + f*x])^ p*(a + b*Sin[e + f*x])^(m + 1)*(a*(m + 1) - b*(m + p + 2)*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)/(b* f*(m + p)) + g^2*(p - 1)/(b*(m + p))* Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^ m*(b + a*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[p, 1] && NeQ[m + p, 0] && IntegersQ[2*m, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*(b - a*Sin[e + f*x])/(f*g*(a^2 - b^2)*(p + 1)) + 1/(g^2*(a^2 - b^2)*(p + 1))* Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^ m*(a^2*(p + 2) - b^2*(m + p + 2) + a*b*(m + p + 3)*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && LtQ[p, -1] && IntegersQ[2*m, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "2*Sqrt[2]*Sqrt[g*Cos[e + f*x]]* Sqrt[(a + b*Sin[e + f*x])/((a - b)*(1 - Sin[e + f*x]))]/ (f*g*Sqrt[a + b*Sin[e + f*x]]* Sqrt[(1 + Cos[e + f*x] + Sin[e + f*x])/(1 + Cos[e + f*x] - Sin[e + f*x])])* Subst[Int[1/Sqrt[1 + (a + b)*x^4/(a - b)], x], x, Sqrt[(1 + Cos[e + f*x] + Sin[e + f*x])/(1 + Cos[e + f*x] - Sin[e + f*x])]]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[g_.*cos[e_. + f_.*x_]]*Sqrt[a_ + b_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "g*(g*Cos[e + f*x])^(p - 1)*(1 - Sin[e + f*x])*(a + b*Sin[e + f*x])^(m + 1)*(-(a - b)*(1 - Sin[e + f*x])/((a + b)*(1 + Sin[e + f*x])))^(m/2)/ (f*(a + b)*(m + 1))* Hypergeometric2F1[m + 1, m/2 + 1, m + 2, 2*(a + b*Sin[e + f*x])/((a + b)*(1 + Sin[e + f*x]))]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, m, p}, x] && NeQ[a^2 - b^2, 0] && EqQ[m + p + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)/(f*g*(a - b)*(p + 1)) + a/(g^2*(a - b))* Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^ m/(1 - Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, m, p}, x] && NeQ[a^2 - b^2, 0] && EqQ[m + p + 2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)/(f*g*(a - b)*(p + 1)) - b*(m + p + 2)/(g^2*(a - b)*(p + 1))* Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m, x] + a/(g^2*(a - b))* Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^ m/(1 - Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, m, p}, x] && NeQ[a^2 - b^2, 0] && ILtQ[m + p + 2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "With[{q = Rt[-a^2 + b^2, 2]}, a*g/(2*b)*Int[1/(Sqrt[g*Cos[e + f*x]]*(q + b*Cos[e + f*x])), x] - a*g/(2*b)* Int[1/(Sqrt[g*Cos[e + f*x]]*(q - b*Cos[e + f*x])), x] + b*g/f* Subst[Int[Sqrt[x]/(g^2*(a^2 - b^2) + b^2*x^2), x], x, g*Cos[e + f*x]]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[g_.*cos[e_. + f_.*x_]]/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "With[{q = Rt[-a^2 + b^2, 2]}, -a/(2*q)* Int[1/(Sqrt[g*Cos[e + f*x]]*(q + b*Cos[e + f*x])), x] - a/(2*q)* Int[1/(Sqrt[g*Cos[e + f*x]]*(q - b*Cos[e + f*x])), x] + b*g/f* Subst[Int[1/(Sqrt[x]*(g^2*(a^2 - b^2) + b^2*x^2)), x], x, g*Cos[e + f*x]]]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[g_.*cos[e_. + f_.*x_]]*(a_ + b_.*sin[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)/ (b* f*(m + p)*(-b*(1 - Sin[e + f*x])/(a + b*Sin[e + f*x]))^((p - 1)/ 2)*(b*(1 + Sin[e + f*x])/(a + b*Sin[e + f*x]))^((p - 1)/2))* AppellF1[-p - m, (1 - p)/2, (1 - p)/2, 1 - p - m, (a + b)/(a + b*Sin[e + f*x]), (a - b)/(a + b*Sin[e + f*x])]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && ILtQ[m, 0] && Not[IGtQ[m + p + 1, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "g*(g*Cos[e + f*x])^(p - 1)/(f*(1 - (a + b*Sin[e + f*x])/(a - b))^((p - 1)/ 2)*(1 - (a + b*Sin[e + f*x])/(a + b))^((p - 1)/2))* Subst[ Int[(-b/(a - b) - b*x/(a - b))^((p - 1)/2)*(b/(a + b) - b*x/(a + b))^((p - 1)/2)*(a + b*x)^m, x], x, Sin[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, m, p}, x] && NeQ[a^2 - b^2, 0] && Not[IGtQ[m, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m",
       "filename": "4.1.1.2 (g cos)^p (a+b sin)^m.m",
            "rhs": "g^(2*IntPart[p])*(g*Cos[e + f*x])^FracPart[p]*(g*Sec[e + f*x])^ FracPart[p]*Int[(a + b*Sin[e + f*x])^m/(g*Cos[e + f*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*sec[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, m, p}, x] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m",
       "filename": "4.1.1.3 (g tan)^p (a+b sin)^m.m",
            "rhs": "1/a*Int[Sec[e + f*x]^2*(g*Tan[e + f*x])^p, x] - 1/(b*g)*Int[Sec[e + f*x]*(g*Tan[e + f*x])^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*tan[e_. + f_.*x_])^p_./(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m",
       "filename": "4.1.1.3 (g tan)^p (a+b sin)^m.m",
            "rhs": "1/f*Subst[Int[x^p*(a + x)^(m - (p + 1)/2)/(a - x)^((p + 1)/2), x], x, b*Sin[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[tan[e_. + f_.*x_]^p_.*(a_ + b_.*sin[e_. + f_.*x_])^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m",
       "filename": "4.1.1.3 (g tan)^p (a+b sin)^m.m",
            "rhs": "a^p*Int[Sin[e + f*x]^p/(a - b*Sin[e + f*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[tan[e_. + f_.*x_]^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p] && EqQ[p, 2*m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m",
       "filename": "4.1.1.3 (g tan)^p (a+b sin)^m.m",
            "rhs": "a^p*Int[ExpandIntegrand[ Sin[e + f*x]^ p*(a + b*Sin[e + f*x])^(m - p/2)/(a - b*Sin[e + f*x])^(p/2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[tan[e_. + f_.*x_]^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p/2] && (LtQ[p, 0] || GtQ[m - p/2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m",
       "filename": "4.1.1.3 (g tan)^p (a+b sin)^m.m",
            "rhs": "Int[ExpandIntegrand[(g*Tan[e + f*x])^p, (a + b*Sin[e + f*x])^m, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*tan[e_. + f_.*x_])^p_.*(a_ + b_.*sin[e_. + f_.*x_])^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m",
       "filename": "4.1.1.3 (g tan)^p (a+b sin)^m.m",
            "rhs": "a^(2*m)*Int[ ExpandIntegrand[(g*Tan[e + f*x])^p* Sec[e + f*x]^(-m), (a*Sec[e + f*x] - b*Tan[e + f*x])^(-m), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*tan[e_. + f_.*x_])^p_.*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m",
       "filename": "4.1.1.3 (g tan)^p (a+b sin)^m.m",
            "rhs": "b*(a + b*Sin[e + f*x])^m/(a*f*(2*m - 1)*Cos[e + f*x]) - 1/(a^2*(2*m - 1))* Int[(a + b*Sin[e + f*x])^(m + 1)*(a*m - b*(2*m - 1)*Sin[e + f*x])/ Cos[e + f*x]^2, x]",
     "rulenumber": 0,
            "lhs": "Int[tan[e_. + f_.*x_]^2*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[m]] && LtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m",
       "filename": "4.1.1.3 (g tan)^p (a+b sin)^m.m",
            "rhs": "-(a + b*Sin[e + f*x])^(m + 1)/(b*f*m*Cos[e + f*x]) + 1/(b*m)* Int[(a + b*Sin[e + f*x])^m*(b*(m + 1) + a*Sin[e + f*x])/ Cos[e + f*x]^2, x]",
     "rulenumber": 0,
            "lhs": "Int[tan[e_. + f_.*x_]^2*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[m]] && Not[LtQ[m, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m",
       "filename": "4.1.1.3 (g tan)^p (a+b sin)^m.m",
            "rhs": "Int[(a + b*Sin[e + f*x])^m, x] - Int[(a + b*Sin[e + f*x])^m*(1 - 2*Sin[e + f*x]^2)/Cos[e + f*x]^4, x]",
     "rulenumber": 0,
            "lhs": "Int[tan[e_. + f_.*x_]^4*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m",
       "filename": "4.1.1.3 (g tan)^p (a+b sin)^m.m",
            "rhs": "-(a + b*Sin[e + f*x])^(m + 1)/(a*f*Tan[e + f*x]) + 1/b^2* Int[(a + b*Sin[e + f*x])^(m + 1)*(b*m - a*(m + 1)*Sin[e + f*x])/ Sin[e + f*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_/tan[e_. + f_.*x_]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m - 1/2] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m",
       "filename": "4.1.1.3 (g tan)^p (a+b sin)^m.m",
            "rhs": "-(a + b*Sin[e + f*x])^m/(f*Tan[e + f*x]) + 1/a*Int[(a + b*Sin[e + f*x])^m*(b*m - a*(m + 1)*Sin[e + f*x])/ Sin[e + f*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_./tan[e_. + f_.*x_]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m - 1/2] && Not[LtQ[m, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m",
       "filename": "4.1.1.3 (g tan)^p (a+b sin)^m.m",
            "rhs": "-2/(a*b)* Int[(a + b*Sin[e + f*x])^(m + 2)/Sin[e + f*x]^3, x] + 1/a^2* Int[(a + b*Sin[e + f*x])^(m + 2)*(1 + Sin[e + f*x]^2)/ Sin[e + f*x]^4, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_/tan[e_. + f_.*x_]^4, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m - 1/2] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m",
       "filename": "4.1.1.3 (g tan)^p (a+b sin)^m.m",
            "rhs": "Int[(a + b*Sin[e + f*x])^m, x] + Int[(a + b*Sin[e + f*x])^m*(1 - 2*Sin[e + f*x]^2)/Sin[e + f*x]^4, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_/tan[e_. + f_.*x_]^4, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m - 1/2] && Not[LtQ[m, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m",
       "filename": "4.1.1.3 (g tan)^p (a+b sin)^m.m",
            "rhs": "Sqrt[a + b*Sin[e + f*x]]* Sqrt[a - b*Sin[e + f*x]]/(b*f*Cos[e + f*x])* Subst[Int[x^p*(a + x)^(m - (p + 1)/2)/(a - x)^((p + 1)/2), x], x, b*Sin[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[tan[e_. + f_.*x_]^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[m]] && IntegerQ[p/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m",
       "filename": "4.1.1.3 (g tan)^p (a+b sin)^m.m",
            "rhs": "(g*Tan[e + f*x])^(p + 1)*(a - b*Sin[e + f*x])^((p + 1)/2)*(a + b*Sin[e + f*x])^((p + 1)/2)/(f* g*(b*Sin[e + f*x])^(p + 1))* Subst[Int[x^p*(a + x)^(m - (p + 1)/2)/(a - x)^((p + 1)/2), x], x, b*Sin[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*tan[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[m]] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m",
       "filename": "4.1.1.3 (g tan)^p (a+b sin)^m.m",
            "rhs": "1/f*Subst[Int[(x^p*(a + x)^m)/(b^2 - x^2)^((p + 1)/2), x], x, b*Sin[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[tan[e_. + f_.*x_]^p_.*(a_ + b_.*sin[e_. + f_.*x_])^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m",
       "filename": "4.1.1.3 (g tan)^p (a+b sin)^m.m",
            "rhs": "Int[ExpandIntegrand[(g*Tan[e + f*x])^p, (a + b*Sin[e + f*x])^m, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*tan[e_. + f_.*x_])^p_.*(a_ + b_.*sin[e_. + f_.*x_])^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m",
       "filename": "4.1.1.3 (g tan)^p (a+b sin)^m.m",
            "rhs": "Int[(a + b*Sin[e + f*x])^m*(1 - Sin[e + f*x]^2)/Sin[e + f*x]^2, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_/tan[e_. + f_.*x_]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m",
       "filename": "4.1.1.3 (g tan)^p (a+b sin)^m.m",
            "rhs": "-Cos[ e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(3*a*f*Sin[e + f*x]^3) - (3*a^2 + b^2*(m - 2))* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(3*a^2*b*f*(m + 1)* Sin[e + f*x]^2) - 1/(3*a^2*b*(m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1)/Sin[e + f*x]^3* Simp[6*a^2 - b^2*(m - 1)*(m - 2) + a*b*(m + 1)*Sin[e + f*x] - (3*a^2 - b^2*m*(m - 2))* Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_/tan[e_. + f_.*x_]^4, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m",
       "filename": "4.1.1.3 (g tan)^p (a+b sin)^m.m",
            "rhs": " -Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)/(3*a*f*Sin[e+f*x]^3) - Cos[e+f*x]*(a+b*Sin[e+f*x])^(m+1)/(b*f*m*Sin[e+f*x]^2) - 1/(3*a*b*m)*Int[(a+b*Sin[e+f*x])^m/Sin[e+f*x]^3* Simp[6*a^2-b^2*m*(m-2)+a*b*(m+3)*Sin[e+f*x]-(3*a^2-b^2*m*(m-1))* Sin[e+f*x]^2,x],x]",
     "rulenumber": 0,
            "lhs": "Int[(a_+b_.*sin[e_.+f_.*x_])^m_/tan[e_.+f_.*x_]^4,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,e,f,m},x] && NeQ[a^2-b^2,0] &&  Not[LtQ[m,-1]] && IntegerQ[2*m] *)"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m",
       "filename": "4.1.1.3 (g tan)^p (a+b sin)^m.m",
            "rhs": "-Cos[ e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(3*a*f*Sin[e + f*x]^3) - b*(m - 2)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(6*a^2*f* Sin[e + f*x]^2) - 1/(6*a^2)*Int[(a + b*Sin[e + f*x])^m/Sin[e + f*x]^2* Simp[8*a^2 - b^2*(m - 1)*(m - 2) + a*b*m*Sin[e + f*x] - (6*a^2 - b^2*m*(m - 2))*Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_/tan[e_. + f_.*x_]^4, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0] && Not[LtQ[m, -1]] && IntegerQ[2*m]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m",
       "filename": "4.1.1.3 (g tan)^p (a+b sin)^m.m",
            "rhs": "-Cos[ e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(5*a*f*Sin[e + f*x]^5) - b*(m - 4)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(20*a^2*f* Sin[e + f*x]^4) + a*Cos[ e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b^2*f*m*(m - 1)* Sin[e + f*x]^3) + Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b*f*m*Sin[e + f*x]^2) + 1/(20*a^2*b^2*m*(m - 1))* Int[(a + b*Sin[e + f*x])^m/Sin[e + f*x]^4* Simp[60*a^4 - 44*a^2*b^2*(m - 1)*m + b^4*m*(m - 1)*(m - 3)*(m - 4) + a*b*m*(20*a^2 - b^2*m*(m - 1))*Sin[e + f*x] - (40*a^4 + b^4*m*(m - 1)*(m - 2)*(m - 4) - 20*a^2*b^2*(m - 1)*(2*m + 1))*Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_/tan[e_. + f_.*x_]^6, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0] && NeQ[m, 1] && IntegerQ[2*m]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m",
       "filename": "4.1.1.3 (g tan)^p (a+b sin)^m.m",
            "rhs": "a/(a^2 - b^2)*Int[(g*Tan[e + f*x])^p/Sin[e + f*x]^2, x] - b*g/(a^2 - b^2)*Int[(g*Tan[e + f*x])^(p - 1)/Cos[e + f*x], x] - a^2*g^2/(a^2 - b^2)* Int[(g*Tan[e + f*x])^(p - 2)/(a + b*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*tan[e_. + f_.*x_])^p_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*p] && GtQ[p, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m",
       "filename": "4.1.1.3 (g tan)^p (a+b sin)^m.m",
            "rhs": "1/a*Int[(g*Tan[e + f*x])^p/Cos[e + f*x]^2, x] - b/(a^2*g)*Int[(g*Tan[e + f*x])^(p + 1)/Cos[e + f*x], x] - (a^2 - b^2)/(a^2*g^2)* Int[(g*Tan[e + f*x])^(p + 2)/(a + b*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*tan[e_. + f_.*x_])^p_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*p] && LtQ[p, -1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m",
       "filename": "4.1.1.3 (g tan)^p (a+b sin)^m.m",
            "rhs": "Sqrt[Cos[e + f*x]]*Sqrt[g*Tan[e + f*x]]/Sqrt[Sin[e + f*x]]* Int[Sqrt[Sin[e + f*x]]/(Sqrt[Cos[e + f*x]]*(a + b*Sin[e + f*x])), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[g_.*tan[e_. + f_.*x_]]/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]"
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  {
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            "rhs": "Sqrt[Sin[e + f*x]]/(Sqrt[Cos[e + f*x]]*Sqrt[g*Tan[e + f*x]])* Int[Sqrt[Cos[e + f*x]]/(Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])), x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m",
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        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[m, p/2]"
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            "rhs": "Unintegrable[(g*Tan[e + f*x])^p*(a + b*Sin[e + f*x])^m, x]",
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         "givens": "FreeQ[{a, b, e, f, g, m, p}, x]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m",
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        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, m, p}, x] && Not[IntegerQ[p]]"
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            "lhs": "Int[(c_. + d_.*x_)^m_.*sin[e_. + f_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e, f}, x] && GtQ[m, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.10 (c+d x)^m (a+b sin)^n.m",
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            "rhs": "(c + d*x)^(m + 1)*Sin[e + f*x]/(d*(m + 1)) - f/(d*(m + 1))*Int[(c + d*x)^(m + 1)*Cos[e + f*x], x]",
     "rulenumber": 0,
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        "comment": false,
         "givens": "FreeQ[{c, d, e, f}, x] && LtQ[m, -1]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.10 (c+d x)^m (a+b sin)^n.m",
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        "comment": false,
         "givens": "FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.10 (c+d x)^m (a+b sin)^n.m",
       "filename": "4.1.10 (c+d x)^m (a+b sin)^n.m",
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            "lhs": "Int[sin[e_. + f_.*x_]/(c_. + d_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]"
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       "filename": "4.1.10 (c+d x)^m (a+b sin)^n.m",
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        "comment": false,
         "givens": "FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0] && NegQ[c*f*fz/d, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.10 (c+d x)^m (a+b sin)^n.m",
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        "comment": false,
         "givens": "FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.10 (c+d x)^m (a+b sin)^n.m",
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            "rhs": "CosIntegral[e - Pi/2 + f*x]/d",
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            "lhs": "Int[sin[e_. + f_.*x_]/(c_. + d_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.10 (c+d x)^m (a+b sin)^n.m",
       "filename": "4.1.10 (c+d x)^m (a+b sin)^n.m",
            "rhs": "Cos[(d*e - c*f)/d]*Int[Sin[c*f/d + f*x]/(c + d*x), x] + Sin[(d*e - c*f)/d]*Int[Cos[c*f/d + f*x]/(c + d*x), x]",
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        "comment": false,
         "givens": "FreeQ[{c, d, e, f}, x] && NeQ[d*e - c*f, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.10 (c+d x)^m (a+b sin)^n.m",
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.10 (c+d x)^m (a+b sin)^n.m",
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            "rhs": "2/d*Subst[Int[Sin[f*x^2/d], x], x, Sqrt[c + d*x]]",
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        "comment": false,
         "givens": "FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.10 (c+d x)^m (a+b sin)^n.m",
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            "rhs": "Cos[(d*e - c*f)/d]*Int[Sin[c*f/d + f*x]/Sqrt[c + d*x], x] + Sin[(d*e - c*f)/d]*Int[Cos[c*f/d + f*x]/Sqrt[c + d*x], x]",
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         "givens": "FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.10 (c+d x)^m (a+b sin)^n.m",
       "filename": "4.1.10 (c+d x)^m (a+b sin)^n.m",
            "rhs": "I/2*Int[(c + d*x)^m*E^(-I*k*Pi)*E^(-I*(e + f*x)), x] - I/2*Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x]",
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        "comment": false,
         "givens": "FreeQ[{c, d, e, f, m}, x] && IntegerQ[2*k]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.10 (c+d x)^m (a+b sin)^n.m",
       "filename": "4.1.10 (c+d x)^m (a+b sin)^n.m",
            "rhs": "I/2*Int[(c + d*x)^m*E^(-I*(e + f*x)), x] - I/2*Int[(c + d*x)^m*E^(I*(e + f*x)), x]",
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        "comment": false,
         "givens": "FreeQ[{c, d, e, f, m}, x]"
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         "givens": "FreeQ[{c, d, e, f, m}, x]"
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            "rhs": "d*(b*Sin[e + f*x])^n/(f^2*n^2) - b*(c + d*x)*Cos[e + f*x]*(b*Sin[e + f*x])^(n - 1)/(f*n) + b^2*(n - 1)/n*Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x]",
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       "filename": "4.1.10 (c+d x)^m (a+b sin)^n.m",
            "rhs": "d*m*(c + d*x)^(m - 1)*(b*Sin[e + f*x])^n/(f^2*n^2) - b*(c + d*x)^m*Cos[e + f*x]*(b*Sin[e + f*x])^(n - 1)/(f*n) + b^2*(n - 1)/n*Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x] - d^2*m*(m - 1)/(f^2*n^2)* Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x]",
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            "lhs": "Int[(c_. + d_.*x_)^m_*(b_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
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         "givens": "FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.10 (c+d x)^m (a+b sin)^n.m",
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        "comment": false,
         "givens": "FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && (Not[RationalQ[m]] || GeQ[m, -1] && LtQ[m, 1])"
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         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] || IGtQ[m, 0] || NeQ[a^2 - b^2, 0])"
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            "rhs": " (2*a)^IntPart[n]*(a+b*Sin[e+f*x])^FracPart[n]/Cos[1/2*(e-Pi*a/(2*b))+ f*x/2]^(2*FracPart[n])* Int[(c+d*x)^m*Cos[1/2*(e-Pi*a/(2*b))+f*x/2]^(2*n),x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.+d_.*x_)^m_.*(a_+b_.*sin[e_.+f_.*x_])^n_,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e,f,m},x] && EqQ[a^2-b^2,0] && IntegerQ[n+1/2] &&  (GtQ[n,0] || IGtQ[m,0]) *)"
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       "filename": "4.1.10 (c+d x)^m (a+b sin)^n.m",
            "rhs": "2*Int[(c + d*x)^m*E^(-I*Pi*(k - 1/2))* E^(-I*e + f*fz*x)/(b + 2*a*E^(-I*Pi*(k - 1/2))*E^(-I*e + f*fz*x) - b*E^(-2*I*k*Pi)*E^(2*(-I*e + f*fz*x))), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^ m_./(a_ + b_.*sin[e_. + k_.*Pi + f_.*Complex[0, fz_]*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, fz}, x] && IntegerQ[2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.10 (c+d x)^m (a+b sin)^n.m",
       "filename": "4.1.10 (c+d x)^m (a+b sin)^n.m",
            "rhs": "2*Int[(c + d*x)^m*E^(I*Pi*(k - 1/2))* E^(I*(e + f*x))/(b + 2*a*E^(I*Pi*(k - 1/2))*E^(I*(e + f*x)) - b*E^(2*I*k*Pi)*E^(2*I*(e + f*x))), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_./(a_ + b_.*sin[e_. + k_.*Pi + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.10 (c+d x)^m (a+b sin)^n.m",
       "filename": "4.1.10 (c+d x)^m (a+b sin)^n.m",
            "rhs": " 2*I*Int[(c+d*x)^m*E^(-I*e+f*fz*x)/(b+2*I*a*E^(-I*e+f*fz*x)-b*E^(2*(-I* e+f*fz*x))),x]",
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            "lhs": "Int[(c_.+d_.*x_)^m_./(a_+b_.*sin[e_.+f_.*Complex[0,fz_]*x_]),x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,fz},x] && NeQ[a^2-b^2,0] &&  IGtQ[m,0] *)"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.10 (c+d x)^m (a+b sin)^n.m",
       "filename": "4.1.10 (c+d x)^m (a+b sin)^n.m",
            "rhs": " -2*I*Int[(c+d*x)^m*E^(I*(e+f*x))/(b-2*I*a*E^(I*(e+f*x))-b*E^(2*I*(e+f* x))),x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.+d_.*x_)^m_./(a_+b_.*sin[e_.+f_.*x_]),x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f},x] && NeQ[a^2-b^2,0] && IGtQ[m,0] *)"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.10 (c+d x)^m (a+b sin)^n.m",
       "filename": "4.1.10 (c+d x)^m (a+b sin)^n.m",
            "rhs": "2*Int[(c + d*x)^m* E^(-I*e + f*fz*x)/(-I*b + 2*a*E^(-I*e + f*fz*x) + I*b*E^(2*(-I*e + f*fz*x))), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_./(a_ + b_.*sin[e_. + f_.*Complex[0, fz_]*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.10 (c+d x)^m (a+b sin)^n.m",
       "filename": "4.1.10 (c+d x)^m (a+b sin)^n.m",
            "rhs": "2*Int[(c + d*x)^m* E^(I*(e + f*x))/(I*b + 2*a*E^(I*(e + f*x)) - I*b*E^(2*I*(e + f*x))), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_./(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.10 (c+d x)^m (a+b sin)^n.m",
       "filename": "4.1.10 (c+d x)^m (a+b sin)^n.m",
            "rhs": "b*(c + d*x)^m*Cos[e + f*x]/(f*(a^2 - b^2)*(a + b*Sin[e + f*x])) + a/(a^2 - b^2)*Int[(c + d*x)^m/(a + b*Sin[e + f*x]), x] - b*d*m/(f*(a^2 - b^2))* Int[(c + d*x)^(m - 1)*Cos[e + f*x]/(a + b*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_./(a_ + b_.*sin[e_. + f_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.10 (c+d x)^m (a+b sin)^n.m",
       "filename": "4.1.10 (c+d x)^m (a+b sin)^n.m",
            "rhs": "-b*(c + d*x)^m* Cos[e + f* x]*(a + b*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(a^2 - b^2)) + a/(a^2 - b^2)* Int[(c + d*x)^m*(a + b*Sin[e + f*x])^(n + 1), x] + b*d*m/(f*(n + 1)*(a^2 - b^2))* Int[(c + d*x)^(m - 1)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(n + 1), x] - b*(n + 2)/((n + 1)*(a^2 - b^2))* Int[(c + d*x)^m*Sin[e + f*x]*(a + b*Sin[e + f*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(a_ + b_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && ILtQ[n, -2] && IGtQ[m, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.10 (c+d x)^m (a+b sin)^n.m",
       "filename": "4.1.10 (c+d x)^m (a+b sin)^n.m",
            "rhs": "Unintegrable[(c + d*x)^m*(a + b*Sin[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(a_. + b_.*sin[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.10 (c+d x)^m (a+b sin)^n.m",
       "filename": "4.1.10 (c+d x)^m (a+b sin)^n.m",
            "rhs": "Int[ExpandToSum[u, x]^m*(a + b*Sin[ExpandToSum[v, x]])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*(a_. + b_.*Sin[v_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && LinearQ[{u, v}, x] && Not[LinearMatchQ[{u, v}, x]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.10 (c+d x)^m (a+b sin)^n.m",
       "filename": "4.1.10 (c+d x)^m (a+b sin)^n.m",
            "rhs": "Int[ExpandToSum[u, x]^m*(a + b*Cos[ExpandToSum[v, x]])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*(a_. + b_.*Cos[v_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && LinearQ[{u, v}, x] && Not[LinearMatchQ[{u, v}, x]]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.11 (e x)^m (a+b x^n)^p sin.m",
       "filename": "4.1.11 (e x)^m (a+b x^n)^p sin.m",
            "rhs": "Int[ExpandIntegrand[Sin[c + d*x], (a + b*x^n)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_.*Sin[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.11 (e x)^m (a+b x^n)^p sin.m",
       "filename": "4.1.11 (e x)^m (a+b x^n)^p sin.m",
            "rhs": "Int[ExpandIntegrand[Cos[c + d*x], (a + b*x^n)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_.*Cos[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.11 (e x)^m (a+b x^n)^p sin.m",
       "filename": "4.1.11 (e x)^m (a+b x^n)^p sin.m",
            "rhs": "x^(-n + 1)*(a + b*x^n)^(p + 1)*Sin[c + d*x]/(b*n*(p + 1)) - (-n + 1)/(b*n*(p + 1))* Int[x^(-n)*(a + b*x^n)^(p + 1)*Sin[c + d*x], x] - d/(b*n*(p + 1))* Int[x^(-n + 1)*(a + b*x^n)^(p + 1)*Cos[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*Sin[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && ILtQ[p, -1] && IGtQ[n, 2]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.11 (e x)^m (a+b x^n)^p sin.m",
       "filename": "4.1.11 (e x)^m (a+b x^n)^p sin.m",
            "rhs": "x^(-n + 1)*(a + b*x^n)^(p + 1)*Cos[c + d*x]/(b*n*(p + 1)) - (-n + 1)/(b*n*(p + 1))* Int[x^(-n)*(a + b*x^n)^(p + 1)*Cos[c + d*x], x] + d/(b*n*(p + 1))* Int[x^(-n + 1)*(a + b*x^n)^(p + 1)*Sin[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*Cos[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && ILtQ[p, -1] && IGtQ[n, 2]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.11 (e x)^m (a+b x^n)^p sin.m",
       "filename": "4.1.11 (e x)^m (a+b x^n)^p sin.m",
            "rhs": "Int[ExpandIntegrand[Sin[c + d*x], (a + b*x^n)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*Sin[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.11 (e x)^m (a+b x^n)^p sin.m",
       "filename": "4.1.11 (e x)^m (a+b x^n)^p sin.m",
            "rhs": "Int[ExpandIntegrand[Cos[c + d*x], (a + b*x^n)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*Cos[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.11 (e x)^m (a+b x^n)^p sin.m",
       "filename": "4.1.11 (e x)^m (a+b x^n)^p sin.m",
            "rhs": "Int[x^(n*p)*(b + a*x^(-n))^p*Sin[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*Sin[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && ILtQ[n, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.11 (e x)^m (a+b x^n)^p sin.m",
       "filename": "4.1.11 (e x)^m (a+b x^n)^p sin.m",
            "rhs": "Int[x^(n*p)*(b + a*x^(-n))^p*Cos[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*Cos[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && ILtQ[n, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.11 (e x)^m (a+b x^n)^p sin.m",
       "filename": "4.1.11 (e x)^m (a+b x^n)^p sin.m",
            "rhs": "Unintegrable[(a + b*x^n)^p*Sin[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*Sin[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.11 (e x)^m (a+b x^n)^p sin.m",
       "filename": "4.1.11 (e x)^m (a+b x^n)^p sin.m",
            "rhs": "Unintegrable[(a + b*x^n)^p*Cos[c + d*x], x]",
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            "lhs": "Int[(a_ + b_.*x_^n_)^p_*Cos[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.11 (e x)^m (a+b x^n)^p sin.m",
       "filename": "4.1.11 (e x)^m (a+b x^n)^p sin.m",
            "rhs": "Int[ExpandIntegrand[Sin[c + d*x], (e*x)^m*(a + b*x^n)^p, x], x]",
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            "lhs": "Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_.*Sin[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.11 (e x)^m (a+b x^n)^p sin.m",
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            "rhs": "Int[ExpandIntegrand[Cos[c + d*x], (e*x)^m*(a + b*x^n)^p, x], x]",
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            "lhs": "Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_.*Cos[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.11 (e x)^m (a+b x^n)^p sin.m",
       "filename": "4.1.11 (e x)^m (a+b x^n)^p sin.m",
            "rhs": "e^m*(a + b*x^n)^(p + 1)*Sin[c + d*x]/(b*n*(p + 1)) - d*e^m/(b*n*(p + 1))*Int[(a + b*x^n)^(p + 1)*Cos[c + d*x], x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, -1] && EqQ[m, n - 1] && (IntegerQ[n] || GtQ[e, 0])"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.11 (e x)^m (a+b x^n)^p sin.m",
       "filename": "4.1.11 (e x)^m (a+b x^n)^p sin.m",
            "rhs": "e^m*(a + b*x^n)^(p + 1)*Cos[c + d*x]/(b*n*(p + 1)) + d*e^m/(b*n*(p + 1))*Int[(a + b*x^n)^(p + 1)*Sin[c + d*x], x]",
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            "lhs": "Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_*Cos[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, -1] && EqQ[m, n - 1] && (IntegerQ[n] || GtQ[e, 0])"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.11 (e x)^m (a+b x^n)^p sin.m",
       "filename": "4.1.11 (e x)^m (a+b x^n)^p sin.m",
            "rhs": "x^(m - n + 1)*(a + b*x^n)^(p + 1)*Sin[c + d*x]/(b*n*(p + 1)) - (m - n + 1)/(b*n*(p + 1))* Int[x^(m - n)*(a + b*x^n)^(p + 1)*Sin[c + d*x], x] - d/(b*n*(p + 1))* Int[x^(m - n + 1)*(a + b*x^n)^(p + 1)*Cos[c + d*x], x]",
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            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_)^p_*Sin[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && ILtQ[p, -1] && IGtQ[n, 0] && (GtQ[m - n + 1, 0] || GtQ[n, 2]) && RationalQ[m]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.11 (e x)^m (a+b x^n)^p sin.m",
       "filename": "4.1.11 (e x)^m (a+b x^n)^p sin.m",
            "rhs": "x^(m - n + 1)*(a + b*x^n)^(p + 1)*Cos[c + d*x]/(b*n*(p + 1)) - (m - n + 1)/(b*n*(p + 1))* Int[x^(m - n)*(a + b*x^n)^(p + 1)*Cos[c + d*x], x] + d/(b*n*(p + 1))* Int[x^(m - n + 1)*(a + b*x^n)^(p + 1)*Sin[c + d*x], x]",
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            "lhs": "Int[x_^m_.*(a_ + b_.*x_^n_)^p_*Cos[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && ILtQ[p, -1] && IGtQ[n, 0] && (GtQ[m - n + 1, 0] || GtQ[n, 2]) && RationalQ[m]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.11 (e x)^m (a+b x^n)^p sin.m",
       "filename": "4.1.11 (e x)^m (a+b x^n)^p sin.m",
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "1/n*Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sin[c + d*x])^p, x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*Sin[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[ Simplify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "1/n*Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Cos[c + d*x])^p, x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*Cos[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[ Simplify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Sin[c + d*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_*x_)^m_*(a_. + b_.*Sin[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Cos[c + d*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_*x_)^m_*(a_. + b_.*Cos[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "2/n*Subst[Int[Sin[a + b*x^2], x], x, x^(n/2)]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Sin[a_. + b_.*x_^n_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && EqQ[m, n/2 - 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "2/n*Subst[Int[Cos[a + b*x^2], x], x, x^(n/2)]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Cos[a_. + b_.*x_^n_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && EqQ[m, n/2 - 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "-e^(n - 1)*(e*x)^(m - n + 1)* Cos[c + d*x^n]/(d*n) + e^n*(m - n + 1)/(d*n)*Int[(e*x)^(m - n)*Cos[c + d*x^n], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Sin[c_. + d_.*x_^n_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[n, m + 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "e^(n - 1)*(e*x)^(m - n + 1)*Sin[c + d*x^n]/(d*n) - e^n*(m - n + 1)/(d*n)*Int[(e*x)^(m - n)*Sin[c + d*x^n], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Cos[c_. + d_.*x_^n_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[n, m + 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "(e*x)^(m + 1)*Sin[c + d*x^n]/(e*(m + 1)) - d*n/(e^n*(m + 1))*Int[(e*x)^(m + n)*Cos[c + d*x^n], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_*Sin[c_. + d_.*x_^n_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "(e*x)^(m + 1)*Cos[c + d*x^n]/(e*(m + 1)) + d*n/(e^n*(m + 1))*Int[(e*x)^(m + n)*Sin[c + d*x^n], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_*Cos[c_. + d_.*x_^n_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "I/2*Int[(e*x)^m*E^(-c*I - d*I*x^n), x] - I/2*Int[(e*x)^m*E^(c*I + d*I*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Sin[c_. + d_.*x_^n_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e, m}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "1/2*Int[(e*x)^m*E^(-c*I - d*I*x^n), x] + 1/2*Int[(e*x)^m*E^(c*I + d*I*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Cos[c_. + d_.*x_^n_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e, m}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "1/2*Int[x^m, x] - 1/2*Int[x^m*Cos[2*a + b*x^n], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Sin[a_. + b_.*x_^n_/2]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "1/2*Int[x^m, x] + 1/2*Int[x^m*Cos[2*a + b*x^n], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Cos[a_. + b_.*x_^n_/2]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "x^(m + 1)*Sin[a + b*x^n]^p/(m + 1) - b*n*p/(m + 1)*Int[Sin[a + b*x^n]^(p - 1)*Cos[a + b*x^n], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Sin[a_. + b_.*x_^n_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IGtQ[p, 1] && EqQ[m + n, 0] && NeQ[n, 1] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "x^(m + 1)*Cos[a + b*x^n]^p/(m + 1) + b*n*p/(m + 1)*Int[Cos[a + b*x^n]^(p - 1)*Sin[a + b*x^n], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Cos[a_. + b_.*x_^n_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IGtQ[p, 1] && EqQ[m + n, 0] && NeQ[n, 1] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "n*Sin[a + b*x^n]^p/(b^2*n^2*p^2) - x^n*Cos[a + b*x^n]*Sin[a + b*x^n]^(p - 1)/(b*n*p) + (p - 1)/p*Int[x^m*Sin[a + b*x^n]^(p - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Sin[a_. + b_.*x_^n_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && EqQ[m - 2*n + 1, 0] && GtQ[p, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "n*Cos[a + b*x^n]^p/(b^2*n^2*p^2) + x^n*Sin[a + b*x^n]*Cos[a + b*x^n]^(p - 1)/(b*n*p) + (p - 1)/p*Int[x^m*Cos[a + b*x^n]^(p - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Cos[a_. + b_.*x_^n_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && EqQ[m - 2*n + 1, 0] && GtQ[p, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "(m - n + 1)*x^(m - 2*n + 1)* Sin[a + b*x^n]^p/(b^2*n^2*p^2) - x^(m - n + 1)*Cos[a + b*x^n]*Sin[a + b*x^n]^(p - 1)/(b*n*p) + (p - 1)/p*Int[x^m*Sin[a + b*x^n]^(p - 2), x] - (m - n + 1)*(m - 2*n + 1)/(b^2*n^2*p^2)* Int[x^(m - 2*n)*Sin[a + b*x^n]^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Sin[a_. + b_.*x_^n_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && GtQ[p, 1] && IGtQ[n, 0] && IGtQ[m, 2*n - 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "(m - n + 1)*x^(m - 2*n + 1)* Cos[a + b*x^n]^p/(b^2*n^2*p^2) + x^(m - n + 1)*Sin[a + b*x^n]*Cos[a + b*x^n]^(p - 1)/(b*n*p) + (p - 1)/p*Int[x^m*Cos[a + b*x^n]^(p - 2), x] - (m - n + 1)*(m - 2*n + 1)/(b^2*n^2*p^2)* Int[x^(m - 2*n)*Cos[a + b*x^n]^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Cos[a_. + b_.*x_^n_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && GtQ[p, 1] && IGtQ[n, 0] && IGtQ[m, 2*n - 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "x^(m + 1)*Sin[a + b*x^n]^p/(m + 1) - b*n*p*x^(m + n + 1)*Cos[a + b*x^n]* Sin[a + b*x^n]^(p - 1)/((m + 1)*(m + n + 1)) - b^2*n^2*p^2/((m + 1)*(m + n + 1))* Int[x^(m + 2*n)*Sin[a + b*x^n]^p, x] + b^2*n^2*p*(p - 1)/((m + 1)*(m + n + 1))* Int[x^(m + 2*n)*Sin[a + b*x^n]^(p - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Sin[a_. + b_.*x_^n_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && GtQ[p, 1] && IGtQ[n, 0] && ILtQ[m, -2*n + 1] && NeQ[m + n + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "x^(m + 1)*Cos[a + b*x^n]^p/(m + 1) + b*n*p*x^(m + n + 1)*Sin[a + b*x^n]* Cos[a + b*x^n]^(p - 1)/((m + 1)*(m + n + 1)) - b^2*n^2*p^2/((m + 1)*(m + n + 1))* Int[x^(m + 2*n)*Cos[a + b*x^n]^p, x] + b^2*n^2*p*(p - 1)/((m + 1)*(m + n + 1))* Int[x^(m + 2*n)*Cos[a + b*x^n]^(p - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Cos[a_. + b_.*x_^n_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && GtQ[p, 1] && IGtQ[n, 0] && ILtQ[m, -2*n + 1] && NeQ[m + n + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "With[{k = Denominator[m]}, k/e*Subst[ Int[x^(k*(m + 1) - 1)*(a + b*Sin[c + d*x^(k*n)/e^n])^p, x], x, (e*x)^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_*(a_. + b_.*Sin[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IntegerQ[p] && IGtQ[n, 0] && FractionQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "With[{k = Denominator[m]}, k/e*Subst[ Int[x^(k*(m + 1) - 1)*(a + b*Cos[c + d*x^(k*n)/e^n])^p, x], x, (e*x)^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_*(a_. + b_.*Cos[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IntegerQ[p] && IGtQ[n, 0] && FractionQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "Int[ExpandTrigReduce[(e*x)^m, (a + b*Sin[c + d*x^n])^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_. + b_.*Sin[c_. + d_.*x_^n_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 1] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "Int[ExpandTrigReduce[(e*x)^m, (a + b*Cos[c + d*x^n])^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_. + b_.*Cos[c_. + d_.*x_^n_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 1] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "x^n*Cos[a + b*x^n]*Sin[a + b*x^n]^(p + 1)/(b*n*(p + 1)) - n*Sin[a + b*x^n]^(p + 2)/(b^2*n^2*(p + 1)*(p + 2)) + (p + 2)/(p + 1)*Int[x^m*Sin[a + b*x^n]^(p + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Sin[a_. + b_.*x_^n_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && EqQ[m - 2*n + 1, 0] && LtQ[p, -1] && NeQ[p, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "-x^n*Sin[a + b*x^n]* Cos[a + b*x^n]^(p + 1)/(b*n*(p + 1)) - n*Cos[a + b*x^n]^(p + 2)/(b^2*n^2*(p + 1)*(p + 2)) + (p + 2)/(p + 1)*Int[x^m*Cos[a + b*x^n]^(p + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Cos[a_. + b_.*x_^n_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && EqQ[m - 2*n + 1, 0] && LtQ[p, -1] && NeQ[p, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "x^(m - n + 1)*Cos[a + b*x^n]*Sin[a + b*x^n]^(p + 1)/(b*n*(p + 1)) - (m - n + 1)*x^(m - 2*n + 1)* Sin[a + b*x^n]^(p + 2)/(b^2*n^2*(p + 1)*(p + 2)) + (p + 2)/(p + 1)*Int[x^m*Sin[a + b*x^n]^(p + 2), x] + (m - n + 1)*(m - 2*n + 1)/(b^2*n^2*(p + 1)*(p + 2))* Int[x^(m - 2*n)*Sin[a + b*x^n]^(p + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Sin[a_. + b_.*x_^n_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && LtQ[p, -1] && NeQ[p, -2] && IGtQ[n, 0] && IGtQ[m, 2*n - 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "-x^(m - n + 1)*Sin[a + b*x^n]* Cos[a + b*x^n]^(p + 1)/(b*n*(p + 1)) - (m - n + 1)*x^(m - 2*n + 1)* Cos[a + b*x^n]^(p + 2)/(b^2*n^2*(p + 1)*(p + 2)) + (p + 2)/(p + 1)*Int[x^m*Cos[a + b*x^n]^(p + 2), x] + (m - n + 1)*(m - 2*n + 1)/(b^2*n^2*(p + 1)*(p + 2))* Int[x^(m - 2*n)*Cos[a + b*x^n]^(p + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Cos[a_. + b_.*x_^n_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && LtQ[p, -1] && NeQ[p, -2] && IGtQ[n, 0] && IGtQ[m, 2*n - 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "-Subst[ Int[(a + b*Sin[c + d*x^(-n)])^p/x^(m + 2), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*Sin[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m] && EqQ[n, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "-Subst[ Int[(a + b*Cos[c + d*x^(-n)])^p/x^(m + 2), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*Cos[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m] && EqQ[n, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "With[{k = Denominator[m]}, -k/e* Subst[Int[(a + b*Sin[c + d/(e^n*x^(k*n))])^p/x^(k*(m + 1) + 1), x], x, 1/(e*x)^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_*(a_. + b_.*Sin[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && ILtQ[n, 0] && FractionQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "With[{k = Denominator[m]}, -k/e* Subst[Int[(a + b*Cos[c + d/(e^n*x^(k*n))])^p/x^(k*(m + 1) + 1), x], x, 1/(e*x)^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_*(a_. + b_.*Cos[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && ILtQ[n, 0] && FractionQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "-(e*x)^m*(x^(-1))^m* Subst[Int[(a + b*Sin[c + d*x^(-n)])^p/x^(m + 2), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_*(a_. + b_.*Sin[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0] && ILtQ[n, 0] && Not[RationalQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "-(e*x)^m*(x^(-1))^m* Subst[Int[(a + b*Cos[c + d*x^(-n)])^p/x^(m + 2), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_*(a_. + b_.*Cos[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0] && ILtQ[n, 0] && Not[RationalQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "Module[{k = Denominator[n]}, k*Subst[Int[x^(k*(m + 1) - 1)*(a + b*Sin[c + d*x^(k*n)])^p, x], x, x^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*Sin[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && IntegerQ[p] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "Module[{k = Denominator[n]}, k*Subst[Int[x^(k*(m + 1) - 1)*(a + b*Cos[c + d*x^(k*n)])^p, x], x, x^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*Cos[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && IntegerQ[p] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Sin[c + d*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_*x_)^m_*(a_. + b_.*Sin[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && IntegerQ[p] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Cos[c + d*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_*x_)^m_*(a_. + b_.*Cos[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && IntegerQ[p] && FractionQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "1/(m + 1)* Subst[Int[(a + b*Sin[c + d*x^Simplify[n/(m + 1)]])^p, x], x, x^(m + 1)]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*Sin[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && IntegerQ[p] && NeQ[m, -1] && IGtQ[Simplify[n/(m + 1)], 0] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "1/(m + 1)* Subst[Int[(a + b*Cos[c + d*x^Simplify[n/(m + 1)]])^p, x], x, x^(m + 1)]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*Cos[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && IntegerQ[p] && NeQ[m, -1] && IGtQ[Simplify[n/(m + 1)], 0] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Sin[c + d*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_*x_)^m_*(a_. + b_.*Sin[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && IntegerQ[p] && NeQ[m, -1] && IGtQ[Simplify[n/(m + 1)], 0] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Cos[c + d*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_*x_)^m_*(a_. + b_.*Cos[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && IntegerQ[p] && NeQ[m, -1] && IGtQ[Simplify[n/(m + 1)], 0] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "I/2*Int[(e*x)^m*E^(-c*I - d*I*x^n), x] - I/2*Int[(e*x)^m*E^(c*I + d*I*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Sin[c_. + d_.*x_^n_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "1/2*Int[(e*x)^m*E^(-c*I - d*I*x^n), x] + 1/2*Int[(e*x)^m*E^(c*I + d*I*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Cos[c_. + d_.*x_^n_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "Int[ExpandTrigReduce[(e*x)^m, (a + b*Sin[c + d*x^n])^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_. + b_.*Sin[c_. + d_.*x_^n_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "Int[ExpandTrigReduce[(e*x)^m, (a + b*Cos[c + d*x^n])^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_. + b_.*Cos[c_. + d_.*x_^n_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "Unintegrable[(e*x)^m*(a + b*Sin[c + d*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_. + b_.*Sin[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "Unintegrable[(e*x)^m*(a + b*Cos[c + d*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_. + b_.*Cos[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "Int[(e*x)^m*(a + b*Sin[ExpandToSum[u, x]])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_*x_)^m_.*(a_. + b_.*Sin[u_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "Int[(e*x)^m*(a + b*Cos[ExpandToSum[u, x]])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_*x_)^m_.*(a_. + b_.*Cos[u_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "1/(n*f)*Subst[ Int[ExpandIntegrand[(a + b*Sin[c + d*x])^p, x^(1/n - 1)*(g - e*h/f + h*x^(1/n)/f)^m, x], x], x, (e + f*x)^n]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)^m_.*(a_. + b_.*Sin[c_. + d_.*(e_. + f_.*x_)^n_])^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p, 0] && IntegerQ[1/n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": "1/(n*f)*Subst[ Int[ExpandIntegrand[(a + b*Cos[c + d*x])^p, x^(1/n - 1)*(g - e*h/f + h*x^(1/n)/f)^m, x], x], x, (e + f*x)^n]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)^m_.*(a_. + b_.*Cos[c_. + d_.*(e_. + f_.*x_)^n_])^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p, 0] && IntegerQ[1/n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": " 1/(n*f^(m+1))*Subst[Int[ExpandIntegrand[(a+b*Sin[c+d*x])^p,x^(1/n-1)*( f*g-e*h+h*x^(1/n))^m,x],x],x,(e+f*x)^n]",
     "rulenumber": 0,
            "lhs": "Int[(g_.+h_.*x_)^m_.*(a_.+b_.*Sin[c_.+d_.*(e_.+f_.*x_)^n_])^p_.,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,g,h},x]  && IGtQ[p,0] && IntegerQ[m] && IntegerQ[1/n] *)"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
       "filename": "4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m",
            "rhs": " 1/(n*f^(m+1))*Subst[Int[ExpandIntegrand[(a+b*Cos[c+d*x])^p,x^(1/n-1)*( f*g-e*h+h*x^(1/n))^m,x],x],x,(e+f*x)^n]",
     "rulenumber": 0,
            "lhs": "Int[(g_.+h_.*x_)^m_.*(a_.+b_.*Cos[c_.+d_.*(e_.+f_.*x_)^n_])^p_.,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,g,h},x]  && IGtQ[p,0] && IntegerQ[m] && IntegerQ[1/n] *)"
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            "rhs": "-e*(d + e*x)^(m - 1)*Cos[a + b*x + c*x^2]/(2*c) - (b*e - 2*c*d)/(2*c)* Int[(d + e*x)^(m - 1)*Sin[a + b*x + c*x^2], x] + e^2*(m - 1)/(2*c)* Int[(d + e*x)^(m - 2)*Cos[a + b*x + c*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*Sin[a_. + b_.*x_ + c_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && GtQ[m, 1]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.13 (d+e x)^m sin(a+b x+c x^2)^n.m",
       "filename": "4.1.13 (d+e x)^m sin(a+b x+c x^2)^n.m",
            "rhs": "e*(d + e*x)^(m - 1)*Sin[a + b*x + c*x^2]/(2*c) - (b*e - 2*c*d)/(2*c)* Int[(d + e*x)^(m - 1)*Cos[a + b*x + c*x^2], x] - e^2*(m - 1)/(2*c)* Int[(d + e*x)^(m - 2)*Sin[a + b*x + c*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_*Cos[a_. + b_.*x_ + c_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && GtQ[m, 1]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.13 (d+e x)^m sin(a+b x+c x^2)^n.m",
       "filename": "4.1.13 (d+e x)^m sin(a+b x+c x^2)^n.m",
            "rhs": "(d + e*x)^(m + 1)* Sin[a + b*x + c*x^2]/(e*(m + 1)) - (b*e - 2*c*d)/(e^2*(m + 1))* Int[(d + e*x)^(m + 1)*Cos[a + b*x + c*x^2], x] - 2*c/(e^2*(m + 1))* Int[(d + e*x)^(m + 2)*Cos[a + b*x + c*x^2], x]",
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            "lhs": "Int[(d_. + e_.*x_)^m_*Sin[a_. + b_.*x_ + c_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && LtQ[m, -1]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.13 (d+e x)^m sin(a+b x+c x^2)^n.m",
       "filename": "4.1.13 (d+e x)^m sin(a+b x+c x^2)^n.m",
            "rhs": "(d + e*x)^(m + 1)*Cos[a + b*x + c*x^2]/(e*(m + 1)) + (b*e - 2*c*d)/(e^2*(m + 1))* Int[(d + e*x)^(m + 1)*Sin[a + b*x + c*x^2], x] + 2*c/(e^2*(m + 1))* Int[(d + e*x)^(m + 2)*Sin[a + b*x + c*x^2], x]",
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            "lhs": "Int[(d_. + e_.*x_)^m_*Cos[a_. + b_.*x_ + c_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && LtQ[m, -1]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.13 (d+e x)^m sin(a+b x+c x^2)^n.m",
       "filename": "4.1.13 (d+e x)^m sin(a+b x+c x^2)^n.m",
            "rhs": "Int[ExpandTrigReduce[(d + e*x)^m, Sin[a + b*x + c*x^2]^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*Sin[a_. + b_.*x_ + c_.*x_^2]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 1]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.13 (d+e x)^m sin(a+b x+c x^2)^n.m",
       "filename": "4.1.13 (d+e x)^m sin(a+b x+c x^2)^n.m",
            "rhs": "Int[ExpandTrigReduce[(d + e*x)^m, Cos[a + b*x + c*x^2]^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*Cos[a_. + b_.*x_ + c_.*x_^2]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 1]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.13 (d+e x)^m sin(a+b x+c x^2)^n.m",
       "filename": "4.1.13 (d+e x)^m sin(a+b x+c x^2)^n.m",
            "rhs": "Unintegrable[(d + e*x)^m*Sin[a + b*x + c*x^2]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*Sin[a_. + b_.*x_ + c_.*x_^2]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.13 (d+e x)^m sin(a+b x+c x^2)^n.m",
       "filename": "4.1.13 (d+e x)^m sin(a+b x+c x^2)^n.m",
            "rhs": "Unintegrable[(d + e*x)^m*Cos[a + b*x + c*x^2]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*Cos[a_. + b_.*x_ + c_.*x_^2]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.13 (d+e x)^m sin(a+b x+c x^2)^n.m",
       "filename": "4.1.13 (d+e x)^m sin(a+b x+c x^2)^n.m",
            "rhs": "Int[ExpandToSum[u, x]^m*Sin[ExpandToSum[v, x]]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*Sin[v_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[m, x] && IGtQ[n, 0] && LinearQ[u, x] && QuadraticQ[v, x] && Not[LinearMatchQ[u, x] && QuadraticMatchQ[v, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.13 (d+e x)^m sin(a+b x+c x^2)^n.m",
       "filename": "4.1.13 (d+e x)^m sin(a+b x+c x^2)^n.m",
            "rhs": "Int[ExpandToSum[u, x]^m*Cos[ExpandToSum[v, x]]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*Cos[v_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[m, x] && IGtQ[n, 0] && LinearQ[u, x] && QuadraticQ[v, x] && Not[LinearMatchQ[u, x] && QuadraticMatchQ[v, x]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "(2*a*c + b*d)*x/2 - (b*c + a*d)*Cos[e + f*x]/f - b*d*Cos[e + f*x]*Sin[e + f*x]/(2*f)",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "b*x/d - (b*c - a*d)/d*Int[1/(c + d*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])/(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "a^m*c^m*Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(c_ + d_.*sin[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && Not[IntegerQ[ n] && (LtQ[m, 0] && GtQ[n, 0] || LtQ[0, n, m] || LtQ[m, n, 0])]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "a*c*Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])* Int[Cos[e + f*x]/(c + d*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]/Sqrt[c_ + d_.*sin[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-2*b* Cos[e + f*x]*(c + d*Sin[e + f*x])^ n/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]])",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[n, -1/2]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-2*b* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^ n/(f*(2*n + 1)) - b*(2*m - 1)/(d*(2*n + 1))* Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[m - 1/2, 0] && LtQ[n, -1] && Not[ILtQ[m + n, 0] && GtQ[2*m + n + 1, 0]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-b* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^ n/(f*(m + n)) + a*(2*m - 1)/(m + n)* Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[m - 1/2, 0] && Not[LtQ[n, -1]] && Not[IGtQ[n - 1/2, 0] && LtQ[n, m]] && Not[ILtQ[m + n, 0] && GtQ[2*m + n + 1, 0]]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])* Int[1/Cos[e + f*x], x]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_ + b_.*sin[e_. + f_.*x_]]* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "b*Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n/(a*f*(2*m + 1))",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && NeQ[m, -1/2]"
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       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "b*Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n/(a*f*(2*m + 1)) + (m + n + 1)/(a*(2*m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + n + 1], 0] && NeQ[m, -1/2] && (SumSimplerQ[m, 1] || Not[SumSimplerQ[n, 1]])"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "b*Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n/(a*f*(2*m + 1)) + (m + n + 1)/(a*(2*m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x]",
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            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && Not[LtQ[m, n, -1]] && IntegersQ[2*m, 2*n]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "a^IntPart[m]* c^IntPart[m]*(a + b*Sin[e + f*x])^ FracPart[m]*(c + d*Sin[e + f*x])^FracPart[m]/ Cos[e + f*x]^(2*FracPart[m])* Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x]",
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            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && (FractionQ[m] || Not[FractionQ[n]])"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-b^2*Cos[e + f*x]/(d*f) + 1/d*Int[Simp[a^2*d - b*(b*c - 2*a*d)*Sin[e + f*x], x]/(c + d*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^2/(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "b/(b*c - a*d)*Int[1/(a + b*Sin[e + f*x]), x] - d/(b*c - a*d)*Int[1/(c + d*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_. + b_.*sin[e_. + f_.*x_])*(c_. + d_.*sin[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "c*Int[(b*Sin[e + f*x])^m, x] + d/b*Int[(b*Sin[e + f*x])^(m + 1), x]",
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            "lhs": "Int[(b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f, m}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-d*Cos[e + f*x]*(a + b*Sin[e + f*x])^m/(f*(m + 1))",
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            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[a*d*m + b*c*(m + 1), 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "(b*c - a*d)* Cos[e + f*x]*(a + b*Sin[e + f*x])^m/(a*f*(2*m + 1)) + (a*d*m + b*c*(m + 1))/(a*b*(2*m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-d*Cos[e + f*x]*(a + b*Sin[e + f*x])^m/(f*(m + 1)) + (a*d*m + b*c*(m + 1))/(b*(m + 1))* Int[(a + b*Sin[e + f*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1/2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "(b*c - a*d)/b*Int[1/Sqrt[a + b*Sin[e + f*x]], x] + d/b*Int[Sqrt[a + b*Sin[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*sin[e_. + f_.*x_])/Sqrt[a_ + b_.*sin[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-d*Cos[e + f*x]*(a + b*Sin[e + f*x])^m/(f*(m + 1)) + 1/(m + 1)* Int[(a + b*Sin[e + f*x])^(m - 1)* Simp[b*d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-(b*c - a*d)* Cos[e + f* x]*(a + b*Sin[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2)) + 1/((m + 1)*(a^2 - b^2))* Int[(a + b*Sin[e + f*x])^(m + 1)* Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + 2)*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "c*Cos[e + f*x]/(f*Sqrt[1 + Sin[e + f*x]]*Sqrt[1 - Sin[e + f*x]])* Subst[Int[(a + b*x)^m*Sqrt[1 + d/c*x]/Sqrt[1 - d/c*x], x], x, Sin[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && Not[IntegerQ[2*m]] && EqQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "(b*c - a*d)/b*Int[(a + b*Sin[e + f*x])^m, x] + d/b*Int[(a + b*Sin[e + f*x])^(m + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Int[ExpandTrig[(a + b*sin[e + f*x])^m*(d*sin[e + f*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(d_.*sin[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0] && RationalQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "b*Cos[e + f*x]*(a + b*Sin[e + f*x])^m/(a*f*(2*m + 1)) - 1/(a^2*(2*m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1)*(a*m - b*(2*m + 1)*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[sin[e_. + f_.*x_]^2*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-Cos[ e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2)) + 1/(b*(m + 2))* Int[(a + b*Sin[e + f*x])^m*(b*(m + 1) - a*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[sin[e_. + f_.*x_]^2*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1/2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "(b*c - a*d)* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])/(a*f*(2*m + 1)) + 1/(a*b*(2*m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1)* Simp[a*c*d*(m - 1) + b*(d^2 + c^2*(m + 1)) + d*(a*d*(m - 1) + b*c*(m + 2))*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*sin[e_. + f_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-d^2* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2)) + 1/(b*(m + 2))* Int[(a + b*Sin[e + f*x])^m* Simp[b*(d^2*(m + 1) + c^2*(m + 2)) - d*(a*d - 2*b*c*(m + 2))*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*sin[e_. + f_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-b^2*(b*c - a*d)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d)) + b^2/(d*(n + 1)*(b*c + a*d))* Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)* Simp[a*c*(m - 2) - b*d*(m - 2*n - 4) - (b*c*(m - 1) - a*d*(m + 2*n + 1))* Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && LtQ[n, -1] && (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || IntegerQ[m] && EqQ[c, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-b^2* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n)) + 1/(d*(m + n))* Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n* Simp[a*b*c*(m - 2) + b^2*d*(n + 1) + a^2*d*(m + n) - b*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && Not[LtQ[n, -1]] && (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || IntegerQ[m] && EqQ[c, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "b*Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n/(a*f*(2*m + 1)) - 1/(a*b*(2*m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)* Simp[a*d*n - b*c*(m + 1) - b*d*(m + n + 1)*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && LtQ[0, n, 1] && (IntegersQ[2*m, 2*n] || IntegerQ[m] && EqQ[c, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "(b*c - a*d)* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n - 1)/(a*f*(2*m + 1)) + 1/(a*b*(2*m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 2)* Simp[b*(c^2*(m + 1) + d^2*(n - 1)) + a*c*d*(m - n + 1) + d*(a*d*(m - n + 1) + b*c*(m + n))*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n] || IntegerQ[m] && EqQ[c, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(a*f*(2*m + 1)*(b*c - a*d)) + 1/(a*(2*m + 1)*(b*c - a*d))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n* Simp[b*c*(m + 1) - a*d*(2*m + n + 2) + b*d*(m + n + 2)*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && Not[GtQ[n, 0]] && (IntegersQ[2*m, 2*n] || IntegerQ[m] && EqQ[c, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-(b*c - a*d)* Cos[e + f*x]*(c + d*Sin[e + f*x])^(n - 1)/(a* f*(a + b*Sin[e + f*x])) - d/(a*b)* Int[(c + d*Sin[e + f*x])^(n - 2)* Simp[b*d*(n - 1) - a*c*n + (b*c*(n - 1) - a*d*n)*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*sin[e_. + f_.*x_])^n_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 1] && (IntegerQ[2*n] || EqQ[c, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-b^2* Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1)/(a* f*(b*c - a*d)*(a + b*Sin[e + f*x])) + d/(a*(b*c - a*d))* Int[(c + d*Sin[e + f*x])^n*(a*n - b*(n + 1)*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*sin[e_. + f_.*x_])^n_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, 0] && (IntegerQ[2*n] || EqQ[c, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-b* Cos[e + f*x]*(c + d*Sin[e + f*x])^n/(a*f*(a + b*Sin[e + f*x])) + d*n/(a*b)* Int[(c + d*Sin[e + f*x])^(n - 1)*(a - b*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*sin[e_. + f_.*x_])^n_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && (IntegerQ[2*n] || EqQ[c, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-2*b* Cos[e + f*x]*(c + d*Sin[e + f*x])^ n/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]]) + 2*n*(b*c + a*d)/(b*(2*n + 1))* Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 0] && IntegerQ[2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-2*b^2* Cos[e + f*x]/(f*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]]* Sqrt[c + d*Sin[e + f*x]])",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]/(c_. + d_.*sin[e_. + f_.*x_])^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "(b*c - a*d)* Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2)* Sqrt[a + b*Sin[e + f*x]]) + (2*n + 3)*(b*c - a*d)/(2*b*(n + 1)*(c^2 - d^2))* Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1] && NeQ[2*n + 3, 0] && IntegerQ[2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-2*b/f* Subst[Int[1/(b*c + a*d - d*x^2), x], x, b*Cos[e + f*x]/Sqrt[a + b*Sin[e + f*x]]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]/(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-2/f* Subst[Int[1/Sqrt[1 - x^2/a], x], x, b*Cos[e + f*x]/Sqrt[a + b*Sin[e + f*x]]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]/Sqrt[d_.*sin[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && EqQ[d, a/b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-2*b/f* Subst[Int[1/(b + d*x^2), x], x, b*Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]* Sqrt[c + d*Sin[e + f*x]])]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]/ Sqrt[c_. + d_.*sin[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "a^2*Cos[e + f*x]/(f*Sqrt[a + b*Sin[e + f*x]]* Sqrt[a - b*Sin[e + f*x]])* Subst[Int[(c + d*x)^n/Sqrt[a - b*x], x], x, Sin[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Not[IntegerQ[2*n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "d/b*Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x] + (b*c - a*d)/b* Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[c_. + d_.*sin[e_. + f_.*x_]]/ Sqrt[a_ + b_.*sin[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-2*d* Cos[e + f*x]*(c + d*Sin[e + f*x])^(n - 1)/(f*(2*n - 1)* Sqrt[a + b*Sin[e + f*x]]) - 1/(b*(2*n - 1))* Int[(c + d*Sin[e + f*x])^(n - 2)/Sqrt[a + b*Sin[e + f*x]]* Simp[a*c*d - b*(2*d^2*(n - 1) + c^2*(2*n - 1)) + d*(a*d - b*c*(4*n - 3))*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*sin[e_. + f_.*x_])^n_/Sqrt[a_ + b_.*sin[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 1] && IntegerQ[2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-d* Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2)* Sqrt[a + b*Sin[e + f*x]]) - 1/(2*b*(n + 1)*(c^2 - d^2))* Int[(c + d*Sin[e + f*x])^(n + 1)* Simp[a*d - 2*b*c*(n + 1) + b*d*(2*n + 3)*Sin[e + f*x], x]/ Sqrt[a + b*Sin[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*sin[e_. + f_.*x_])^n_/Sqrt[a_ + b_.*sin[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1] && IntegerQ[2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "b/(b*c - a*d)*Int[1/Sqrt[a + b*Sin[e + f*x]], x] - d/(b*c - a*d)* Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[ a_ + b_.*sin[e_. + f_.*x_]]*(c_. + d_.*sin[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-Sqrt[2]/(Sqrt[a]*f)* Subst[Int[1/Sqrt[1 - x^2], x], x, b*Cos[e + f*x]/(a + b*Sin[e + f*x])]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*Sqrt[d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && EqQ[d, a/b] && GtQ[a, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-2*a/f* Subst[Int[1/(2*b^2 - (a*c - b*d)*x^2), x], x, b*Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]* Sqrt[c + d*Sin[e + f*x]])]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_ + b_.*sin[e_. + f_.*x_]]* Sqrt[c_. + d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-d* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n - 1)/(f*(m + n)) + 1/(b*(m + n))* Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n - 2)* Simp[d*(a*c*m + b*d*(n - 1)) + b*c^2*(m + n) + d*(a*d*m + b*c*(m + 2*n - 1))*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 1] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "a^m*Cos[e + f*x]/(f*Sqrt[1 + Sin[e + f*x]]*Sqrt[1 - Sin[e + f*x]])* Subst[Int[(1 + b/a*x)^(m - 1/2)*(c + d*x)^n/Sqrt[1 - b/a*x], x], x, Sin[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-b*(d/b)^n* Cos[e + f*x]/(f*Sqrt[a + b*Sin[e + f*x]]* Sqrt[a - b*Sin[e + f*x]])* Subst[Int[(a - x)^n*(2*a - x)^(m - 1/2)/Sqrt[x], x], x, a - b*Sin[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[m]] && GtQ[a, 0] && GtQ[d/b, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "(d/b)^ IntPart[n]*(d*Sin[e + f*x])^FracPart[n]/(b*Sin[e + f*x])^ FracPart[n]*Int[(a + b*Sin[e + f*x])^m*(b*Sin[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(d_.*sin[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[m]] && GtQ[a, 0] && Not[GtQ[d/b, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "a^IntPart[m]*(a + b*Sin[e + f*x])^FracPart[m]/(1 + b/a*Sin[e + f*x])^ FracPart[m]* Int[(1 + b/a*Sin[e + f*x])^m*(d*Sin[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(d_.*sin[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[m]] && Not[GtQ[a, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "a^2*Cos[e + f*x]/(f*Sqrt[a + b*Sin[e + f*x]]* Sqrt[a - b*Sin[e + f*x]])* Subst[Int[(a + b*x)^(m - 1/2)*(c + d*x)^n/Sqrt[a - b*x], x], x, Sin[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*sin[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "2*c*d/b*Int[(b*Sin[e + f*x])^(m + 1), x] + Int[(b*Sin[e + f*x])^m*(c^2 + d^2*Sin[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*sin[e_. + f_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f, m}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-(b^2*c^2 - 2*a*b*c*d + a^2*d^2)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b* f*(m + 1)*(a^2 - b^2)) - 1/(b*(m + 1)*(a^2 - b^2))*Int[(a + b*Sin[e + f*x])^(m + 1)* Simp[b*(m + 1)*(2*b*c*d - a*(c^2 + d^2)) + (a^2*d^2 - 2*a*b*c*d*(m + 2) + b^2*(d^2*(m + 1) + c^2*(m + 2)))* Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-d^2* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2)) + 1/(b*(m + 2))* Int[(a + b*Sin[e + f*x])^m* Simp[b*(d^2*(m + 1) + c^2*(m + 2)) - d*(a*d - 2*b*c*(m + 2))*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && Not[LtQ[m, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Int[ExpandTrig[(a+b*sin[e+f*x])^m*(d*sin[e+f*x])^n,x],x] ",
     "rulenumber": 0,
            "lhs": "Int[(a_+b_.*sin[e_.+f_.*x_])^m_.*(d_.*sin[e_.+f_.*x_])^n_.,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,d,e,f,n},x] && NeQ[a^2-b^2,0] && IGtQ[m,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-(b^2*c^2 - 2*a*b*c*d + a^2*d^2)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2)) + 1/(d*(n + 1)*(c^2 - d^2))* Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^(n + 1)* Simp[b*(m - 2)*(b*c - a*d)^2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 + c*d*(a^2 + b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - d^2) - m*(b*c - a*d)^2 + d*n*(2*a*b*c - d*(a^2 + b^2)))*Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-b^2* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n)) + 1/(d*(m + n))* Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^n* Simp[a^3*d*(m + n) + b^2*(b*c*(m - 2) + a*d*(n + 1)) - b*(a*b*c - b^2*d*(m + n - 1) - 3*a^2*d*(m + n))* Sin[e + f*x] - b^2*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && Not[IGtQ[n, 2] && (Not[IntegerQ[m]] || EqQ[a, 0] && NeQ[c, 0])]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-2*a*d* Cos[e + f*x]/(f*(a^2 - b^2)*Sqrt[a + b*Sin[e + f*x]]* Sqrt[d*Sin[e + f*x]]) - d^2/(a^2 - b^2)* Int[Sqrt[a + b*Sin[e + f*x]]/(d*Sin[e + f*x])^(3/2), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[d_.*sin[e_. + f_.*x_]]/(a_ + b_.*sin[e_. + f_.*x_])^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "(c - d)/(a - b)* Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x] - (b*c - a*d)/(a - b)* Int[(1 + Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)* Sqrt[c + d*Sin[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[c_ + d_.*sin[e_. + f_.*x_]]/(a_. + b_.*sin[e_. + f_.*x_])^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-b* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^ n/(f*(m + 1)*(a^2 - b^2)) + 1/((m + 1)*(a^2 - b^2))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)* Simp[a*c*(m + 1) + b*d*n + (a*d*(m + 1) - b*c*(m + 2))*Sin[e + f*x] - b*d*(m + n + 2)*Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && LtQ[0, n, 1] && IntegersQ[2*m, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "d/b*Int[Sqrt[d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]], x] - a*d/b*Int[Sqrt[d*Sin[e + f*x]]/(a + b*Sin[e + f*x])^(3/2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sin[e_. + f_.*x_])^(3/2)/(a_ + b_.*sin[e_. + f_.*x_])^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "d^2/b^2*Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x] + (b*c - a*d)/b^2* Int[Simp[b*c + a*d + 2*b*d*Sin[e + f*x], x]/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_ + d_.*sin[e_. + f_.*x_])^(3/2)/(a_. + b_.*sin[e_. + f_.*x_])^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-(b*c - a*d)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)/(f*(m + 1)*(a^2 - b^2)) + 1/((m + 1)*(a^2 - b^2))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 2)* Simp[c*(a*c - b*d)*(m + 1) + d*(b*c - a*d)*(n - 1) + (d*(a*c - b*d)*(m + 1) - c*(b*c - a*d)*(m + 2))*Sin[e + f*x] - d*(b*c - a*d)*(m + n + 1)*Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && LtQ[1, n, 2] && IntegersQ[2*m, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "2*b*Cos[e + f*x]/(f*(a^2 - b^2)*Sqrt[a + b*Sin[e + f*x]]* Sqrt[d*Sin[e + f*x]]) + d/(a^2 - b^2)* Int[(b + a*Sin[e + f*x])/(Sqrt[ a + b*Sin[e + f*x]]*(d*Sin[e + f*x])^(3/2)), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_ + b_.*sin[e_. + f_.*x_])^(3/2)* Sqrt[d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "1/(a - b)* Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x] - b/(a - b)* Int[(1 + Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)* Sqrt[c + d*Sin[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_. + b_.*sin[e_. + f_.*x_])^(3/2)* Sqrt[c_. + d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-b^2* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)) + 1/((m + 1)*(b*c - a*d)*(a^2 - b^2))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n* Simp[a*(b*c - a*d)*(m + 1) + b^2*d*(m + n + 2) - (b^2*c + b*(b*c - a*d)*(m + 1))* Sin[e + f*x] - b^2*d*(m + n + 3)*Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n] && (EqQ[a, 0] && IntegerQ[m] && Not[IntegerQ[n]] || Not[IntegerQ[2*n] && LtQ[n, -1] && (IntegerQ[n] && Not[IntegerQ[m]] || EqQ[a, 0])])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "d/b*Int[1/Sqrt[c + d*Sin[e + f*x]], x] + (b*c - a*d)/b* Int[1/((a + b*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[c_. + d_.*sin[e_. + f_.*x_]]/(a_. + b_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "b/d*Int[Sqrt[a + b*Sin[e + f*x]], x] - (b*c - a*d)/d* Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^(3/2)/(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "2/(f*(a + b)*Sqrt[c + d])* EllipticPi[2*b/(a + b), 1/2*(e - Pi/2 + f*x), 2*d/(c + d)]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_. + b_.*sin[e_. + f_.*x_])* Sqrt[c_. + d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "2/(f*(a - b)*Sqrt[c - d])* EllipticPi[-2*b/(a - b), 1/2*(e + Pi/2 + f*x), -2*d/(c - d)]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_. + b_.*sin[e_. + f_.*x_])* Sqrt[c_. + d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c - d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt[c + d*Sin[e + f*x]]* Int[1/((a + b*Sin[e + f*x])* Sqrt[c/(c + d) + d/(c + d)*Sin[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_. + b_.*sin[e_. + f_.*x_])* Sqrt[c_. + d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Not[GtQ[c + d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "2*c*Rt[b*(c + d), 2]*Tan[e + f*x]*Sqrt[1 + Csc[e + f*x]]* Sqrt[1 - Csc[e + f*x]]/(d*f*Sqrt[c^2 - d^2])* EllipticPi[(c + d)/d, ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/ Rt[(c + d)/b, 2]], -(c + d)/(c - d)]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[b_.*sin[e_. + f_.*x_]]/Sqrt[c_ + d_.*sin[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f}, x] && GtQ[c^2 - d^2, 0] && PosQ[(c + d)/b] && GtQ[c^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "2*b*Tan[e + f*x]/(d*f)*Rt[(c + d)/b, 2]* Sqrt[c*(1 + Csc[e + f*x])/(c - d)]* Sqrt[c*(1 - Csc[e + f*x])/(c + d)]* EllipticPi[(c + d)/d, ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/ Rt[(c + d)/b, 2]], -(c + d)/(c - d)]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[b_.*sin[e_. + f_.*x_]]/Sqrt[c_ + d_.*sin[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] && PosQ[(c + d)/b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Sqrt[b*Sin[e + f*x]]/Sqrt[-b*Sin[e + f*x]]* Int[Sqrt[-b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[b_.*sin[e_. + f_.*x_]]/Sqrt[c_ + d_.*sin[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] && NegQ[(c + d)/b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "a*Int[1/(Sqrt[a+b*Sin[e+f*x]]*Sqrt[d*Sin[e+f*x]]),x] + b/d*Int[Sqrt[d*Sin[e+f*x]]/Sqrt[a+b*Sin[e+f*x]],x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_+b_.*sin[e_.+f_.*x_]]/Sqrt[d_.*sin[e_.+f_.*x_]],x_ Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,d,e,f},x] && NeQ[a^2-b^2,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": " 2*(a+b*Sin[e+f*x])/(d*f*Rt[(a+b)/d,2]*Cos[e+f*x])*Sqrt[a*(1-Sin[e+f*x] )/(a+b*Sin[e+f*x])]*Sqrt[a*(1+Sin[e+f*x])/(a+b*Sin[e+f*x])]* EllipticPi[b/(a+b),ArcSin[Rt[(a+b)/d,2]*(Sqrt[d*Sin[e+f*x]]/Sqrt[ a+b*Sin[e+f*x]])],-(a-b)/(a+b)]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_+b_.*sin[e_.+f_.*x_]]/Sqrt[d_.*sin[e_.+f_.*x_]],x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,d,e,f},x] &&  NeQ[a^2-b^2,0] && PosQ[(a+b)/d] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "2*(a + b*Sin[e + f*x])/(d*f*Rt[(a + b)/(c + d), 2]*Cos[e + f*x])* Sqrt[(b*c - a*d)*(1 + Sin[e + f*x])/((c - d)*(a + b*Sin[e + f*x]))]* Sqrt[-(b*c - a*d)*(1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + f*x]))]* EllipticPi[b*(c + d)/(d*(a + b)), ArcSin[Rt[(a + b)/(c + d), 2]* Sqrt[c + d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]]], (a - b)*(c + d)/((a + b)*(c - d))]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]/ Sqrt[c_. + d_.*sin[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && PosQ[(a + b)/(c + d)]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Sqrt[-c - d*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]]* Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[-c - d*Sin[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]/ Sqrt[c_. + d_.*sin[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NegQ[(a + b)/(c + d)]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-2*d/(f*Sqrt[a + b*d])* EllipticF[ ArcSin[Cos[e + f*x]/(1 + d*Sin[e + f*x])], -(a - b*d)/(a + b*d)]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*Sqrt[d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && LtQ[a^2 - b^2, 0] && EqQ[d^2, 1] && GtQ[b*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Sqrt[Sign[b]*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]* Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[Sign[b]*Sin[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*Sqrt[d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && LtQ[a^2 - b^2, 0] && GtQ[b^2, 0] && Not[EqQ[d^2, 1] && GtQ[b*d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-2*Sqrt[a^2]* Sqrt[-Cot[e + f*x]^2]/(a*f*Sqrt[a^2 - b^2]*Cot[e + f*x])* Rt[(a + b)/d, 2]* EllipticF[ ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/ Rt[(a + b)/d, 2]], -(a + b)/(a - b)]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*Sqrt[d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && GtQ[a^2 - b^2, 0] && PosQ[(a + b)/d] && GtQ[a^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-2*Tan[e + f*x]/(a*f)*Rt[(a + b)/d, 2]* Sqrt[a*(1 - Csc[e + f*x])/(a + b)]* Sqrt[a*(1 + Csc[e + f*x])/(a - b)]* EllipticF[ ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/ Rt[(a + b)/d, 2]], -(a + b)/(a - b)]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*Sqrt[d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && PosQ[(a + b)/d]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Sqrt[-d*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]* Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[-d*Sin[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*Sqrt[d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && NegQ[(a + b)/d]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "2*(c + d*Sin[e + f*x])/(f*(b*c - a*d)*Rt[(c + d)/(a + b), 2]* Cos[e + f*x])* Sqrt[(b*c - a*d)*(1 - Sin[e + f*x])/((a + b)*(c + d*Sin[e + f*x]))]* Sqrt[-(b*c - a*d)*(1 + Sin[e + f*x])/((a - b)*(c + d*Sin[e + f*x]))]* EllipticF[ ArcSin[Rt[(c + d)/(a + b), 2]*(Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]])], (a + b)*(c - d)/((a - b)*(c + d))]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_ + b_.*sin[e_. + f_.*x_]]* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && PosQ[(c + d)/(a + b)]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Sqrt[-a - b*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]]* Int[1/(Sqrt[-a - b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_. + b_.*sin[e_. + f_.*x_]]* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NegQ[(c + d)/(a + b)]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-a*d/(2*b)* Int[Sqrt[d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]], x] + d/(2*b)* Int[Sqrt[d*Sin[e + f*x]]*(a + 2*b*Sin[e + f*x])/ Sqrt[a + b*Sin[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sin[e_. + f_.*x_])^(3/2)/Sqrt[a_. + b_.*sin[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-b* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^ n/(f*(m + n)) + 1/(d*(m + n))* Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n - 1)* Simp[a^2*c*d*(m + n) + b*d*(b*c*(m - 1) + a*d*n) + (a*d*(2*b*c + a*d)*(m + n) - b*d*(a*c - b*d*(m + n - 1)))*Sin[e + f*x] + b*d*(b*c*n + a*d*(2*m + n - 1))*Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[0, m, 2] && LtQ[-1, n, 2] && NeQ[m + n, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "b/d*Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1), x] - (b*c - a*d)/d* Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "a*Int[(d*Sin[e + f*x])^n/(a^2 - b^2*Sin[e + f*x]^2), x] - b/d*Int[(d*Sin[e + f*x])^(n + 1)/(a^2 - b^2*Sin[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sin[e_. + f_.*x_])^n_./(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Int[ExpandTrig[(d*sin[e + f*x])^ n*(a - b*sin[e + f*x])^(-m)/(a^2 - b^2*sin[e + f*x]^2)^(-m), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(d_.*sin[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0] && ILtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Unintegrable[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "d^m*Int[(d*Csc[e+f*x])^(n-m)*(b+a*Csc[e+f*x])^m,x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.+b_.*sin[e_.+f_.*x_])^m_.*(d_./sin[e_.+f_.*x_])^n_,x_ Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,d,e,f,n},x] && Not[IntegerQ[n]] && IntegerQ[m] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "d^m*Int[(d*Sec[e+f*x])^(n-m)*(b+a*Sec[e+f*x])^m,x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.+b_.*cos[e_.+f_.*x_])^m_.*(d_./cos[e_.+f_.*x_])^n_,x_ Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,d,e,f,n},x] && Not[IntegerQ[n]] && IntegerQ[m] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "c^IntPart[n]*(c*(d*Sin[e + f*x])^p)^ FracPart[n]/(d*Sin[e + f*x])^(p*FracPart[n])* Int[(a + b*Sin[e + f*x])^m*(d*Sin[e + f*x])^(n*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^ m_.*(c_.*(d_.*sin[e_. + f_.*x_])^p_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "c^IntPart[n]*(c*(d*Cos[e + f*x])^p)^ FracPart[n]/(d*Cos[e + f*x])^(p*FracPart[n])* Int[(a + b*Cos[e + f*x])^m*(d*Cos[e + f*x])^(n*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*cos[e_. + f_.*x_])^ m_.*(c_.*(d_.*cos[e_. + f_.*x_])^p_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Int[(a + b*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n/Sin[e + f*x]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Int[(b + a*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^n/Csc[e + f*x]^m, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && Not[IntegerQ[n]] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Int[(b + a*Sec[e + f*x])^m*(c + d*Sec[e + f*x])^n/Sec[e + f*x]^m, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*cos[e_. + f_.*x_])^m_.*(c_ + d_.*sec[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && Not[IntegerQ[n]] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Sin[e + f*x]^n*(c + d*Csc[e + f*x])^n/(d + c*Sin[e + f*x])^n* Int[(a + b*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n/Sin[e + f*x]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && Not[IntegerQ[n]] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.1 (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Cos[e + f*x]^n*(c + d*Sec[e + f*x])^n/(d + c*Cos[e + f*x])^n* Int[(a + b*Cos[e + f*x])^m*(d + c*Cos[e + f*x])^n/Cos[e + f*x]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*cos[e_. + f_.*x_])^m_*(c_ + d_.*sec[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && Not[IntegerQ[n]] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "1/(b*f)*Subst[Int[(a + x)^m*(c + d/b*x)^n, x], x, b*Sin[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_. + d_.*sin[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "a*Int[Cos[e + f*x]^p*(d*Sin[e + f*x])^n, x] + b/d*Int[Cos[e + f*x]^p*(d*Sin[e + f*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^p_*(d_.*sin[e_. + f_.*x_])^ n_.*(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2] && IntegerQ[ n] && (LtQ[p, 0] && NeQ[a^2 - b^2, 0] || LtQ[0, n, p - 1] || LtQ[p + 1, -n, 2*p + 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "1/a*Int[Cos[e + f*x]^(p - 2)*(d*Sin[e + f*x])^n, x] - 1/(b*d)*Int[Cos[e + f*x]^(p - 2)*(d*Sin[e + f*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^ p_*(d_.*sin[e_. + f_.*x_])^n_./(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && IntegerQ[ n] && (LtQ[0, n, (p + 1)/2] || LeQ[p, -n] && LtQ[-n, 2*p - 3] || GtQ[n, 0] && LeQ[n, -p])"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "1/(b^p*f)* Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + d/b*x)^ n, x], x, b*Sin[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_. + d_.*sin[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, c, d, m, n}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "1/(b^p*f)* Subst[Int[(a + x)^m*(c + d/b*x)^n*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_. + d_.*sin[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "a*Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n, x] + b/d*Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(d_.*sin[e_. + f_.*x_])^ n_.*(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, g, n, p}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "g^2/a*Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x] - g^2/(b*d)* Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^ p_*(d_.*sin[e_. + f_.*x_])^n_./(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "a^m*c^m/g^(2*m)* Int[(g*Cos[e + f*x])^(2*m + p)*(c + d*Sin[e + f*x])^(n - m), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_ + d_.*sin[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && Not[IntegerQ[n] && LtQ[n^2, m^2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "1/(a^(p/2)*c^(p/2))* Int[(a + b*Sin[e + f*x])^(m + p/2)*(c + d*Sin[e + f*x])^(n + p/2), x]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_ + d_.*sin[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[p/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "g*Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])* Int[(g*Cos[e + f*x])^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^ p_/(Sqrt[a_ + b_.*sin[e_. + f_.*x_]]* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "a^IntPart[m]* c^IntPart[m]*(a + b*Sin[e + f*x])^ FracPart[m]*(c + d*Sin[e + f*x])^FracPart[m]/ (g^(2*IntPart[m])*(g*Cos[e + f*x])^(2*FracPart[m]))* Int[(g*Cos[e + f*x])^(2*m + p)/(c + d*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && EqQ[m - n - 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n/(f*g*(m - n - 1))",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m - n - 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-2* b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n/(f*g*(2*n + p + 1)) - b*(2*m + p - 1)/(d*(2*n + p + 1))* Int[(g*Cos[e + f*x])^ p*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[Simplify[m + p/2 - 1/2], 0] && LtQ[n, -1] && NeQ[2*n + p + 1, 0] && Not[ILtQ[Simplify[m + n + p], 0] && GtQ[Simplify[2*m + n + 3*p/2 + 1], 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^ n/(f*g*(m + n + p)) + a*(2*m + p - 1)/(m + n + p)* Int[(g*Cos[e + f*x])^ p*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[Simplify[m + p/2 - 1/2], 0] && Not[LtQ[n, -1]] && Not[IGtQ[Simplify[n + p/2 - 1/2], 0] && GtQ[m - n, 0]] && Not[ILtQ[Simplify[m + n + p], 0] && GtQ[Simplify[2*m + n + 3*p/2 + 1], 0]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "a^IntPart[m]* c^IntPart[m]*(a + b*Sin[e + f*x])^ FracPart[m]*(c + d*Sin[e + f*x])^FracPart[m]/ (g^(2*IntPart[m])*(g*Cos[e + f*x])^(2*FracPart[m]))* Int[(g*Cos[e + f*x])^(2*m + p), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[2*m + p + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n/(a*f*g*(m - n))",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + p + 1, 0] && NeQ[m, n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n/(a*f*g*(2*m + p + 1)) + (m + n + p + 1)/(a*(2*m + p + 1))* Int[(g*Cos[e + f*x])^ p*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + n + p + 1], 0] && NeQ[2*m + p + 1, 0] && (SumSimplerQ[m, 1] || Not[SumSimplerQ[n, 1]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-2* b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n/(f*g*(2*n + p + 1)) - b*(2*m + p - 1)/(d*(2*n + p + 1))* Int[(g*Cos[e + f*x])^ p*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && LtQ[n, -1] && NeQ[2*n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^ n/(f*g*(m + n + p)) + a*(2*m + p - 1)/(m + n + p)* Int[(g*Cos[e + f*x])^ p*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && NeQ[m + n + p, 0] && Not[LtQ[0, n, m]] && IntegersQ[2*m, 2*n, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n/(a*f*g*(2*m + p + 1)) + (m + n + p + 1)/(a*(2*m + p + 1))* Int[(g*Cos[e + f*x])^ p*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && NeQ[2*m + p + 1, 0] && Not[LtQ[m, n, -1]] && IntegersQ[2*m, 2*n, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "a^IntPart[m]* c^IntPart[m]*(a + b*Sin[e + f*x])^ FracPart[m]*(c + d*Sin[e + f*x])^FracPart[m]/ (g^(2*IntPart[m])*(g*Cos[e + f*x])^(2*FracPart[m]))* Int[(g*Cos[e + f*x])^(2*m + p)*(c + d*Sin[e + f*x])^(n - m), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && (FractionQ[m] || Not[FractionQ[n]])"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-d*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ m/(f*g*(m + p + 1))",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[a*d*m + b*c*(m + p + 1), 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-(b*c + a*d)*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m/(a*f*g*(p + 1)) + b*(a*d*m + b*c*(m + p + 1))/(a*g^2*(p + 1))* Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, -1] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-d*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ m/(f*g*(m + p + 1)) + (a*d*m + b*c*(m + p + 1))/(b*(m + p + 1))* Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[Simplify[(2*m + p + 1)/2], 0] && NeQ[m + p + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "2*(b*c - a*d)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(2*m + 3)) + 1/(b^3*(2*m + 3))* Int[(a + b*Sin[e + f*x])^(m + 2)*(b*c + 2*a*d*(m + 1) - b*d*(2*m + 3)*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^2*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -3/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "d*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 2)/(b^2*f*(m + 3)) - 1/(b^2*(m + 3))* Int[(a + b*Sin[e + f*x])^(m + 1)*(b*d*(m + 2) - a*c*(m + 3) + (b*c*(m + 3) - a*d*(m + 4))*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^2*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GeQ[m, -3/2] && LtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "(b*c - a*d)*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ m/(a*f*g*(2*m + p + 1)) + (a*d*m + b*c*(m + p + 1))/(a*b*(2*m + p + 1))* Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1] || ILtQ[Simplify[m + p], 0]) && NeQ[2*m + p + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-d*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ m/(f*g*(m + p + 1)) + (a*d*m + b*c*(m + p + 1))/(b*(m + p + 1))* Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[m + p + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ m*(d + c*Sin[e + f*x])/(f*g*(p + 1)) + 1/(g^2*(p + 1))* Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1)* Simp[a*c*(p + 2) + b*d*m + b*c*(m + p + 2)*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && LtQ[p, -1] && IntegerQ[2*m] && Not[EqQ[m, 1] && NeQ[c^2 - d^2, 0] && SimplerQ[c + d*x, a + b*x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-d*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ m/(f*g*(m + p + 1)) + 1/(m + p + 1)* Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)* Simp[a*c*(m + p + 1) + b*d*m + (a*d*m + b*c*(m + p + 1))*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && Not[LtQ[p, -1]] && IntegerQ[2*m] && Not[EqQ[m, 1] && NeQ[c^2 - d^2, 0] && SimplerQ[c + d*x, a + b*x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*(b*c*(m + p + 1) - a*d*p + b*d*(m + 1)*Sin[e + f*x])/(b^2* f*(m + 1)*(m + p + 1)) + g^2*(p - 1)/(b^2*(m + 1)*(m + p + 1))* Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1)* Simp[b*d*(m + 1) + (b*c*(m + p + 1) - a*d*p)*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && NeQ[m + p + 1, 0] && IntegerQ[2*m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-(b*c - a*d)*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)/(f*g*(a^2 - b^2)*(m + 1)) + 1/((a^2 - b^2)*(m + 1))* Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1)* Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + p + 2)*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*(b*c*(m + p + 1) - a*d*p + b*d*(m + p)*Sin[e + f*x])/(b^2* f*(m + p)*(m + p + 1)) + g^2*(p - 1)/(b^2*(m + p)*(m + p + 1))* Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^m* Simp[b*(a*d*m + b*c*(m + p + 1)) + (a*b*c*(m + p + 1) - d*(a^2*p - b^2*(m + p)))*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[p, 1] && NeQ[m + p, 0] && NeQ[m + p + 1, 0] && IntegerQ[2*m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*(b*c - a*d - (a*c - b*d)*Sin[e + f*x])/(f* g*(a^2 - b^2)*(p + 1)) + 1/(g^2*(a^2 - b^2)*(p + 1))* Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m* Simp[c*(a^2*(p + 2) - b^2*(m + p + 2)) + a*b*d*m + b*(a*c - b*d)*(m + p + 3)*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && LtQ[p, -1] && IntegerQ[2*m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "d/b*Int[(g*Cos[e + f*x])^p, x] + (b*c - a*d)/b* Int[(g*Cos[e + f*x])^p/(a + b*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^ p_*(c_. + d_.*sin[e_. + f_.*x_])/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "c*g*(g*Cos[e + f*x])^(p - 1)/(f*(1 + Sin[e + f*x])^((p - 1)/2)*(1 - Sin[e + f*x])^((p - 1)/2))* Subst[ Int[(1 + d/c*x)^((p + 1)/2)*(1 - d/c*x)^((p - 1)/2)*(a + b*x)^m, x], x, Sin[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p}, x] && NeQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "a^(2*m)*Int[(d*Sin[e + f*x])^n/(a - b*Sin[e + f*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p] && EqQ[2*m + p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)/(2*b*f*g*(m + 1)) + a/(2*g^2)* Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_* sin[e_. + f_.*x_]^2*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[m - p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m/(a*f*g*m) - 1/g^2*Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_* sin[e_. + f_.*x_]^2*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[m + p + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "1/a^p*Int[ ExpandTrig[(d*sin[e + f*x])^ n*(a - b*sin[e + f*x])^(p/2)*(a + b*sin[e + f*x])^(m + p/2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^p_*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, n, p/2] && (GtQ[m, 0] && GtQ[p, 0] && LtQ[-m - p, n, -1] || GtQ[m, 2] && LtQ[p, 0] && GtQ[m + p/2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Int[ExpandTrig[(g*cos[e + f*x])^ p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "1/b^2*Int[(d*Sin[e + f*x])^ n*(a + b*Sin[e + f*x])^(m + 1)*(a - b*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^2*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && (ILtQ[m, 0] || Not[IGtQ[n, 0]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "(a/g)^(2*m)* Int[(g*Cos[e + f*x])^(2*m + p)*(d*Sin[e + f*x])^ n/(a - b*Sin[e + f*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "(a/g)^(2*m)* Int[(g*Cos[e + f*x])^(2*m + p)*(d*Sin[e + f*x])^ n/(a - b*Sin[e + f*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, g, n}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && RationalQ[p] && (EqQ[2*m + p, 0] || GtQ[2*m + p, 0] && LtQ[p, -1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ m/(a*f*g*(2*m + p + 1)) - 1/(a^2*(2*m + p + 1))* Int[(g*Cos[e + f*x])^ p*(a + b*Sin[e + f*x])^(m + 1)*(a*m - b*(2*m + p + 1)*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_* sin[e_. + f_.*x_]^2*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && LeQ[m, -1/2] && NeQ[2*m + p + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)/(b*f*g*(m + p + 2)) + 1/(b*(m + p + 2))* Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^ m*(b*(m + 1) - a*(p + 1)*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_* sin[e_. + f_.*x_]^2*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[m + p + 2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "1/b^2*Int[(d*Sin[e + f*x])^ n*(a + b*Sin[e + f*x])^(m + 1)*(a - b*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^2*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-2/(a*b*d)* Int[(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 2), x] + 1/a^2* Int[(d*Sin[e + f*x])^ n*(a + b*Sin[e + f*x])^(m + 2)*(1 + Sin[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^4*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "1/d^4*Int[(d*Sin[e + f*x])^(n + 4)*(a + b*Sin[e + f*x])^m, x] + Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^ m*(1 - 2*Sin[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^4*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && Not[IGtQ[m, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "a^m*Cos[e + f*x]/(f*Sqrt[1 + Sin[e + f*x]]*Sqrt[1 - Sin[e + f*x]])* Subst[ Int[(d*x)^n*(1 + b/a*x)^(m + (p - 1)/2)*(1 - b/a*x)^((p - 1)/2), x], x, Sin[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^p_*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[p/2] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Cos[e + f*x]/(a^(p - 2)*f*Sqrt[a + b*Sin[e + f*x]]* Sqrt[a - b*Sin[e + f*x]])* Subst[ Int[(d*x)^n (a + b*x)^(m + p/2 - 1/2)*(a - b*x)^(p/2 - 1/2), x], x, Sin[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^p_*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[p/2] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Int[ExpandTrig[(g*cos[e + f*x])^ p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0] && (IntegerQ[p] || IGtQ[n, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "a^m*g*(g*Cos[e + f*x])^(p - 1)/(f*(1 + Sin[e + f*x])^((p - 1)/2)*(1 - Sin[e + f*x])^((p - 1)/2))* Subst[ Int[(d*x)^n*(1 + b/a*x)^(m + (p - 1)/2)*(1 - b/a*x)^((p - 1)/2), x], x, Sin[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, n, p}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "g*(g*Cos[e + f*x])^(p - 1)/(f*(a + b*Sin[e + f*x])^((p - 1)/2)*(a - b*Sin[e + f*x])^((p - 1)/2))* Subst[ Int[(d*x)^n*(a + b*x)^(m + (p - 1)/2)*(a - b*x)^((p - 1)/2), x], x, Sin[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n, p}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-g*(g*Cos[e + f*x])^(p - 1)* Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^(m + 1)/(a*d* f*(m + 1)) + g^2*(2*m + 3)/(2*a*(m + 1))* Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1)/ Sqrt[d*Sin[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_/ Sqrt[d_.*sin[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && EqQ[m + p + 1/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "2*(g*Cos[e + f*x])^(p + 1)* Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^m/(d*f*g*(2*m + 1)) + 2*a*m/(g^2*(2*m + 1))* Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1)/ Sqrt[d*Sin[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^m_/ Sqrt[d_.*sin[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && EqQ[m + p + 3/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m*(1 - Sin[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^2*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": " (a^2-b^2)*Cos[e+f*x]*Sin[e+f*x]^(n+1)*(a+b*Sin[e+f*x])^(m+1)/(a*b^2*d* (m+1)) - (a^2*(n+1)-b^2*(m+n+2))*Cos[e+f*x]*Sin[e+f*x]^(n+1)*(a+b*Sin[e+f*x]) ^(m+2)/(a^2*b^2*d*(n+1)*(m+1)) + 1/(a^2*b*(n+1)*(m+1))*Int[Sin[e+f*x]^(n+1)*(a+b*Sin[e+f*x])^(m+1)* Simp[a^2*(n+1)*(n+2)-b^2*(m+n+2)*(m+n+3)+a*b*(m+1)*Sin[e+f*x]-(a^ 2*(n+1)*(n+3)-b^2*(m+n+2)*(m+n+4))*Sin[e+f*x]^2,x],x]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_.+f_.*x_]^4*sin[e_.+f_.*x_]^n_*(a_+b_.*sin[e_.+f_.*x_])^ m_,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,d,e,f},x] && NeQ[a^2-b^2,0] && IntegersQ[2*m,2*n] &&  LtQ[m,-1] && LtQ[n,-1] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Cos[e + f*x]*(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 1)/(a*d*f*(n + 1)) - (a^2*(n + 1) - b^2*(m + n + 2))* Cos[e + f*x]*(d*Sin[e + f*x])^(n + 2)*(a + b*Sin[e + f*x])^(m + 1)/(a^2*b*d^2*f*(n + 1)*(m + 1)) + 1/(a^2*b*d*(n + 1)*(m + 1))* Int[(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 1)* Simp[a^2*(n + 1)*(n + 2) - b^2*(m + n + 2)*(m + n + 3) + a*b*(m + 1)* Sin[e + f*x] - (a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m + n + 4))*Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^4*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && LtQ[m, -1] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "(a^2 - b^2)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^(n + 1)/(a*b^2*d*f*(m + 1)) + (a^2*(n - m + 1) - b^2*(m + n + 2))* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 2)*(d*Sin[e + f*x])^(n + 1)/(a^2*b^2*d*f*(m + 1)*(m + 2)) - 1/(a^2*b^2*(m + 1)*(m + 2))* Int[(a + b*Sin[e + f*x])^(m + 2)*(d*Sin[e + f*x])^n* Simp[a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m + n + 3) + a*b*(m + 2)* Sin[e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*(m + n + 2)*(m + n + 4))*Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^4*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && LtQ[m, -1] && Not[LtQ[n, -1]] && (LtQ[m, -2] || EqQ[m + n + 4, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "(a^2 - b^2)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^(n + 1)/(a*b^2*d*f*(m + 1)) - Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 2)*(d*Sin[e + f*x])^(n + 1)/(b^2* d*f*(m + n + 4)) - 1/(a*b^2*(m + 1)*(m + n + 4))* Int[(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^n* Simp[a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m + n + 4) + a*b*(m + 1)* Sin[e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*(m + n + 3)*(m + n + 4))*Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^4*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && LtQ[m, -1] && Not[LtQ[n, -1]] && NeQ[m + n + 4, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^(n + 1)/(a*d*f*(n + 1)) - b*(m + n + 2)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^(n + 2)/(a^2*d^2*f*(n + 1)*(n + 2)) - 1/(a^2*d^2*(n + 1)*(n + 2))* Int[(a + b*Sin[e + f*x])^m*(d*Sin[e + f*x])^(n + 2)* Simp[a^2*n*(n + 2) - b^2*(m + n + 2)*(m + n + 3) + a*b*m*Sin[ e + f*x] - (a^2*(n + 1)*(n + 2) - b^2*(m + n + 2)*(m + n + 4))*Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^4*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n]) && Not[m < -1] && LtQ[n, -1] && (LtQ[n, -2] || EqQ[m + n + 4, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^(n + 1)/(a*d*f*(n + 1)) - Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^(n + 2)/(b* d^2*f*(m + n + 4)) + 1/(a*b*d*(n + 1)*(m + n + 4))* Int[(a + b*Sin[e + f*x])^m*(d*Sin[e + f*x])^(n + 1)* Simp[a^2*(n + 1)*(n + 2) - b^2*(m + n + 2)*(m + n + 4) + a*b*(m + 3)* Sin[e + f*x] - (a^2*(n + 1)*(n + 3) - b^2*(m + n + 3)*(m + n + 4))*Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^4*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n]) && Not[m < -1] && LtQ[n, -1] && NeQ[m + n + 4, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "a*(n + 3)* Cos[e + f*x]*(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 1)/(b^2*d* f*(m + n + 3)*(m + n + 4)) - Cos[e + f*x]*(d*Sin[e + f*x])^(n + 2)*(a + b*Sin[e + f*x])^(m + 1)/(b* d^2*f*(m + n + 4)) - 1/(b^2*(m + n + 3)*(m + n + 4))* Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m* Simp[a^2*(n + 1)*(n + 3) - b^2*(m + n + 3)*(m + n + 4) + a*b*m*Sin[ e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*(m + n + 3)*(m + n + 5))*Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^4*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n]) && Not[m < -1] && Not[LtQ[n, -1]] && NeQ[m + n + 3, 0] && NeQ[m + n + 4, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Cos[e + f*x]*(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 1)/(a*d*f*(n + 1)) - b*(m + n + 2)* Cos[e + f*x]*(d*Sin[e + f*x])^(n + 2)*(a + b*Sin[e + f*x])^(m + 1)/(a^2*d^2*f*(n + 1)*(n + 2)) - a*(n + 5)* Cos[e + f*x]*(d*Sin[e + f*x])^(n + 3)*(a + b*Sin[e + f*x])^(m + 1)/(b^2*d^3* f*(m + n + 5)*(m + n + 6)) + Cos[e + f*x]*(d*Sin[e + f*x])^(n + 4)*(a + b*Sin[e + f*x])^(m + 1)/(b* d^4*f*(m + n + 6)) + 1/(a^2*b^2*d^2*(n + 1)*(n + 2)*(m + n + 5)*(m + n + 6))* Int[(d*Sin[e + f*x])^(n + 2)*(a + b*Sin[e + f*x])^m* Simp[a^4*(n + 1)*(n + 2)*(n + 3)*(n + 5) - a^2*b^2*(n + 2)*(2*n + 1)*(m + n + 5)*(m + n + 6) + b^4*(m + n + 2)*(m + n + 3)*(m + n + 5)*(m + n + 6) + a*b*m*(a^2*(n + 1)*(n + 2) - b^2*(m + n + 5)*(m + n + 6))* Sin[e + f*x] - (a^4*(n + 1)*(n + 2)*(4 + n)*(n + 5) + b^4*(m + n + 2)*(m + n + 4)*(m + n + 5)*(m + n + 6) - a^2*b^2*(n + 1)*(n + 2)*(m + n + 5)*(2*n + 2*m + 13))* Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^6*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && NeQ[n, -1] && NeQ[n, -2] && NeQ[m + n + 5, 0] && NeQ[m + n + 6, 0] && Not[IGtQ[m, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Int[ExpandTrig[(d*sin[e + f*x])^n*(a + b*sin[e + f*x])^ m*(1 - sin[e + f*x]^2)^(p/2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^p_*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[m, 2*n, p/2] && (LtQ[m, -1] || EqQ[m, -1] && GtQ[p, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Int[ExpandTrig[(g*cos[e + f*x])^p, sin[e + f*x]^n/(a + b*sin[e + f*x]), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_* sin[e_. + f_.*x_]^n_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[n] && (LtQ[n, 0] || IGtQ[p + 1/2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "g^2/a*Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x] - b*g^2/(a^2*d)* Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x] - g^2*(a^2 - b^2)/(a^2*d^2)* Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 2)/(a + b*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^ p_*(d_.*sin[e_. + f_.*x_])^n_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && GtQ[p, 1] && (LeQ[n, -2] || EqQ[n, -3/2] && EqQ[p, 3/2])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "g^2/(a*b)* Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^ n*(b - a*Sin[e + f*x]), x] + g^2*(a^2 - b^2)/(a*b*d)* Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1)/(a + b*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^ p_*(d_.*sin[e_. + f_.*x_])^n_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && GtQ[p, 1] && (LtQ[n, -1] || EqQ[p, 3/2] && EqQ[n, -1/2])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "g^2/b^2*Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^ n*(a - b*Sin[e + f*x]), x] - g^2*(a^2 - b^2)/b^2* Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^ n/(a + b*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^ p_*(d_.*sin[e_. + f_.*x_])^n_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && GtQ[p, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": " g^2*Int[(g*Cos[e+f*x])^(p-2)*(d*Sin[e+f*x])^n/(a+b*Sin[e+f*x]),x] - g^2/d^2*Int[(g*Cos[e+f*x])^(p-2)*(d*Sin[e+f*x])^(n+2)/(a+b*Sin[e+f* x]),x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_.+f_.*x_])^p_*(d_.*sin[e_.+f_.*x_])^n_/(a_+b_.*sin[ e_.+f_.*x_]),x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,d,e,f,g},x] && NeQ[a^2-b^2,0] &&  IntegersQ[2*n,2*p] && GtQ[p,1] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "a*d^2/(a^2 - b^2)* Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 2), x] - b*d/(a^2 - b^2)* Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 1), x] - a^2*d^2/(g^2*(a^2 - b^2))* Int[(g*Cos[e + f*x])^(p + 2)*(d*Sin[e + f*x])^(n - 2)/(a + b*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^ p_*(d_.*sin[e_. + f_.*x_])^n_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && LtQ[p, -1] && GtQ[n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-d/(a^2 - b^2)* Int[(g*Cos[e + f*x])^ p*(d*Sin[e + f*x])^(n - 1)*(b - a*Sin[e + f*x]), x] + a*b*d/(g^2*(a^2 - b^2))* Int[(g*Cos[e + f*x])^(p + 2)*(d*Sin[e + f*x])^(n - 1)/(a + b*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^ p_*(d_.*sin[e_. + f_.*x_])^n_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && LtQ[p, -1] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "1/(a^2 - b^2)* Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n*(a - b*Sin[e + f*x]), x] - b^2/(g^2*(a^2 - b^2))* Int[(g*Cos[e + f*x])^(p + 2)*(d*Sin[e + f*x])^ n/(a + b*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^ p_*(d_.*sin[e_. + f_.*x_])^n_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-4*Sqrt[2]*g/f* Subst[Int[x^2/(((a + b)*g^2 + (a - b)*x^4)*Sqrt[1 - x^4/g^2]), x], x, Sqrt[g*Cos[e + f*x]]/Sqrt[1 + Sin[e + f*x]]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[g_.* cos[e_. + f_.*x_]]/(Sqrt[ sin[e_. + f_.*x_]]*(a_ + b_.*sin[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Sqrt[Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]* Int[Sqrt[g*Cos[e + f*x]]/(Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[g_.* cos[e_. + f_.*x_]]/(Sqrt[ d_*sin[e_. + f_.*x_]]*(a_ + b_.*sin[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "With[{q = Rt[-a^2 + b^2, 2]}, 2*Sqrt[2]*d*(b + q)/(f*q)* Subst[Int[1/((d*(b + q) + a*x^2)*Sqrt[1 - x^4/d^2]), x], x, Sqrt[d*Sin[e + f*x]]/Sqrt[1 + Cos[e + f*x]]] - 2*Sqrt[2]*d*(b - q)/(f*q)* Subst[Int[1/((d*(b - q) + a*x^2)*Sqrt[1 - x^4/d^2]), x], x, Sqrt[d*Sin[e + f*x]]/Sqrt[1 + Cos[e + f*x]]]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[d_.* sin[e_. + f_.*x_]]/(Sqrt[ cos[e_. + f_.*x_]]*(a_ + b_.*sin[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Sqrt[Cos[e + f*x]]/Sqrt[g*Cos[e + f*x]]* Int[Sqrt[d*Sin[e + f*x]]/(Sqrt[Cos[e + f*x]]*(a + b*Sin[e + f*x])), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[d_.* sin[e_. + f_.*x_]]/(Sqrt[ g_.*cos[e_. + f_.*x_]]*(a_ + b_.*sin[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "d/b*Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 1), x] - a*d/b* Int[(g*Cos[e + f*x])^ p*(d*Sin[e + f*x])^(n - 1)/(a + b*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^ p_*(d_.*sin[e_. + f_.*x_])^n_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && LtQ[-1, p, 1] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "1/a*Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n, x] - b/(a*d)* Int[(g*Cos[e + f*x])^ p*(d*Sin[e + f*x])^(n + 1)/(a + b*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^ p_*(d_.*sin[e_. + f_.*x_])^n_/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && LtQ[-1, p, 1] && LtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "2*a*b/d*Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n + 1), x] + Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^ n*(a^2 + b^2*Sin[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, g, n, p}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Int[ExpandTrig[(g*cos[e + f*x])^ p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, g, n, p}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[m] && (GtQ[m, 0] || IntegerQ[n])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "g^2/a*Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^ n*(a + b*Sin[e + f*x])^(m + 1), x] - b*g^2/(a^2*d)* Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 1), x] - g^2*(a^2 - b^2)/(a^2*d^2)* Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 2)*(a + b*Sin[e + f*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(d_.*sin[e_. + f_.*x_])^ n_*(a_ + b_.*sin[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[m, 2*n, 2*p] && LtQ[m, 0] && GtQ[p, 1] && (LeQ[n, -2] || EqQ[m, -1] && EqQ[n, -3/2] && EqQ[p, 3/2])"
  },
  {
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       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "a^(2*m)*Int[(c + d*Sin[e + f*x])^n/(a - b*Sin[e + f*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p] && EqQ[2*m + p, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "(a/g)^(2*m)* Int[(g*Cos[e + f*x])^(2*m + p)*(c + d*Sin[e + f*x])^ n/(a - b*Sin[e + f*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && (EqQ[2*m + p, 0] || GtQ[2*m + p, 0] && LtQ[p, -1])"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "1/b^2*Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^ n*(a - b*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^2*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "a^m*Cos[e + f*x]/(f*Sqrt[1 + Sin[e + f*x]]*Sqrt[1 - Sin[e + f*x]])* Subst[ Int[(1 + b/a*x)^(m + (p - 1)/2)*(1 - b/a*x)^((p - 1)/2)*(c + d*x)^ n, x], x, Sin[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[p/2] && IntegerQ[m]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Cos[e + f*x]/(a^(p - 2)*f*Sqrt[a + b*Sin[e + f*x]]* Sqrt[a - b*Sin[e + f*x]])* Subst[ Int[(a + b*x)^(m + p/2 - 1/2)*(a - b*x)^(p/2 - 1/2)*(c + d*x)^n, x], x, Sin[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[p/2] && Not[IntegerQ[m]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Int[ExpandTrig[(g*cos[e + f*x])^ p, (a + b*sin[e + f*x])^m*(c + d*sin[e + f*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0] && (IntegerQ[p] || IGtQ[n, 0])"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "a^m*g*(g*Cos[e + f*x])^(p - 1)/(f*(1 + Sin[e + f*x])^((p - 1)/2)*(1 - Sin[e + f*x])^((p - 1)/2))* Subst[ Int[(1 + b/a*x)^(m + (p - 1)/2)*(1 - b/a*x)^((p - 1)/2)*(c + d*x)^ n, x], x, Sin[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "g*(g*Cos[e + f*x])^(p - 1)/(f*(a + b*Sin[e + f*x])^((p - 1)/2)*(a - b*Sin[e + f*x])^((p - 1)/2))* Subst[ Int[(a + b*x)^(m + (p - 1)/2)*(a - b*x)^((p - 1)/2)*(c + d*x)^n, x], x, Sin[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[m]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^ n*(1 - Sin[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^2*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Int[ExpandTrig[(a + b*sin[e + f*x])^m*(c + d*sin[e + f*x])^ n*(1 - sin[e + f*x]^2)^(p/2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && IGtQ[p/2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Int[ExpandTrig[(g*cos[e + f*x])^p*(a + b*sin[e + f*x])^ m*(c + d*sin[e + f*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Unintegrable[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_. + d_.*sin[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "g^(2*IntPart[p])*(g*Cos[e + f*x])^FracPart[p]*(g*Sec[e + f*x])^ FracPart[p]* Int[(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n/(g*Cos[e + f*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*sec[e_. + f_.*x_])^p_*(a_. + b_.*sin[e_. + f_.*x_])^ m_.*(c_. + d_.*sin[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "g^(2*IntPart[p])*(g*Sin[e + f*x])^FracPart[p]*(g*Csc[e + f*x])^ FracPart[p]* Int[(a + b*Cos[e + f*x])^ m*(c + d*Cos[e + f*x])^n/(g*Sin[e + f*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*csc[e_. + f_.*x_])^p_*(a_. + b_.*cos[e_. + f_.*x_])^ m_.*(c_. + d_.*cos[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "g/d*Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[g*Sin[e + f*x]], x] - c*g/d* Int[Sqrt[ a + b*Sin[e + f*x]]/(Sqrt[ g*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[g_.*sin[e_. + f_.*x_]]* Sqrt[a_ + b_.*sin[e_. + f_.*x_]]/(c_ + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "b/d*Int[Sqrt[g*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]], x] - (b*c - a*d)/d* Int[Sqrt[ g*Sin[e + f*x]]/(Sqrt[ a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[g_.*sin[e_. + f_.*x_]]* Sqrt[a_ + b_.*sin[e_. + f_.*x_]]/(c_ + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-2*b/f* Subst[Int[1/(b*c + a*d + c*g*x^2), x], x, b*Cos[e + f*x]/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]/(Sqrt[ g_.*sin[e_. + f_.*x_]]*(c_ + d_.*sin[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-Sqrt[a + b]/(c*f)* EllipticE[ ArcSin[Cos[e + f*x]/(1 + Sin[e + f*x])], -(a - b)/(a + b)]",
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            "lhs": "Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]/(Sqrt[ sin[e_. + f_.*x_]]*(c_ + d_.*sin[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[d, c] && GtQ[b^2 - a^2, 0] && GtQ[b, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-Sqrt[a + b*Sin[e + f*x]]* Sqrt[d*Sin[e + f*x]/(c + d*Sin[e + f*x])]/ (d*f*Sqrt[g*Sin[e + f*x]]* Sqrt[c^2*(a + b*Sin[e + f*x])/((a*c + b*d)*(c + d*Sin[e + f*x]))])* EllipticE[ ArcSin[c* Cos[e + f*x]/(c + d*Sin[e + f*x])], (b*c - a*d)/(b*c + a*d)]",
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            "lhs": "Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]/(Sqrt[ g_.*sin[e_. + f_.*x_]]*(c_ + d_.*sin[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]"
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       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "a/c*Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]), x] + (b*c - a*d)/(c*g)* Int[Sqrt[ g*Sin[e + f*x]]/(Sqrt[ a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x]",
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            "lhs": "Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]/(Sqrt[ g_.*sin[e_. + f_.*x_]]*(c_ + d_.*sin[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
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       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "1/c*Int[Sqrt[a + b*Sin[e + f*x]]/Sin[e + f*x], x] - d/c*Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x]), x]",
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            "lhs": "Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]/(sin[ e_. + f_.*x_]*(c_ + d_.*sin[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]"
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       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "a/c*Int[1/(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]), x] + (b*c - a*d)/c* Int[1/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x]",
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            "lhs": "Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]/(sin[ e_. + f_.*x_]*(c_ + d_.*sin[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-a*g/(b*c - a*d)* Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]), x] + c*g/(b*c - a*d)* Int[Sqrt[ a + b*Sin[e + f*x]]/(Sqrt[ g*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "2*Sqrt[-Cot[e + f*x]^2]* Sqrt[g*Sin[e + f*x]]/(f*(c + d)*Cot[e + f*x]* Sqrt[a + b*Sin[e + f*x]])*Sqrt[(b + a*Csc[e + f*x])/(a + b)]* EllipticPi[2*c/(c + d), ArcSin[Sqrt[1 - Csc[e + f*x]]/Sqrt[2]], 2*a/(a + b)]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[g_.* sin[e_. + f_.*x_]]/(Sqrt[ a_ + b_.*sin[e_. + f_.*x_]]*(c_ + d_.*sin[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "b/(b*c - a*d)* Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]), x] - d/(b*c - a*d)* Int[Sqrt[ a + b*Sin[e + f*x]]/(Sqrt[ g*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[g_.*sin[e_. + f_.*x_]]* Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*(c_ + d_.*sin[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "1/c*Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]), x] - d/(c*g)* Int[Sqrt[ g*Sin[e + f*x]]/(Sqrt[ a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[g_.*sin[e_. + f_.*x_]]* Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*(c_ + d_.*sin[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "d^2/(c*(b*c - a*d))* Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x]), x] + 1/(c*(b*c - a*d))* Int[(b*c - a*d - b*d*Sin[e + f*x])/(Sin[e + f*x]* Sqrt[a + b*Sin[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(sin[e_. + f_.*x_]* Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*(c_ + d_.*sin[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "1/c*Int[1/(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]), x] - d/c*Int[1/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(sin[e_. + f_.*x_]* Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*(c_ + d_.*sin[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-d/c* Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x] + 1/c*Int[ Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]/Sin[e + f*x], x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]/(sin[e_. + f_.*x_]* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[b*c + a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-2*a/f* Subst[Int[1/(1 - a*c*x^2), x], x, Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]* Sqrt[c + d*Sin[e + f*x]])]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]/(sin[e_. + f_.*x_]* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[b*c + a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "(b*c - a*d)/c* Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x] + a/c*Int[ Sqrt[c + d*Sin[e + f*x]]/(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]/(sin[e_. + f_.*x_]* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-2*(a + b*Sin[e + f*x])/(c*f*Rt[(a + b)/(c + d), 2]* Cos[e + f*x])* Sqrt[-(b*c - a*d)*(1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + f*x]))]* Sqrt[(b*c - a*d)*(1 + Sin[e + f*x])/((c - d)*(a + b*Sin[e + f*x]))]* EllipticPi[a*(c + d)/(c*(a + b)), ArcSin[Rt[(a + b)/(c + d), 2]* Sqrt[c + d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]]], (a - b)*(c + d)/((a + b)*(c - d))]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]/(sin[e_. + f_.*x_]* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])* Int[1/(Cos[e + f*x]*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(sin[e_. + f_.*x_]*Sqrt[a_ + b_.*sin[e_. + f_.*x_]]* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "-b/a* Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x] + 1/a*Int[ Sqrt[a + b*Sin[e + f*x]]/(Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(sin[e_. + f_.*x_]*Sqrt[a_ + b_.*sin[e_. + f_.*x_]]* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && (NeQ[a^2 - b^2, 0] || NeQ[c^2 - d^2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]/Cos[e + f*x]* Int[Cot[e + f*x], x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]/sin[e_. + f_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "d*Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x] + c*Int[ Sqrt[a + b*Sin[e + f*x]]/(Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]/sin[e_. + f_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && (NeQ[a^2 - b^2, 0] || NeQ[c^2 - d^2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "a^n*c^n*Int[Tan[e + f*x]^p*(a + b*Sin[e + f*x])^(m - n), x]",
     "rulenumber": 0,
            "lhs": "Int[sin[e_. + f_.*x_]^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_ + d_.*sin[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[p + 2*n, 0] && IntegerQ[n]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Sqrt[a - b*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]/(f*Cos[e + f*x])* Subst[ Int[(g*x)^p*(a + b*x)^(m - 1/2)*(c + d*x)^n/Sqrt[a - b*x], x], x, Sin[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*sin[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && IntegerQ[m - 1/2]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Int[ExpandTrig[(g*sin[e + f*x])^p*(a + b*sin[e + f*x])^ m*(c + d*sin[e + f*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*sin[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[b*c - a*d, 0] && (IntegersQ[m, n] || IntegersQ[m, p] || IntegersQ[n, p]) && NeQ[p, 2]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Unintegrable[(g*Sin[e + f*x])^p*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*sin[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[p, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "g^(m + n)* Int[(g*Sin[e + f*x])^(p - m - n)*(b + a*Sin[e + f*x])^ m*(d + c*Sin[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*sin[e_. + f_.*x_])^p_.*(a_. + b_.*csc[e_. + f_.*x_])^ m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b*c - a*d, 0] && Not[IntegerQ[p]] && IntegerQ[m] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "(g*Csc[e + f*x])^p*(g*Sin[e + f*x])^p* Int[(a + b*Csc[e + f*x])^ m*(c + d*Csc[e + f*x])^n/(g*Csc[e + f*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*sin[e_. + f_.*x_])^p_.*(a_. + b_.*csc[e_. + f_.*x_])^ m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[b*c - a*d, 0] && Not[IntegerQ[p]] && Not[IntegerQ[m] && IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "g^n*Int[(g*Sin[e + f*x])^(p - n)*(a + b*Sin[e + f*x])^ m*(d + c*Sin[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*sin[e_. + f_.*x_])^p_.*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IntegerQ[n]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Int[(b + a*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^n/ Csc[e + f*x]^(m + p), x]",
     "rulenumber": 0,
            "lhs": "Int[sin[e_. + f_.*x_]^p_.*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && Not[IntegerQ[n]] && IntegerQ[m] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Csc[e + f*x]^p*(g*Sin[e + f*x])^p* Int[(b + a*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^n/ Csc[e + f*x]^(m + p), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*sin[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p}, x] && Not[IntegerQ[n]] && IntegerQ[m] && Not[IntegerQ[p]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "(g*Sin[e + f*x])^ n*(c + d*Csc[e + f*x])^n/(d + c*Sin[e + f*x])^n* Int[(g*Sin[e + f*x])^(p - n)*(a + b*Sin[e + f*x])^ m*(d + c*Sin[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*sin[e_. + f_.*x_])^p_.*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && Not[IntegerQ[n]] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "g^(m + n)* Int[(g*Csc[e + f*x])^(p - m - n)*(b + a*Csc[e + f*x])^ m*(d + c*Csc[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*csc[e_. + f_.*x_])^p_.*(a_. + b_.*sin[e_. + f_.*x_])^ m_.*(c_ + d_.*sin[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b*c - a*d, 0] && Not[IntegerQ[p]] && IntegerQ[m] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "(g*Csc[e + f*x])^p*(g*Sin[e + f*x])^p* Int[(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n/(g*Sin[e + f*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*csc[e_. + f_.*x_])^p_.*(a_. + b_.*sin[e_. + f_.*x_])^ m_.*(c_ + d_.*sin[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[b*c - a*d, 0] && Not[IntegerQ[p]] && Not[IntegerQ[m] && IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "g^m*Int[(g*Csc[e + f*x])^(p - m)*(b + a*Csc[e + f*x])^ m*(c + d*Csc[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*csc[e_. + f_.*x_])^p_.*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Int[(a + b*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n/ Sin[e + f*x]^(n + p), x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]^p_.*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && Not[IntegerQ[m]] && IntegerQ[n] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "Sin[e + f*x]^p*(g*Csc[e + f*x])^p* Int[(a + b*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n/ Sin[e + f*x]^(n + p), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*csc[e_. + f_.*x_])^p_*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, p}, x] && Not[IntegerQ[m]] && IntegerQ[n] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
       "filename": "4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m",
            "rhs": "(a + b*Sin[e + f*x])^ m*(g*Csc[e + f*x])^m/(b + a*Csc[e + f*x])^m* Int[(g*Csc[e + f*x])^(p - m)*(b + a*Csc[e + f*x])^ m*(c + d*Csc[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*csc[e_. + f_.*x_])^p_.*(a_ + b_.*sin[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && Not[IntegerQ[m]] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "Int[ExpandTrig[ sin[e + f*x]^n*(a + b*sin[e + f*x])^m*(A + B*sin[e + f*x]), x], x]",
     "rulenumber": 0,
            "lhs": "Int[sin[e_. + f_.*x_]^n_.*(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B}, x] && EqQ[A*b + a*B, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "a^m*c^m*Int[ Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m)*(A + B*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(c_ + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && Not[IntegerQ[ n] && (LtQ[m, 0] && GtQ[n, 0] || LtQ[0, n, m] || LtQ[m, n, 0])]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "Int[(a + b*Sin[e + f*x])^ m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^ m_.*(c_. + d_.*sin[e_. + f_.*x_])*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "(A*b + a*B)/(2*a*b)* Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x] + (B*c + A*d)/(2*c*d)* Int[Sqrt[c + d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*sin[e_. + f_.*x_])/(Sqrt[a_ + b_.*sin[e_. + f_.*x_]]* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "-B* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n/(f*(m + n + 1))",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[A*b*(m + n + 1) + a*B*(m - n), 0] && NeQ[m, -1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "B/d*Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), x] - (B*c - A*d)/d* Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_. + b_.*sin[e_. + f_.*x_]]*(c_ + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "(A*b - a*B)* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n/(a*f*(2*m + 1)) + (a*B*(m - n) + A*b*(m + n + 1))/(a*b*(2*m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_ + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1/2] || ILtQ[m + n, 0] && Not[SumSimplerQ[n, 1]]) && NeQ[2*m + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "-B* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n/(f*(m + n + 1)) - (B*c*(m - n) - A*d*(m + n + 1))/(d*(m + n + 1))* Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(c_ + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1/2]] && NeQ[m + n + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "(B*c - A*d)* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2))",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[m + n + 2, 0] && EqQ[A*(a*d*m + b*c*(n + 1)) - B*(a*c*m + b*d*(n + 1)), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "-b^2*(B*c - A*d)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d)) - b/(d*(n + 1)*(b*c + a*d))* Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)* Simp[a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "-b*B* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 1)) + 1/(d*(m + n + 1))* Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n* Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && Not[LtQ[n, -1]] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "(A*b - a*B)* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n/(a*f*(2*m + 1)) - 1/(a*b*(2*m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)* Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1/2] && GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "b*(A*b - a*B)* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(a*f*(2*m + 1)*(b*c - a*d)) + 1/(a*(2*m + 1)*(b*c - a*d))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n* Simp[B*(a*c*m + b*d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*(m + n + 2)*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1/2] && Not[GtQ[n, 0]] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "-2*b*B* Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(2*n + 3)* Sqrt[a + b*Sin[e + f*x]])",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)), 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "-b^2*(B*c - A*d)* Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1)/(d* f*(n + 1)*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]]) + (A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(2* d*(n + 1)*(b*c + a*d))* Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "-2*b*B* Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(2*n + 3)* Sqrt[a + b*Sin[e + f*x]]) + (A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(b*d*(2*n + 3))* Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]]*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Not[LtQ[n, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "(A*b - a*B)/b* Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x] + B/b*Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*sin[e_. + f_.*x_])/(Sqrt[a_ + b_.*sin[e_. + f_.*x_]]* Sqrt[c_. + d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "-B* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^n/(f*(m + n + 1)) + 1/(b*(m + n + 1))* Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n - 1)* Simp[A*b*c*(m + n + 1) + B*(a*c*m + b*d*n) + (A*b*d*(m + n + 1) + B*(a*d*m + b*c*n))* Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 0] && (IntegerQ[n] || EqQ[m + 1/2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "(B*c - A*d)* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2)) + 1/(b*(n + 1)*(c^2 - d^2))* Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)* Simp[A*(a*d*m + b*c*(n + 1)) - B*(a*c*m + b*d*(n + 1)) + b*(B*c - A*d)*(m + n + 2)*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1] && (IntegerQ[n] || EqQ[m + 1/2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "(A*b - a*B)/(b*c - a*d)* Int[1/Sqrt[a + b*Sin[e + f*x]], x] + (B*c - A*d)/(b*c - a*d)* Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*sin[e_. + f_.*x_])/(Sqrt[ a_ + b_.*sin[e_. + f_.*x_]]*(c_. + d_.*sin[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "B/d*Int[(a + b*Sin[e + f*x])^m, x] - (B*c - A*d)/d* Int[(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^ m_*(A_. + B_.*sin[e_. + f_.*x_])/(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NeQ[m + 1/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "(A*b - a*B)/b* Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x] + B/b*Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NeQ[A*b + a*B, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "(B*c - A*d)*(b*c - a*d)^2* Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1)/(f* d^2*(n + 1)*(c^2 - d^2)) - 1/(d^2*(n + 1)*(c^2 - d^2))*Int[(c + d*Sin[e + f*x])^(n + 1)* Simp[d*(n + 1)*(B*(b*c - a*d)^2 - A*d*(a^2*c + b^2*c - 2*a*b*d)) - ((B*c - A*d)*(a^2*d^2*(n + 2) + b^2*(c^2 + d^2*(n + 1))) + 2*a*b*d*(A*c*d*(n + 2) - B*(c^2 + d^2*(n + 1))))* Sin[e + f*x] - b^2*B*d*(n + 1)*(c^2 - d^2)*Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^2*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "-(b*c - a*d)*(B*c - A*d)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2)) + 1/(d*(n + 1)*(c^2 - d^2))* Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)* Simp[b*(b*c - a*d)*(B*c - A*d)*(m - 1) + a*d*(a*A*c + b*B*c - (A*b + a*B)*d)*(n + 1) + (b*(b*d*(B*c - A*d) + a*(A*c*d + B*(c^2 - 2*d^2)))*(n + 1) - a*(b*c - a*d)*(B*c - A*d)*(n + 2))*Sin[e + f*x] + b*(d*(A*b*c + a*B*c - a*A*d)*(m + n + 1) - b*B*(c^2*m + d^2*(n + 1)))*Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "-b*B* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 1)) + 1/(d*(m + n + 1))* Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n* Simp[a^2*A*d*(m + n + 1) + b*B*(b*c*(m - 1) + a*d*(n + 1)) + (a*d*(2*A*b + a*B)*(m + n + 1) - b*B*(a*c - b*d*(m + n)))*Sin[e + f*x] + b*(A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))* Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && Not[IGtQ[n, 1] && (Not[IntegerQ[m]] || EqQ[a, 0] && NeQ[c, 0])]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "B*d/b^2*Int[Sqrt[b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x] + Int[(A*c + (B*c + A*d)*Sin[e + f*x])/((b*Sin[e + f*x])^(3/2)* Sqrt[c + d*Sin[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[c_ + d_.*sin[e_. + f_.*x_]]*(A_. + B_.*sin[e_. + f_.*x_])/(b_.*sin[e_. + f_.*x_])^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "B/b*Int[Sqrt[c + d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]], x] + (A*b - a*B)/b* Int[Sqrt[c + d*Sin[e + f*x]]/(a + b*Sin[e + f*x])^(3/2), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[c_. + d_.*sin[e_. + f_.*x_]]*(A_. + B_.*sin[e_. + f_.*x_])/(a_ + b_.*sin[e_. + f_.*x_])^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "2*(A*b - a*B)* Cos[e + f*x]/(f*(a^2 - b^2)*Sqrt[a + b*Sin[e + f*x]]* Sqrt[d*Sin[e + f*x]]) + d/(a^2 - b^2)* Int[(A*b - a*B + (a*A - b*B)*Sin[e + f*x])/(Sqrt[ a + b*Sin[e + f*x]]*(d*Sin[e + f*x])^(3/2)), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*sin[e_. + f_.*x_])/((a_ + b_.*sin[e_. + f_.*x_])^(3/2)* Sqrt[d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "-2*A*(c - d)*Tan[e + f*x]/(f*b*c^2)* Rt[(c + d)/b, 2]*Sqrt[c*(1 + Csc[e + f*x])/(c - d)]* Sqrt[c*(1 - Csc[e + f*x])/(c + d)]* EllipticE[ ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/ Rt[(c + d)/b, 2]], -(c + d)/(c - d)]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*sin[e_. + f_.*x_])/((b_.*sin[e_. + f_.*x_])^(3/2)* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(c + d)/b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "-Sqrt[-b*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]* Int[(A + B*Sin[e + f*x])/((-b*Sin[e + f*x])^(3/2)* Sqrt[c + d*Sin[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*sin[e_. + f_.*x_])/((b_.*sin[e_. + f_.*x_])^(3/2)* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && NegQ[(c + d)/b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "-2* A*(c - d)*(a + b*Sin[e + f*x])/(f*(b*c - a*d)^2* Rt[(a + b)/(c + d), 2]*Cos[e + f*x])* Sqrt[(b*c - a*d)*(1 + Sin[e + f*x])/((c - d)*(a + b*Sin[e + f*x]))]* Sqrt[-(b*c - a*d)*(1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + f*x]))]* EllipticE[ ArcSin[Rt[(a + b)/(c + d), 2]* Sqrt[c + d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]]], (a - b)*(c + d)/((a + b)*(c - d))]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*sin[e_. + f_.*x_])/((a_ + b_.*sin[e_. + f_.*x_])^(3/2)* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(a + b)/(c + d)]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "Sqrt[-c - d*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]]* Int[(A + B*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)* Sqrt[-c - d*Sin[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*sin[e_. + f_.*x_])/((a_ + b_.*sin[e_. + f_.*x_])^(3/2)* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && NegQ[(a + b)/(c + d)]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "(A - B)/(a - b)* Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x] - (A*b - a*B)/(a - b)* Int[(1 + Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)* Sqrt[c + d*Sin[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*sin[e_. + f_.*x_])/((a_. + b_.*sin[e_. + f_.*x_])^(3/2)* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NeQ[A, B]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "(B*a - A*b)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^ n/(f*(m + 1)*(a^2 - b^2)) + 1/((m + 1)*(a^2 - b^2))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)* Simp[c*(a*A - b*B)*(m + 1) + d*n*(A*b - a*B) + (d*(a*A - b*B)*(m + 1) - c*(A*b - a*B)*(m + 2))*Sin[e + f*x] - d*(A*b - a*B)*(m + n + 2)*Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "-(A*b^2 - a*b*B)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(1 + n)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)) + 1/((m + 1)*(b*c - a*d)*(a^2 - b^2))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n* Simp[(a*A - b*B)*(b*c - a*d)*(m + 1) + b*d*(A*b - a*B)*(m + n + 2) + (A*b - a*B)*(a*d*(m + 1) - b*c*(m + 2))*Sin[e + f*x] - b*d*(A*b - a*B)*(m + n + 3)*Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && RationalQ[m] && m < -1 && (EqQ[a, 0] && IntegerQ[m] && Not[IntegerQ[n]] || Not[IntegerQ[2*n] && LtQ[n, -1] && (IntegerQ[n] && Not[IntegerQ[m]] || EqQ[a, 0])])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "(A*b - a*B)/(b*c - a*d)* Int[1/(a + b*Sin[e + f*x]), x] + (B*c - A*d)/(b*c - a*d)* Int[1/(c + d*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*sin[e_. + f_.*x_])/((a_. + b_.*sin[e_. + f_.*x_])*(c_. + d_.*sin[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "B/d*Int[(a + b*Sin[e + f*x])^m, x] - (B*c - A*d)/d* Int[(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^ m_*(A_. + B_.*sin[e_. + f_.*x_])/(c_. + d_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "-2*B*Cos[e + f*x]* Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^n/(f*(2*n + 3)) + 1/(2*n + 3)* Int[(c + d*Sin[e + f*x])^(n - 1)/Sqrt[a + b*Sin[e + f*x]]* Simp[a*A*c*(2*n + 3) + B*(b*c + 2*a*d*n) + (B*(a*c + b*d)*(2*n + 1) + A*(b*c + a*d)*(2*n + 3))* Sin[e + f*x] + (A*b*d*(2*n + 3) + B*(a*d + 2*b*c*n))*Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_. + b_.*sin[e_. + f_.*x_]]*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[n^2, 1/4]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "4*A/(f*Sqrt[a + b])* EllipticPi[-1, -ArcSin[ Cos[e + f*x]/(1 + Sin[e + f*x])], -(a - b)/(a + b)]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*sin[e_. + f_.*x_])/(Sqrt[sin[e_. + f_.*x_]]* Sqrt[a_ + b_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B}, x] && GtQ[b, 0] && GtQ[b^2 - a^2, 0] && EqQ[A, B]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "Sqrt[Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]* Int[(A + B*Sin[e + f*x])/(Sqrt[Sin[e + f*x]]* Sqrt[a + b*Sin[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*sin[e_. + f_.*x_])/(Sqrt[a_ + b_.*sin[e_. + f_.*x_]]* Sqrt[d_*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, d, A, B}, x] && GtQ[b, 0] && GtQ[b^2 - a^2, 0] && EqQ[A, B]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "B/d*Int[Sqrt[c + d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]], x] - (B*c - A*d)/d* Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*sin[e_. + f_.*x_])/(Sqrt[a_ + b_.*sin[e_. + f_.*x_]]* Sqrt[c_. + d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "Unintegrable[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^ n*(A + B*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + B_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": " a^m*c^m*Int[Cos[e+f*x]^(2*m)*(c+d*Sin[e+f*x])^(n-m)*(A+B*Sin[e+f*x])^ p,x]",
     "rulenumber": 0,
            "lhs": "Int[(a_+b_.*sin[e_.+f_.*x_])^m_*(c_+d_.*sin[e_.+f_.*x_])^n_*(A_.+B_ .*sin[e_.+f_.*x_])^p_,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,A,B,n,p},x] && EqQ[b*c+a*d,0] &&  EqQ[a^2-b^2,0] && IntegerQ[m] && Not[IntegerQ[n] && (LtQ[m,0] && GtQ[n,0] || LtQ[0,n,m] ||  LtQ[m,n,0])] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": " a^m*c^m*Int[Sin[e+f*x]^(2*m)*(c+d*Cos[e+f*x])^(n-m)*(A+B*Cos[e+f*x])^ p,x]",
     "rulenumber": 0,
            "lhs": "Int[(a_+b_.*cos[e_.+f_.*x_])^m_*(c_+d_.*cos[e_.+f_.*x_])^n_*(A_.+B_ .*cos[e_.+f_.*x_])^p_,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,A,B,n,p},x] && EqQ[b*c+a*d,0] &&  EqQ[a^2-b^2,0] && IntegerQ[m] && Not[IntegerQ[n] && (LtQ[m,0] && GtQ[n,0] || LtQ[0,n,m] ||  LtQ[m,n,0])] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]/(f*Cos[e + f*x])* Subst[ Int[(a + b*x)^(m - 1/2)*(c + d*x)^(n - 1/2)*(A + B*x)^p, x], x, Sin[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(c_ + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + B_.*sin[e_. + f_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
       "filename": "4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m",
            "rhs": "-Sqrt[a + b*Cos[e + f*x]]* Sqrt[c + d*Cos[e + f*x]]/(f*Sin[e + f*x])* Subst[ Int[(a + b*x)^(m - 1/2)*(c + d*x)^(n - 1/2)*(A + B*x)^p, x], x, Cos[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*cos[e_. + f_.*x_])^m_.*(c_ + d_.*cos[e_. + f_.*x_])^ n_.*(A_. + B_.*cos[e_. + f_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
       "filename": "4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
            "rhs": "1/b*Int[(b*Sin[e + f*x])^(m + 1)*(B + C*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*sin[e_. + f_.*x_])^ m_.*(B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, e, f, B, C, m}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
       "filename": "4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
            "rhs": "A*Cos[e + f*x]*(b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1))",
     "rulenumber": 0,
            "lhs": "Int[(b_.*sin[e_. + f_.*x_])^m_.*(A_ + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, e, f, A, C, m}, x] && EqQ[A*(m + 2) + C*(m + 1), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
       "filename": "4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
            "rhs": "A*Cos[e + f*x]*(b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1)) + (A*(m + 2) + C*(m + 1))/(b^2*(m + 1))*Int[(b*Sin[e + f*x])^(m + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*sin[e_. + f_.*x_])^m_*(A_ + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, e, f, A, C}, x] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
       "filename": "4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
            "rhs": "-1/f* Subst[Int[(1 - x^2)^((m - 1)/2)*(A + C - C*x^2), x], x, Cos[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[sin[e_. + f_.*x_]^m_.*(A_ + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{e, f, A, C}, x] && IGtQ[(m + 1)/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
       "filename": "4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
            "rhs": "-C* Cos[e + f*x]*(b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2)) + (A*(m + 2) + C*(m + 1))/(m + 2)*Int[(b*Sin[e + f*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*sin[e_. + f_.*x_])^m_.*(A_ + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, e, f, A, C, m}, x] && Not[LtQ[m, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
       "filename": "4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
            "rhs": "1/b^2*Int[(a + b*Sin[e + f*x])^(m + 1)* Simp[b*B - a*C + b*C*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B, C, m}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
       "filename": "4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
            "rhs": "C/b^2*Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[-a + b*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A*b^2 + a^2*C, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
       "filename": "4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
            "rhs": "(A - C)* Int[(a + b*Sin[e + f*x])^m*(1 + Sin[e + f*x]), x] + C*Int[(a + b*Sin[e + f*x])^m*(1 + Sin[e + f*x])^2, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^ m_.*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B, C, m}, x] && EqQ[A - B + C, 0] && Not[IntegerQ[2*m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
       "filename": "4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
            "rhs": "(A - C)* Int[(a + b*Sin[e + f*x])^m*(1 + Sin[e + f*x]), x] + C*Int[(a + b*Sin[e + f*x])^m*(1 + Sin[e + f*x])^2, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A + C, 0] && Not[IntegerQ[2*m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
       "filename": "4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
            "rhs": "(A*b - a*B + b*C)* Cos[e + f*x]*(a + b*Sin[e + f*x])^m/(a*f*(2*m + 1)) + 1/(a^2*(2*m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1)* Simp[a*A*(m + 1) + m*(b*B - a*C) + b*C*(2*m + 1)*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^ m_*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
       "filename": "4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
            "rhs": "b*(A + C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m/(a*f*(2*m + 1)) + 1/(a^2*(2*m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1)* Simp[a*A*(m + 1) - a*C*m + b*C*(2*m + 1)*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, C}, x] && LtQ[m, -1] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
       "filename": "4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
            "rhs": "-(A*b^2 - a*b*B + a^2*C)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b* f*(m + 1)*(a^2 - b^2)) + 1/(b*(m + 1)*(a^2 - b^2))* Int[(a + b*Sin[e + f*x])^(m + 1)* Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^ m_*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
       "filename": "4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
            "rhs": "-(A*b^2 + a^2*C)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b* f*(m + 1)*(a^2 - b^2)) + 1/(b*(m + 1)*(a^2 - b^2))* Int[(a + b*Sin[e + f*x])^(m + 1)* Simp[a*b*(A + C)*(m + 1) - (A*b^2 + a^2*C + b^2*(A + C)*(m + 1))*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
       "filename": "4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
            "rhs": "-C* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2)) + 1/(b*(m + 2))* Int[(a + b*Sin[e + f*x])^m* Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^ m_.*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B, C, m}, x] && Not[LtQ[m, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
       "filename": "4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
            "rhs": "-C* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2)) + 1/(b*(m + 2))* Int[(a + b*Sin[e + f*x])^m* Simp[A*b*(m + 2) + b*C*(m + 1) - a*C*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, C, m}, x] && Not[LtQ[m, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
       "filename": "4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
            "rhs": "(b*Sin[e + f*x]^p)^m/(b*Sin[e + f*x])^(m*p)* Int[(b*Sin[e + f*x])^(m*p)*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*sin[e_. + f_.*x_]^p_)^ m_*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, e, f, A, B, C, m, p}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
       "filename": "4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
            "rhs": "(b*Cos[e + f*x]^p)^m/(b*Cos[e + f*x])^(m*p)* Int[(b*Cos[e + f*x])^(m*p)*(A + B*Cos[e + f*x] + C*Cos[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*cos[e_. + f_.*x_]^p_)^ m_*(A_. + B_.*cos[e_. + f_.*x_] + C_.*cos[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, e, f, A, B, C, m, p}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
       "filename": "4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
            "rhs": "(b*Sin[e + f*x]^p)^m/(b*Sin[e + f*x])^(m*p)* Int[(b*Sin[e + f*x])^(m*p)*(A + C*Sin[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*sin[e_. + f_.*x_]^p_)^m_*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, e, f, A, C, m, p}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
       "filename": "4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m",
            "rhs": "(b*Cos[e + f*x]^p)^m/(b*Cos[e + f*x])^(m*p)* Int[(b*Cos[e + f*x])^(m*p)*(A + C*Cos[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*cos[e_. + f_.*x_]^p_)^m_*(A_. + C_.*cos[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, e, f, A, C, m, p}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "1/b^2*Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^ n*(b*B - a*C + b*C*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^m_.*(c_. + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - a*b*B + a^2*C, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "-C/b^2* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^ n*(a - b*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^m_.*(c_. + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 + a^2*C, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "-(b*c - a*d)*(A*b^2 - a*b*B + a^2*C)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b^2* f*(m + 1)*(a^2 - b^2)) - 1/(b^2*(m + 1)*(a^2 - b^2))*Int[(a + b*Sin[e + f*x])^(m + 1)* Simp[b*(m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b* B*(a^2*d + b^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))* Sin[e + f*x] - b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^ m_*(c_. + d_.*sin[e_. + f_.*x_])*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "-(b*c - a*d)*(A*b^2 + a^2*C)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b^2* f*(m + 1)*(a^2 - b^2)) + 1/(b^2*(m + 1)*(a^2 - b^2))*Int[(a + b*Sin[e + f*x])^(m + 1)* Simp[b*(m + 1)*(a*C*(b*c - a*d) + A*b*(a*c - b*d)) - ((b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))* Sin[e + f*x] + b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^ m_*(c_. + d_.*sin[e_. + f_.*x_])*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "-C*d*Cos[e + f*x]* Sin[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 3)) + 1/(b*(m + 3))*Int[(a + b*Sin[e + f*x])^m* Simp[a*C*d + A*b*c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))* Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))* Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^ m_.*(c_ + d_.*sin[e_. + f_.*x_])*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && Not[LtQ[m, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "-C*d*Cos[e + f*x]* Sin[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 3)) + 1/(b*(m + 3))*Int[(a + b*Sin[e + f*x])^m* Simp[a*C*d + A*b*c*(m + 3) + b*d*(C*(m + 2) + A*(m + 3))* Sin[e + f*x] - (2*a*C*d - b*c*C*(m + 3))*Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^ m_.*(c_ + d_.*sin[e_. + f_.*x_])*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && Not[LtQ[m, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "(a*A - b*B + a*C)* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(2*b*c*f*(2*m + 1)) - 1/(2*b*c*d*(2*m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n* Simp[A*(c^2*(m + 1) + d^2*(2*m + n + 2)) - B*c*d*(m - n - 1) - C*(c^2*m - d^2*(n + 1)) + d*((A*c + B*d)*(m + n + 2) - c*C*(3*m - n))*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1/2] || EqQ[m + n + 2, 0] && NeQ[2*m + 1, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "(a*A + a*C)* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(2*b*c*f*(2*m + 1)) - 1/(2*b*c*d*(2*m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n* Simp[A*(c^2*(m + 1) + d^2*(2*m + n + 2)) - C*(c^2*m - d^2*(n + 1)) + d*(A*c*(m + n + 2) - c*C*(3*m - n))*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1/2] || EqQ[m + n + 2, 0] && NeQ[2*m + 1, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "-2*C* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b*f*(2*m + 3)* Sqrt[c + d*Sin[e + f*x]]) + Int[(a + b*Sin[e + f*x])^m* Simp[A + C + B*Sin[e + f*x], x]/Sqrt[c + d*Sin[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^ m_.*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2)/ Sqrt[c_. + d_.*sin[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1/2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "-2*C* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)/(b*f*(2*m + 3)* Sqrt[c + d*Sin[e + f*x]]) + (A + C)*Int[(a + b*Sin[e + f*x])^m/Sqrt[c + d*Sin[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^m_.*(A_. + C_.*sin[e_. + f_.*x_]^2)/ Sqrt[c_. + d_.*sin[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1/2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "-C* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2)) + 1/(b*d*(m + n + 2))* Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n* Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + (b*B*d*(m + n + 2) - b*c*C*(2*m + 1))*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(c_. + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1/2]] && NeQ[m + n + 2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "-C* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2)) + 1/(b*d*(m + n + 2))* Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n* Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) - b*c*C*(2*m + 1)*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(c_. + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1/2]] && NeQ[m + n + 2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "(a*A - b*B + a*C)* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(f*(b*c - a*d)*(2*m + 1)) + 1/(b*(b*c - a*d)*(2*m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n* Simp[A*(a*c*(m + 1) - b*d*(2*m + n + 2)) + B*(b*c*m + a*d*(n + 1)) - C*(a*c*m + b*d*(n + 1)) + (d*(a*A - b*B)*(m + n + 2) + C*(b*c*(2*m + 1) - a*d*(m - n - 1)))*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "a*(A + C)* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(f*(b*c - a*d)*(2*m + 1)) + 1/(b*(b*c - a*d)*(2*m + 1))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n* Simp[A*(a*c*(m + 1) - b*d*(2*m + n + 2)) - C*(a*c*m + b*d*(n + 1)) + (a*A*d*(m + n + 2) + C*(b*c*(2*m + 1) - a*d*(m - n - 1)))*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "-(c^2*C - B*c*d + A*d^2)* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2)) + 1/(b*d*(n + 1)*(c^2 - d^2))* Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)* Simp[A*d*(a*d*m + b*c*(n + 1)) + (c*C - B*d)*(a*c*m + b*d*(n + 1)) + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))* Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Not[LtQ[m, -1/2]] && (LtQ[n, -1] || EqQ[m + n + 2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "-(c^2*C + A*d^2)* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2)) + 1/(b*d*(n + 1)*(c^2 - d^2))* Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)* Simp[A*d*(a*d*m + b*c*(n + 1)) + c*C*(a*c*m + b*d*(n + 1)) - b*(A*d^2*(m + n + 2) + C*(c^2*(m + 1) + d^2*(n + 1)))* Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Not[LtQ[m, -1/2]] && (LtQ[n, -1] || EqQ[m + n + 2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "-C* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2)) + 1/(b*d*(m + n + 2))* Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n* Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(c_. + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Not[LtQ[m, -1/2]] && NeQ[m + n + 2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "-C* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2)) + 1/(b*d*(m + n + 2))* Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n* Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_])^m_.*(c_. + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Not[LtQ[m, -1/2]] && NeQ[m + n + 2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "-(c^2*C - B*c*d + A*d^2)* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2)) + 1/(d*(n + 1)*(c^2 - d^2))* Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)* Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B*d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))* Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "-(c^2*C + A*d^2)* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2)) + 1/(d*(n + 1)*(c^2 - d^2))* Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)* Simp[A*d*(b*d*m + a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b*c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*(A*d^2*(m + n + 2) + C*(c^2*(m + 1) + d^2*(n + 1)))* Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "-C* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2)) + 1/(d*(m + n + 2))* Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n* Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))* Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))* Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^m_.*(c_. + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && Not[IGtQ[n, 0] && (Not[IntegerQ[m]] || EqQ[a, 0] && NeQ[c, 0])]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "-C* Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*(c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2)) + 1/(d*(m + n + 2))* Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n* Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^m_.*(c_. + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && Not[IGtQ[n, 0] && (Not[IntegerQ[m]] || EqQ[a, 0] && NeQ[c, 0])]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "C/(b*d)*Int[Sqrt[d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]], x] + 1/b*Int[(A*b + (b*B - a*C)* Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)* Sqrt[d*Sin[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2)/((a_ + b_.*sin[e_. + f_.*x_])^(3/2)* Sqrt[d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "C/(b*d)*Int[Sqrt[d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]], x] + 1/b*Int[(A*b - a*C*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)* Sqrt[d*Sin[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + C_.*sin[e_. + f_.*x_]^2)/((a_ + b_.*sin[e_. + f_.*x_])^(3/2)* Sqrt[d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, C}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "C/b^2*Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x] + 1/b^2* Int[(A*b^2 - a^2*C + b*(b*B - 2*a*C)*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)* Sqrt[c + d*Sin[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2)/((a_. + b_.*sin[e_. + f_.*x_])^(3/2)* Sqrt[c_. + d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "C/b^2*Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x] + 1/b^2* Int[(A*b^2 - a^2*C - 2*a*b*C*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)* Sqrt[c + d*Sin[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + C_.*sin[e_. + f_.*x_]^2)/((a_. + b_.*sin[e_. + f_.*x_])^(3/2)* Sqrt[c_. + d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "-(A*b^2 - a*b*B + a^2*C)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)) + 1/((m + 1)*(b*c - a*d)*(a^2 - b^2))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n* Simp[(m + 1)*(b*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && (EqQ[a, 0] && IntegerQ[m] && Not[IntegerQ[n]] || Not[IntegerQ[2*n] && LtQ[n, -1] && (IntegerQ[n] && Not[IntegerQ[m]] || EqQ[a, 0])])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "-(A*b^2 + a^2*C)* Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)) + 1/((m + 1)*(b*c - a*d)*(a^2 - b^2))* Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n* Simp[a*(m + 1)*(b*c - a*d)*(A + C) + d*(A*b^2 + a^2*C)*(m + n + 2) - (c*(A*b^2 + a^2*C) + b*(m + 1)*(b*c - a*d)*(A + C))* Sin[e + f*x] - d*(A*b^2 + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && (EqQ[a, 0] && IntegerQ[m] && Not[IntegerQ[n]] || Not[IntegerQ[2*n] && LtQ[n, -1] && (IntegerQ[n] && Not[IntegerQ[m]] || EqQ[a, 0])])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "C*x/(b*d) + (A*b^2 - a*b*B + a^2*C)/(b*(b*c - a*d))* Int[1/(a + b*Sin[e + f*x]), x] - (c^2*C - B*c*d + A*d^2)/(d*(b*c - a*d))* Int[1/(c + d*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2)/((a_ + b_.*sin[e_. + f_.*x_])*(c_. + d_.*sin[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "C*x/(b*d) + (A*b^2 + a^2*C)/(b*(b*c - a*d))* Int[1/(a + b*Sin[e + f*x]), x] - (c^2*C + A*d^2)/(d*(b*c - a*d))*Int[1/(c + d*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + C_.*sin[e_. + f_.*x_]^2)/((a_ + b_.*sin[e_. + f_.*x_])*(c_. + d_.*sin[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "C/(b*d)*Int[Sqrt[a + b*Sin[e + f*x]], x] - 1/(b*d)* Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2)/(Sqrt[ a_. + b_.*sin[e_. + f_.*x_]]*(c_. + d_.*sin[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "C/(b*d)*Int[Sqrt[a + b*Sin[e + f*x]], x] - 1/(b*d)* Int[Simp[a*c*C - A*b*d + (b*c*C + a*C*d)*Sin[e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + C_.*sin[e_. + f_.*x_]^2)/(Sqrt[ a_. + b_.*sin[e_. + f_.*x_]]*(c_. + d_.*sin[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "-C*Cos[e + f*x]* Sqrt[c + d*Sin[e + f*x]]/(d*f*Sqrt[a + b*Sin[e + f*x]]) + 1/(2*d)* Int[1/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]])* Simp[2*a*A*d - C*(b*c - a*d) - 2*(a*c*C - d*(A*b + a*B))* Sin[e + f*x] + (2*b*B*d - C*(b*c + a*d))*Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2)/(Sqrt[a_. + b_.*sin[e_. + f_.*x_]]* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "-C*Cos[e + f*x]* Sqrt[c + d*Sin[e + f*x]]/(d*f*Sqrt[a + b*Sin[e + f*x]]) + 1/(2*d)* Int[1/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]])* Simp[2*a*A*d - C*(b*c - a*d) - 2*(a*c*C - A*b*d)*Sin[e + f*x] - C*(b*c + a*d)*Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + C_.*sin[e_. + f_.*x_]^2)/(Sqrt[a_. + b_.*sin[e_. + f_.*x_]]* Sqrt[c_ + d_.*sin[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "(b*B - a*C)/b^2*Int[(d*Sin[e + f*x])^n, x] + C/(b*d)*Int[(d*Sin[e + f*x])^(n + 1), x] + (A*b^2 - a*b*B + a^2*C)/b^2* Int[(d*Sin[e + f*x])^n/(a + b*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sin[e_. + f_.*x_])^ n_.*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2)/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, C, n}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "-a*C/b^2*Int[(d*Sin[e + f*x])^n, x] + C/(b*d)*Int[(d*Sin[e + f*x])^(n + 1), x] + (A*b^2 + a^2*C)/b^2* Int[(d*Sin[e + f*x])^n/(a + b*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sin[e_. + f_.*x_])^ n_.*(A_. + C_.*sin[e_. + f_.*x_]^2)/(a_ + b_.*sin[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, C, n}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "Unintegrable[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^ n*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "Unintegrable[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^ n*(A + C*Sin[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_])^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "(b*Sin[e + f*x]^p)^m/(b*Sin[e + f*x])^(m*p)* Int[(b*Sin[e + f*x])^(m*p)*(c + d*Sin[e + f*x])^ n*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*sin[e_. + f_.*x_]^p_)^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + B_.*sin[e_. + f_.*x_] + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f, A, B, C, m, n, p}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "(b*Cos[e + f*x]^p)^m/(b*Cos[e + f*x])^(m*p)* Int[(b*Cos[e + f*x])^(m*p)*(c + d*Cos[e + f*x])^ n*(A + B*Cos[e + f*x] + C*Cos[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*cos[e_. + f_.*x_]^p_)^m_*(c_. + d_.*cos[e_. + f_.*x_])^ n_.*(A_. + B_.*cos[e_. + f_.*x_] + C_.*cos[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f, A, B, C, m, n, p}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "(b*Sin[e + f*x]^p)^m/(b*Sin[e + f*x])^(m*p)* Int[(b*Sin[e + f*x])^(m*p)*(c + d*Sin[e + f*x])^ n*(A + C*Sin[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*sin[e_. + f_.*x_]^p_)^m_*(c_. + d_.*sin[e_. + f_.*x_])^ n_.*(A_. + C_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f, A, C, m, n, p}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
       "filename": "4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m",
            "rhs": "(b*Cos[e + f*x]^p)^m/(b*Cos[e + f*x])^(m*p)* Int[(b*Cos[e + f*x])^(m*p)*(c + d*Cos[e + f*x])^ n*(A + C*Cos[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*cos[e_. + f_.*x_]^p_)^m_*(c_. + d_.*cos[e_. + f_.*x_])^ n_.*(A_. + C_.*cos[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f, A, C, m, n, p}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "a*(a*Cos[c + d*x] + b*Sin[c + d*x])^n/(b*d*n)",
     "rulenumber": 0,
            "lhs": "Int[(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "-1/d* Subst[Int[(a^2 + b^2 - x^2)^((n - 1)/2), x], x, b*Cos[c + d*x] - a*Sin[c + d*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && IGtQ[(n - 1)/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "-(b*Cos[c + d*x] - a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 1)/(d* n) + (n - 1)*(a^2 + b^2)/n* Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && Not[IntegerQ[(n - 1)/2]] && GtQ[n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "-1/d* Subst[Int[1/(a^2 + b^2 - x^2), x], x, b*Cos[c + d*x] - a*Sin[c + d*x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "Sin[c + d*x]/(a*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))",
     "rulenumber": 0,
            "lhs": "Int[1/(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "(b*Cos[c + d*x] - a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1)/(d*(n + 1)*(a^2 + b^2)) + (n + 2)/((n + 1)*(a^2 + b^2))* Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1] && NeQ[n, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "(a^2 + b^2)^(n/2)* Int[(Cos[c + d*x - ArcTan[a, b]])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && Not[GeQ[n, 1] || LeQ[n, -1]] && GtQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "(a*Cos[c + d*x] + b*Sin[c + d*x])^ n/((a*Cos[c + d*x] + b*Sin[c + d*x])/Sqrt[a^2 + b^2])^n* Int[Cos[c + d*x - ArcTan[a, b]]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && Not[GeQ[n, 1] || LeQ[n, -1]] && Not[GtQ[a^2 + b^2, 0] || EqQ[a^2 + b^2, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "-a*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 1)/(d*(n - 1)*Sin[c + d*x]^(n - 1)) + 2*b*Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 1)/ Sin[c + d*x]^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[sin[c_. + d_.*x_]^ m_*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && EqQ[a^2 + b^2, 0] && GtQ[n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "b*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 1)/(d*(n - 1)* Cos[c + d*x]^(n - 1)) + 2*a*Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 1)/ Cos[c + d*x]^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[cos[c_. + d_.*x_]^ m_*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && EqQ[a^2 + b^2, 0] && GtQ[n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "a*(a*Cos[c + d*x] + b*Sin[c + d*x])^n/(2*b*d*n*Sin[c + d*x]^n) + 1/(2*b)* Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1)/ Sin[c + d*x]^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[sin[c_. + d_.*x_]^ m_.*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && EqQ[a^2 + b^2, 0] && LtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "-b*(a*Cos[c + d*x] + b*Sin[c + d*x])^ n/(2*a*d*n*Cos[c + d*x]^n) + 1/(2*a)* Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1)/ Cos[c + d*x]^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[cos[c_. + d_.*x_]^ m_.*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && EqQ[a^2 + b^2, 0] && LtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "a*(a*Cos[c + d*x] + b*Sin[c + d*x])^n/(2*b*d*n*Sin[c + d*x]^n)* Hypergeometric2F1[1, n, n + 1, (b + a*Cot[c + d*x])/(2*b)]",
     "rulenumber": 0,
            "lhs": "Int[sin[c_. + d_.*x_]^ m_.*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && EqQ[m + n, 0] && EqQ[a^2 + b^2, 0] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "-b*(a*Cos[c + d*x] + b*Sin[c + d*x])^ n/(2*a*d*n*Cos[c + d*x]^n)* Hypergeometric2F1[1, n, n + 1, (a + b*Tan[c + d*x])/(2*a)]",
     "rulenumber": 0,
            "lhs": "Int[cos[c_. + d_.*x_]^ m_.*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && EqQ[m + n, 0] && EqQ[a^2 + b^2, 0] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "Int[(b + a*Cot[c + d*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[sin[c_. + d_.*x_]^ m_*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && IntegerQ[n] && NeQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "Int[(a + b*Tan[c + d*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[cos[c_. + d_.*x_]^ m_*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && IntegerQ[n] && NeQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "1/d*Subst[Int[x^m*(a + b*x)^n/(1 + x^2)^((m + n + 2)/2), x], x, Tan[c + d*x]]",
     "rulenumber": 0,
            "lhs": "Int[sin[c_. + d_.*x_]^ m_.*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IntegerQ[n] && IntegerQ[(m + n)/2] && NeQ[n, -1] && Not[GtQ[n, 0] && GtQ[m, 1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "-1/d* Subst[Int[x^m*(b + a*x)^n/(1 + x^2)^((m + n + 2)/2), x], x, Cot[c + d*x]]",
     "rulenumber": 0,
            "lhs": "Int[cos[c_. + d_.*x_]^ m_.*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IntegerQ[n] && IntegerQ[(m + n)/2] && NeQ[n, -1] && Not[GtQ[n, 0] && GtQ[m, 1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "Int[ExpandTrig[sin[c + d*x]^m*(a*cos[c + d*x] + b*sin[c + d*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[sin[c_. + d_.*x_]^ m_.*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IntegerQ[m] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "Int[ExpandTrig[cos[c + d*x]^m*(a*cos[c + d*x] + b*sin[c + d*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[cos[c_. + d_.*x_]^ m_.*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IntegerQ[m] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "a^n*b^n*Int[Sin[c + d*x]^m*(b*Cos[c + d*x] + a*Sin[c + d*x])^(-n), x]",
     "rulenumber": 0,
            "lhs": "Int[sin[c_. + d_.*x_]^ m_.*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && EqQ[a^2 + b^2, 0] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "a^n*b^n*Int[Cos[c + d*x]^m*(b*Cos[c + d*x] + a*Sin[c + d*x])^(-n), x]",
     "rulenumber": 0,
            "lhs": "Int[cos[c_. + d_.*x_]^ m_.*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && EqQ[a^2 + b^2, 0] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "a*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 1)/(d*(n - 1)) + b*Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 1), x] + a^2*Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 2)/Sin[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_/ sin[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "-b*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 1)/(d*(n - 1)) + a*Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 1), x] + b^2*Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 2)/Cos[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_/ cos[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "-(a^2 + b^2)* Int[Sin[c + d*x]^(m + 2)*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 2), x] + 2*b*Int[ Sin[c + d*x]^(m + 1)*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 1), x] + a^2*Int[Sin[c + d*x]^m*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[sin[c_. + d_.*x_]^ m_*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && GtQ[n, 1] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "-(a^2 + b^2)* Int[Cos[c + d*x]^(m + 2)*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 2), x] + 2*a*Int[ Cos[c + d*x]^(m + 1)*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 1), x] + b^2*Int[Cos[c + d*x]^m*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[cos[c_. + d_.*x_]^ m_*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && GtQ[n, 1] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "b*x/(a^2 + b^2) - a/(a^2 + b^2)* Int[(b*Cos[c + d*x] - a*Sin[c + d*x])/(a*Cos[c + d*x] + b*Sin[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[sin[c_. + d_.*x_]/(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "a*x/(a^2 + b^2) + b/(a^2 + b^2)* Int[(b*Cos[c + d*x] - a*Sin[c + d*x])/(a*Cos[c + d*x] + b*Sin[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[cos[c_. + d_.*x_]/(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "-a*Sin[c + d*x]^(m - 1)/(d*(a^2 + b^2)*(m - 1)) + b/(a^2 + b^2)*Int[Sin[c + d*x]^(m - 1), x] + a^2/(a^2 + b^2)* Int[Sin[c + d*x]^(m - 2)/(a*Cos[c + d*x] + b*Sin[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[sin[c_. + d_.*x_]^ m_/(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && GtQ[m, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "b*Cos[c + d*x]^(m - 1)/(d*(a^2 + b^2)*(m - 1)) + a/(a^2 + b^2)*Int[Cos[c + d*x]^(m - 1), x] + b^2/(a^2 + b^2)* Int[Cos[c + d*x]^(m - 2)/(a*Cos[c + d*x] + b*Sin[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[cos[c_. + d_.*x_]^ m_/(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && GtQ[m, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "1/a*Int[Cot[c + d*x], x] - 1/a*Int[(b*Cos[c + d*x] - a*Sin[c + d*x])/(a*Cos[c + d*x] + b*Sin[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(sin[ c_. + d_.*x_]*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "1/b*Int[Tan[c + d*x], x] + 1/b*Int[(b*Cos[c + d*x] - a*Sin[c + d*x])/(a*Cos[c + d*x] + b*Sin[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(cos[ c_. + d_.*x_]*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "Sin[c + d*x]^(m + 1)/(a*d*(m + 1)) - b/a^2*Int[Sin[c + d*x]^(m + 1), x] + (a^2 + b^2)/a^2* Int[Sin[c + d*x]^(m + 2)/(a*Cos[c + d*x] + b*Sin[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[sin[c_. + d_.*x_]^ m_/(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "-Cos[c + d*x]^(m + 1)/(b*d*(m + 1)) - a/b^2*Int[Cos[c + d*x]^(m + 1), x] + (a^2 + b^2)/b^2* Int[Cos[c + d*x]^(m + 2)/(a*Cos[c + d*x] + b*Sin[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[cos[c_. + d_.*x_]^ m_/(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "-(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1)/(a* d*(n + 1)) - b/a^2*Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1), x] + 1/a^2* Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 2)/Sin[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_/ sin[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1)/(b* d*(n + 1)) - a/b^2*Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1), x] + 1/b^2* Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 2)/Cos[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_/ cos[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "(a^2 + b^2)/a^2* Int[Sin[c + d*x]^(m + 2)*(a*Cos[c + d*x] + b*Sin[c + d*x])^n, x] - 2*b/a^2* Int[Sin[c + d*x]^(m + 1)*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1), x] + 1/a^2* Int[Sin[c + d*x]^m*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[sin[c_. + d_.*x_]^ m_*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "(a^2 + b^2)/b^2* Int[Cos[c + d*x]^(m + 2)*(a*Cos[c + d*x] + b*Sin[c + d*x])^n, x] - 2*a/b^2* Int[Cos[c + d*x]^(m + 1)*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1), x] + 1/b^2* Int[Cos[c + d*x]^m*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[cos[c_. + d_.*x_]^ m_*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "Int[ExpandTrig[ cos[c + d*x]^m*sin[c + d*x]^n*(a*cos[c + d*x] + b*sin[c + d*x])^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[cos[c_. + d_.*x_]^m_.* sin[c_. + d_.*x_]^ n_.*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "a^p*b^p*Int[ Cos[c + d*x]^m* Sin[c + d*x]^n*(b*Cos[c + d*x] + a*Sin[c + d*x])^(-p), x]",
     "rulenumber": 0,
            "lhs": "Int[cos[c_. + d_.*x_]^m_.* sin[c_. + d_.*x_]^ n_.*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && EqQ[a^2 + b^2, 0] && ILtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "b/(a^2 + b^2)*Int[Cos[c + d*x]^m*Sin[c + d*x]^(n - 1), x] + a/(a^2 + b^2)*Int[Cos[c + d*x]^(m - 1)*Sin[c + d*x]^n, x] - a*b/(a^2 + b^2)* Int[Cos[c + d*x]^(m - 1)* Sin[c + d*x]^(n - 1)/(a*Cos[c + d*x] + b*Sin[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[cos[c_. + d_.*x_]^m_.* sin[c_. + d_.*x_]^ n_./(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "Int[ExpandTrig[ cos[c + d*x]^m*sin[c + d*x]^n/(a*cos[c + d*x] + b*sin[c + d*x]), x], x]",
     "rulenumber": 0,
            "lhs": "Int[cos[c_. + d_.*x_]^m_.* sin[c_. + d_.*x_]^ n_./(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && IntegersQ[m, n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.5 trig^m (a cos+b sin)^n.m",
       "filename": "4.1.5 trig^m (a cos+b sin)^n.m",
            "rhs": "b/(a^2 + b^2)* Int[Cos[c + d*x]^m* Sin[c + d*x]^(n - 1)*(a*Cos[c + d*x] + b*Sin[c + d*x])^(p + 1), x] + a/(a^2 + b^2)* Int[Cos[c + d*x]^(m - 1)* Sin[c + d*x]^n*(a*Cos[c + d*x] + b*Sin[c + d*x])^(p + 1), x] - a*b/(a^2 + b^2)* Int[Cos[c + d*x]^(m - 1)* Sin[c + d*x]^(n - 1)*(a*Cos[c + d*x] + b*Sin[c + d*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[cos[c_. + d_.*x_]^m_.* sin[c_. + d_.*x_]^ n_.*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0] && IGtQ[n, 0] && ILtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "-2*(c*Cos[d + e*x] - b*Sin[d + e*x])/(e* Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 - c^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "-(c*Cos[d + e*x] - b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1)/(e*n) + a*(2*n - 1)/n* Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 - c^2, 0] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "-(c - a*Sin[d + e*x])/(c* e*(c*Cos[d + e*x] - b*Sin[d + e*x]))",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 - c^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "Int[1/Sqrt[a + Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]], x]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 - c^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "(c*Cos[d + e*x] - b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^ n/(a*e*(2*n + 1)) + (n + 1)/(a*(2*n + 1))* Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 - c^2, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "b/(c*e)*Subst[Int[Sqrt[a + x]/x, x], x, b*Cos[d + e*x] + c*Sin[d + e*x]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[b^2 + c^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "Int[Sqrt[a + Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]], x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 + c^2, 0] && GtQ[a + Sqrt[b^2 + c^2], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]/ Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])]* Int[ Sqrt[a/(a + Sqrt[b^2 + c^2]) + Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2])* Cos[d + e*x - ArcTan[b, c]]], x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[b^2 + c^2, 0] && Not[GtQ[a + Sqrt[b^2 + c^2], 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "-(c*Cos[d + e*x] - b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1)/(e*n) + 1/n*Int[ Simp[n*a^2 + (n - 1)*(b^2 + c^2) + a*b*(2*n - 1)*Cos[d + e*x] + a*c*(2*n - 1)*Sin[d + e*x], x]* (a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && GtQ[n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": " x/Sqrt[a^2-b^2-c^2] + 2/(e*Sqrt[a^2-b^2-c^2])*ArcTan[(c*Cos[d+e*x]-b*Sin[d+e*x])/(a+Sqrt[ a^2-b^2-c^2]+b*Cos[d+e*x]+c*Sin[d+e*x])]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_]),x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e},x] &&  GtQ[a^2-b^2-c^2,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": " Log[RemoveContent[b^2+c^2+(a*b-c*Rt[-a^2+b^2+c^2,2])*Cos[d+e*x]+(a*c+ b*Sqrt[-a^2+b^2+c^2])*Sin[d+e*x],x]]/ (2*e*Rt[-a^2+b^2+c^2,2]) - Log[RemoveContent[b^2+c^2+(a*b+c*Rt[-a^2+b^2+c^2,2])*Cos[d+e*x]+(a* c-b*Sqrt[-a^2+b^2+c^2])*Sin[d+e*x],x]]/ (2*e*Rt[-a^2+b^2+c^2,2])",
     "rulenumber": 0,
            "lhs": "Int[1/(a_+b_.*cos[d_.+e_.*x_]+c_.*sin[d_.+e_.*x_]),x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e},x] &&  LtQ[a^2-b^2-c^2,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "Module[{f = FreeFactors[Cot[(d + e*x)/2], x]}, -f/e*Subst[Int[1/(a + c*f*x), x], x, Cot[(d + e*x)/2]/f]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[a + b, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "Module[{f = FreeFactors[Tan[(d + e*x)/2 + Pi/4], x]}, f/e*Subst[Int[1/(a + b*f*x), x], x, Tan[(d + e*x)/2 + Pi/4]/f]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[a + c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "Module[{f = FreeFactors[Cot[(d + e*x)/2 + Pi/4], x]}, -f/e* Subst[Int[1/(a + b*f*x), x], x, Cot[(d + e*x)/2 + Pi/4]/f]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[a - c, 0] && NeQ[a - b, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "Module[{f = FreeFactors[Tan[(d + e*x)/2], x]}, 2*f/e* Subst[Int[1/(a + b + 2*c*f*x + (a - b)*f^2*x^2), x], x, Tan[(d + e*x)/2]/f]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "b/(c*e)*Subst[Int[1/(x*Sqrt[a + x]), x], x, b*Cos[d + e*x] + c*Sin[d + e*x]]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[b^2 + c^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "Int[1/Sqrt[a + Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]], x]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 + c^2, 0] && GtQ[a + Sqrt[b^2 + c^2], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])]/ Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]* Int[ 1/Sqrt[a/(a + Sqrt[b^2 + c^2]) + Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2])* Cos[d + e*x - ArcTan[b, c]]], x]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[b^2 + c^2, 0] && Not[GtQ[a + Sqrt[b^2 + c^2], 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "2*(c*Cos[d + e*x] - b*Sin[d + e*x])/(e*(a^2 - b^2 - c^2)* Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]) + 1/(a^2 - b^2 - c^2)* Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "(-c*Cos[d + e*x] + b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)/(e*(n + 1)*(a^2 - b^2 - c^2)) + 1/((n + 1)*(a^2 - b^2 - c^2))* Int[(a*(n + 1) - b*(n + 2)*Cos[d + e*x] - c*(n + 2)*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && LtQ[n, -1] && NeQ[n, -3/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "(2*a*A - b*B - c*C)* x/(2*a^2) - (b*B + c*C)*(b*Cos[d + e*x] - c*Sin[d + e*x])/(2*a*b*c*e) + (a^2*(b*B - c*C) - 2*a*A*b^2 + b^2*(b*B + c*C))* Log[RemoveContent[a + b*Cos[d + e*x] + c*Sin[d + e*x], x]]/(2*a^2* b*c*e)",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*cos[d_. + e_.*x_] + C_.*sin[d_. + e_.*x_])/(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B, C}, x] && EqQ[b^2 + c^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "(2*a*A - c*C)*x/(2*a^2) - C*Cos[d + e*x]/(2*a*e) + c*C*Sin[d + e*x]/(2*a*b*e) + (-a^2*C + 2*a*c*A + b^2*C)* Log[RemoveContent[a + b*Cos[d + e*x] + c*Sin[d + e*x], x]]/(2*a^2* b*e)",
     "rulenumber": 0,
            "lhs": "Int[(A_. + C_.*sin[d_. + e_.*x_])/(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, C}, x] && EqQ[b^2 + c^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "(2*a*A - b*B)*x/(2*a^2) - b*B*Cos[d + e*x]/(2*a*c*e) + B*Sin[d + e*x]/(2*a*e) + (a^2*B - 2*a*b*A + b^2*B)* Log[RemoveContent[a + b*Cos[d + e*x] + c*Sin[d + e*x], x]]/(2*a^2* c*e)",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*cos[d_. + e_.*x_])/(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 + c^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "(b*B + c*C)*x/(b^2 + c^2) + (c*B - b*C)* Log[a + b*Cos[d + e*x] + c*Sin[d + e*x]]/(e*(b^2 + c^2))",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*cos[d_. + e_.*x_] + C_.*sin[d_. + e_.*x_])/(a_. + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B, C}, x] && NeQ[b^2 + c^2, 0] && EqQ[A*(b^2 + c^2) - a*(b*B + c*C), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "c*C*x/(b^2 + c^2) - b*C*Log[a + b*Cos[d + e*x] + c*Sin[d + e*x]]/(e*(b^2 + c^2))",
     "rulenumber": 0,
            "lhs": "Int[(A_. + C_.*sin[d_. + e_.*x_])/(a_. + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, C}, x] && NeQ[b^2 + c^2, 0] && EqQ[A*(b^2 + c^2) - a*c*C, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "b*B*x/(b^2 + c^2) + c*B*Log[a + b*Cos[d + e*x] + c*Sin[d + e*x]]/(e*(b^2 + c^2))",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*cos[d_. + e_.*x_])/(a_. + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 + c^2, 0] && EqQ[A*(b^2 + c^2) - a*b*B, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "(b*B + c*C)*x/(b^2 + c^2) + (c*B - b*C)* Log[a + b*Cos[d + e*x] + c*Sin[d + e*x]]/(e*(b^2 + c^2)) + (A*(b^2 + c^2) - a*(b*B + c*C))/(b^2 + c^2)* Int[1/(a + b*Cos[d + e*x] + c*Sin[d + e*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*cos[d_. + e_.*x_] + C_.*sin[d_. + e_.*x_])/(a_. + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B, C}, x] && NeQ[b^2 + c^2, 0] && NeQ[A*(b^2 + c^2) - a*(b*B + c*C), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "c*C*(d + e*x)/(e*(b^2 + c^2)) - b*C*Log[a + b*Cos[d + e*x] + c*Sin[d + e*x]]/(e*(b^2 + c^2)) + (A*(b^2 + c^2) - a*c*C)/(b^2 + c^2)* Int[1/(a + b*Cos[d + e*x] + c*Sin[d + e*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + C_.*sin[d_. + e_.*x_])/(a_. + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, C}, x] && NeQ[b^2 + c^2, 0] && NeQ[A*(b^2 + c^2) - a*c*C, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "b*B*(d + e*x)/(e*(b^2 + c^2)) + c*B*Log[a + b*Cos[d + e*x] + c*Sin[d + e*x]]/(e*(b^2 + c^2)) + (A*(b^2 + c^2) - a*b*B)/(b^2 + c^2)* Int[1/(a + b*Cos[d + e*x] + c*Sin[d + e*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*cos[d_. + e_.*x_])/(a_. + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 + c^2, 0] && NeQ[A*(b^2 + c^2) - a*b*B, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "(B*c - b*C - a*C*Cos[d + e*x] + a*B*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^ n/(a*e*(n + 1))",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*cos[d_. + e_.*x_] + C_.*sin[d_. + e_.*x_])*(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B, C, n}, x] && NeQ[n, -1] && EqQ[a^2 - b^2 - c^2, 0] && EqQ[(b*B + c*C)*n + a*A*(n + 1), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "-(b*C + a*C*Cos[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^ n/(a*e*(n + 1))",
     "rulenumber": 0,
            "lhs": "Int[(A_. + C_.*sin[d_. + e_.*x_])*(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, C, n}, x] && NeQ[n, -1] && EqQ[a^2 - b^2 - c^2, 0] && EqQ[c*C*n + a*A*(n + 1), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "(B*c + a*B*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^ n/(a*e*(n + 1))",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*cos[d_. + e_.*x_])*(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B, n}, x] && NeQ[n, -1] && EqQ[a^2 - b^2 - c^2, 0] && EqQ[b*B*n + a*A*(n + 1), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "(B*c - b*C - a*C*Cos[d + e*x] + a*B*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^ n/(a*e*(n + 1)) + ((b*B + c*C)*n + a*A*(n + 1))/(a*(n + 1))* Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*cos[d_. + e_.*x_] + C_.*sin[d_. + e_.*x_])*(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B, C, n}, x] && NeQ[n, -1] && EqQ[a^2 - b^2 - c^2, 0] && NeQ[(b*B + c*C)*n + a*A*(n + 1), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "-(b*C + a*C*Cos[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^ n/(a*e*(n + 1)) + (c*C*n + a*A*(n + 1))/(a*(n + 1))* Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + C_.*sin[d_. + e_.*x_])*(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, C, n}, x] && NeQ[n, -1] && EqQ[a^2 - b^2 - c^2, 0] && NeQ[c*C*n + a*A*(n + 1), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "(B*c + a*B*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^ n/(a*e*(n + 1)) + (b*B*n + a*A*(n + 1))/(a*(n + 1))* Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*cos[d_. + e_.*x_])*(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B, n}, x] && NeQ[n, -1] && EqQ[a^2 - b^2 - c^2, 0] && NeQ[b*B*n + a*A*(n + 1), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "(c*B - b*C)*(b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)/(e*(n + 1)*(b^2 + c^2))",
     "rulenumber": 0,
            "lhs": "Int[(B_.*cos[d_. + e_.*x_] + C_.*sin[d_. + e_.*x_])*(b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, B, C}, x] && NeQ[n, -1] && NeQ[b^2 + c^2, 0] && EqQ[b*B + c*C, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "(B*c - b*C - a*C*Cos[d + e*x] + a*B*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^ n/(a*e*(n + 1)) + 1/(a*(n + 1))*Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1)* Simp[a*(b*B + c*C)*n + a^2*A*(n + 1) + (n*(a^2*B - B*c^2 + b*c*C) + a*b*A*(n + 1))* Cos[d + e*x] + (n*(b*B*c + a^2*C - b^2*C) + a*c*A*(n + 1))* Sin[d + e*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*cos[d_. + e_.*x_] + C_.*sin[d_. + e_.*x_])*(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B, C}, x] && GtQ[n, 0] && NeQ[a^2 - b^2 - c^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "-(b*C + a*C*Cos[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^ n/(a*e*(n + 1)) + 1/(a*(n + 1))*Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1)* Simp[a*c*C*n + a^2*A*(n + 1) + (c*b*C*n + a*b*A*(n + 1))* Cos[d + e*x] + (a^2*C*n - b^2*C*n + a*c*A*(n + 1))* Sin[d + e*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + C_.*sin[d_. + e_.*x_])*(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, C}, x] && GtQ[n, 0] && NeQ[a^2 - b^2 - c^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "(B*c + a*B*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^ n/(a*e*(n + 1)) + 1/(a*(n + 1))*Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1)* Simp[a*b*B*n + a^2*A*(n + 1) + (a^2*B*n - c^2*B*n + a*b*A*(n + 1))* Cos[d + e*x] + (b*c*B*n + a*c*A*(n + 1))*Sin[d + e*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*cos[d_. + e_.*x_])*(a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && GtQ[n, 0] && NeQ[a^2 - b^2 - c^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "B/b*Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x] + (A*b - a*B)/b* Int[1/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*cos[d_. + e_.*x_] + C_.*sin[d_. + e_.*x_])/ Sqrt[a_ + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B, C}, x] && EqQ[B*c - b*C, 0] && NeQ[A*b - a*B, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "(c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B - b*A)*Sin[d + e*x])/ (e*(a^2 - b^2 - c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x]))",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*cos[d_. + e_.*x_] + C_.*sin[d_. + e_.*x_])/(a_. + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B, C}, x] && NeQ[a^2 - b^2 - c^2, 0] && EqQ[a*A - b*B - c*C, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "-(b*C + (a*C - c*A)*Cos[d + e*x] + b*A*Sin[d + e*x])/(e*(a^2 - b^2 - c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x]))",
     "rulenumber": 0,
            "lhs": "Int[(A_. + C_.*sin[d_. + e_.*x_])/(a_. + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, C}, x] && NeQ[a^2 - b^2 - c^2, 0] && EqQ[a*A - c*C, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "(c*B + c*A*Cos[d + e*x] + (a*B - b*A)* Sin[d + e*x])/(e*(a^2 - b^2 - c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x]))",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*cos[d_. + e_.*x_])/(a_. + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[a^2 - b^2 - c^2, 0] && EqQ[a*A - b*B, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "(c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B - b*A)*Sin[d + e*x])/ (e*(a^2 - b^2 - c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])) + (a*A - b*B - c*C)/(a^2 - b^2 - c^2)* Int[1/(a + b*Cos[d + e*x] + c*Sin[d + e*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*cos[d_. + e_.*x_] + C_.*sin[d_. + e_.*x_])/(a_. + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B, C}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[a*A - b*B - c*C, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "-(b*C + (a*C - c*A)*Cos[d + e*x] + b*A*Sin[d + e*x])/(e*(a^2 - b^2 - c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])) + (a*A - c*C)/(a^2 - b^2 - c^2)* Int[1/(a + b*Cos[d + e*x] + c*Sin[d + e*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + C_.*sin[d_. + e_.*x_])/(a_. + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, C}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[a*A - c*C, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "(c*B + c*A*Cos[d + e*x] + (a*B - b*A)* Sin[d + e*x])/(e*(a^2 - b^2 - c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])) + (a*A - b*B)/(a^2 - b^2 - c^2)* Int[1/(a + b*Cos[d + e*x] + c*Sin[d + e*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*cos[d_. + e_.*x_])/(a_. + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[a*A - b*B, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "-(c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B - b*A)*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)/ (e*(n + 1)*(a^2 - b^2 - c^2)) + 1/((n + 1)*(a^2 - b^2 - c^2))* Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)* Simp[(n + 1)*(a*A - b*B - c*C) + (n + 2)*(a*B - b*A)* Cos[d + e*x] + (n + 2)*(a*C - c*A)*Sin[d + e*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*cos[d_. + e_.*x_] + C_.*sin[d_. + e_.*x_])*(a_. + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B, C}, x] && LtQ[n, -1] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[n, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "(b*C + (a*C - c*A)*Cos[d + e*x] + b*A*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)/ (e*(n + 1)*(a^2 - b^2 - c^2)) + 1/((n + 1)*(a^2 - b^2 - c^2))* Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)* Simp[(n + 1)*(a*A - c*C) - (n + 2)*b*A* Cos[d + e*x] + (n + 2)*(a*C - c*A)*Sin[d + e*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + C_.*sin[d_. + e_.*x_])*(a_. + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, C}, x] && LtQ[n, -1] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[n, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "-(c*B + c*A*Cos[d + e*x] + (a*B - b*A)*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)/ (e*(n + 1)*(a^2 - b^2 - c^2)) + 1/((n + 1)*(a^2 - b^2 - c^2))* Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)* Simp[(n + 1)*(a*A - b*B) + (n + 2)*(a*B - b*A)* Cos[d + e*x] - (n + 2)*c*A*Sin[d + e*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*cos[d_. + e_.*x_])*(a_. + b_.*cos[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && LtQ[n, -1] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[n, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "Int[Cos[d + e*x]/(b + a*Cos[d + e*x] + c*Sin[d + e*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_. + b_.*sec[d_. + e_.*x_] + c_.*tan[d_. + e_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "Int[Sin[d + e*x]/(b + a*Sin[d + e*x] + c*Cos[d + e*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_. + b_.*csc[d_. + e_.*x_] + c_.*cot[d_. + e_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "Int[(b + a*Cos[d + e*x] + c*Sin[d + e*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[cos[d_. + e_.*x_]^ n_.*(a_. + b_.*sec[d_. + e_.*x_] + c_.*tan[d_. + e_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "Int[(b + a*Sin[d + e*x] + c*Cos[d + e*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[sin[d_. + e_.*x_]^ n_.*(a_. + b_.*csc[d_. + e_.*x_] + c_.*cot[d_. + e_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "Cos[d + e*x]^ n*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^ n/(b + a*Cos[d + e*x] + c*Sin[d + e*x])^n* Int[(b + a*Cos[d + e*x] + c*Sin[d + e*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[cos[d_. + e_.*x_]^ n_*(a_. + b_.*sec[d_. + e_.*x_] + c_.*tan[d_. + e_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "Sin[d + e*x]^ n*(a + b*Csc[d + e*x] + c*Cot[d + e*x])^ n/(b + a*Sin[d + e*x] + c*Cos[d + e*x])^n* Int[(b + a*Sin[d + e*x] + c*Cos[d + e*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[sin[d_. + e_.*x_]^ n_*(a_. + b_.*csc[d_. + e_.*x_] + c_.*cot[d_. + e_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "Int[1/(b + a*Cos[d + e*x] + c*Sin[d + e*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[sec[d_. + e_.*x_]^ n_.*(a_. + b_.*sec[d_. + e_.*x_] + c_.*tan[d_. + e_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[m + n, 0] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "Int[1/(b + a*Sin[d + e*x] + c*Cos[d + e*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[csc[d_. + e_.*x_]^ n_.*(a_. + b_.*csc[d_. + e_.*x_] + c_.*cot[d_. + e_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[m + n, 0] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "Sec[d + e*x]^ n*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^ n/(a + b*Sec[d + e*x] + c*Tan[d + e*x])^n* Int[1/(b + a*Cos[d + e*x] + c*Sin[d + e*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[sec[d_. + e_.*x_]^ n_.*(a_. + b_.*sec[d_. + e_.*x_] + c_.*tan[d_. + e_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[m + n, 0] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.6 (a+b cos+c sin)^n.m",
       "filename": "4.1.6 (a+b cos+c sin)^n.m",
            "rhs": "Csc[d + e*x]^ n*(b + a*Sin[d + e*x] + c*Cos[d + e*x])^ n/(a + b*Csc[d + e*x] + c*Cot[d + e*x])^n* Int[1/(b + a*Sin[d + e*x] + c*Cos[d + e*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[csc[d_. + e_.*x_]^ n_.*(a_. + b_.*csc[d_. + e_.*x_] + c_.*cot[d_. + e_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[m + n, 0] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "(4*A*(2*a + b) + B*(4*a + 3*b))*x/8 - (4*A*b + B*(4*a + 3*b))*Cos[e + f*x]*Sin[e + f*x]/(8*f) - b*B*Cos[e + f*x]*Sin[e + f*x]^3/(4*f)",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_]^2)*(A_. + B_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "-B*Cos[e + f*x]* Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^p/(2*f*(p + 1)) + 1/(2*(p + 1))*Int[(a + b*Sin[e + f*x]^2)^(p - 1)* Simp[a*B + 2*a*A*(p + 1) + (2*A*b*(p + 1) + B*(b + 2*a*p + 2*b*p))* Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_]^2)^p_*(A_. + B_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B}, x] && GtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "B*x/b + (A*b - a*B)/b*Int[1/(a + b*Sin[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*sin[e_. + f_.*x_]^2)/(a_ + b_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "B/b*Int[Sqrt[a + b*Sin[e + f*x]^2], x] + (A*b - a*B)/b* Int[1/Sqrt[a + b*Sin[e + f*x]^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*sin[e_. + f_.*x_]^2)/ Sqrt[a_ + b_.*sin[e_. + f_.*x_]^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "-(A*b - a*B)*Cos[e + f*x]* Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(p + 1)/(2*a* f*(a + b)*(p + 1)) - 1/(2*a*(a + b)*(p + 1))*Int[(a + b*Sin[e + f*x]^2)^(p + 1)* Simp[a*B - A*(2*a*(p + 1) + b*(2*p + 3)) + 2*(A*b - a*B)*(p + 2)*Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_]^2)^p_*(A_. + B_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B}, x] && LtQ[p, -1] && NeQ[a + b, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, ff*(a + b*Sin[e + f*x]^2)^ p*(Sec[e + f*x]^2)^p/(f*(a + (a + b)*Tan[e + f*x]^2)^p)* Subst[ Int[(a + (a + b)*ff^2*x^2)^ p*(A + (A + B)*ff^2*x^2)/(1 + ff^2*x^2)^(p + 2), x], x, Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[e_. + f_.*x_]^2)^ p_*(A_. + B_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B}, x] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "a^p*Int[ActivateTrig[u*cos[e + f*x]^(2*p)], x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "Int[ActivateTrig[u*(a*cos[e + f*x]^2)^p], x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "Sqrt[a]/f*EllipticE[e + f*x, -b/a]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && GtQ[a, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[1 + b*Sin[e + f*x]^2/a]* Int[Sqrt[1 + (b*Sin[e + f*x]^2)/a], x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*sin[e_. + f_.*x_]^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && Not[GtQ[a, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "(8*a^2 + 8*a*b + 3*b^2)*x/8 - b*(8*a + 3*b)*Cos[e + f*x]*Sin[e + f*x]/(8*f) - b^2*Cos[e + f*x]*Sin[e + f*x]^3/(4*f)",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_]^2)^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "-b*Cos[e + f*x]* Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(p - 1)/(2*f*p) + 1/(2*p)* Int[(a + b*Sin[e + f*x]^2)^(p - 2)* Simp[a*(b + 2*a*p) + b*(2*a + b)*(2*p - 1)*Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_]^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && NeQ[a + b, 0] && GtQ[p, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[1/(a + (a + b)*ff^2*x^2), x], x, Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*sin[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "1/(Sqrt[a]*f)*EllipticF[e + f*x, -b/a]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*sin[e_. + f_.*x_]^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && GtQ[a, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "Sqrt[1 + b*Sin[e + f*x]^2/a]/Sqrt[a + b*Sin[e + f*x]^2]* Int[1/Sqrt[1 + (b*Sin[e + f*x]^2)/a], x]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*sin[e_. + f_.*x_]^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && Not[GtQ[a, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "-b*Cos[e + f*x]* Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(p + 1)/(2*a* f*(p + 1)*(a + b)) + 1/(2*a*(p + 1)*(a + b))* Int[(a + b*Sin[e + f*x]^2)^(p + 1)* Simp[2*a*(p + 1) + b*(2*p + 3) - 2*b*(p + 2)*Sin[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_]^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && NeQ[a + b, 0] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Sin[e + f*x], x]}, ff*Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x])* Subst[Int[(a + b*ff^2*x^2)^p/Sqrt[1 - ff^2*x^2], x], x, Sin[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, p}, x] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Cos[e + f*x], x]}, -ff/f* Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[sin[e_. + f_.*x_]^m_.*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, ff^(m + 1)/f* Subst[Int[ x^m*(a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p + 1), x], x, Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[sin[e_. + f_.*x_]^m_*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Sin[e + f*x], x]}, ff^(m + 1)*Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x])* Subst[Int[x^m*(a + b*ff^2*x^2)^p/Sqrt[1 - ff^2*x^2], x], x, Sin[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[sin[e_. + f_.*x_]^m_*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Cos[e + f*x], x]}, -ff*d^(2*IntPart[(m - 1)/2] + 1)*(d*Sin[e + f*x])^(2* FracPart[(m - 1)/2])/(f*(Sin[e + f*x]^2)^ FracPart[(m - 1)/2])* Subst[ Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sin[e_. + f_.*x_])^m_*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, p}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Sin[e + f*x], x]}, ff/f* Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^m_.*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[(a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p + 1), x], x, Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^m_*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Sin[e + f*x], x]}, ff*Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x])* Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^m_*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Sin[e + f*x], x]}, ff*d^(2*IntPart[(m - 1)/2] + 1)*(d*Cos[e + f*x])^(2* FracPart[(m - 1)/2])/(f*(Cos[e + f*x]^2)^ FracPart[(m - 1)/2])* Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*cos[e_. + f_.*x_])^m_*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, p}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Sin[e + f*x]^2, x]}, ff^((m + 1)/2)/(2*f)* Subst[Int[x^((m - 1)/2)*(a + b*ff*x)^p/(1 - ff*x)^((m + 1)/2), x], x, Sin[e + f*x]^2/ff]]",
     "rulenumber": 0,
            "lhs": "Int[tan[e_. + f_.*x_]^m_.*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[(d*ff*x)^ m*(a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(p + 1), x], x, Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*tan[e_. + f_.*x_])^m_*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m}, x] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Sin[e + f*x], x]}, ff^(m + 1)*Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x])* Subst[ Int[x^m*(a + b*ff^2*x^2)^p/(1 - ff^2*x^2)^((m + 1)/2), x], x, Sin[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[tan[e_. + f_.*x_]^m_*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Sin[e + f*x], x]}, ff*(d*Tan[e + f*x])^(m + 1)*(Cos[e + f*x]^2)^((m + 1)/2)/(d*f* Sin[e + f*x]^(m + 1))* Subst[ Int[(ff*x)^m*(a + b*ff^2*x^2)^p/(1 - ff^2*x^2)^((m + 1)/2), x], x, Sin[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*tan[e_. + f_.*x_])^m_*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, p}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Sin[e + f*x], x]}, ff/f* Subst[Int[(d*ff*x)^n*(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^ p, x], x, Sin[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^m_.*(d_.*sin[e_. + f_.*x_])^ n_.*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(m - 1)/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Cos[e + f*x], x]}, -ff/f* Subst[Int[(c*ff*x)^ m*(1 - ff^2*x^2)^((n - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*sin[e_. + f_.*x_])^m_* sin[e_. + f_.*x_]^n_.*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, e, f, m, p}, x] && IntegerQ[(n - 1)/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, ff^(n + 1)/f* Subst[Int[ x^n*(a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^((m + n)/2 + p + 1), x], x, Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^m_* sin[e_. + f_.*x_]^n_*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[n/2] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Sin[e + f*x], x]}, ff*Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x])* Subst[Int[(d*ff*x)^n*(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^ p, x], x, Sin[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^m_*(d_.*sin[e_. + f_.*x_])^ n_.*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[m/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Sin[e + f*x], x]}, ff*c^(2*IntPart[(m - 1)/2] + 1)*(c*Cos[e + f*x])^(2* FracPart[(m - 1)/2])/(f*(Cos[e + f*x]^2)^ FracPart[(m - 1)/2])* Subst[ Int[(d*ff*x)^n*(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*cos[e_. + f_.*x_])^m_*(d_.*sin[e_. + f_.*x_])^ n_.*(a_ + b_.*sin[e_. + f_.*x_]^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "-Cot[e + f*x]*(b*Sin[e + f*x]^2)^p/(2*f*p) + b*(2*p - 1)/(2*p)*Int[(b*Sin[e + f*x]^2)^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*sin[e_. + f_.*x_]^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, e, f}, x] && Not[IntegerQ[p]] && GtQ[p, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "Cot[e + f*x]*(b*Sin[e + f*x]^2)^(p + 1)/(b*f*(2*p + 1)) + 2*(p + 1)/(b*(2*p + 1))*Int[(b*Sin[e + f*x]^2)^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*sin[e_. + f_.*x_]^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, e, f}, x] && Not[IntegerQ[p]] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Sin[e + f*x]^2, x]}, ff^((m + 1)/2)/(2*f)* Subst[Int[ x^((m - 1)/2)*(b*ff^(n/2)*x^(n/2))^p/(1 - ff*x)^((m + 1)/2), x], x, Sin[e + f*x]^2/ff]]",
     "rulenumber": 0,
            "lhs": "Int[tan[e_. + f_.*x_]^m_.*(b_.*sin[e_. + f_.*x_]^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, e, f, p}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Sin[e + f*x], x]}, ff^(m + 1)/f* Subst[Int[x^m*(b*(c*ff*x)^n)^p/(1 - ff^2*x^2)^((m + 1)/2), x], x, Sin[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[tan[e_. + f_.*x_]^m_.*(b_.*(c_.*sin[e_. + f_.*x_])^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, e, f, n, p}, x] && ILtQ[(m - 1)/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Sin[e + f*x], x]}, (b*ff^n)^ IntPart[p]*(b*Sin[e + f*x]^n)^ FracPart[p]/(Sin[e + f*x]/ff)^(n*FracPart[p])* Int[ActivateTrig[u]*(Sin[e + f*x]/ff)^(n*p), x]] /; FreeQ[{b, e, f, n, p}, x] && Not[IntegerQ[p]] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, (d_.*trig_[e + f*x])^m_.",
     "rulenumber": 0,
            "lhs": "Int[u_.*(b_.*sin[e_. + f_.*x_]^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "b^IntPart[p]*(b*(c*Sin[e + f*x])^n)^ FracPart[p]/(c*Sin[e + f*x])^(n*FracPart[p])* Int[ActivateTrig[u]*(c*Sin[e + f*x])^(n*p), x] /; FreeQ[{b, c, e, f, n, p}, x] && Not[IntegerQ[p]] && Not[IntegerQ[n]] && (EqQ[u, 1] || MatchQ[u, (d_.*trig_[e + f*x])^m_.",
     "rulenumber": 0,
            "lhs": "Int[u_.*(b_.*(c_.*sin[e_. + f_.*x_])^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": " With[{ff=FreeFactors[Tan[e+f*x],x]}, -ff/f*Subst[Int[(a+b+2*a*ff^2*x^2+a*ff^4*x^4)^p/(1+ff^2*x^2)^(2*p+1) ,x],x,Cot[e+f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[(a_+b_.*sin[e_.+f_.*x_]^4)^p_.,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,e,f},x] && IntegerQ[p] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^ p/(1 + ff^2*x^2)^(2*p + 1), x], x, Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_]^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, ff*(a + b*Sin[e + f*x]^4)^ p*(Sec[e + f*x]^2)^(2* p)/(f*(a + 2*a*Tan[e + f*x]^2 + (a + b)*Tan[e + f*x]^4)^p)* Subst[ Int[(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^ p/(1 + ff^2*x^2)^(2*p + 1), x], x, Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_]^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, p}, x] && IntegerQ[p - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "Module[{k}, Dist[2/(a*n), Sum[Int[1/(1 - Sin[e + f*x]^2/((-1)^(4*k/n)*Rt[-a/b, n/2])), x], {k, 1, n/2}], x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*sin[e_. + f_.*x_]^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && IntegerQ[n/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": " With[{ff=FreeFactors[Tan[e+f*x],x]}, -ff/f*Subst[Int[(b+a*(1+ff^2*x^2)^(n/2))^p/(1+ff^2*x^2)^(n*p/2+1),x] ,x,Cot[e+f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[(a_+b_.*sin[e_.+f_.*x_]^n_)^p_,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,e,f},x] && IntegerQ[n/2] &&  IGtQ[p,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[(b*ff^n*x^n + a*(1 + ff^2*x^2)^(n/2))^ p/(1 + ff^2*x^2)^(n*p/2 + 1), x], x, Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sin[e_. + f_.*x_]^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && IntegerQ[n/2] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "Int[ExpandTrig[(a + b*(c*sin[e + f*x])^n)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*(c_.*sin[e_. + f_.*x_])^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, e, f, n}, x] && (IGtQ[p, 0] || EqQ[p, -1] && IntegerQ[n])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "Unintegrable[(a + b*(c*Sin[e + f*x])^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*(c_.*sin[e_. + f_.*x_])^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, e, f, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Cos[e + f*x], x]}, -ff/f* Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4)^p, x], x, Cos[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[sin[e_. + f_.*x_]^m_.*(a_ + b_.*sin[e_. + f_.*x_]^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Cos[e + f*x], x]}, -ff/f* Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*(1 - ff^2*x^2)^(n/2))^ p, x], x, Cos[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[sin[e_. + f_.*x_]^m_.*(a_ + b_.*sin[e_. + f_.*x_]^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, ff^(m + 1)/f* Subst[Int[ x^m*(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^ p/(1 + ff^2*x^2)^(m/2 + 2*p + 1), x], x, Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[sin[e_. + f_.*x_]^m_*(a_ + b_.*sin[e_. + f_.*x_]^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, ff^(m + 1)/f* Subst[Int[ x^m*(a*(1 + ff^2*x^2)^(n/2) + b*ff^n*x^n)^ p/(1 + ff^2*x^2)^(m/2 + n*p/2 + 1), x], x, Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[sin[e_. + f_.*x_]^m_*(a_ + b_.*sin[e_. + f_.*x_]^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[n/2] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, ff^(m + 1)*(a + b*Sin[e + f*x]^4)^ p*(Sec[e + f*x]^2)^(2*p)/(f* Apart[a*(1 + Tan[e + f*x]^2)^2 + b*Tan[e + f*x]^4]^p)* Subst[ Int[x^m*ExpandToSum[a*(1 + ff^2*x^2)^2 + b*ff^4*x^4, x]^ p/(1 + ff^2*x^2)^(m/2 + 2*p + 1), x], x, Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[sin[e_. + f_.*x_]^m_*(a_ + b_.*sin[e_. + f_.*x_]^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && IntegerQ[p - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "Int[ExpandTrig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[sin[e_. + f_.*x_]^m_.*(a_ + b_.*sin[e_. + f_.*x_]^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4] || GtQ[p, 0] || EqQ[p, -1] && IntegerQ[n])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "Int[ExpandTrig[(d*sin[e + f*x])^m*(a + b*(c*sin[e + f*x])^n)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sin[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*sin[e_. + f_.*x_])^n_)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "Unintegrable[(d*Sin[e + f*x])^m*(a + b*(c*Sin[e + f*x])^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sin[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*sin[e_. + f_.*x_])^n_)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Sin[e + f*x], x]}, ff/f* Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p, x], x, Sin[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^m_.*(a_ + b_.*(c_.*sin[e_. + f_.*x_])^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (EqQ[n, 4] || GtQ[m, 0] || IGtQ[p, 0] || IntegersQ[m, p])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^ p/(1 + ff^2*x^2)^(m/2 + 2*p + 1), x], x, Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^m_*(a_ + b_.*sin[e_. + f_.*x_]^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[(b*ff^n*x^n + a*(1 + ff^2*x^2)^(n/2))^ p/(1 + ff^2*x^2)^(m/2 + n*p/2 + 1), x], x, Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^m_*(a_ + b_.*sin[e_. + f_.*x_]^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[n/2] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "Int[Expand[(1 - Sin[e + f*x]^2)^(m/2)/(a + b*Sin[e + f*x]^n), x], x]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^m_/(a_ + b_.*sin[e_. + f_.*x_]^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && IGtQ[m/2, 0] && IntegerQ[(n - 1)/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "  Int[ExpandTrig[(1-sin[e+f*x]^2)^(m/2)*(a+b*sin[e+f*x]^n)^p,x],x]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_.+f_.*x_]^m_*(a_+b_.*sin[e_.+f_.*x_]^n_)^p_,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,e,f},x] && IntegerQ[m/2] && IntegerQ[(n-1)/2] &&  ILtQ[p,-1] && LtQ[m,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "Int[ExpandTrig[(d*cos[e + f*x])^m*(a + b*(c*sin[e + f*x])^n)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*cos[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*sin[e_. + f_.*x_])^n_)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "Unintegrable[(d*Cos[e + f*x])^m*(a + b*(c*Sin[e + f*x])^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*cos[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*sin[e_. + f_.*x_])^n_)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Sin[e + f*x]^2, x]}, ff^((m + 1)/2)/(2*f)* Subst[Int[ x^((m - 1)/2)*(a + b*ff^(n/2)*x^(n/2))^p/(1 - ff*x)^((m + 1)/2), x], x, Sin[e + f*x]^2/ff]]",
     "rulenumber": 0,
            "lhs": "Int[tan[e_. + f_.*x_]^m_.*(a_ + b_.*sin[e_. + f_.*x_]^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Sin[e + f*x], x]}, ff^(m + 1)/f* Subst[Int[x^m*(a + b*(c*ff*x)^n)^p/(1 - ff^2*x^2)^((m + 1)/2), x], x, Sin[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[tan[e_. + f_.*x_]^m_.*(a_ + b_.*(c_.*sin[e_. + f_.*x_])^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, e, f, n, p}, x] && ILtQ[(m - 1)/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[(d*ff*x)^m* ExpandToSum[a*(1 + ff^2*x^2)^2 + b*ff^4*x^4, x]^ p/(1 + ff^2*x^2)^(2*p + 1), x], x, Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*tan[e_. + f_.*x_])^m_*(a_ + b_.*sin[e_. + f_.*x_]^4)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m}, x] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, ff*(a + b*Sin[e + f*x]^4)^ p*(Sec[e + f*x]^2)^(2*p)/(f* Apart[a*(1 + Tan[e + f*x]^2)^2 + b*Tan[e + f*x]^4]^p)* Subst[ Int[(d*ff*x)^m* ExpandToSum[a*(1 + ff^2*x^2)^2 + b*ff^4*x^4, x]^ p/(1 + ff^2*x^2)^(2*p + 1), x], x, Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*tan[e_. + f_.*x_])^m_*(a_ + b_.*sin[e_. + f_.*x_]^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m}, x] && IntegerQ[p - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, ff^(m + 1)/f* Subst[Int[(d*x)^ m*(b*ff^n*x^n + a*(1 + ff^2*x^2)^(n/2))^ p/(1 + ff^2*x^2)^(n*p/2 + 1), x], x, Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*tan[e_. + f_.*x_])^m_*(a_ + b_.*sin[e_. + f_.*x_]^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m}, x] && IntegerQ[n/2] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "Int[ExpandTrig[(d*tan[e + f*x])^m*(a + b*(c*sin[e + f*x])^n)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*tan[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*sin[e_. + f_.*x_])^n_)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "Unintegrable[(d*Tan[e + f*x])^m*(a + b*(c*Sin[e + f*x])^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*tan[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*sin[e_. + f_.*x_])^n_)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "(d*Cot[e + f*x])^FracPart[m]*(Tan[e + f*x]/d)^ FracPart[m]* Int[(Tan[e + f*x]/d)^(-m)*(a + b*(c*Sin[e + f*x])^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*cot[e_. + f_.*x_])^m_*(a_ + b_.*(c_.*sin[e_. + f_.*x_])^n_)^ p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "(d*Sec[e + f*x])^FracPart[m]*(Cos[e + f*x]/d)^ FracPart[m]* Int[(Cos[e + f*x]/d)^(-m)*(a + b*(c*Sin[e + f*x])^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sec[e_. + f_.*x_])^m_*(a_ + b_.*(c_.*sin[e_. + f_.*x_])^n_)^ p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "d^(n*p)*Int[(d*Csc[e + f*x])^(m - n*p)*(b + a*Csc[e + f*x]^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*csc[e_. + f_.*x_])^m_*(a_ + b_.*sin[e_. + f_.*x_]^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n, p}, x] && Not[IntegerQ[m]] && IntegersQ[n, p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
       "filename": "4.1.7 (d trig)^m (a+b (c sin)^n)^p.m",
            "rhs": "(d*Csc[e + f*x])^FracPart[m]*(Sin[e + f*x]/d)^ FracPart[m]* Int[(Sin[e + f*x]/d)^(-m)*(a + b*(c*Sin[e + f*x])^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*csc[e_. + f_.*x_])^m_*(a_ + b_.*(c_.*sin[e_. + f_.*x_])^n_)^ p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.8 trig^m (a+b cos^p+c sin^q)^n.m",
       "filename": "4.1.8 trig^m (a+b cos^p+c sin^q)^n.m",
            "rhs": "Module[{f = FreeFactors[Cot[d + e*x], x]}, -f/e* Subst[Int[ ExpandToSum[ c + b*(1 + f^2*x^2)^(q/2 - p/2) + a*(1 + f^2*x^2)^(q/2), x]^ n/(1 + f^2*x^2)^(m/2 + n*q/2 + 1), x], x, Cot[d + e*x]/f]]",
     "rulenumber": 0,
            "lhs": "Int[sin[d_. + e_.*x_]^ m_*(a_ + b_.*cos[d_. + e_.*x_]^p_ + c_.*sin[d_. + e_.*x_]^q_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IntegerQ[m/2] && IntegerQ[p/2] && IntegerQ[q/2] && IntegerQ[n] && GtQ[p, 0] && LeQ[p, q]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.8 trig^m (a+b cos^p+c sin^q)^n.m",
       "filename": "4.1.8 trig^m (a+b cos^p+c sin^q)^n.m",
            "rhs": "Module[{f = FreeFactors[Tan[d + e*x], x]}, f/e*Subst[ Int[ExpandToSum[ c + b*(1 + f^2*x^2)^(q/2 - p/2) + a*(1 + f^2*x^2)^(q/2), x]^ n/(1 + f^2*x^2)^(m/2 + n*q/2 + 1), x], x, Tan[d + e*x]/f]]",
     "rulenumber": 0,
            "lhs": "Int[cos[d_. + e_.*x_]^ m_*(a_ + b_.*sin[d_. + e_.*x_]^p_ + c_.*cos[d_. + e_.*x_]^q_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IntegerQ[m/2] && IntegerQ[p/2] && IntegerQ[q/2] && IntegerQ[n] && GtQ[p, 0] && LeQ[p, q]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.8 trig^m (a+b cos^p+c sin^q)^n.m",
       "filename": "4.1.8 trig^m (a+b cos^p+c sin^q)^n.m",
            "rhs": "Module[{f = FreeFactors[Cot[d + e*x], x]}, -f/e* Subst[Int[ ExpandToSum[ a*(1 + f^2*x^2)^(p/2) + b*f^p*x^p + c*(1 + f^2*x^2)^(p/2 - q/2), x]^ n/(1 + f^2*x^2)^(m/2 + n*p/2 + 1), x], x, Cot[d + e*x]/f]]",
     "rulenumber": 0,
            "lhs": "Int[sin[d_. + e_.*x_]^ m_*(a_ + b_.*cos[d_. + e_.*x_]^p_ + c_.*sin[d_. + e_.*x_]^q_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IntegerQ[m/2] && IntegerQ[p/2] && IntegerQ[q/2] && IntegerQ[n] && LtQ[0, q, p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.8 trig^m (a+b cos^p+c sin^q)^n.m",
       "filename": "4.1.8 trig^m (a+b cos^p+c sin^q)^n.m",
            "rhs": "Module[{f = FreeFactors[Tan[d + e*x], x]}, f/e*Subst[ Int[ExpandToSum[ a*(1 + f^2*x^2)^(p/2) + b*f^p*x^p + c*(1 + f^2*x^2)^(p/2 - q/2), x]^ n/(1 + f^2*x^2)^(m/2 + n*p/2 + 1), x], x, Tan[d + e*x]/f]]",
     "rulenumber": 0,
            "lhs": "Int[cos[d_. + e_.*x_]^ m_*(a_ + b_.*sin[d_. + e_.*x_]^p_ + c_.*cos[d_. + e_.*x_]^q_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IntegerQ[m/2] && IntegerQ[p/2] && IntegerQ[q/2] && IntegerQ[n] && LtQ[0, q, p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.8 trig^m (a+b cos^p+c sin^q)^n.m",
       "filename": "4.1.8 trig^m (a+b cos^p+c sin^q)^n.m",
            "rhs": "Module[{f = FreeFactors[Cot[d + e*x], x]}, -f/e* Subst[Int[ ExpandToSum[ c + b*(1 + f^2*x^2)^(q/2 - p/2) + a*(1 + f^2*x^2)^(q/2), x]^ n/(1 + f^2*x^2)^(m/2 + n*q/2 + 1), x], x, Cot[d + e*x]/f]]",
     "rulenumber": 0,
            "lhs": "Int[sin[d_. + e_.*x_]^ m_*(a_ + b_.*cos[d_. + e_.*x_]^p_ + c_.*sin[d_. + e_.*x_]^q_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IntegerQ[m/2] && IntegerQ[p/2] && IntegerQ[q/2] && IntegerQ[n] && GtQ[p, 0] && LeQ[p, q]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.8 trig^m (a+b cos^p+c sin^q)^n.m",
       "filename": "4.1.8 trig^m (a+b cos^p+c sin^q)^n.m",
            "rhs": "Module[{f = FreeFactors[Tan[d + e*x], x]}, f/e*Subst[ Int[ExpandToSum[ c + b*(1 + f^2*x^2)^(q/2 - p/2) + a*(1 + f^2*x^2)^(q/2), x]^ n/(1 + f^2*x^2)^(m/2 + n*q/2 + 1), x], x, Tan[d + e*x]/f]]",
     "rulenumber": 0,
            "lhs": "Int[cos[d_. + e_.*x_]^ m_*(a_ + b_.*sin[d_. + e_.*x_]^p_ + c_.*cos[d_. + e_.*x_]^q_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IntegerQ[m/2] && IntegerQ[p/2] && IntegerQ[q/2] && IntegerQ[n] && GtQ[p, 0] && LeQ[p, q]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.8 trig^m (a+b cos^p+c sin^q)^n.m",
       "filename": "4.1.8 trig^m (a+b cos^p+c sin^q)^n.m",
            "rhs": "Module[{f = FreeFactors[Cot[d + e*x], x]}, -f/e* Subst[Int[ ExpandToSum[ a*(1 + f^2*x^2)^(p/2) + b*f^p*x^p + c*(1 + f^2*x^2)^(p/2 - q/2), x]^ n/(1 + f^2*x^2)^(m/2 + n*p/2 + 1), x], x, Cot[d + e*x]/f]]",
     "rulenumber": 0,
            "lhs": "Int[sin[d_. + e_.*x_]^ m_*(a_ + b_.*cos[d_. + e_.*x_]^p_ + c_.*sin[d_. + e_.*x_]^q_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IntegerQ[m/2] && IntegerQ[p/2] && IntegerQ[q/2] && IntegerQ[n] && LtQ[0, q, p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.8 trig^m (a+b cos^p+c sin^q)^n.m",
       "filename": "4.1.8 trig^m (a+b cos^p+c sin^q)^n.m",
            "rhs": "Module[{f = FreeFactors[Tan[d + e*x], x]}, f/e*Subst[ Int[ExpandToSum[ a*(1 + f^2*x^2)^(p/2) + b*f^p*x^p + c*(1 + f^2*x^2)^(p/2 - q/2), x]^ n/(1 + f^2*x^2)^(m/2 + n*p/2 + 1), x], x, Tan[d + e*x]/f]]",
     "rulenumber": 0,
            "lhs": "Int[cos[d_. + e_.*x_]^ m_*(a_ + b_.*sin[d_. + e_.*x_]^p_ + c_.*cos[d_. + e_.*x_]^q_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IntegerQ[m/2] && IntegerQ[p/2] && IntegerQ[q/2] && IntegerQ[n] && LtQ[0, q, p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
       "filename": "4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
            "rhs": "1/(4^p*c^p)*Int[(b + 2*c*Sin[d + e*x]^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sin[d_. + e_.*x_]^n_. + c_.*sin[d_. + e_.*x_]^n2_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
       "filename": "4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
            "rhs": "1/(4^p*c^p)*Int[(b + 2*c*Cos[d + e*x]^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*cos[d_. + e_.*x_]^n_. + c_.*cos[d_. + e_.*x_]^n2_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
       "filename": "4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
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       "filename": "4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
            "rhs": "1/(4^p*c^p)*Int[Tan[d + e*x]^m*(b + 2*c*Sin[d + e*x]^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[tan[d_. + e_.*x_]^ m_*(a_. + b_.*sin[d_. + e_.*x_]^n_. + c_.*sin[d_. + e_.*x_]^n2_.)^ p_., x_Symbol]",
        "comment": false,
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       "filename": "4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
            "rhs": "1/(4^p*c^p)*Int[Cot[d + e*x]^m*(b + 2*c*Cos[d + e*x]^n)^(2*p), x]",
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            "lhs": "Int[cot[d_. + e_.*x_]^ m_*(a_. + b_.*cos[d_. + e_.*x_]^n_. + c_.*cos[d_. + e_.*x_]^n2_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && Not[IntegerQ[(m - 1)/2]] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]"
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       "filename": "4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
            "rhs": "(a + b*Sin[d + e*x]^n + c*Sin[d + e*x]^(2*n))^ p/(b + 2*c*Sin[d + e*x]^n)^(2*p)* Int[Tan[d + e*x]^m*(b + 2*c*Sin[d + e*x]^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[tan[d_. + e_.*x_]^ m_*(a_. + b_.*sin[d_. + e_.*x_]^n_. + c_.*sin[d_. + e_.*x_]^n2_.)^ p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && Not[IntegerQ[(m - 1)/2]] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
       "filename": "4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
            "rhs": "(a + b*Cos[d + e*x]^n + c*Cos[d + e*x]^(2*n))^ p/(b + 2*c*Cos[d + e*x]^n)^(2*p)* Int[Cot[d + e*x]^m*(b + 2*c*Cos[d + e*x]^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[cot[d_. + e_.*x_]^ m_*(a_. + b_.*cos[d_. + e_.*x_]^n_. + c_.*cos[d_. + e_.*x_]^n2_.)^ p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && Not[IntegerQ[(m - 1)/2]] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]"
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       "filename": "4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
            "rhs": "Module[{f = FreeFactors[Tan[d + e*x], x]}, f^(m + 1)/e* Subst[Int[ x^m*ExpandToSum[ c*x^(2*n) + b*x^n*(1 + x^2)^(n/2) + a*(1 + x^2)^n, x]^ p/(1 + f^2*x^2)^(n*p + 1), x], x, Tan[d + e*x]/f]]",
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            "lhs": "Int[tan[d_. + e_.*x_]^ m_.*(a_. + b_.*sin[d_. + e_.*x_]^n_ + c_.*sin[d_. + e_.*x_]^n2_)^ p_., x_Symbol]",
        "comment": false,
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            "rhs": "Module[{f = FreeFactors[Cot[d + e*x], x]}, -f^(m + 1)/e* Subst[Int[ x^m*ExpandToSum[ c*x^(2*n) + b*x^n*(1 + x^2)^(n/2) + a*(1 + x^2)^n, x]^ p/(1 + f^2*x^2)^(n*p + 1), x], x, Cot[d + e*x]/f]]",
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            "rhs": "Int[ExpandTrig[ sin[d + e*x]^ m*(a + b*sin[d + e*x]^n + c*sin[d + e*x]^(2*n))^ p/(1 - sin[d + e*x]^2)^(m/2), x], x]",
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            "lhs": "Int[tan[d_. + e_.*x_]^ m_.*(a_. + b_.*sin[d_. + e_.*x_]^n_. + c_.*sin[d_. + e_.*x_]^n2_.)^p_., x_Symbol]",
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         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && NeQ[b^2 - 4*a*c, 0] && IntegersQ[n, p]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
       "filename": "4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
            "rhs": "Int[ExpandTrig[ cos[d + e*x]^ m*(a + b*cos[d + e*x]^n + c*cos[d + e*x]^(2*n))^ p/(1 - cos[d + e*x]^2)^(m/2), x], x]",
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            "lhs": "Int[cot[d_. + e_.*x_]^ m_.*(a_. + b_.*cos[d_. + e_.*x_]^n_. + c_.*cos[d_. + e_.*x_]^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && NeQ[b^2 - 4*a*c, 0] && IntegersQ[n, p]"
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       "filename": "4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
            "rhs": "Module[{g = FreeFactors[Sin[d + e*x], x]}, g^(m + 1)/e* Subst[Int[(1 - g^2*x^2)^((m - 1)/ 2)*(a + b*(f*g*x)^n + c*(f*g*x)^(2*n))^p/x^m, x], x, Sin[d + e*x]/g]]",
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            "lhs": "Int[cot[d_. + e_.*x_]^ m_.*(a_ + b_.*(f_.*sin[d_. + e_.*x_])^n_ + c_.*(f_.*sin[d_. + e_.*x_])^n2_.)^p_., x_Symbol]",
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         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && IntegerQ[(m - 1)/2] && IntegerQ[2*p]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
       "filename": "4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
            "rhs": "Module[{g = FreeFactors[Cos[d + e*x], x]}, -g^(m + 1)/e* Subst[Int[(1 - g^2*x^2)^((m - 1)/ 2)*(a + b*(f*g*x)^n + c*(f*g*x)^(2*n))^p/x^m, x], x, Cos[d + e*x]/g]]",
     "rulenumber": 0,
            "lhs": "Int[tan[d_. + e_.*x_]^ m_.*(a_ + b_.*(f_.*cos[d_. + e_.*x_])^n_ + c_.*(f_.*cos[d_. + e_.*x_])^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && IntegerQ[(m - 1)/2] && IntegerQ[2*p]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
       "filename": "4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
            "rhs": "1/(4^p*c^p)*Int[Cot[d + e*x]^m*(b + 2*c*Sin[d + e*x]^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[cot[d_. + e_.*x_]^ m_*(a_. + b_.*sin[d_. + e_.*x_]^n_. + c_.*sin[d_. + e_.*x_]^n2_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && Not[IntegerQ[(m - 1)/2]] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
       "filename": "4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
            "rhs": "1/(4^p*c^p)*Int[Tan[d + e*x]^m*(b + 2*c*Cos[d + e*x]^n)^(2*p), x]",
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            "lhs": "Int[tan[d_. + e_.*x_]^ m_*(a_. + b_.*cos[d_. + e_.*x_]^n_. + c_.*cos[d_. + e_.*x_]^n2_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && Not[IntegerQ[(m - 1)/2]] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
       "filename": "4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
            "rhs": "(a + b*Sin[d + e*x]^n + c*Sin[d + e*x]^(2*n))^ p/(b + 2*c*Sin[d + e*x]^n)^(2*p)* Int[Cot[d + e*x]^m*(b + 2*c*Sin[d + e*x]^n)^(2*p), x]",
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            "lhs": "Int[cot[d_. + e_.*x_]^ m_*(a_. + b_.*sin[d_. + e_.*x_]^n_. + c_.*sin[d_. + e_.*x_]^n2_.)^ p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && Not[IntegerQ[(m - 1)/2]] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]"
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       "filename": "4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
            "rhs": "(a + b*Cos[d + e*x]^n + c*Cos[d + e*x]^(2*n))^ p/(b + 2*c*Cos[d + e*x]^n)^(2*p)* Int[Tan[d + e*x]^m*(b + 2*c*Cos[d + e*x]^n)^(2*p), x]",
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            "lhs": "Int[tan[d_. + e_.*x_]^ m_*(a_. + b_.*cos[d_. + e_.*x_]^n_. + c_.*cos[d_. + e_.*x_]^n2_.)^ p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && Not[IntegerQ[(m - 1)/2]] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
       "filename": "4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
            "rhs": "Module[{f = FreeFactors[Cot[d + e*x], x]}, -f^(m + 1)/e* Subst[Int[ x^m*ExpandToSum[c + b*(1 + f^2*x^2)^(n/2) + a*(1 + f^2*x^2)^n, x]^p/(1 + f^2*x^2)^(n*p + 1), x], x, Cot[d + e*x]/f]]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && EqQ[n2, 2*n] && IntegerQ[n/2] && IntegerQ[p]"
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            "rhs": "Int[ExpandTrig[(1 - cos[d + e*x]^2)^(m/ 2)*(a + b*cos[d + e*x]^n + c*cos[d + e*x]^(2*n))^p/ cos[d + e*x]^m, x], x]",
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         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && NeQ[b^2 - 4*a*c, 0] && IntegersQ[n, p]"
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            "rhs": "1/(4^n*c^n)* Int[(A + B*Cos[d + e*x])*(b + 2*c*Cos[d + e*x])^(2*n), x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[n]"
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       "filename": "4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
            "rhs": "(a + b*Sin[d + e*x] + c*Sin[d + e*x]^2)^ n/(b + 2*c*Sin[d + e*x])^(2*n)* Int[(A + B*Sin[d + e*x])*(b + 2*c*Sin[d + e*x])^(2*n), x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[n]]"
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       "filename": "4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
            "rhs": "(a + b*Cos[d + e*x] + c*Cos[d + e*x]^2)^ n/(b + 2*c*Cos[d + e*x])^(2*n)* Int[(A + B*Cos[d + e*x])*(b + 2*c*Cos[d + e*x])^(2*n), x]",
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        "comment": false,
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            "rhs": "Module[{q = Rt[b^2 - 4*a*c, 2]}, (B + (b*B - 2*A*c)/q)*Int[1/(b + q + 2*c*Cos[d + e*x]), x] + (B - (b*B - 2*A*c)/q)*Int[1/(b - q + 2*c*Cos[d + e*x]), x]]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*cos[d_. + e_.*x_])/(a_. + b_.*cos[d_. + e_.*x_] + c_.*cos[d_. + e_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
       "filename": "4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
            "rhs": "Int[ExpandTrig[(A + B*sin[d + e*x])*(a + b*sin[d + e*x] + c*sin[d + e*x]^2)^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*sin[d_. + e_.*x_])*(a_. + b_.*sin[d_. + e_.*x_] + c_.*sin[d_. + e_.*x_]^2)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.1 Sine/4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
       "filename": "4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m",
            "rhs": "Int[ExpandTrig[(A + B*cos[d + e*x])*(a + b*cos[d + e*x] + c*cos[d + e*x]^2)^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*cos[d_. + e_.*x_])*(a_. + b_.*cos[d_. + e_.*x_] + c_.*cos[d_. + e_.*x_]^2)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.1 (a+b tan)^n.m",
       "filename": "4.3.1.1 (a+b tan)^n.m",
            "rhs": "b*(b*Tan[c + d*x])^(n - 1)/(d*(n - 1)) - b^2*Int[(b*Tan[c + d*x])^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*tan[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d}, x] && GtQ[n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.1 (a+b tan)^n.m",
       "filename": "4.3.1.1 (a+b tan)^n.m",
            "rhs": "(b*Tan[c + d*x])^(n + 1)/(b*d*(n + 1)) - 1/b^2*Int[(b*Tan[c + d*x])^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*tan[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d}, x] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.1 (a+b tan)^n.m",
       "filename": "4.3.1.1 (a+b tan)^n.m",
            "rhs": "-Log[RemoveContent[Cos[c + d*x], x]]/d",
     "rulenumber": 0,
            "lhs": "Int[tan[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.1 (a+b tan)^n.m",
       "filename": "4.3.1.1 (a+b tan)^n.m",
            "rhs": " Log[RemoveContent[Sin[c+d*x],x]]/d",
     "rulenumber": 0,
            "lhs": "Int[1/tan[c_.+d_.*x_],x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c,d},x] *)"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.1 (a+b tan)^n.m",
       "filename": "4.3.1.1 (a+b tan)^n.m",
            "rhs": "b/d*Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*tan[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, n}, x] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.1 (a+b tan)^n.m",
       "filename": "4.3.1.1 (a+b tan)^n.m",
            "rhs": "(a^2 - b^2)*x + b^2*Tan[c + d*x]/d + 2*a*b*Int[Tan[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[c_. + d_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.1 (a+b tan)^n.m",
       "filename": "4.3.1.1 (a+b tan)^n.m",
            "rhs": " Int[ExpandIntegrand[(a+b*Tan[c+d*x])^n,x],x]",
     "rulenumber": 0,
            "lhs": "Int[(a_+b_.*tan[c_.+d_.*x_])^n_,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d},x] &&  IGtQ[n,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.1 (a+b tan)^n.m",
       "filename": "4.3.1.1 (a+b tan)^n.m",
            "rhs": "b*(a + b*Tan[c + d*x])^(n - 1)/(d*(n - 1)) + 2*a*Int[(a + b*Tan[c + d*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && GtQ[n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.1 (a+b tan)^n.m",
       "filename": "4.3.1.1 (a+b tan)^n.m",
            "rhs": "a*(a + b*Tan[c + d*x])^n/(2*b*d*n) + 1/(2*a)*Int[(a + b*Tan[c + d*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && LtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.1 (a+b tan)^n.m",
       "filename": "4.3.1.1 (a+b tan)^n.m",
            "rhs": "-2*b/d* Subst[Int[1/(2*a - x^2), x], x, Sqrt[a + b*Tan[c + d*x]]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*tan[c_. + d_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.1 (a+b tan)^n.m",
       "filename": "4.3.1.1 (a+b tan)^n.m",
            "rhs": "-b/d* Subst[Int[(a + x)^(n - 1)/(a - x), x], x, b*Tan[c + d*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.1 (a+b tan)^n.m",
       "filename": "4.3.1.1 (a+b tan)^n.m",
            "rhs": "b*(a + b*Tan[c + d*x])^(n - 1)/(d*(n - 1)) + Int[(a^2 - b^2 + 2*a*b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && GtQ[n, 1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.1 (a+b tan)^n.m",
       "filename": "4.3.1.1 (a+b tan)^n.m",
            "rhs": "b*(a + b*Tan[c + d*x])^(n + 1)/(d*(n + 1)*(a^2 + b^2)) + 1/(a^2 + b^2)* Int[(a - b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.1 (a+b tan)^n.m",
       "filename": "4.3.1.1 (a+b tan)^n.m",
            "rhs": "a*x/(a^2 + b^2) + b/(a^2 + b^2)*Int[(b - a*Tan[c + d*x])/(a + b*Tan[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*tan[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.1 (a+b tan)^n.m",
       "filename": "4.3.1.1 (a+b tan)^n.m",
            "rhs": "b/d*Subst[Int[(a + x)^n/(b^2 + x^2), x], x, b*Tan[c + d*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 + b^2, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "b*(d*Sec[e + f*x])^m/(f*m) + a*Int[(d*Sec[e + f*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2*m] || NeQ[a^2 + b^2, 0])"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "1/(a^(m - 2)*b*f)* Subst[Int[(a - x)^(m/2 - 1)*(a + x)^(n + m/2 - 1), x], x, b*Tan[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[sec[e_. + f_.*x_]^m_*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, n}, x] && EqQ[a^2 + b^2, 0] && IntegerQ[m/2]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "b*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n/(a*f*m)",
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            "lhs": "Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] && EqQ[Simplify[m + n], 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "-2*a/(b*f)* Subst[Int[1/(2 - a*x^2), x], x, Sec[e + f*x]/Sqrt[a + b*Tan[e + f*x]]]",
     "rulenumber": 0,
            "lhs": "Int[sec[e_. + f_.*x_]/Sqrt[a_ + b_.*tan[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && EqQ[a^2 + b^2, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "b*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n/(a*f*m) + a/(2*d^2)* Int[(d*Sec[e + f*x])^(m + 2)*(a + b*Tan[e + f*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && EqQ[m/2 + n, 0] && GtQ[n, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "2*d^2*(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1)/(b* f*(m - 2)) + 2*d^2/a* Int[(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && EqQ[m/2 + n, 0] && LtQ[n, -1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "(a/d)^(2*IntPart[n])*(a + b*Tan[e + f*x])^ FracPart[n]*(a - b*Tan[e + f*x])^ FracPart[n]/(d*Sec[e + f*x])^(2*FracPart[n])* Int[1/(a - b*Tan[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sec[e_. + f_.*x_])^m_*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] && EqQ[Simplify[m/2 + n], 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "2*b*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1)/(f*m)",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] && EqQ[Simplify[m/2 + n - 1], 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "b*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1)) + a*(m + 2*n - 2)/(m + n - 1)* Int[(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] && IGtQ[Simplify[m/2 + n - 1], 0] && Not[IntegerQ[n]]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "-4*b*d^2/f* Subst[Int[x^2/(a^2 + d^2*x^4), x], x, Sqrt[a + b*Tan[e + f*x]]/Sqrt[d*Sec[e + f*x]]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[d_.*sec[e_. + f_.*x_]]*Sqrt[a_ + b_.*tan[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "2*b*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1)/(f*m) - b^2*(m + 2*n - 2)/(d^2*m)* Int[(d*Sec[e + f*x])^(m + 2)*(a + b*Tan[e + f*x])^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sec[e_. + f_.*x_])^m_*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && GtQ[n, 1] && (IGtQ[n/2, 0] && ILtQ[m - 1/2, 0] || EqQ[n, 2] && LtQ[m, 0] || LeQ[m, -1] && GtQ[m + n, 0] || ILtQ[m, 0] && LtQ[m/2 + n - 1, 0] && IntegerQ[n] || EqQ[n, 3/2] && EqQ[m, -1/2]) && IntegerQ[2*m]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "b*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n/(a*f*m) + a*(m + n)/(m*d^2)* Int[(d*Sec[e + f*x])^(m + 2)*(a + b*Tan[e + f*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sec[e_. + f_.*x_])^m_*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && GtQ[n, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "b*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1)) + a*(m + 2*n - 2)/(m + n - 1)* Int[(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && GtQ[n, 0] && NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "d*Sec[e + f*x]/(Sqrt[a - b*Tan[e + f*x]]*Sqrt[a + b*Tan[e + f*x]])* Int[Sqrt[d*Sec[e + f*x]]*Sqrt[a - b*Tan[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sec[e_. + f_.*x_])^(3/2)/Sqrt[a_ + b_.*tan[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "2*d^2*(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1)/(b* f*(m + 2*n)) - d^2*(m - 2)/(b^2*(m + 2*n))* Int[(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sec[e_. + f_.*x_])^m_*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && LtQ[n, -1] && (ILtQ[n/2, 0] && IGtQ[m - 1/2, 0] || EqQ[n, -2] || IGtQ[m + n, 0] || IntegersQ[n, m + 1/2] && GtQ[2*m + n + 1, 0]) && IntegerQ[2*m]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "d^2*(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1)/(b* f*(m + n - 1)) + d^2*(m - 2)/(a*(m + n - 1))* Int[(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && LtQ[n, 0] && GtQ[m, 1] && Not[ILtQ[m + n, 0]] && NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "a*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n/(b*f*(m + 2*n)) + Simplify[m + n]/(a*(m + 2*n))* Int[(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && LtQ[n, 0] && NeQ[m + 2*n, 0] && IntegersQ[2*m, 2*n]"
  },
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "b*(d*Sec[e + f*x])^ m*(a + b*Tan[e + f*x])^(n - 1)/(f*Simplify[m + n - 1]) + a*(m + 2*n - 2)/Simplify[m + n - 1]* Int[(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] && IGtQ[Simplify[m + n - 1], 0] && RationalQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "a*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n/(b*f*(m + 2*n)) + Simplify[m + n]/(a*(m + 2*n))* Int[(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] && ILtQ[Simplify[m + n], 0] && NeQ[m + 2*n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "a^n*(d*Sec[e+f*x])^m/(b*f*(Sec[e+f*x]^2)^(m/2))*Subst[Int[(1+x/ a)^(n+m/2-1)*(1-x/a)^(m/2-1),x],x,b*Tan[e+f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sec[e_.+f_.*x_])^m_.*(a_+b_.*tan[e_.+f_.*x_])^n_,x_Symbol] ",
        "comment": false,
         "givens": " FreeQ[{a,b,d,e,f,m},x] && EqQ[a^2+b^2,0] && IntegerQ[n] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "(d*Sec[e+f*x])^m/(b*f*(Sec[e+f*x]^2)^(m/2))*Subst[Int[(a+x)^n*( 1+x^2/b^2)^(m/2-1),x],x,b*Tan[e+f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sec[e_.+f_.*x_])^m_.*(a_+b_.*tan[e_.+f_.*x_])^n_,x_Symbol] ",
        "comment": false,
         "givens": "FreeQ[{a,b,d,e,f,m,n},x] &&  EqQ[a^2+b^2,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "(d*Sec[e + f*x])^ m/((a + b*Tan[e + f*x])^(m/2)*(a - b*Tan[e + f*x])^(m/2))* Int[(a + b*Tan[e + f*x])^(m/2 + n)*(a - b*Tan[e + f*x])^(m/2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "1/(b*f)*Subst[Int[(a + x)^n*(1 + x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[sec[e_. + f_.*x_]^m_*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, n}, x] && NeQ[a^2 + b^2, 0] && IntegerQ[m/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "b^2*ArcTanh[Sin[e + f*x]]/f - 2*a*b*Cos[e + f*x]/f + (a^2 - b^2)*Sin[e + f*x]/f",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^2/sec[e_. + f_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && NeQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "b*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])/(f*(m + 1)) + 1/(m + 1)* Int[(d*Sec[e + f*x])^ m*(a^2*(m + 1) - b^2 + a*b*(m + 2)*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*tan[e_. + f_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m}, x] && NeQ[a^2 + b^2, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "-1/f* Subst[Int[1/(a^2 + b^2 - x^2), x], x, (b - a*Tan[e + f*x])/Sec[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[sec[e_. + f_.*x_]/(a_ + b_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && NeQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "-d^2/b^2* Int[(d*Sec[e + f*x])^(m - 2)*(a - b*Tan[e + f*x]), x] + d^2*(a^2 + b^2)/b^2* Int[(d*Sec[e + f*x])^(m - 2)/(a + b*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sec[e_. + f_.*x_])^m_/(a_ + b_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 + b^2, 0] && IGtQ[m, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "1/(a^2 + b^2)*Int[(d*Sec[e + f*x])^m*(a - b*Tan[e + f*x]), x] + b^2/(d^2*(a^2 + b^2))* Int[(d*Sec[e + f*x])^(m + 2)/(a + b*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sec[e_. + f_.*x_])^m_/(a_ + b_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 + b^2, 0] && ILtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "d^(2*IntPart[m/2])*(d*Sec[e + f*x])^(2*FracPart[m/2])/(b* f*(Sec[e + f*x]^2)^FracPart[m/2])* Subst[Int[(a + x)^n*(1 + x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n}, x] && NeQ[a^2 + b^2, 0] && Not[IntegerQ[m/2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "-4*b/f* Subst[Int[x^2/(a^2*d^2 + x^4), x], x, Sqrt[d*Cos[e + f*x]]*Sqrt[a + b*Tan[e + f*x]]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*tan[e_. + f_.*x_]]/Sqrt[d_. cos[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "1/(d*Cos[e + f*x]*Sqrt[a - b*Tan[e + f*x]]*Sqrt[a + b*Tan[e + f*x]])* Int[Sqrt[a - b*Tan[e + f*x]]/Sqrt[d*Cos[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[1/((d_. cos[e_. + f_.*x_])^(3/2)* Sqrt[a_ + b_.*tan[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m",
       "filename": "4.3.1.2 (d sec)^m (a+b tan)^n.m",
            "rhs": "(d*Cos[e + f*x])^m*(d*Sec[e + f*x])^m* Int[(a + b*Tan[e + f*x])^n/(d*Sec[e + f*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*cos[e_. + f_.*x_])^m_*(a_ + b_.*tan[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.3 (d sin)^m (a+b tan)^n.m",
       "filename": "4.3.1.3 (d sin)^m (a+b tan)^n.m",
            "rhs": "b/f*Subst[Int[x^m*(a + x)^n/(b^2 + x^2)^(m/2 + 1), x], x, b*Tan[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[sin[e_. + f_.*x_]^m_*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, n}, x] && IntegerQ[m/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.3 (d sin)^m (a+b tan)^n.m",
       "filename": "4.3.1.3 (d sin)^m (a+b tan)^n.m",
            "rhs": "Int[Expand[Sin[e + f*x]^m*(a + b*Tan[e + f*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[sin[e_. + f_.*x_]^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.3 (d sin)^m (a+b tan)^n.m",
       "filename": "4.3.1.3 (d sin)^m (a+b tan)^n.m",
            "rhs": "Int[Sin[e + f*x]^m*(a*Cos[e + f*x] + b*Sin[e + f*x])^n/ Cos[e + f*x]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[sin[e_. + f_.*x_]^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && ILtQ[n, 0] && (LtQ[m, 5] && GtQ[n, -4] || EqQ[m, 5] && EqQ[n, -1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.3 (d sin)^m (a+b tan)^n.m",
       "filename": "4.3.1.3 (d sin)^m (a+b tan)^n.m",
            "rhs": "(d*Csc[e + f*x])^FracPart[m]*(Sin[e + f*x]/d)^ FracPart[m]*Int[(a + b*Tan[e + f*x])^n/(Sin[e + f*x]/d)^m, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*csc[e_. + f_.*x_])^m_*(a_. + b_.*tan[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n}, x] && Not[IntegerQ[m]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.3 (d sin)^m (a+b tan)^n.m",
       "filename": "4.3.1.3 (d sin)^m (a+b tan)^n.m",
            "rhs": "Int[Cos[e + f*x]^(m - n)* Sin[e + f*x]^p*(a*Cos[e + f*x] + b*Sin[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^m_.* sin[e_. + f_.*x_]^p_.*(a_ + b_.*tan[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m, p}, x] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.1.3 (d sin)^m (a+b tan)^n.m",
       "filename": "4.3.1.3 (d sin)^m (a+b tan)^n.m",
            "rhs": "Int[Sin[e + f*x]^(m - n)* Cos[e + f*x]^p*(a*Sin[e + f*x] + b*Cos[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[sin[e_. + f_.*x_]^m_.* cos[e_. + f_.*x_]^p_.*(a_ + b_.*cot[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m, p}, x] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.10 (c+d x)^m (a+b tan)^n.m",
       "filename": "4.3.10 (c+d x)^m (a+b tan)^n.m",
            "rhs": "-I*(c + d*x)^(m + 1)/(d*(m + 1)) + 2*I*Int[(c + d*x)^m*E^(-2*I*k*Pi)* E^(2*(-I*e + f*fz*x))/(1 + E^(-2*I*k*Pi)*E^(2*(-I*e + f*fz*x))), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*tan[e_. + k_.*Pi + f_.*Complex[0, fz_]*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.10 (c+d x)^m (a+b tan)^n.m",
       "filename": "4.3.10 (c+d x)^m (a+b tan)^n.m",
            "rhs": "I*(c + d*x)^(m + 1)/(d*(m + 1)) - 2*I*Int[(c + d*x)^m*E^(2*I*k*Pi)* E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x))), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*tan[e_. + k_.*Pi + f_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.10 (c+d x)^m (a+b tan)^n.m",
       "filename": "4.3.10 (c+d x)^m (a+b tan)^n.m",
            "rhs": "-I*(c + d*x)^(m + 1)/(d*(m + 1)) + 2*I*Int[(c + d*x)^m* E^(2*(-I*e + f*fz*x))/(1 + E^(2*(-I*e + f*fz*x))), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*tan[e_. + f_.*Complex[0, fz_]*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.10 (c+d x)^m (a+b tan)^n.m",
       "filename": "4.3.10 (c+d x)^m (a+b tan)^n.m",
            "rhs": "I*(c + d*x)^(m + 1)/(d*(m + 1)) - 2*I*Int[(c + d*x)^m*E^(2*I*(e + f*x))/(1 + E^(2*I*(e + f*x))), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*tan[e_. + f_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.10 (c+d x)^m (a+b tan)^n.m",
       "filename": "4.3.10 (c+d x)^m (a+b tan)^n.m",
            "rhs": "b*(c + d*x)^m*(b*Tan[e + f*x])^(n - 1)/(f*(n - 1)) - b*d*m/(f*(n - 1))* Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1), x] - b^2*Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.10 (c+d x)^m (a+b tan)^n.m",
       "filename": "4.3.10 (c+d x)^m (a+b tan)^n.m",
            "rhs": "(c + d*x)^m*(b*Tan[e + f*x])^(n + 1)/(b*f*(n + 1)) - d*m/(b*f*(n + 1))* Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n + 1), x] - 1/b^2*Int[(c + d*x)^m*(b*Tan[e + f*x])^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f}, x] && LtQ[n, -1] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.10 (c+d x)^m (a+b tan)^n.m",
       "filename": "4.3.10 (c+d x)^m (a+b tan)^n.m",
            "rhs": "Int[ExpandIntegrand[(c + d*x)^m, (a + b*Tan[e + f*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.10 (c+d x)^m (a+b tan)^n.m",
       "filename": "4.3.10 (c+d x)^m (a+b tan)^n.m",
            "rhs": "(c + d*x)^(m + 1)/(2*a*d*(m + 1)) - a*(c + d*x)^m/(2*b*f*(a + b*Tan[e + f*x])) + a*d*m/(2*b*f)*Int[(c + d*x)^(m - 1)/(a + b*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_./(a_ + b_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.10 (c+d x)^m (a+b tan)^n.m",
       "filename": "4.3.10 (c+d x)^m (a+b tan)^n.m",
            "rhs": "-1/(d*(c + d*x)*(a + b*Tan[e + f*x])) + f/(b*d)*Int[Cos[2*e + 2*f*x]/(c + d*x), x] - f/(a*d)*Int[Sin[2*e + 2*f*x]/(c + d*x), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((c_. + d_.*x_)^2*(a_ + b_.*tan[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.10 (c+d x)^m (a+b tan)^n.m",
       "filename": "4.3.10 (c+d x)^m (a+b tan)^n.m",
            "rhs": "f*(c + d*x)^(m + 2)/(b*d^2*(m + 1)*(m + 2)) + (c + d*x)^(m + 1)/(d*(m + 1)*(a + b*Tan[e + f*x])) + 2*b*f/(a*d*(m + 1))* Int[(c + d*x)^(m + 1)/(a + b*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_/(a_ + b_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0] && LtQ[m, -1] && NeQ[m, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.10 (c+d x)^m (a+b tan)^n.m",
       "filename": "4.3.10 (c+d x)^m (a+b tan)^n.m",
            "rhs": " (c+d*x)^(m+1)/(d*(m+1)*(a+b*Tan[e+f*x])) + f/(b*d*(m+1))*Int[(c+d*x)^(m+1),x] + 2*b*f/(a*d*(m+1))*Int[(c+d*x)^(m+1)/(a+b*Tan[e+f*x]),x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.+d_.*x_)^m_/(a_+b_.*tan[e_.+f_.*x_]),x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e,f},x] && EqQ[a^2+b^2,0] && LtQ[m,-1] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.10 (c+d x)^m (a+b tan)^n.m",
       "filename": "4.3.10 (c+d x)^m (a+b tan)^n.m",
            "rhs": "Log[c + d*x]/(2*a*d) + 1/(2*a)*Int[Cos[2*e + 2*f*x]/(c + d*x), x] + 1/(2*b)*Int[Sin[2*e + 2*f*x]/(c + d*x), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((c_. + d_.*x_)*(a_ + b_.*tan[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.10 (c+d x)^m (a+b tan)^n.m",
       "filename": "4.3.10 (c+d x)^m (a+b tan)^n.m",
            "rhs": "(c + d*x)^(m + 1)/(2*a*d*(m + 1)) + 1/(2*a)*Int[(c + d*x)^m*E^(2*a/b*(e + f*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_/(a_ + b_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.10 (c+d x)^m (a+b tan)^n.m",
       "filename": "4.3.10 (c+d x)^m (a+b tan)^n.m",
            "rhs": "Int[ExpandIntegrand[(c + d*x)^ m, (1/(2*a) + Cos[2*e + 2*f*x]/(2*a) + Sin[2*e + 2*f*x]/(2*b))^(-n), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0] && ILtQ[m, 0] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.10 (c+d x)^m (a+b tan)^n.m",
       "filename": "4.3.10 (c+d x)^m (a+b tan)^n.m",
            "rhs": "Int[ExpandIntegrand[(c + d*x)^ m, (1/(2*a) + E^(2*a/b*(e + f*x))/(2*a))^(-n), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.10 (c+d x)^m (a+b tan)^n.m",
       "filename": "4.3.10 (c+d x)^m (a+b tan)^n.m",
            "rhs": "With[{u = IntHide[(a + b*Tan[e + f*x])^n, x]}, Dist[(c + d*x)^m, u, x] - d*m*Int[Dist[(c + d*x)^(m - 1), u, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0] && ILtQ[n, -1] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.10 (c+d x)^m (a+b tan)^n.m",
       "filename": "4.3.10 (c+d x)^m (a+b tan)^n.m",
            "rhs": "(c + d*x)^(m + 1)/(d*(m + 1)*(a + I*b)) + 2*I*b* Int[(c + d*x)^m*E^(2*I*k*Pi)* E^Simp[2*I*(e + f*x), x]/((a + I*b)^2 + (a^2 + b^2)*E^(2*I*k*Pi)* E^Simp[2*I*(e + f*x), x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_./(a_ + b_.*tan[e_. + k_.*Pi + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[4*k] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.10 (c+d x)^m (a+b tan)^n.m",
       "filename": "4.3.10 (c+d x)^m (a+b tan)^n.m",
            "rhs": "(c + d*x)^(m + 1)/(d*(m + 1)*(a + I*b)) + 2*I*b* Int[(c + d*x)^m* E^Simp[2*I*(e + f*x), x]/((a + I*b)^2 + (a^2 + b^2)*E^Simp[2*I*(e + f*x), x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_./(a_ + b_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.10 (c+d x)^m (a+b tan)^n.m",
       "filename": "4.3.10 (c+d x)^m (a+b tan)^n.m",
            "rhs": "-(c + d*x)^2/(2*d*(a^2 + b^2)) - b*(c + d*x)/(f*(a^2 + b^2)*(a + b*Tan[e + f*x])) + 1/(f*(a^2 + b^2))* Int[(b*d + 2*a*c*f + 2*a*d*f*x)/(a + b*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)/(a_ + b_.*tan[e_. + f_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.10 (c+d x)^m (a+b tan)^n.m",
       "filename": "4.3.10 (c+d x)^m (a+b tan)^n.m",
            "rhs": "Int[ExpandIntegrand[(c + d*x)^ m, (1/(a - I*b) - 2*I*b/(a^2 + b^2 + (a - I*b)^2*E^(2*I*(e + f*x))))^(-n), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(a_ + b_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 + b^2, 0] && ILtQ[n, 0] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.10 (c+d x)^m (a+b tan)^n.m",
       "filename": "4.3.10 (c+d x)^m (a+b tan)^n.m",
            "rhs": "If[MatchQ[f, f1_.*Complex[0, j_]], If[MatchQ[e, e1_. + Pi/2], I^n*Unintegrable[(c + d*x)^m*Coth[-I*(e - Pi/2) - I*f*x]^n, x], I^n*Unintegrable[(c + d*x)^m*Tanh[-I*e - I*f*x]^n, x]], If[MatchQ[e, e1_. + Pi/2], (-1)^n*Unintegrable[(c + d*x)^m*Cot[e - Pi/2 + f*x]^n, x], Unintegrable[(c + d*x)^m*Tan[e + f*x]^n, x]]]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*tan[e_. + f_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e, f, m, n}, x] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.10 (c+d x)^m (a+b tan)^n.m",
       "filename": "4.3.10 (c+d x)^m (a+b tan)^n.m",
            "rhs": "Unintegrable[(c + d*x)^m*(a + b*Tan[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(a_. + b_.*tan[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.10 (c+d x)^m (a+b tan)^n.m",
       "filename": "4.3.10 (c+d x)^m (a+b tan)^n.m",
            "rhs": "Int[ExpandToSum[u, x]^m*(a + b*Tan[ExpandToSum[v, x]])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*(a_. + b_.*Tan[v_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && LinearQ[{u, v}, x] && Not[LinearMatchQ[{u, v}, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.10 (c+d x)^m (a+b tan)^n.m",
       "filename": "4.3.10 (c+d x)^m (a+b tan)^n.m",
            "rhs": "Int[ExpandToSum[u, x]^m*(a + b*Cot[ExpandToSum[v, x]])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*(a_. + b_.*Cot[v_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && LinearQ[{u, v}, x] && Not[LinearMatchQ[{u, v}, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.11 (e x)^m (a+b tan(c+d x^n))^p.m",
       "filename": "4.3.11 (e x)^m (a+b tan(c+d x^n))^p.m",
            "rhs": "1/n*Subst[Int[x^(1/n - 1)*(a + b*Tan[c + d*x])^p, x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Tan[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, p}, x] && IGtQ[1/n, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.11 (e x)^m (a+b tan(c+d x^n))^p.m",
       "filename": "4.3.11 (e x)^m (a+b tan(c+d x^n))^p.m",
            "rhs": "1/n*Subst[Int[x^(1/n - 1)*(a + b*Cot[c + d*x])^p, x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Cot[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, p}, x] && IGtQ[1/n, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.11 (e x)^m (a+b tan(c+d x^n))^p.m",
       "filename": "4.3.11 (e x)^m (a+b tan(c+d x^n))^p.m",
            "rhs": "Unintegrable[(a + b*Tan[c + d*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Tan[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.11 (e x)^m (a+b tan(c+d x^n))^p.m",
       "filename": "4.3.11 (e x)^m (a+b tan(c+d x^n))^p.m",
            "rhs": "Unintegrable[(a + b*Cot[c + d*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Cot[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.11 (e x)^m (a+b tan(c+d x^n))^p.m",
       "filename": "4.3.11 (e x)^m (a+b tan(c+d x^n))^p.m",
            "rhs": "1/Coefficient[u, x, 1]* Subst[Int[(a + b*Tan[c + d*x^n])^p, x], x, u]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Tan[c_. + d_.*u_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x] && NeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.11 (e x)^m (a+b tan(c+d x^n))^p.m",
       "filename": "4.3.11 (e x)^m (a+b tan(c+d x^n))^p.m",
            "rhs": "1/Coefficient[u, x, 1]* Subst[Int[(a + b*Cot[c + d*x^n])^p, x], x, u]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Cot[c_. + d_.*u_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x] && NeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.11 (e x)^m (a+b tan(c+d x^n))^p.m",
       "filename": "4.3.11 (e x)^m (a+b tan(c+d x^n))^p.m",
            "rhs": "Int[(a + b*Tan[ExpandToSum[u, x]])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Tan[u_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.11 (e x)^m (a+b tan(c+d x^n))^p.m",
       "filename": "4.3.11 (e x)^m (a+b tan(c+d x^n))^p.m",
            "rhs": "Int[(a + b*Cot[ExpandToSum[u, x]])^p, x]",
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         "givens": "FreeQ[{a,b},x] && LtQ[0,n,m+1] &&  LtQ[p,-1] *)"
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         "givens": "FreeQ[{a,b},x] && LtQ[0,n,m+1] &&  LtQ[p,-1] *)"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.11 (e x)^m (a+b tan(c+d x^n))^p.m",
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            "rhs": "Unintegrable[x^m*(a + b*Tan[c + d*x^n])^p, x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p}, x]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.11 (e x)^m (a+b tan(c+d x^n))^p.m",
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         "givens": "FreeQ[{a, b, c, d, m, n, p}, x]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.11 (e x)^m (a+b tan(c+d x^n))^p.m",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x]"
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         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.11 (e x)^m (a+b tan(c+d x^n))^p.m",
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         "givens": "FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.11 (e x)^m (a+b tan(c+d x^n))^p.m",
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.11 (e x)^m (a+b tan(c+d x^n))^p.m",
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        "comment": false,
         "givens": "FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m, n] && EqQ[q, 1]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.11 (e x)^m (a+b tan(c+d x^n))^p.m",
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        "comment": false,
         "givens": "FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m, n] && EqQ[q, 1]"
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         "givens": "FreeQ[{a, b, c, n}, x]"
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        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x]"
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         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0]"
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         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0] && IntegerQ[m] && Not[IGtQ[n, 0] && (LtQ[m, 0] || GtQ[m, n])]"
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0]"
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c + a*d, 0]"
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            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])*(c_. + d_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[b*c + a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "-(b*c - a*d)*(a + b*Tan[e + f*x])^m/(2*a*f*m) + (b*c + a*d)/(2*a*b)*Int[(a + b*Tan[e + f*x])^(m + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "d*(a + b*Tan[e + f*x])^m/(f*m) + (b*c + a*d)/b* Int[(a + b*Tan[e + f*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && Not[LtQ[m, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "d*(a + b*Tan[e + f*x])^m/(f*m) + Int[(a + b*Tan[e + f*x])^(m - 1)* Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "(b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 + b^2)) + 1/(a^2 + b^2)* Int[(a + b*Tan[e + f*x])^(m + 1)* Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "c/(b*f)*Log[RemoveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(c_ + d_.*tan[e_. + f_.*x_])/(a_ + b_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "(a*c + b*d)*x/(a^2 + b^2) + (b*c - a*d)/(a^2 + b^2)* Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*tan[e_. + f_.*x_])/(a_. + b_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[a*c + b*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "-2*d^2/f* Subst[Int[1/(2*c*d + b*x^2), x], x, (c - d*Tan[e + f*x])/Sqrt[b*Tan[e + f*x]]]",
     "rulenumber": 0,
            "lhs": "Int[(c_ + d_.*tan[e_. + f_.*x_])/Sqrt[b_.*tan[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f}, x] && EqQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "  (c+d)/2*Int[(1+Tan[e+f*x])/Sqrt[b*Tan[e+f*x]],x] + (c-d)/2*Int[(1-Tan[e+f*x])/Sqrt[b*Tan[e+f*x]],x]",
     "rulenumber": 0,
            "lhs": "Int[(c_+d_.*tan[e_.+f_.*x_])/Sqrt[b_.*tan[e_.+f_.*x_]],x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{b,c,d,e,f},x] && NeQ[c^2+d^2,0] && NeQ[c^2-d^2,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "2*c^2/f*Subst[Int[1/(b*c - d*x^2), x], x, Sqrt[b*Tan[e + f*x]]]",
     "rulenumber": 0,
            "lhs": "Int[(c_ + d_.*tan[e_. + f_.*x_])/Sqrt[b_.*tan[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f}, x] && EqQ[c^2 + d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "  (c+I*d)/2*Int[(1-I*Tan[e+f*x])/Sqrt[b*Tan[e+f*x]],x] +  (c-I*d)/2*Int[(1+I*Tan[e+f*x])/Sqrt[b*Tan[e+f*x]],x]",
     "rulenumber": 0,
            "lhs": "Int[(c_+d_.*tan[e_.+f_.*x_])/Sqrt[b_.*tan[e_.+f_.*x_]],x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{b,c,d,e,f},x] && NeQ[c^2-d^2,0] && NeQ[c^2+d^2,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "2/f*Subst[Int[(b*c + d*x^2)/(b^2 + x^4), x], x, Sqrt[b*Tan[e + f*x]]]",
     "rulenumber": 0,
            "lhs": "Int[(c_ + d_.*tan[e_. + f_.*x_])/Sqrt[b_.*tan[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] && NeQ[c^2 + d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "-2*d^2/f* Subst[Int[1/(2*b*c*d - 4*a*d^2 + x^2), x], x, (b*c - 2*a*d - b*d*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]]]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*tan[e_. + f_.*x_])/Sqrt[a_ + b_.*tan[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && EqQ[2*a*c*d - b*(c^2 - d^2), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "With[{q = Rt[a^2 + b^2, 2]}, 1/(2*q)* Int[(a*c + b*d + c*q + (b*c - a*d + d*q)*Tan[e + f*x])/ Sqrt[a + b*Tan[e + f*x]], x] - 1/(2*q)* Int[(a*c + b*d - c*q + (b*c - a*d - d*q)*Tan[e + f*x])/ Sqrt[a + b*Tan[e + f*x]], x]]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*tan[e_. + f_.*x_])/Sqrt[a_ + b_.*tan[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && NeQ[2*a*c*d - b*(c^2 - d^2), 0] && (PerfectSquareQ[a^2 + b^2] || RationalQ[a, b, c, d])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "c*d/f*Subst[Int[(a + b/d*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_ + d_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "c*Int[(b*Tan[e + f*x])^m, x] + d/b*Int[(b*Tan[e + f*x])^(m + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*tan[e_. + f_.*x_])^m_*(c_ + d_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f, m}, x] && NeQ[c^2 + d^2, 0] && Not[IntegerQ[2*m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "(c + I*d)/2* Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x] + (c - I*d)/2*Int[(a + b*Tan[e + f*x])^m*(1 + I*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "-b*(a*c + b*d)^2*(a + b*Tan[e + f*x])^ m/(2*a^3*f*m) + 1/(2*a^2)* Int[(a + b*Tan[e + f*x])^(m + 1)* Simp[a*c^2 - 2*b*c*d + a*d^2 - 2*b*d^2*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LeQ[m, -1] && EqQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "d*(2*b*c - a*d)*x/b^2 + d^2/b*Int[Tan[e + f*x], x] + (b*c - a*d)^2/b^2* Int[1/(a + b*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*tan[e_. + f_.*x_])^2/(a_. + b_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m + 1)/(b* f*(m + 1)*(a^2 + b^2)) + 1/(a^2 + b^2)* Int[(a + b*Tan[e + f*x])^(m + 1)* Simp[a*c^2 + 2*b*c*d - a*d^2 - (b*c^2 - 2*a*c*d - b*d^2)*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "d^2*(a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)) + Int[(a + b*Tan[e + f*x])^m* Simp[c^2 - d^2 + 2*c*d*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && Not[LeQ[m, -1]] && Not[EqQ[m, 2] && EqQ[a, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "-2*a*b/f* Subst[Int[1/(a*c - b*d - 2*a^2*x^2), x], x, Sqrt[c + d*Tan[e + f*x]]/Sqrt[a + b*Tan[e + f*x]]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*tan[e_. + f_.*x_]]/ Sqrt[c_. + d_.*tan[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "a*b*(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)/(f*(m - 1)*(a*c - b*d)) + 2*a^2/(a*c - b*d)* Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && EqQ[m + n, 0] && GtQ[m, 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "a*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n/(2*b*f*m) - (a*c - b*d)/(2*b^2)* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && EqQ[m + n, 0] && LeQ[m, -1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "a*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(2*f*m*(b*c - a*d)) + 1/(2*a)* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && EqQ[m + n + 1, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "-d*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(f*m*(c^2 + d^2)) + a/(a*c - b*d)* Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && EqQ[m + n + 1, 0] && Not[LtQ[m, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "-(a*c + b*d)*(c + d*Tan[e + f*x])^ n/(2*(b*c - a*d)*f*(a + b*Tan[e + f*x])) + 1/(2*a*(b*c - a*d))* Int[(c + d*Tan[e + f*x])^(n - 1)* Simp[a*c*d*(n - 1) + b*c^2 + b*d^2*n - d*(b*c - a*d)*(n - 1)*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*tan[e_. + f_.*x_])^n_/(a_ + b_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[0, n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "(b*c - a*d)*(c + d*Tan[e + f*x])^(n - 1)/(2*a*f*(a + b*Tan[e + f*x])) + 1/(2*a^2)* Int[(c + d*Tan[e + f*x])^(n - 2)* Simp[a*c^2 + a*d^2*(n - 1) - b*c*d*n - d*(a*c*(n - 2) + b*d*n)*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*tan[e_. + f_.*x_])^n_/(a_ + b_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "b/(b*c - a*d)*Int[1/(a + b*Tan[e + f*x]), x] - d/(b*c - a*d)*Int[1/(c + d*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_. + b_.*tan[e_. + f_.*x_])*(c_. + d_.*tan[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "-a*(c + d*Tan[e + f*x])^(n + 1)/(2* f*(b*c - a*d)*(a + b*Tan[e + f*x])) + 1/(2*a*(b*c - a*d))* Int[(c + d*Tan[e + f*x])^n* Simp[b*c + a*d*(n - 1) - b*d*n*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*tan[e_. + f_.*x_])^n_/(a_ + b_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Not[GtQ[n, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "-a^2*(b*c - a*d)*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1)/(d*f*(b*c + a*d)*(n + 1)) + a/(d*(b*c + a*d)*(n + 1))* Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1)* Simp[b*(b*c*(m - 2) - a*d*(m - 2*n - 4)) + (a*b*c*(m - 2) + b^2*d*(n + 1) - a^2*d*(m + n - 1))*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "2*a^2/(a*c - b*d)*Int[Sqrt[a + b*Tan[e + f*x]], x] - (2*b*c*d + a*(c^2 - d^2))/(a*(c^2 + d^2))* Int[(a - b*Tan[e + f*x])* Sqrt[a + b*Tan[e + f*x]]/(c + d*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^(3/2)/(c_. + d_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "2*a*Int[Sqrt[a + b*Tan[e + f*x]]/Sqrt[c + d*Tan[e + f*x]], x] + b/a*Int[(b + a*Tan[e + f*x])* Sqrt[a + b*Tan[e + f*x]]/Sqrt[c + d*Tan[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^(3/2)/ Sqrt[c_. + d_.*tan[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "b^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1)/(d* f*(m + n - 1)) + a/(d*(m + n - 1))* Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^n* Simp[b*c*(m - 2) + a*d*(m + 2*n) + (a*c*(m - 2) + b*d*(3*m + 2*n - 4))* Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && GtQ[m, 1] && NeQ[m + n - 1, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "-b*(a + b*Tan[e + f*x])^m* Sqrt[c + d*Tan[e + f*x]]/(2*a*f*m) + 1/(4*a^2*m)* Int[(a + b*Tan[e + f*x])^(m + 1)* Simp[2*a*c*m + b*d + a*d*(2*m + 1)*Tan[e + f*x], x]/ Sqrt[c + d*Tan[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*Sqrt[c_. + d_.*tan[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, 0] && IntegersQ[2*m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "-(b*c - a*d)*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n - 1)/(2*a*f*m) + 1/(2*a^2*m)* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 2)* Simp[c*(a*c*m + b*d*(n - 1)) - d*(b*c*m + a*d*(n - 1)) - d*(b*d*(m - n + 1) - a*c*(m + n - 1))*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, 0] && GtQ[n, 1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "a*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(2*f*m*(b*c - a*d)) + 1/(2*a*m*(b*c - a*d))* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n* Simp[b*c*m - a*d*(2*m + n + 1) + b*d*(m + n + 1)*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "d*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n - 1)/(f*(m + n - 1)) - 1/(a*(m + n - 1))* Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n - 2)* Simp[d*(b*c*m + a*d*(-1 + n)) - a*c^2*(m + n - 1) + d*(b*d*m - a*c*(m + 2*n - 2))*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[n, 1] && NeQ[m + n - 1, 0] && (IntegerQ[n] || IntegersQ[2*m, 2*n])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "d*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 + d^2)) - 1/(a*(c^2 + d^2)*(n + 1))* Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1)* Simp[b*d*m - a*c*(n + 1) + a*d*(m + n + 1)*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[n, -1] && (IntegerQ[n] || IntegersQ[2*m, 2*n])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "a/(a*c - b*d)*Int[(a + b*Tan[e + f*x])^m, x] - d/(a*c - b*d)* Int[(a + b*Tan[e + f*x])^ m*(b + a*Tan[e + f*x])/(c + d*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_/(c_. + d_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "(a*c - b*d)/a* Int[Sqrt[a + b*Tan[e + f*x]]/Sqrt[c + d*Tan[e + f*x]], x] + d/a*Int[ Sqrt[a + b*Tan[e + f*x]]*(b + a*Tan[e + f*x])/ Sqrt[c + d*Tan[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*tan[e_. + f_.*x_]]* Sqrt[c_. + d_.*tan[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "a*b/f*Subst[Int[(a + x)^(m - 1)*(c + d/b*x)^n/(b^2 + a*x), x], x, b*Tan[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2)) - 1/(d*(n + 1)*(c^2 + d^2))* Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)* Simp[a^2*d*(b*d*(m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d)* Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 2] && LtQ[n, -1] && IntegerQ[2*m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "b^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1)/(d* f*(m + n - 1)) + 1/(d*(m + n - 1))* Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n* Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || IntegerQ[m]) && Not[IGtQ[n, 2] && (Not[IntegerQ[m]] || EqQ[c, 0] && NeQ[a, 0])]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "(b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1)/(f*(m + 1)*(a^2 + b^2)) + 1/((m + 1)*(a^2 + b^2))* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 2)* Simp[a*c^2*(m + 1) + a*d^2*(n - 1) + b*c*d*(m - n + 2) - (b*c^2 - 2*a*c*d - b*d^2)*(m + 1)* Tan[e + f*x] - d*(b*c - a*d)*(m + n)*Tan[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && LtQ[1, n, 2] && IntegerQ[2*m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "b*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^ n/(f*(m + 1)*(a^2 + b^2)) + 1/((m + 1)*(a^2 + b^2))* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1)* Simp[a*c*(m + 1) - b*d*n - (b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(m + n + 1)*Tan[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[2*m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "b^2*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d)) + 1/((m + 1)*(a^2 + b^2)*(b*c - a*d))* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n* Simp[a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || IntegerQ[m]) && Not[ILtQ[n, -1] && (Not[IntegerQ[m]] || EqQ[c, 0] && NeQ[a, 0])]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "b*(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^ n/(f*(m + n - 1)) + 1/(m + n - 1)* Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n - 1)* Simp[a^2*c*(m + n - 1) - b*(b*c*(m - 1) + a*d*n) + (2*a*b*c + a^2*d - b^2*d)*(m + n - 1)*Tan[e + f*x] + b*(b*c*n + a*d*(2*m + n - 2))*Tan[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] && GtQ[n, 0] && IntegerQ[2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "(a*c - b*d)*x/((a^2 + b^2)*(c^2 + d^2)) + b^2/((b*c - a*d)*(a^2 + b^2))* Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x] - d^2/((b*c - a*d)*(c^2 + d^2))* Int[(d - c*Tan[e + f*x])/(c + d*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/((a_ + b_.*tan[e_. + f_.*x_])*(c_. + d_.*tan[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "1/(c^2 + d^2)* Int[Simp[a*c + b*d + (b*c - a*d)*Tan[e + f*x], x]/ Sqrt[a + b*Tan[e + f*x]], x] - d*(b*c - a*d)/(c^2 + d^2)* Int[(1 + Tan[e + f*x]^2)/(Sqrt[ a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_. + b_.*tan[e_. + f_.*x_]]/(c_. + d_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "1/(c^2 + d^2)* Int[Simp[ a^2*c - b^2*c + 2*a*b*d + (2*a*b*c - a^2*d + b^2*d)*Tan[e + f*x], x]/ Sqrt[a + b*Tan[e + f*x]], x] + (b*c - a*d)^2/(c^2 + d^2)* Int[(1 + Tan[e + f*x]^2)/(Sqrt[ a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^(3/2)/(c_. + d_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "1/(c^2 + d^2)*Int[(a + b*Tan[e + f*x])^m*(c - d*Tan[e + f*x]), x] + d^2/(c^2 + d^2)* Int[(a + b*Tan[e + f*x])^ m*(1 + Tan[e + f*x]^2)/(c + d*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_/(c_. + d_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[(a + b*ff*x)^m*(c + d*ff*x)^n/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_ + d_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "d^m*Int[(b + a*Cot[e + f*x])^m*(d*Cot[e + f*x])^(n - m), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_.*(d_./tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, n}, x] && Not[IntegerQ[n]] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "d^m*Int[(b + a*Tan[e + f*x])^m*(d*Tan[e + f*x])^(n - m), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*cot[e_. + f_.*x_])^m_.*(d_./cot[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, n}, x] && Not[IntegerQ[n]] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "c^IntPart[n]*(c*(d*Tan[e + f*x])^p)^ FracPart[n]/(d*Tan[e + f*x])^(p*FracPart[n])* Int[(a + b*Tan[e + f*x])^m*(d*Tan[e + f*x])^(n*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^ m_.*(c_.*(d_.*tan[e_. + f_.*x_])^p_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[n]] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.1 (a+b tan)^m (c+d tan)^n.m",
            "rhs": "c^IntPart[n]*(c*(d*Cot[e + f*x])^p)^ FracPart[n]/(d*Cot[e + f*x])^(p*FracPart[n])* Int[(a + b*Cot[e + f*x])^m*(d*Cot[e + f*x])^(n*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*cot[e_. + f_.*x_])^ m_.*(c_.*(d_.*cot[e_. + f_.*x_])^p_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[n]] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.3 (g tan)^p (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.3 (g tan)^p (a+b tan)^m (c+d tan)^n.m",
            "rhs": "Unintegrable[(g*Tan[e + f*x])^p*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*tan[e_. + f_.*x_])^p_.*(a_ + b_.*tan[e_. + f_.*x_])^ m_*(c_ + d_.*tan[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.3 (g tan)^p (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.3 (g tan)^p (a+b tan)^m (c+d tan)^n.m",
            "rhs": "g^(m + n)* Int[(g*Cot[e + f*x])^(p - m - n)*(b + a*Cot[e + f*x])^ m*(d + c*Cot[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cot[e_. + f_.*x_])^p_*(a_. + b_.*tan[e_. + f_.*x_])^ m_.*(c_ + d_.*tan[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, p}, x] && Not[IntegerQ[p]] && IntegerQ[m] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.3 (g tan)^p (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.3 (g tan)^p (a+b tan)^m (c+d tan)^n.m",
            "rhs": "g^(m + n)* Int[(g*Tan[e + f*x])^(p - m - n)*(b + a*Tan[e + f*x])^ m*(d + c*Tan[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*tan[e_. + f_.*x_])^p_*(a_. + b_.*cot[e_. + f_.*x_])^ m_.*(c_ + d_.*cot[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, p}, x] && Not[IntegerQ[p]] && IntegerQ[m] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.3 (g tan)^p (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.3 (g tan)^p (a+b tan)^m (c+d tan)^n.m",
            "rhs": "(g*Tan[e + f*x]^q)^p/(g*Tan[e + f*x])^(p*q)* Int[(g*Tan[e + f*x])^(p*q)*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*tan[e_. + f_.*x_]^q_)^p_*(a_. + b_.*tan[e_. + f_.*x_])^ m_.*(c_ + d_.*tan[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p, q}, x] && Not[IntegerQ[p]] && Not[IntegerQ[m] && IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.3 (g tan)^p (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.3 (g tan)^p (a+b tan)^m (c+d tan)^n.m",
            "rhs": "g^n*Int[(g*Tan[e + f*x])^(p - n)*(a + b*Tan[e + f*x])^ m*(d + c*Tan[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*tan[e_. + f_.*x_])^p_.*(a_ + b_.*tan[e_. + f_.*x_])^ m_.*(c_ + d_.*cot[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.3 (g tan)^p (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.3 (g tan)^p (a+b tan)^m (c+d tan)^n.m",
            "rhs": "Int[(b + a*Cot[e + f*x])^m*(c + d*Cot[e + f*x])^n/ Cot[e + f*x]^(m + p), x]",
     "rulenumber": 0,
            "lhs": "Int[tan[e_. + f_.*x_]^p_.*(a_ + b_.*tan[e_. + f_.*x_])^ m_.*(c_ + d_.*cot[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && Not[IntegerQ[n]] && IntegerQ[m] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.3 (g tan)^p (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.3 (g tan)^p (a+b tan)^m (c+d tan)^n.m",
            "rhs": "Cot[e + f*x]^p*(g*Tan[e + f*x])^p* Int[(b + a*Cot[e + f*x])^m*(c + d*Cot[e + f*x])^n/ Cot[e + f*x]^(m + p), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*tan[e_. + f_.*x_])^p_*(a_ + b_.*tan[e_. + f_.*x_])^ m_.*(c_ + d_.*cot[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p}, x] && Not[IntegerQ[n]] && IntegerQ[m] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.2.3 (g tan)^p (a+b tan)^m (c+d tan)^n.m",
       "filename": "4.3.2.3 (g tan)^p (a+b tan)^m (c+d tan)^n.m",
            "rhs": "(g*Tan[e + f*x])^ n*(c + d*Cot[e + f*x])^n/(d + c*Tan[e + f*x])^n* Int[(g*Tan[e + f*x])^(p - n)*(a + b*Tan[e + f*x])^ m*(d + c*Tan[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*tan[e_. + f_.*x_])^p_.*(a_ + b_.*tan[e_. + f_.*x_])^ m_*(c_ + d_.*cot[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && Not[IntegerQ[n]] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": "a*c/f*Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^(n - 1)*(A + B*x), x], x, Tan[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_.*(c_ + d_.*tan[e_. + f_.*x_])^ n_.*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": "B*d/b*Int[Tan[e + f*x], x] + 1/b*Int[Simp[A*b*c + (A*b*d + B*(b*c - a*d))*Tan[e + f*x], x]/(a + b*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*tan[e_. + f_.*x_])*(A_. + B_.*tan[e_. + f_.*x_])/(a_. + b_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": "-(A*b - a*B)*(a*c + b*d)*(a + b*Tan[e + f*x])^ m/(2*a^2*f*m) + 1/(2*a*b)* Int[(a + b*Tan[e + f*x])^(m + 1)* Simp[A*b*c + a*B*c + a*A*d + b*B*d + 2*a*B*d*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^ m_*(c_. + d_.*tan[e_. + f_.*x_])*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && EqQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": "(b*c - a*d)*(A*b - a*B)*(a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2)) + 1/(a^2 + b^2)* Int[(a + b*Tan[e + f*x])^(m + 1)* Simp[a*A*c + b*B*c + A*b*d - a*B*d - (A*b*c - a*B*c - a*A*d - b*B*d)*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^ m_*(c_. + d_.*tan[e_. + f_.*x_])*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": "B*d*(a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)) + Int[(a + b*Tan[e + f*x])^m* Simp[A*c - B*d + (B*c + A*d)*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^ m_.*(c_. + d_.*tan[e_. + f_.*x_])*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && Not[LeQ[m, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": "-a^2*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)/(d*f*(b*c + a*d)*(n + 1)) - a/(d*(b*c + a*d)*(n + 1))* Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)* Simp[A*b*d*(m - n - 2) - B*(b*c*(m - 1) + a*d*(n + 1)) + (a*A*d*(m + n) - B*(a*c*(m - 1) + b*d*(n + 1)))*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && GtQ[m, 1] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": "b*B*(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)/(d* f*(m + n)) + 1/(d*(m + n))* Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n* Simp[a*A*d*(m + n) + B*(a*c*(m - 1) - b*d*(n + 1)) - (B*(b*c - a*d)*(m - 1) - d*(A*b + a*B)*(m + n))*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && GtQ[m, 1] && Not[LtQ[n, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": "-(A*b - a*B)*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^n/(2*a*f*m) + 1/(2*a^2*m)* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1)* Simp[A*(a*c*m + b*d*n) - B*(b*c*m + a*d*n) - d*(b*B*(m - n) - a*A*(m + n))*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": "(a*A + b*B)*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(2*f*m*(b*c - a*d)) + 1/(2*a*m*(b*c - a*d))* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n* Simp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m - b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] && Not[GtQ[n, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": "B*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n/(f*(m + n)) + 1/(a*(m + n))* Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n - 1)* Simp[a*A*c*(m + n) - B*(b*c*m + a*d*n) + (a*A*d*(m + n) - B*(b*d*m - a*c*n))* Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": "(A*d - B*c)*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 + d^2)) - 1/(a*(n + 1)*(c^2 + d^2))* Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1)* Simp[A*(b*d*m - a*c*(n + 1)) - B*(b*c*m + a*d*(n + 1)) - a*(B*c - A*d)*(m + n + 1)*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": "b*B/f*Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^n, x], x, Tan[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && EqQ[A*b + a*B, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": "(A*b + a*B)/(b*c + a*d)* Int[(a + b*Tan[e + f*x])^m, x] - (B*c - A*d)/(b*c + a*d)* Int[(a + b*Tan[e + f*x])^ m*(a - b*Tan[e + f*x])/(c + d*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^ m_*(A_. + B_.*tan[e_. + f_.*x_])/(c_. + d_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[A*b + a*B, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": " (A*b-a*B)/b*Int[(a+b*Tan[e+f*x])^m*(c+d*Tan[e+f*x])^n,x] + B/b*Int[(a+b*Tan[e+f*x])^(m+1)*(c+d*Tan[e+f*x])^n,x]",
     "rulenumber": 0,
            "lhs": "Int[(a_+b_.*tan[e_.+f_.*x_])^m_*(c_.+d_.*tan[e_.+f_.*x_])^n_*(A_.+ B_.*tan[e_.+f_.*x_]),x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e,f,A,B,m},x] && NeQ[b*c-a*d,0] && EqQ[a^2+b^2,0] &&  NeQ[c^2+d^2,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": "(A*b + a*B)/b* Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n, x] - B/b*Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^ n*(a - b*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[A*b + a*B, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": "A^2/f*Subst[Int[(a + b*x)^m*(c + d*x)^n/(A - B*x), x], x, Tan[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_ + B_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && Not[IntegersQ[2*m, 2*n]] && EqQ[A^2 + B^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": "(A + I*B)/2* Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^ n*(1 - I*Tan[e + f*x]), x] + (A - I*B)/2* Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^ n*(1 + I*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && Not[IntegersQ[2*m, 2*n]] && NeQ[A^2 + B^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": "-(B*c - A*d)*(b*c - a*d)^2*(c + d*Tan[e + f*x])^(n + 1)/(f* d^2*(n + 1)*(c^2 + d^2)) + 1/(d*(c^2 + d^2))*Int[(c + d*Tan[e + f*x])^(n + 1)* Simp[B*(b*c - a*d)^2 + A*d*(a^2*c - b^2*c + 2*a*b*d) + d*(B*(a^2*c - b^2*c + 2*a*b*d) + A*(2*a*b*c - a^2*d + b^2*d))* Tan[e + f*x] + b^2*B*(c^2 + d^2)*Tan[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^2*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": "(b*c - a*d)*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2)) - 1/(d*(n + 1)*(c^2 + d^2))* Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1)* Simp[a*A* d*(b*d*(m - 1) - a*c*(n + 1)) + (b*B*c - (A*b + a*B)*d)*(b*c*(m - 1) + a*d*(n + 1)) - d*((a*A - b*B)*(b*c - a*d) + (A*b + a*B)*(a*c + b*d))*(n + 1)* Tan[e + f*x] - b*(d*(A*b*c + a*B*c - a*A*d)*(m + n) - b*B*(c^2*(m - 1) - d^2*(n + 1)))*Tan[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": "b^2*B*Tan[e + f*x]/(d*f) + 1/d*Int[(a^2*A*d - b^2*B*c + (2*a*A*b + B*(a^2 - b^2))*d* Tan[e + f*x] + (A*b^2*d - b*B*(b*c - 2*a*d))* Tan[e + f*x]^2)/(c + d*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^2*(A_. + B_.*tan[e_. + f_.*x_])/(c_. + d_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": "b*B*(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)/(d* f*(m + n)) + 1/(d*(m + n))* Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^n* Simp[a^2*A*d*(m + n) - b*B*(b*c*(m - 1) + a*d*(n + 1)) + d*(m + n)*(2*a*A*b + B*(a^2 - b^2))*Tan[e + f*x] - (b*B*(b*c - a*d)*(m - 1) - b*(A*b + a*B)*d*(m + n))* Tan[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && Not[IGtQ[n, 1] && (Not[IntegerQ[m]] || EqQ[c, 0] && NeQ[a, 0])]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": "(A*b - a*B)*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^ n/(f*(m + 1)*(a^2 + b^2)) + 1/(b*(m + 1)*(a^2 + b^2))* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1)* Simp[b*B*(b*c*(m + 1) + a*d*n) + A*b*(a*c*(m + 1) - b*d*n) - b*(A*(b*c - a*d) - B*(a*c + b*d))*(m + 1)*Tan[e + f*x] - b*d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && LtQ[0, n, 1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": "b*(A*b - a*B)*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2)) + 1/((m + 1)*(b*c - a*d)*(a^2 + b^2))* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n* Simp[b*B*(b*c*(m + 1) + a*d*(n + 1)) + A*(a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2)) - (A*b - a*B)*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(A*b - a*B)*(m + n + 2)*Tan[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && Not[ILtQ[n, -1] && (Not[IntegerQ[m]] || EqQ[c, 0] && NeQ[a, 0])]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": "B*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n/(f*(m + n)) + 1/(m + n)* Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n - 1)* Simp[a*A*c*(m + n) - B*(b*c*m + a*d*n) + (A*b*c + a*B*c + a*A*d - b*B*d)*(m + n)* Tan[e + f*x] + (A*b*d*(m + n) + B*(a*d*m + b*c*n))* Tan[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[0, m, 1] && LtQ[0, n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": "(B*(b*c + a*d) + A*(a*c - b*d))* x/((a^2 + b^2)*(c^2 + d^2)) + b*(A*b - a*B)/((b*c - a*d)*(a^2 + b^2))* Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x] + d*(B*c - A*d)/((b*c - a*d)*(c^2 + d^2))* Int[(d - c*Tan[e + f*x])/(c + d*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*tan[e_. + f_.*x_])/((a_ + b_.*tan[e_. + f_.*x_])*(c_. + d_.*tan[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": "1/(a^2 + b^2)* Int[Simp[ A*(a*c + b*d) + B*(b*c - a*d) - (A*(b*c - a*d) - B*(a*c + b*d))*Tan[e + f*x], x]/Sqrt[c + d*Tan[e + f*x]], x] - (b*c - a*d)*(B*a - A*b)/(a^2 + b^2)* Int[(1 + Tan[e + f*x]^2)/((a + b*Tan[e + f*x])* Sqrt[c + d*Tan[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[c_. + d_.*tan[e_. + f_.*x_]]*(A_. + B_.*tan[e_. + f_.*x_])/(a_. + b_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": "1/(a^2 + b^2)* Int[(c + d*Tan[e + f*x])^n* Simp[a*A + b*B - (A*b - a*B)*Tan[e + f*x], x], x] + b*(A*b - a*B)/(a^2 + b^2)* Int[(c + d*Tan[e + f*x])^ n*(1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_])/(a_. + b_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": "Int[Simp[a*A - b*B + (A*b + a*B)*Tan[e + f*x], x]/(Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]), x] + b*B*Int[(1 + Tan[e + f*x]^2)/(Sqrt[a + b*Tan[e + f*x]]* Sqrt[c + d*Tan[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_. + b_.*tan[e_. + f_.*x_]]*(A_. + B_.*tan[e_. + f_.*x_])/ Sqrt[c_. + d_.*tan[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": " A^2/f*Subst[Int[1/((A-B*x)*Sqrt[a+b*x]*Sqrt[c+d*x]),x],x,Tan[e+f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(A_.+B_.*tan[e_.+f_.*x_])/(Sqrt[a_.+b_.*tan[e_.+f_.*x_]]*Sqrt[ c_.+d_.*tan[e_.+f_.*x_]]),x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e,f,A,B},x] && NeQ[b*c-a*d,0] && NeQ[a^2+b^2,0] &&  NeQ[c^2+d^2,0] && EqQ[A^2+B^2,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": " (A+I*B)/2*Int[(1-I*Tan[e+f*x])/(Sqrt[a+b*Tan[e+f*x]]*Sqrt[c+d*Tan[e+f* x]]),x] + (A-I*B)/2*Int[(1+I*Tan[e+f*x])/(Sqrt[a+b*Tan[e+f*x]]*Sqrt[c+d*Tan[e+ f*x]]),x]",
     "rulenumber": 0,
            "lhs": "Int[(A_.+B_.*tan[e_.+f_.*x_])/(Sqrt[a_.+b_.*tan[e_.+f_.*x_]]*Sqrt[ c_.+d_.*tan[e_.+f_.*x_]]),x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,A,B},x] && NeQ[b*c-a*d,0] &&  NeQ[a^2+b^2,0] && NeQ[c^2+d^2,0] && NeQ[A^2+B^2,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": "A^2/f*Subst[Int[(a + b*x)^m*(c + d*x)^n/(A - B*x), x], x, Tan[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[A^2 + B^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
       "filename": "4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m",
            "rhs": "(A + I*B)/2* Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^ n*(1 - I*Tan[e + f*x]), x] + (A - I*B)/2* Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^ n*(1 + I*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[A^2 + B^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m",
       "filename": "4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m",
            "rhs": "A/(b*f)*Subst[Int[(a + x)^m, x], x, b*Tan[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_.*(A_ + C_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A, C]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m",
       "filename": "4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m",
            "rhs": "-A/(b*f)* Subst[Int[(a + x)^m, x], x, b*Cot[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*cot[e_. + f_.*x_])^m_.*(A_ + C_.*cot[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A, C]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m",
       "filename": "4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m",
            "rhs": "1/b^2*Int[(a + b*Tan[e + f*x])^(m + 1)* Simp[b*B - a*C + b*C*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^ m_.*(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B, C, m}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m",
       "filename": "4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m",
            "rhs": "-C/b^2* Int[(a + b*Tan[e + f*x])^(m + 1)*(a - b*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_.*(A_. + C_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A*b^2 + a^2*C, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m",
       "filename": "4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m",
            "rhs": "-(a*A + b*B - a*C)* Tan[e + f*x]*(a + b*Tan[e + f*x])^m/(2*a*f*m) + 1/(2*a^2*m)* Int[(a + b*Tan[e + f*x])^(m + 1)* Simp[(b*B - a*C) + a*A*(2*m + 1) - (b*C*(m - 1) + (A*b - a*B)*(m + 1))* Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^ m_.*(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && LeQ[m, -1] && EqQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m",
       "filename": "4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m",
            "rhs": "-(a*A - a*C)* Tan[e + f*x]*(a + b*Tan[e + f*x])^m/(2*a*f*m) + 1/(2*a^2*m)* Int[(a + b*Tan[e + f*x])^(m + 1)* Simp[-a*C + a*A*(2*m + 1) - (b*C*(m - 1) + A*b*(m + 1))*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_.*(A_. + C_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, C}, x] && NeQ[A*b^2 + a^2*C, 0] && LeQ[m, -1] && EqQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m",
       "filename": "4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m",
            "rhs": "(a*A + b*B - a*C)*x/(a^2 + b^2) + (A*b^2 - a*b*B + a^2*C)/(a^2 + b^2)* Int[(1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2)/(a_. + b_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 + b^2, 0] && EqQ[A*b - a*B - b*C, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m",
       "filename": "4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m",
            "rhs": "B*x + A*Int[1/Tan[e + f*x], x] + C*Int[Tan[e + f*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2)/ tan[e_. + f_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{e, f, A, B, C}, x] && NeQ[A, C]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m",
       "filename": "4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m",
            "rhs": "A*Int[1/Tan[e + f*x], x] + C*Int[Tan[e + f*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + C_.*tan[e_. + f_.*x_]^2)/tan[e_. + f_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{e, f, A, C}, x] && NeQ[A, C]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m",
       "filename": "4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m",
            "rhs": "(a*A + b*B - a*C)*x/(a^2 + b^2) - (A*b - a*B - b*C)/(a^2 + b^2)*Int[Tan[e + f*x], x] + (A*b^2 - a*b*B + a^2*C)/(a^2 + b^2)* Int[(1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2)/(a_. + b_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && NeQ[a^2 + b^2, 0] && NeQ[A*b - a*B - b*C, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m",
       "filename": "4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m",
            "rhs": "a*(A - C)*x/(a^2 + b^2) - b*(A - C)/(a^2 + b^2)*Int[Tan[e + f*x], x] + (a^2*C + A*b^2)/(a^2 + b^2)* Int[(1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + C_.*tan[e_. + f_.*x_]^2)/(a_ + b_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, C}, x] && NeQ[a^2*C + A*b^2, 0] && NeQ[a^2 + b^2, 0] && NeQ[A, C]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m",
       "filename": "4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m",
            "rhs": "(A*b^2 - a*b*B + a^2*C)*(a + b*Tan[e + f*x])^(m + 1)/(b* f*(m + 1)*(a^2 + b^2)) + 1/(a^2 + b^2)* Int[(a + b*Tan[e + f*x])^(m + 1)* Simp[b*B + a*(A - C) - (A*b - a*B - b*C)*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^ m_*(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m",
       "filename": "4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m",
            "rhs": "(A*b^2 + a^2*C)*(a + b*Tan[e + f*x])^(m + 1)/(b* f*(m + 1)*(a^2 + b^2)) + 1/(a^2 + b^2)* Int[(a + b*Tan[e + f*x])^(m + 1)* Simp[a*(A - C) - (A*b - b*C)*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(A_. + C_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, C}, x] && NeQ[A*b^2 + a^2*C, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m",
       "filename": "4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m",
            "rhs": "C*(a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)) + Int[(a + b*Tan[e + f*x])^m*Simp[A - C + B*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^ m_.*(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && Not[LeQ[m, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m",
       "filename": "4.3.4.1 (a+b tan)^m (A+B tan+C tan^2).m",
            "rhs": "C*(a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)) + (A - C)* Int[(a + b*Tan[e + f*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_.*(A_. + C_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, C, m}, x] && NeQ[A*b^2 + a^2*C, 0] && Not[LeQ[m, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
       "filename": "4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
            "rhs": "1/b^2*Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^ n*(b*B - a*C + b*C*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_.*(c_. + d_.*tan[e_. + f_.*x_])^ n_.*(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - a*b*B + a^2*C, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
       "filename": "4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
            "rhs": "-C/b^2* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^ n*(a - b*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_.*(c_. + d_.*tan[e_. + f_.*x_])^ n_.*(A_. + C_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 + a^2*C, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
       "filename": "4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
            "rhs": "A/f*Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_.*(c_. + d_.*tan[e_. + f_.*x_])^ n_.*(A_ + C_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
       "filename": "4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
            "rhs": "-(b*c - a*d)*(c^2*C - B*c*d + A*d^2)*(c + d*Tan[e + f*x])^(n + 1)/(d^2* f*(n + 1)*(c^2 + d^2)) + 1/(d*(c^2 + d^2))*Int[(c + d*Tan[e + f*x])^(n + 1)* Simp[a*d*(A*c - c*C + B*d) + b*(c^2*C - B*c*d + A*d^2) + d*(A*b*c + a*B*c - b*c*C - a*A*d + b*B*d + a*C*d)* Tan[e + f*x] + b*C*(c^2 + d^2)*Tan[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
       "filename": "4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
            "rhs": "-(b*c - a*d)*(c^2*C + A*d^2)*(c + d*Tan[e + f*x])^(n + 1)/(d^2* f*(n + 1)*(c^2 + d^2)) + 1/(d*(c^2 + d^2))*Int[(c + d*Tan[e + f*x])^(n + 1)* Simp[a*d*(A*c - c*C) + b*(c^2*C + A*d^2) + d*(A*b*c - b*c*C - a*A*d + a*C*d)*Tan[e + f*x] + b*C*(c^2 + d^2)*Tan[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + C_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
       "filename": "4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
            "rhs": "b*C*Tan[e + f*x]*(c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 2)) - 1/(d*(n + 2))*Int[(c + d*Tan[e + f*x])^n* Simp[b*c*C - a*A*d*(n + 2) - (A*b + a*B - b*C)*d*(n + 2)* Tan[e + f*x] - (a*C*d*(n + 2) - b*(c*C - B*d*(n + 2)))* Tan[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])*(c_. + d_.*tan[e_. + f_.*x_])^ n_.*(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] && Not[LtQ[n, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
       "filename": "4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
            "rhs": "b*C*Tan[e + f*x]*(c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 2)) - 1/(d*(n + 2))*Int[(c + d*Tan[e + f*x])^n* Simp[b*c*C - a*A*d*(n + 2) - (A*b - b*C)*d*(n + 2)* Tan[e + f*x] - (a*C*d*(n + 2) - b*c*C)*Tan[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])*(c_. + d_.*tan[e_. + f_.*x_])^ n_.*(A_. + C_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] && Not[LtQ[n, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
       "filename": "4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
            "rhs": "(a*A + b*B - a*C)*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(2*f*m*(b*c - a*d)) + 1/(2*a*m*(b*c - a*d))* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n* Simp[b*(c*(A + C)*m - B*d*(n + 1)) + a*(B*c*m + C*d*(n + 1) - A*d*(2*m + n + 1)) + (b*C*d*(m - n - 1) + A*b*d*(m + n + 1) + a*(2*c*C*m - B*d*(m + n + 1)))*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_.*(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && (LtQ[m, 0] || EqQ[m + n + 1, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
       "filename": "4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
            "rhs": "a*(A - C)*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(2*f*m*(b*c - a*d)) + 1/(2*a*m*(b*c - a*d))* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n* Simp[b*c*(A + C)*m + a*(C*d*(n + 1) - A*d*(2*m + n + 1)) + (b*C*d*(m - n - 1) + A*b*d*(m + n + 1) + 2*a*c*C*m)*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_.*(A_. + C_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && (LtQ[m, 0] || EqQ[m + n + 1, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
       "filename": "4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
            "rhs": "(c^2*C - B*c*d + A*d^2)*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2)) - 1/(a*d*(n + 1)*(c^2 + d^2))* Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1)* Simp[b*(c^2*C - B*c*d + A*d^2)*m - a*d*(n + 1)*(A*c - c*C + B*d) - a*(d*(B*c - A*d)*(m + n + 1) - C*(c^2*m - d^2*(n + 1)))* Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_.*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && Not[LtQ[m, 0]] && LtQ[n, -1] && NeQ[c^2 + d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
       "filename": "4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
            "rhs": "(c^2*C + A*d^2)*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2)) - 1/(a*d*(n + 1)*(c^2 + d^2))* Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1)* Simp[b*(c^2*C + A*d^2)*m - a*d*(n + 1)*(A*c - c*C) - a*(-A*d^2*(m + n + 1) - C*(c^2*m - d^2*(n + 1)))*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_.*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + C_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && Not[LtQ[m, 0]] && LtQ[n, -1] && NeQ[c^2 + d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
       "filename": "4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
            "rhs": "C*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n + 1)) + 1/(b*d*(m + n + 1))* Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n* Simp[A*b*d*(m + n + 1) + C*(a*c*m - b*d*(n + 1)) - (C*m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_.*(c_. + d_.*tan[e_. + f_.*x_])^ n_.*(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && Not[LtQ[m, 0]] && NeQ[m + n + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
       "filename": "4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
            "rhs": "C*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n + 1)) + 1/(b*d*(m + n + 1))* Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n* Simp[A*b*d*(m + n + 1) + C*(a*c*m - b*d*(n + 1)) - C*m*(b*c - a*d)*Tan[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[e_. + f_.*x_])^m_.*(c_. + d_.*tan[e_. + f_.*x_])^ n_.*(A_. + C_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && Not[LtQ[m, 0]] && NeQ[m + n + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
       "filename": "4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
            "rhs": "(A*d^2 + c*(c*C - B*d))*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2)) - 1/(d*(n + 1)*(c^2 + d^2))* Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)* Simp[A*d*(b*d*m - a*c*(n + 1)) + (c*C - B*d)*(b*c*m + a*d*(n + 1)) - d*(n + 1)*((A - C)*(b*c - a*d) + B*(a*c + b*d))*Tan[e + f*x] - b*(d*(B*c - A*d)*(m + n + 1) - C*(c^2*m - d^2*(n + 1)))* Tan[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
       "filename": "4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
            "rhs": "(A*d^2 + c^2*C)*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2)) - 1/(d*(n + 1)*(c^2 + d^2))* Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)* Simp[A*d*(b*d*m - a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - d*(n + 1)*((A - C)*(b*c - a*d))*Tan[e + f*x] + b*(A*d^2*(m + n + 1) + C*(c^2*m - d^2*(n + 1)))* Tan[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + C_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
       "filename": "4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
            "rhs": "C*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n + 1)) + 1/(d*(m + n + 1))* Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n* Simp[a*A*d*(m + n + 1) - C*(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)* Tan[e + f*x] - (C*m*(b*c - a*d) - b*B*d*(m + n + 1))* Tan[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_.*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && Not[IGtQ[n, 0] && (Not[IntegerQ[m]] || EqQ[c, 0] && NeQ[a, 0])]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
       "filename": "4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
            "rhs": "C*(a + b*Tan[e + f*x])^ m*(c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n + 1)) + 1/(d*(m + n + 1))* Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n* Simp[a*A*d*(m + n + 1) - C*(b*c*m + a*d*(n + 1)) + d*(A*b - b*C)*(m + n + 1)*Tan[e + f*x] - C*m*(b*c - a*d)*Tan[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_.*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + C_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && Not[IGtQ[n, 0] && (Not[IntegerQ[m]] || EqQ[c, 0] && NeQ[a, 0])]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
       "filename": "4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
            "rhs": "(A*b^2 - a*(b*B - a*C))*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2)) + 1/((m + 1)*(b*c - a*d)*(a^2 + b^2))* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n* Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && Not[ILtQ[n, -1] && (Not[IntegerQ[m]] || EqQ[c, 0] && NeQ[a, 0])]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
       "filename": "4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
            "rhs": "(A*b^2 + a^2*C)*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2)) + 1/((m + 1)*(b*c - a*d)*(a^2 + b^2))* Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n* Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2)) - a*C*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - b*C)*Tan[e + f*x] - d*(A*b^2 + a^2*C)*(m + n + 2)*Tan[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + C_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && Not[ILtQ[n, -1] && (Not[IntegerQ[m]] || EqQ[c, 0] && NeQ[a, 0])]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
       "filename": "4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
            "rhs": "(a*(A*c - c*C + B*d) + b*(B*c - A*d + C*d))* x/((a^2 + b^2)*(c^2 + d^2)) + (A*b^2 - a*b*B + a^2*C)/((b*c - a*d)*(a^2 + b^2))* Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x] - (c^2*C - B*c*d + A*d^2)/((b*c - a*d)*(c^2 + d^2))* Int[(d - c*Tan[e + f*x])/(c + d*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2)/((a_ + b_.*tan[e_. + f_.*x_])*(c_. + d_.*tan[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
       "filename": "4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
            "rhs": "(a*(A*c - c*C) - b*(A*d - C*d))* x/((a^2 + b^2)*(c^2 + d^2)) + (A*b^2 + a^2*C)/((b*c - a*d)*(a^2 + b^2))* Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x] - (c^2*C + A*d^2)/((b*c - a*d)*(c^2 + d^2))* Int[(d - c*Tan[e + f*x])/(c + d*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + C_.*tan[e_. + f_.*x_]^2)/((a_ + b_.*tan[e_. + f_.*x_])*(c_. + d_.*tan[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
       "filename": "4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
            "rhs": "1/(a^2 + b^2)* Int[(c + d*Tan[e + f*x])^n* Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x] + (A*b^2 - a*b*B + a^2*C)/(a^2 + b^2)* Int[(c + d*Tan[e + f*x])^ n*(1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2)/(a_. + b_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Not[GtQ[n, 0]] && Not[LeQ[n, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
       "filename": "4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
            "rhs": "1/(a^2 + b^2)* Int[(c + d*Tan[e + f*x])^n* Simp[a*(A - C) - (A*b - b*C)*Tan[e + f*x], x], x] + (A*b^2 + a^2*C)/(a^2 + b^2)* Int[(c + d*Tan[e + f*x])^ n*(1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + C_.*tan[e_. + f_.*x_]^2)/(a_. + b_.*tan[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Not[GtQ[n, 0]] && Not[LeQ[n, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
       "filename": "4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[(a + b*ff*x)^m*(c + d*ff*x)^ n*(A + B*ff*x + C*ff^2*x^2)/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + B_.*tan[e_. + f_.*x_] + C_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
       "filename": "4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[(a + b*ff*x)^m*(c + d*ff*x)^ n*(A + C*ff^2*x^2)/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*tan[e_. + f_.*x_])^m_*(c_. + d_.*tan[e_. + f_.*x_])^ n_*(A_. + C_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
       "filename": "4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
            "rhs": "Int[ActivateTrig[u*(a*sec[e + f*x]^2)^p], x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_ + b_.*tan[e_. + f_.*x_]^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, p}, x] && EqQ[a, b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
       "filename": "4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, (b*ff^n)^ IntPart[p]*(b*Tan[e + f*x]^n)^ FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p])* Int[ActivateTrig[u]*(Tan[e + f*x]/ff)^(n*p), x]] /; FreeQ[{b, e, f, n, p}, x] && Not[IntegerQ[p]] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, (d_.*trig_[e + f*x])^m_.",
     "rulenumber": 0,
            "lhs": "Int[u_.*(b_.*tan[e_. + f_.*x_]^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
       "filename": "4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
            "rhs": "b^IntPart[p]*(b*(c*Tan[e + f*x])^n)^ FracPart[p]/(c*Tan[e + f*x])^(n*FracPart[p])* Int[ActivateTrig[u]*(c*Tan[e + f*x])^(n*p), x] /; FreeQ[{b, c, e, f, n, p}, x] && Not[IntegerQ[p]] && Not[IntegerQ[n]] && (EqQ[u, 1] || MatchQ[u, (d_.*trig_[e + f*x])^m_.",
     "rulenumber": 0,
            "lhs": "Int[u_.*(b_.*(c_.*tan[e_. + f_.*x_])^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
       "filename": "4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
            "rhs": "x/(a - b) - b/(a - b)*Int[Sec[e + f*x]^2/(a + b*Tan[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*tan[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && NeQ[a, b]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
       "filename": "4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, c*ff/f* Subst[Int[(a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, c*Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*(c_.*tan[e_. + f_.*x_])^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
       "filename": "4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
            "rhs": "Unintegrable[(a + b*(c*Tan[e + f*x])^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*(c_.*tan[e_. + f_.*x_])^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, e, f, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
       "filename": "4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, c*ff^(m + 1)/f* Subst[Int[x^m*(a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2)^(m/2 + 1), x], x, c*Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[sin[e_. + f_.*x_]^m_*(a_ + b_.*(c_.*tan[e_. + f_.*x_])^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
       "filename": "4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Sec[e + f*x], x]}, 1/(f*ff^m)* Subst[Int[(-1 + ff^2*x^2)^((m - 1)/2)*(a - b + b*ff^2*x^2)^p/ x^(m + 1), x], x, Sec[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[sin[e_. + f_.*x_]^m_.*(a_ + b_.*tan[e_. + f_.*x_]^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
       "filename": "4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Sec[e + f*x], x]}, 1/(f*ff^m)* Subst[Int[(-1 + ff^2*x^2)^((m - 1)/ 2)*(a + b*(-1 + ff^2*x^2)^(n/2))^p/x^(m + 1), x], x, Sec[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[sin[e_. + f_.*x_]^m_.*(a_ + b_.*tan[e_. + f_.*x_]^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
       "filename": "4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
            "rhs": "Int[ExpandTrig[(d*sin[e + f*x])^m*(a + b*(c*tan[e + f*x])^n)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sin[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*tan[e_. + f_.*x_])^n_)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
       "filename": "4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, ff*(d*Sin[e + f*x])^m*(Sec[e + f*x]^2)^(m/2)/(f*Tan[e + f*x]^m)* Subst[ Int[(ff*x)^m*(a + b*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + 1), x], x, Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sin[e_. + f_.*x_])^m_*(a_ + b_.*tan[e_. + f_.*x_]^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, p}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
       "filename": "4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
            "rhs": "Unintegrable[(d*Sin[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sin[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*tan[e_. + f_.*x_])^n_)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
       "filename": "4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
            "rhs": "(d*Cos[e + f*x])^FracPart[m]*(Sec[e + f*x]/d)^ FracPart[m]* Int[(Sec[e + f*x]/d)^(-m)*(a + b*(c*Tan[e + f*x])^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*cos[e_. + f_.*x_])^m_*(a_ + b_.*(c_.*tan[e_. + f_.*x_])^n_)^ p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
       "filename": "4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, c*ff/f* Subst[Int[(d*ff*x/c)^m*(a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, c*Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*tan[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*tan[e_. + f_.*x_])^n_)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ[n, 2] || EqQ[n, 4] || IntegerQ[p] && RationalQ[n])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
       "filename": "4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
            "rhs": "Int[ExpandTrig[(d*tan[e + f*x])^m*(a + b*(c*tan[e + f*x])^n)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*tan[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*tan[e_. + f_.*x_])^n_)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
       "filename": "4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
            "rhs": "Unintegrable[(d*Tan[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*tan[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*tan[e_. + f_.*x_])^n_)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
       "filename": "4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
            "rhs": "d^(n*p)*Int[(d*Cot[e + f*x])^(m - n*p)*(b + a*Cot[e + f*x]^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*cot[e_. + f_.*x_])^m_*(a_ + b_.*tan[e_. + f_.*x_]^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n, p}, x] && Not[IntegerQ[m]] && IntegersQ[n, p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
       "filename": "4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
            "rhs": "(d*Cot[e + f*x])^FracPart[m]*(Tan[e + f*x]/d)^ FracPart[m]* Int[(Tan[e + f*x]/d)^(-m)*(a + b*(c*Tan[e + f*x])^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*cot[e_. + f_.*x_])^m_*(a_ + b_.*(c_.*tan[e_. + f_.*x_])^n_)^ p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
       "filename": "4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/(c^(m - 1)*f)* Subst[Int[(c^2 + ff^2*x^2)^(m/2 - 1)*(a + b*(ff*x)^n)^p, x], x, c*Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[sec[e_. + f_.*x_]^m_*(a_ + b_.*(c_.*tan[e_. + f_.*x_])^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[ m/2] && (IntegersQ[n, p] || IGtQ[m, 0] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
       "filename": "4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Sin[e + f*x], x]}, ff/f* Subst[Int[ ExpandToSum[b*(ff*x)^n + a*(1 - ff^2*x^2)^(n/2), x]^ p/(1 - ff^2*x^2)^((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[sec[e_. + f_.*x_]^m_.*(a_ + b_.*tan[e_. + f_.*x_]^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
       "filename": "4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Sin[e + f*x], x]}, ff/f* Subst[Int[ 1/(1 - ff^2*x^2)^((m + 1)/ 2)*((b*(ff*x)^n + a*(1 - ff^2*x^2)^(n/2))/(1 - ff^2*x^2)^(n/2))^p, x], x, Sin[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[sec[e_. + f_.*x_]^m_.*(a_ + b_.*tan[e_. + f_.*x_]^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
       "filename": "4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
            "rhs": "Int[ExpandTrig[(d*sec[e + f*x])^m*(a + b*(c*tan[e + f*x])^n)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*tan[e_. + f_.*x_])^n_)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
       "filename": "4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, ff*(d*Sec[e + f*x])^m/(f*(Sec[e + f*x]^2)^(m/2))* Subst[Int[(1 + ff^2*x^2)^(m/2 - 1)*(a + b*ff^2*x^2)^p, x], x, Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sec[e_. + f_.*x_])^m_*(a_ + b_.*tan[e_. + f_.*x_]^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, p}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
       "filename": "4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
            "rhs": "Unintegrable[(d*Sec[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*tan[e_. + f_.*x_])^n_)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
       "filename": "4.3.7 (d trig)^m (a+b (c tan)^n)^p.m",
            "rhs": "(d*Csc[e + f*x])^FracPart[m]*(Sin[e + f*x]/d)^ FracPart[m]* Int[(Sin[e + f*x]/d)^(-m)*(a + b*(c*Tan[e + f*x])^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*csc[e_. + f_.*x_])^m_*(a_ + b_.*(c_.*tan[e_. + f_.*x_])^n_)^ p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "1/(4^p*c^p)*Int[(b + 2*c*Tan[d + e*x]^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[d_. + e_.*x_]^n_. + c_.*tan[d_. + e_.*x_]^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "1/(4^p*c^p)*Int[(b + 2*c*Cot[d + e*x]^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*cot[d_. + e_.*x_]^n_. + c_.*cot[d_. + e_.*x_]^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "(a + b*Tan[d + e*x]^n + c*Tan[d + e*x]^(2*n))^ p/(b + 2*c*Tan[d + e*x]^n)^(2*p)* Int[(b + 2*c*Tan[d + e*x]^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*tan[d_. + e_.*x_]^n_. + c_.*tan[d_. + e_.*x_]^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "(a + b*Cot[d + e*x]^n + c*Cot[d + e*x]^(2*n))^ p/(b + 2*c*Cot[d + e*x]^n)^(2*p)* Int[(b + 2*c*Cot[d + e*x]^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*cot[d_. + e_.*x_]^n_. + c_.*cot[d_. + e_.*x_]^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "Module[{q = Rt[b^2 - 4*a*c, 2]}, 2*c/q*Int[1/(b - q + 2*c*Tan[d + e*x]^n), x] - 2*c/q*Int[1/(b + q + 2*c*Tan[d + e*x]^n), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_. + b_.*tan[d_. + e_.*x_]^n_. + c_.*tan[d_. + e_.*x_]^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "Module[{q = Rt[b^2 - 4*a*c, 2]}, 2*c/q*Int[1/(b - q + 2*c*Cot[d + e*x]^n), x] - 2*c/q*Int[1/(b + q + 2*c*Cot[d + e*x]^n), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_. + b_.*cot[d_. + e_.*x_]^n_. + c_.*cot[d_. + e_.*x_]^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "f/e*Subst[ Int[x^m*(a + b*x^n + c*x^(2*n))^p/(f^2 + x^2)^(m/2 + 1), x], x, f*Tan[d + e*x]]",
     "rulenumber": 0,
            "lhs": "Int[sin[d_. + e_.*x_]^ m_*(a_. + b_.*(f_.*tan[d_. + e_.*x_])^n_. + c_.*(f_.*tan[d_. + e_.*x_])^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[n2, 2*n] && IntegerQ[m/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "-f/e* Subst[Int[x^m*(a + b*x^n + c*x^(2*n))^p/(f^2 + x^2)^(m/2 + 1), x], x, f*Cot[d + e*x]]",
     "rulenumber": 0,
            "lhs": "Int[cos[d_. + e_.*x_]^ m_*(a_. + b_.*(f_.*cot[d_. + e_.*x_])^n_. + c_.*(f_.*cot[d_. + e_.*x_])^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[n2, 2*n] && IntegerQ[m/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "Module[{g = FreeFactors[Cos[d + e*x], x]}, -g/e* Subst[Int[(1 - g^2*x^2)^((m - 1)/2)* ExpandToSum[ a*(g*x)^(2*n) + b*(g*x)^n*(1 - g^2*x^2)^(n/2) + c*(1 - g^2*x^2)^n, x]^p/(g*x)^(2*n*p), x], x, Cos[d + e*x]/g]]",
     "rulenumber": 0,
            "lhs": "Int[sin[d_. + e_.*x_]^ m_.*(a_. + b_.*tan[d_. + e_.*x_]^n_. + c_.*tan[d_. + e_.*x_]^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "Module[{g = FreeFactors[Sin[d + e*x], x]}, g/e*Subst[ Int[(1 - g^2*x^2)^((m - 1)/2)* ExpandToSum[ a*(g*x)^(2*n) + b*(g*x)^n*(1 - g^2*x^2)^(n/2) + c*(1 - g^2*x^2)^n, x]^p/(g*x)^(2*n*p), x], x, Sin[d + e*x]/g]]",
     "rulenumber": 0,
            "lhs": "Int[cos[d_. + e_.*x_]^ m_.*(a_. + b_.*cot[d_. + e_.*x_]^n_. + c_.*tan[d_. + e_.*x_]^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "f^(m + 1)/e* Subst[Int[(a + b*x^n + c*x^(2*n))^p/(f^2 + x^2)^(m/2 + 1), x], x, f*Tan[d + e*x]]",
     "rulenumber": 0,
            "lhs": "Int[cos[d_. + e_.*x_]^ m_*(a_. + b_.*(f_.*tan[d_. + e_.*x_])^n_. + c_.*(f_.*tan[d_. + e_.*x_])^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[n2, 2*n] && IntegerQ[m/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "-f^(m + 1)/e* Subst[Int[(a + b*x^n + c*x^(2*n))^p/(f^2 + x^2)^(m/2 + 1), x], x, f*Cot[d + e*x]]",
     "rulenumber": 0,
            "lhs": "Int[sin[d_. + e_.*x_]^ m_*(a_. + b_.*(f_.*cot[d_. + e_.*x_])^n_. + c_.*(f_.*cot[d_. + e_.*x_])^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[n2, 2*n] && IntegerQ[m/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "Module[{g = FreeFactors[Sin[d + e*x], x]}, g/e*Subst[ Int[(1 - g^2*x^2)^((m - 2*n*p - 1)/2)* ExpandToSum[c*x^(2*n) + b*x^n*(1 - x^2)^(n/2) + a*(1 - x^2)^n, x]^p, x], x, Sin[d + e*x]/g]]",
     "rulenumber": 0,
            "lhs": "Int[cos[d_. + e_.*x_]^ m_*(a_. + b_.*tan[d_. + e_.*x_]^n_. + c_.*tan[d_. + e_.*x_]^n2_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "Module[{g = FreeFactors[Cos[d + e*x], x]}, -g/e* Subst[Int[(1 - g^2*x^2)^((m - 2*n*p - 1)/2)* ExpandToSum[c*x^(2*n) + b*x^n*(1 - x^2)^(n/2) + a*(1 - x^2)^n, x]^p, x], x, Cos[d + e*x]/g]]",
     "rulenumber": 0,
            "lhs": "Int[sin[d_. + e_.*x_]^ m_*(a_. + b_.*cot[d_. + e_.*x_]^n_. + c_.*cot[d_. + e_.*x_]^n2_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "1/(4^p*c^p)*Int[Tan[d + e*x]^m*(b + 2*c*Tan[d + e*x]^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[tan[d_. + e_.*x_]^ m_.*(a_. + b_.*tan[d_. + e_.*x_]^n_. + c_.*tan[d_. + e_.*x_]^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "1/(4^p*c^p)*Int[Cot[d + e*x]^m*(b + 2*c*Cot[d + e*x]^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[cot[d_. + e_.*x_]^ m_.*(a_. + b_.*cot[d_. + e_.*x_]^n_. + c_.*cot[d_. + e_.*x_]^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "(a + b*Tan[d + e*x]^n + c*Tan[d + e*x]^(2*n))^ p/(b + 2*c*Tan[d + e*x]^n)^(2*p)* Int[Tan[d + e*x]^m*(b + 2*c*Tan[d + e*x]^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[tan[d_. + e_.*x_]^ m_.*(a_. + b_.*tan[d_. + e_.*x_]^n_. + c_.*tan[d_. + e_.*x_]^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "(a + b*Cot[d + e*x]^n + c*Cot[d + e*x]^(2*n))^ p/(b + 2*c*Cot[d + e*x]^n)^(2*p)* Int[Cot[d + e*x]^m*(b + 2*c*Cot[d + e*x]^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[cot[d_. + e_.*x_]^ m_.*(a_. + b_.*cot[d_. + e_.*x_]^n_. + c_.*cot[d_. + e_.*x_]^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "f/e*Subst[Int[(x/f)^m*(a + b*x^n + c*x^(2*n))^p/(f^2 + x^2), x], x, f*Tan[d + e*x]]",
     "rulenumber": 0,
            "lhs": "Int[tan[d_. + e_.*x_]^ m_.*(a_. + b_.*(f_.*tan[d_. + e_.*x_])^n_. + c_.*(f_.*tan[d_. + e_.*x_])^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "-f/e* Subst[Int[(x/f)^m*(a + b*x^n + c*x^(2*n))^p/(f^2 + x^2), x], x, f*Cot[d + e*x]]",
     "rulenumber": 0,
            "lhs": "Int[cot[d_. + e_.*x_]^ m_.*(a_. + b_.*(f_.*cot[d_. + e_.*x_])^n_. + c_.*(f_.*cot[d_. + e_.*x_])^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "1/(4^p*c^p)*Int[Cot[d + e*x]^m*(b + 2*c*Tan[d + e*x]^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[cot[d_. + e_.*x_]^ m_.*(a_. + b_.*tan[d_. + e_.*x_]^n_. + c_.*tan[d_. + e_.*x_]^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "1/(4^p*c^p)*Int[Tan[d + e*x]^m*(b + 2*c*Cot[d + e*x]^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[tan[d_. + e_.*x_]^ m_.*(a_. + b_.*cot[d_. + e_.*x_]^n_. + c_.*cot[d_. + e_.*x_]^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "(a + b*Tan[d + e*x]^n + c*Tan[d + e*x]^(2*n))^ p/(b + 2*c*Tan[d + e*x]^n)^(2*p)* Int[Cot[d + e*x]^m*(b + 2*c*Tan[d + e*x]^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[cot[d_. + e_.*x_]^ m_.*(a_. + b_.*tan[d_. + e_.*x_]^n_. + c_.*tan[d_. + e_.*x_]^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "(a + b*Cot[d + e*x]^n + c*Cot[d + e*x]^(2*n))^ p/(b + 2*c*Cot[d + e*x]^n)^(2*p)* Int[Tan[d + e*x]^m*(b + 2*c*Cot[d + e*x]^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[tan[d_. + e_.*x_]^ m_.*(a_. + b_.*cot[d_. + e_.*x_]^n_. + c_.*cot[d_. + e_.*x_]^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "Module[{g = FreeFactors[Cot[d + e*x], x]}, g/e*Subst[ Int[(g*x)^(m - 2*n*p)*(c + b*(g*x)^n + a*(g*x)^(2*n))^ p/(1 + g^2*x^2), x], x, Cot[d + e*x]/g]]",
     "rulenumber": 0,
            "lhs": "Int[cot[d_. + e_.*x_]^ m_.*(a_. + b_.*tan[d_. + e_.*x_]^n_ + c_.*tan[d_. + e_.*x_]^n2_)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "Module[{g = FreeFactors[Tan[d + e*x], x]}, -g/e* Subst[Int[(g*x)^(m - 2*n*p)*(c + b*(g*x)^n + a*(g*x)^(2*n))^ p/(1 + g^2*x^2), x], x, Tan[d + e*x]/g]]",
     "rulenumber": 0,
            "lhs": "Int[tan[d_. + e_.*x_]^ m_.*(a_. + b_.*cot[d_. + e_.*x_]^n_ + c_.*cot[d_. + e_.*x_]^n2_)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "1/(4^n*c^n)* Int[(A + B*Tan[d + e*x])*(b + 2*c*Tan[d + e*x])^(2*n), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*tan[d_. + e_.*x_])*(a_ + b_.*tan[d_. + e_.*x_] + c_.*tan[d_. + e_.*x_]^2)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "1/(4^n*c^n)* Int[(A + B*Cot[d + e*x])*(b + 2*c*Cot[d + e*x])^(2*n), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*cot[d_. + e_.*x_])*(a_ + b_.*cot[d_. + e_.*x_] + c_.*cot[d_. + e_.*x_]^2)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^ n/(b + 2*c*Tan[d + e*x])^(2*n)* Int[(A + B*Tan[d + e*x])*(b + 2*c*Tan[d + e*x])^(2*n), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*tan[d_. + e_.*x_])*(a_ + b_.*tan[d_. + e_.*x_] + c_.*tan[d_. + e_.*x_]^2)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^ n/(b + 2*c*Cot[d + e*x])^(2*n)* Int[(A + B*Cot[d + e*x])*(b + 2*c*Cot[d + e*x])^(2*n), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*cot[d_. + e_.*x_])*(a_ + b_.*cot[d_. + e_.*x_] + c_.*cot[d_. + e_.*x_]^2)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "Module[{q = Rt[b^2 - 4*a*c, 2]}, (B + (b*B - 2*A*c)/q)* Int[1/Simp[b + q + 2*c*Tan[d + e*x], x], x] + (B - (b*B - 2*A*c)/q)* Int[1/Simp[b - q + 2*c*Tan[d + e*x], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*tan[d_. + e_.*x_])/(a_. + b_.*tan[d_. + e_.*x_] + c_.*tan[d_. + e_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "Module[{q = Rt[b^2 - 4*a*c, 2]}, (B + (b*B - 2*A*c)/q)* Int[1/Simp[b + q + 2*c*Cot[d + e*x], x], x] + (B - (b*B - 2*A*c)/q)* Int[1/Simp[b - q + 2*c*Cot[d + e*x], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*cot[d_. + e_.*x_])/(a_. + b_.*cot[d_. + e_.*x_] + c_.*cot[d_. + e_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "Int[ExpandTrig[(A + B*tan[d + e*x])*(a + b*tan[d + e*x] + c*tan[d + e*x]^2)^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*tan[d_. + e_.*x_])*(a_. + b_.*tan[d_. + e_.*x_] + c_.*tan[d_. + e_.*x_]^2)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
       "filename": "4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m",
            "rhs": "Int[ExpandTrig[(A + B*cot[d + e*x])*(a + b*cot[d + e*x] + c*cot[d + e*x]^2)^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*cot[d_. + e_.*x_])*(a_. + b_.*cot[d_. + e_.*x_] + c_.*cot[d_. + e_.*x_]^2)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.1 (a+b sec)^n.m",
       "filename": "4.5.1.1 (a+b sec)^n.m",
            "rhs": "-1/d* Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]]",
     "rulenumber": 0,
            "lhs": "Int[csc[c_. + d_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d}, x] && IGtQ[n/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.1 (a+b sec)^n.m",
       "filename": "4.5.1.1 (a+b sec)^n.m",
            "rhs": "-b* Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1)/(d*(n - 1)) + b^2*(n - 2)/(n - 1)*Int[(b*Csc[c + d*x])^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*csc[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.1 (a+b sec)^n.m",
       "filename": "4.5.1.1 (a+b sec)^n.m",
            "rhs": "Cos[c + d*x]*(b*Csc[c + d*x])^(n + 1)/(b*d*n) + (n + 1)/(b^2*n)*Int[(b*Csc[c + d*x])^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*csc[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.1 (a+b sec)^n.m",
       "filename": "4.5.1.1 (a+b sec)^n.m",
            "rhs": "(* -ArcCoth[Cos[c+d*x]]/d /; *) -ArcTanh[ Cos[c + d*x]]/d",
     "rulenumber": 0,
            "lhs": "Int[csc[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.1 (a+b sec)^n.m",
       "filename": "4.5.1.1 (a+b sec)^n.m",
            "rhs": "-Cos[c+d*x]/d",
     "rulenumber": 0,
            "lhs": "Int[1/csc[c_.+d_.*x_],x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{c,d},x] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.1 (a+b sec)^n.m",
       "filename": "4.5.1.1 (a+b sec)^n.m",
            "rhs": "(b*Csc[c + d*x])^n*Sin[c + d*x]^n* Int[1/Sin[c + d*x]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*csc[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.1 (a+b sec)^n.m",
       "filename": "4.5.1.1 (a+b sec)^n.m",
            "rhs": "(b*Csc[c + d*x])^(n - 1)*((Sin[c + d*x]/b)^(n - 1)* Int[1/(Sin[c + d*x]/b)^n, x])",
     "rulenumber": 0,
            "lhs": "Int[(b_.*csc[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, n}, x] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.1 (a+b sec)^n.m",
       "filename": "4.5.1.1 (a+b sec)^n.m",
            "rhs": "a^2*x + 2*a*b*Int[Csc[c + d*x], x] + b^2*Int[Csc[c + d*x]^2, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[c_. + d_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.1 (a+b sec)^n.m",
       "filename": "4.5.1.1 (a+b sec)^n.m",
            "rhs": "-2*b/d* Subst[Int[1/(a + x^2), x], x, b*Cot[c + d*x]/Sqrt[a + b*Csc[c + d*x]]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*csc[c_. + d_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.1 (a+b sec)^n.m",
       "filename": "4.5.1.1 (a+b sec)^n.m",
            "rhs": "-b^2* Cot[c + d*x]*(a + b*Csc[c + d*x])^(n - 2)/(d*(n - 1)) + a/(n - 1)* Int[(a + b*Csc[c + d*x])^(n - 2)*(a*(n - 1) + b*(3*n - 4)*Csc[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && GtQ[n, 1] && IntegerQ[2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.1 (a+b sec)^n.m",
       "filename": "4.5.1.1 (a+b sec)^n.m",
            "rhs": "1/a*Int[Sqrt[a + b*Csc[c + d*x]], x] - b/a*Int[Csc[c + d*x]/Sqrt[a + b*Csc[c + d*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*csc[c_. + d_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.1 (a+b sec)^n.m",
       "filename": "4.5.1.1 (a+b sec)^n.m",
            "rhs": "-Cot[c + d*x]*(a + b*Csc[c + d*x])^n/(d*(2*n + 1)) + 1/(a^2*(2*n + 1))* Int[(a + b*Csc[c + d*x])^(n + 1)*(a*(2*n + 1) - b*(n + 1)*Csc[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LeQ[n, -1] && IntegerQ[2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.1 (a+b sec)^n.m",
       "filename": "4.5.1.1 (a+b sec)^n.m",
            "rhs": "a^n*Cot[c + d*x]/(d*Sqrt[1 + Csc[c + d*x]]*Sqrt[1 - Csc[c + d*x]])* Subst[Int[(1 + b*x/a)^(n - 1/2)/(x*Sqrt[1 - b*x/a]), x], x, Csc[c + d*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[2*n]] && GtQ[a, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.1 (a+b sec)^n.m",
       "filename": "4.5.1.1 (a+b sec)^n.m",
            "rhs": "a^IntPart[n]*(a + b*Csc[c + d*x])^FracPart[n]/(1 + b/a*Csc[c + d*x])^ FracPart[n]*Int[(1 + b/a*Csc[c + d*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[2*n]] && Not[GtQ[a, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.1 (a+b sec)^n.m",
       "filename": "4.5.1.1 (a+b sec)^n.m",
            "rhs": "2*(a + b*Csc[c + d*x])/(d*Rt[a + b, 2]*Cot[c + d*x])* Sqrt[b*(1 + Csc[c + d*x])/(a + b*Csc[c + d*x])]* Sqrt[-b*(1 - Csc[c + d*x])/(a + b*Csc[c + d*x])]* EllipticPi[a/(a + b), ArcSin[Rt[a + b, 2]/Sqrt[a + b*Csc[c + d*x]]], (a - b)/(a + b)]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*csc[c_. + d_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.1 (a+b sec)^n.m",
       "filename": "4.5.1.1 (a+b sec)^n.m",
            "rhs": "Int[(a^2 + b*(2*a - b)*Csc[c + d*x])/Sqrt[a + b*Csc[c + d*x]], x] + b^2*Int[Csc[c + d*x]*(1 + Csc[c + d*x])/Sqrt[a + b*Csc[c + d*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[c_. + d_.*x_])^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.1 (a+b sec)^n.m",
       "filename": "4.5.1.1 (a+b sec)^n.m",
            "rhs": "-b^2* Cot[c + d*x]*(a + b*Csc[c + d*x])^(n - 2)/(d*(n - 1)) + 1/(n - 1)*Int[(a + b*Csc[c + d*x])^(n - 3)* Simp[a^3*(n - 1) + (b*(b^2*(n - 2) + 3*a^2*(n - 1)))* Csc[c + d*x] + (a*b^2*(3*n - 4))*Csc[c + d*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 2] && IntegerQ[2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.1 (a+b sec)^n.m",
       "filename": "4.5.1.1 (a+b sec)^n.m",
            "rhs": "x/a - 1/a*Int[1/(1 + a/b*Sin[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*csc[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.1 (a+b sec)^n.m",
       "filename": "4.5.1.1 (a+b sec)^n.m",
            "rhs": "2*Rt[a + b, 2]/(a*d*Cot[c + d*x])*Sqrt[b*(1 - Csc[c + d*x])/(a + b)]* Sqrt[-b*(1 + Csc[c + d*x])/(a - b)]* EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Csc[c + d*x]]/Rt[a + b, 2]], (a + b)/(a - b)]",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_ + b_.*csc[c_. + d_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.1 (a+b sec)^n.m",
       "filename": "4.5.1.1 (a+b sec)^n.m",
            "rhs": "b^2*Cot[c + d*x]*(a + b*Csc[c + d*x])^(n + 1)/(a*d*(n + 1)*(a^2 - b^2)) + 1/(a*(n + 1)*(a^2 - b^2))* Int[(a + b*Csc[c + d*x])^(n + 1)* Simp[(a^2 - b^2)*(n + 1) - a*b*(n + 1)*Csc[c + d*x] + b^2*(n + 2)*Csc[c + d*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.1 (a+b sec)^n.m",
       "filename": "4.5.1.1 (a+b sec)^n.m",
            "rhs": "Unintegrable[(a + b*Csc[c + d*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 - b^2, 0] && Not[IntegerQ[2*n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "a*Int[(d*Csc[e + f*x])^n, x] + b/d*Int[(d*Csc[e + f*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])*(d_.*csc[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "2*a*b/d*Int[(d*Csc[e + f*x])^(n + 1), x] + Int[(d*Csc[e + f*x])^n*(a^2 + b^2*Csc[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^2*(d_.*csc[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "1/b*Int[Csc[e + f*x], x] - a/b*Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]^2/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-Cot[e + f*x]/(b*f) - a/b*Int[Csc[e + f*x]^2/(a + b*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]^3/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "Int[ExpandTrig[(a + b*csc[e + f*x])^m*(d*csc[e + f*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0] && RationalQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-2*b*Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]])",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-b* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)/(f*m) + a*(2*m - 1)/m*Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 1/2] && IntegerQ[2*m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-Cot[e + f*x]/(f*(b + a*Csc[e + f*x]))",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-2/f* Subst[Int[1/(2*a + x^2), x], x, b*Cot[e + f*x]/Sqrt[a + b*Csc[e + f*x]]]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "b*Cot[e + f*x]*(a + b*Csc[e + f*x])^ m/(a*f*(2*m + 1)) + (m + 1)/(a*(2*m + 1))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1/2] && IntegerQ[2*m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(f*(2*m + 1)) + m/(b*(2*m + 1))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]^2*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(f*(m + 1)) + a*m/(b*(m + 1))*Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]^2*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1/2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "b*Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(a*f*(2*m + 1)) - 1/(a^2*(2*m + 1))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(a*m - b*(2*m + 1)*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]^3*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-Cot[ e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 2)) + 1/(b*(m + 2))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^ m*(b*(m + 1) - a*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]^3*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1/2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-2*a/(b*f)*Sqrt[a*d/b]* Subst[Int[1/Sqrt[1 + x^2/a], x], x, b*Cot[e + f*x]/Sqrt[a + b*Csc[e + f*x]]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*Sqrt[d_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[a*d/b, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-2*b*d/f* Subst[Int[1/(b - d*x^2), x], x, b*Cot[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[d*Csc[e + f*x]])]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*Sqrt[d_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && Not[GtQ[a*d/b, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-2*b*d* Cot[e + f*x]*(d*Csc[e + f*x])^(n - 1)/(f*(2*n - 1)* Sqrt[a + b*Csc[e + f*x]]) + 2*a*d*(n - 1)/(b*(2*n - 1))* Int[Sqrt[a + b*Csc[e + f*x]]*(d*Csc[e + f*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[n, 1] && IntegerQ[2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-2*a* Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[d*Csc[e + f*x]])",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]/Sqrt[d_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "a*Cot[e + f*x]*(d*Csc[e + f*x])^n/(f*n*Sqrt[a + b*Csc[e + f*x]]) + a*(2*n + 1)/(2*b*d*n)* Int[Sqrt[a + b*Csc[e + f*x]]*(d*Csc[e + f*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1/2] && IntegerQ[2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "a^2*d*Cot[ e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x]])* Subst[Int[(d*x)^(n - 1)/Sqrt[a - b*x], x], x, Csc[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-Sqrt[2]*Sqrt[a]/(b*f)* Subst[Int[1/Sqrt[1 + x^2], x], x, b*Cot[e + f*x]/(a + b*Csc[e + f*x])]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[d_.*csc[e_. + f_.*x_]]/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && EqQ[d - a/b, 0] && GtQ[a, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-2*b*d/(a*f)* Subst[Int[1/(2*b - d*x^2), x], x, b*Cot[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[d*Csc[e + f*x]])]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[d_.*csc[e_. + f_.*x_]]/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-a* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^ n/(f*m) + b*(2*m - 1)/(d*m)* Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && EqQ[m + n, 0] && GtQ[m, 1/2] && IntegerQ[2*m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "b*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^(n - 1)/(a*f*(2*m + 1)) + d*(m + 1)/(b*(2*m + 1))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && EqQ[m + n, 0] && LtQ[m, -1/2] && IntegerQ[2*m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^n/(f*(2*m + 1)) + m/(a*(2*m + 1))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && LtQ[m, -1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^n/(f*(m + 1)) + a*m/(b*d*(m + 1))* Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && Not[LtQ[m, -1/2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "b^2*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^ n/(f*n) - a/(d*n)* Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^(n + 1)*(b*(m - 2*n - 2) - a*(m + 2*n - 1)*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 1] && (LtQ[n, -1] || EqQ[m, 3/2] && EqQ[n, -1/2]) && IntegerQ[2*m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-b^2* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^ n/(f*(m + n - 1)) + b/(m + n - 1)* Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^ n*(b*(m + 2*n - 1) + a*(3*m + 2*n - 4)*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 1] && NeQ[m + n - 1, 0] && IntegerQ[2*m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "b*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^(n - 1)/(a*f*(2*m + 1)) - d/(a*b*(2*m + 1))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)*(a*(n - 1) - b*(m + n)*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && LtQ[1, n, 2] && (IntegersQ[2*m, 2*n] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-d^2* Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^(n - 2)/(f*(2*m + 1)) + d^2/(a*b*(2*m + 1))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 2)*(b*(n - 2) + a*(m - n + 2)*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[n, 2] && (IntegersQ[2*m, 2*n] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^n/(f*(2*m + 1)) + 1/(a^2*(2*m + 1))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^ n*(a*(2*m + n + 1) - b*(m + n + 1)*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && (IntegersQ[2*m, 2*n] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "d^2*Cot[e + f*x]*(d*Csc[e + f*x])^(n - 2)/(f*(a + b*Csc[e + f*x])) - d^2/(a*b)* Int[(d*Csc[e + f*x])^(n - 2)*(b*(n - 2) - a*(n - 1)*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*csc[e_. + f_.*x_])^n_/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "Cot[e + f*x]*(d*Csc[e + f*x])^n/(f*(a + b*Csc[e + f*x])) - 1/a^2*Int[(d*Csc[e + f*x])^n*(a*(n - 1) - b*n*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*csc[e_. + f_.*x_])^n_/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-b*d* Cot[e + f*x]*(d*Csc[e + f*x])^(n - 1)/(a*f*(a + b*Csc[e + f*x])) + d*(n - 1)/(a*b)* Int[(d*Csc[e + f*x])^(n - 1)*(a - b*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*csc[e_. + f_.*x_])^n_/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "d/b*Int[Sqrt[a + b*Csc[e + f*x]]*Sqrt[d*Csc[e + f*x]], x] - a*d/b*Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*csc[e_. + f_.*x_])^(3/2)/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-2*d^2* Cot[e + f*x]*(d*Csc[e + f*x])^(n - 2)/(f*(2*n - 3)* Sqrt[a + b*Csc[e + f*x]]) + d^2/(b*(2*n - 3))* Int[(d*Csc[e + f*x])^(n - 2)*(2*b*(n - 2) - a*Csc[e + f*x])/ Sqrt[a + b*Csc[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*csc[e_. + f_.*x_])^n_/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[n, 2] && IntegerQ[2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "Cot[e + f*x]*(d*Csc[e + f*x])^n/(f*n*Sqrt[a + b*Csc[e + f*x]]) + 1/(2*b*d*n)* Int[(d*Csc[e + f*x])^(n + 1)*(a + b*(2*n + 1)*Csc[e + f*x])/ Sqrt[a + b*Csc[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*csc[e_. + f_.*x_])^n_/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, 0] && IntegerQ[2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-d^2* Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^(n - 2)/(f*(m + n - 1)) + d^2/(b*(m + n - 1))* Int[(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^(n - 2)*(b*(n - 2) + a*m*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m}, x] && EqQ[a^2 - b^2, 0] && GtQ[n, 2] && NeQ[m + n - 1, 0] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-(a*d/b)^n* Cot[e + f*x]/(a^(n - 2)*f*Sqrt[a + b*Csc[e + f*x]]* Sqrt[a - b*Csc[e + f*x]])* Subst[Int[(a - x)^(n - 1)*(2*a - x)^(m - 1/2)/Sqrt[x], x], x, a - b*Csc[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[m]] && GtQ[a, 0] && Not[IntegerQ[n]] && GtQ[a*d/b, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-(-a*d/b)^n* Cot[e + f*x]/(a^(n - 1)*f*Sqrt[a + b*Csc[e + f*x]]* Sqrt[a - b*Csc[e + f*x]])* Subst[Int[x^(m - 1/2)*(a - x)^(n - 1)/Sqrt[2*a - x], x], x, a + b*Csc[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[m]] && GtQ[a, 0] && Not[IntegerQ[n]] && LtQ[a*d/b, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "a^2*d*Cot[ e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x]])* Subst[Int[(d*x)^(n - 1)*(a + b*x)^(m - 1/2)/Sqrt[a - b*x], x], x, Csc[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[m]] && GtQ[a, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "a^IntPart[m]*(a + b*Csc[e + f*x])^FracPart[m]/(1 + b/a*Csc[e + f*x])^ FracPart[m]*Int[(1 + b/a*Csc[e + f*x])^m*(d*Csc[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[m]] && Not[GtQ[a, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "(a - b)* Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x] + b*Int[Csc[e + f*x]*(1 + Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-b*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)/(f*m) + 1/m*Int[ Csc[e + f*x]*(a + b*Csc[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*m + a*b*(2*m - 1)*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && IntegerQ[2*m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": " -2/f*Subst[Int[1/(a+b-(a-b)*x^2),x],x,Cot[e+f*x]/(1+Csc[e+f*x])]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_.+f_.*x_]/(a_+b_.*csc[e_.+f_.*x_]),x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,e,f},x] && NeQ[a^2-b^2,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "1/b*Int[1/(1 + a/b*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-2*Rt[a + b, 2]/(b*f*Cot[e + f*x])* Sqrt[(b*(1 - Csc[e + f*x]))/(a + b)]* Sqrt[-b*(1 + Csc[e + f*x])/(a - b)]* EllipticF[ ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-b* Cot[e + f* x]*(a + b*Csc[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2)) + 1/((m + 1)*(a^2 - b^2))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(a*(m + 1) - b*(m + 2)*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "Cot[e + f*x]/(f*Sqrt[1 + Csc[e + f*x]]*Sqrt[1 - Csc[e + f*x]])* Subst[Int[(a + b*x)^m/(Sqrt[1 + x]*Sqrt[1 - x]), x], x, Csc[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0] && Not[IntegerQ[2*m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(f*(m + 1)) + m/(m + 1)* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*(b + a*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]^2*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "a*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2)) - 1/((m + 1)*(a^2 - b^2))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(b*(m + 1) - a*(m + 2)*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]^2*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x] + Int[Csc[e + f*x]*(1 + Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]^2/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-a/b*Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m, x] + 1/b*Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]^2*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-a^2* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(b* f*(m + 1)*(a^2 - b^2)) + 1/(b*(m + 1)*(a^2 - b^2))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)* Simp[a*b*(m + 1) - (a^2 + b^2*(m + 1))*Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]^3*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-Cot[ e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 2)) + 1/(b*(m + 2))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^ m*(b*(m + 1) - a*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]^3*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0] && Not[LtQ[m, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "a^2*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^ n/(f*n) - 1/(d*n)*Int[(a + b*Csc[e + f*x])^(m - 3)*(d*Csc[e + f*x])^(n + 1)* Simp[a^2*b*(m - 2*n - 2) - a*(3*b^2*n + a^2*(n + 1))*Csc[e + f*x] - b*(b^2*n + a^2*(m + n - 1))*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 2] && (IntegerQ[m] && LtQ[n, -1] || IntegersQ[m + 1/2, 2*n] && LeQ[n, -1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-b^2* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^ n/(f*(m + n - 1)) + 1/(d*(m + n - 1))* Int[(a + b*Csc[e + f*x])^(m - 3)*(d*Csc[e + f*x])^n* Simp[a^3*d*(m + n - 1) + a*b^2*d*n + b*(b^2*d*(m + n - 2) + 3*a^2*d*(m + n - 1))*Csc[e + f*x] + a*b^2*d*(3*m + 2*n - 4)*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 2] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && Not[IGtQ[n, 2] && Not[IntegerQ[m]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-b*d* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)/(f*(m + 1)*(a^2 - b^2)) + 1/((m + 1)*(a^2 - b^2))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)* Simp[b*d*(n - 1) + a*d*(m + 1)*Csc[e + f*x] - b*d*(m + n + 1)*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && LtQ[0, n, 1] && IntegersQ[2*m, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "a*d^2*Cot[ e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 2)/(f*(m + 1)*(a^2 - b^2)) - d^2/((m + 1)*(a^2 - b^2))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 2)*(a*(n - 2) + b*(m + 1)*Csc[e + f*x] - a*(m + n)*Csc[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && LtQ[1, n, 2] && IntegersQ[2*m, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-a^2*d^3* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 3)/(b*f*(m + 1)*(a^2 - b^2)) + d^3/(b*(m + 1)*(a^2 - b^2))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 3)* Simp[a^2*(n - 3) + a*b*(m + 1)*Csc[e + f*x] - (a^2*(n - 2) + b^2*(m + 1))* Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && (IGtQ[n, 3] || IntegersQ[n + 1/2, 2*m] && GtQ[n, 2])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^ n/(a*f*n) - 1/(a*d*n)*Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)* Simp[b*(m + n + 1) - a*(n + 1)*Csc[e + f*x] - b*(m + n + 2)*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && ILtQ[m + 1/2, 0] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "b^2*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^ n/(a*f*(m + 1)*(a^2 - b^2)) + 1/(a*(m + 1)*(a^2 - b^2))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n* (a^2*(m + 1) - b^2*(m + n + 1) - a*b*(m + 1)*Csc[e + f*x] + b^2*(m + n + 2)*Csc[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "Sqrt[d*Sin[e + f*x]]*Sqrt[d*Csc[e + f*x]]/d* Int[Sqrt[d*Sin[e + f*x]]/(b + a*Sin[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[d_.*csc[e_. + f_.*x_]]/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "d*Sqrt[d*Sin[e + f*x]]*Sqrt[d*Csc[e + f*x]]* Int[1/(Sqrt[d*Sin[e + f*x]]*(b + a*Sin[e + f*x])), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*csc[e_. + f_.*x_])^(3/2)/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "d/b*Int[(d*Csc[e + f*x])^(3/2), x] - a*d/b*Int[(d*Csc[e + f*x])^(3/2)/(a + b*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*csc[e_. + f_.*x_])^(5/2)/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-d^3* Cot[e + f*x]*(d*Csc[e + f*x])^(n - 3)/(b*f*(n - 2)) + d^3/(b*(n - 2))* Int[(d*Csc[e + f*x])^(n - 3)* Simp[a*(n - 3) + b*(n - 3)*Csc[e + f*x] - a*(n - 2)*Csc[e + f*x]^2, x]/(a + b*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*csc[e_. + f_.*x_])^n_/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 3]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "b^2/(a^2*d^2)*Int[(d*Csc[e + f*x])^(3/2)/(a + b*Csc[e + f*x]), x] + 1/a^2*Int[(a - b*Csc[e + f*x])/Sqrt[d*Csc[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[d_.*csc[e_. + f_.*x_]]*(a_ + b_.*csc[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "Cot[e + f*x]*(d*Csc[e + f*x])^n/(a*f*n) - 1/(a*d*n)*Int[(d*Csc[e + f*x])^(n + 1)/(a + b*Csc[e + f*x])* Simp[b*n - a*(n + 1)*Csc[e + f*x] - b*(n + 1)*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*csc[e_. + f_.*x_])^n_/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1] && IntegerQ[2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "a*Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x] + b/d*Int[(d*Csc[e + f*x])^(3/2)/Sqrt[a + b*Csc[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*Sqrt[d_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-2*d*Cos[e + f*x]* Sqrt[a + b*Csc[e + f*x]]*(d*Csc[e + f*x])^(n - 1)/(f*(2*n - 1)) + d^2/(2*n - 1)* Int[(d*Csc[e + f*x])^(n - 2)* Simp[2*a*(n - 2) + b*(2*n - 3)*Csc[e + f*x] + a*Csc[e + f*x]^2, x]/Sqrt[a + b*Csc[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 1] && IntegerQ[2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "Sqrt[a + b*Csc[e + f*x]]/(Sqrt[d*Csc[e + f*x]]* Sqrt[b + a*Sin[e + f*x]])*Int[Sqrt[b + a*Sin[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]/Sqrt[d_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "Cot[e + f*x]*Sqrt[a + b*Csc[e + f*x]]*(d*Csc[e + f*x])^n/(f*n) - 1/(2*d*n)* Int[(d*Csc[e + f*x])^(n + 1)* Simp[b - 2*a*(n + 1)*Csc[e + f*x] - b*(2*n + 3)*Csc[e + f*x]^2, x]/Sqrt[a + b*Csc[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1] && IntegerQ[2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "Sqrt[d*Csc[e + f*x]]* Sqrt[b + a*Sin[e + f*x]]/Sqrt[a + b*Csc[e + f*x]]* Int[1/Sqrt[b + a*Sin[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[d_.*csc[e_. + f_.*x_]]/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "d*Sqrt[d*Csc[e + f*x]]* Sqrt[b + a*Sin[e + f*x]]/Sqrt[a + b*Csc[e + f*x]]* Int[1/(Sin[e + f*x]*Sqrt[b + a*Sin[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*csc[e_. + f_.*x_])^(3/2)/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-2*d^2*Cos[e + f*x]*(d*Csc[e + f*x])^(n - 2)* Sqrt[a + b*Csc[e + f*x]]/(b*f*(2*n - 3)) + d^3/(b*(2*n - 3))* Int[(d*Csc[e + f*x])^(n - 3)/Sqrt[a + b*Csc[e + f*x]]* Simp[2*a*(n - 3) + b*(2*n - 5)*Csc[e + f*x] - 2*a*(n - 2)*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*csc[e_. + f_.*x_])^n_/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 2] && IntegerQ[2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-Cos[e + f*x]*Sqrt[a + b*Csc[e + f*x]]/(a*f) - b/(2*a)*Int[(1 + Csc[e + f*x]^2)/Sqrt[a + b*Csc[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[1/(csc[e_. + f_.*x_]*Sqrt[a_ + b_.*csc[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "1/a*Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x] - b/(a*d)*Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*Sqrt[d_.*csc[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "Cos[e + f*x]*(d*Csc[e + f*x])^(n + 1)* Sqrt[a + b*Csc[e + f*x]]/(a*d*f*n) + 1/(2*a*d*n)*Int[(d*Csc[e + f*x])^(n + 1)/Sqrt[a + b*Csc[e + f*x]]* Simp[-b*(2*n + 1) + 2*a*(n + 1)*Csc[e + f*x] + b*(2*n + 3)*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*csc[e_. + f_.*x_])^n_/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "a*Cot[e + f*x]*Sqrt[a + b*Csc[e + f*x]]*(d*Csc[e + f*x])^n/(f*n) + 1/(2*d*n)*Int[(d*Csc[e + f*x])^(n + 1)/Sqrt[a + b*Csc[e + f*x]]* Simp[a*b*(2*n - 1) + 2*(b^2*n + a^2*(n + 1))*Csc[e + f*x] + a*b*(2*n + 3)*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^(3/2)*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1] && IntegersQ[2*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-d^3* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 3)/(b*f*(m + n - 1)) + d^3/(b*(m + n - 1))* Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 3)* Simp[a*(n - 3) + b*(m + n - 2)*Csc[e + f*x] - a*(n - 2)*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 3] && (IntegerQ[n] || IntegersQ[2*m, 2*n]) && Not[IGtQ[m, 2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-b*d* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n - 1)/(f*(m + n - 1)) + d/(m + n - 1)* Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^(n - 1)* Simp[a*b*(n - 1) + (b^2*(m + n - 2) + a^2*(m + n - 1))* Csc[e + f*x] + a*b*(2*m + n - 2)*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[0, m, 2] && LtQ[0, n, 3] && NeQ[m + n - 1, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "-d^2* Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^(n - 2)/(f*(m + n - 1)) + d^2/(b*(m + n - 1))* Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n - 2)* Simp[a*b*(n - 2) + b^2*(m + n - 2)*Csc[e + f*x] + a*b*m*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[-1, m, 2] && LtQ[1, n, 3] && NeQ[m + n - 1, 0] && (IntegerQ[n] || IntegersQ[2*m, 2*n])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "a*Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x] + b/d*Int[Sqrt[a + b*Csc[e + f*x]]*Sqrt[d*Csc[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^(3/2)/Sqrt[d_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "Sin[e + f*x]^n*(d*Csc[e + f*x])^n* Int[(b + a*Sin[e + f*x])^m/Sin[e + f*x]^(m + n), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*csc[e_. + f_.*x_])^n_.*(a_ + b_.*csc[e_. + f_.*x_])^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "Unintegrable[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(d_.*csc[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m",
       "filename": "4.5.1.2 (d sec)^n (a+b sec)^m.m",
            "rhs": "(d*Cos[e + f*x])^FracPart[m]*(Sec[e + f*x]/d)^ FracPart[m]*Int[(Sec[e + f*x]/d)^(-m)*(a + b*Sec[e + f*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*cos[e_. + f_.*x_])^m_*(a_. + b_.*sec[e_. + f_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, p}, x] && Not[IntegerQ[m]] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.3 (d sin)^n (a+b sec)^m.m",
       "filename": "4.5.1.3 (d sin)^n (a+b sec)^m.m",
            "rhs": "Int[(g*Cos[e + f*x])^p*(b + a*Sin[e + f*x])^m/Sin[e + f*x]^m, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_.*(a_ + b_.*csc[e_. + f_.*x_])^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.3 (d sin)^n (a+b sec)^m.m",
       "filename": "4.5.1.3 (d sin)^n (a+b sec)^m.m",
            "rhs": "-1/(f*b^(p - 1))* Subst[Int[(-a + b*x)^((p - 1)/2)*(a + b*x)^(m + (p - 1)/2)/ x^(p + 1), x], x, Csc[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^p_.*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.3 (d sin)^n (a+b sec)^m.m",
       "filename": "4.5.1.3 (d sin)^n (a+b sec)^m.m",
            "rhs": "-1/f* Subst[Int[(-1 + x)^((p - 1)/2)*(1 + x)^((p - 1)/2)*(a + b*x)^m/ x^(p + 1), x], x, Csc[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[cos[e_. + f_.*x_]^p_.*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.3 (d sin)^n (a+b sec)^m.m",
       "filename": "4.5.1.3 (d sin)^n (a+b sec)^m.m",
            "rhs": "Tan[e + f*x]*(a + b*Csc[e + f*x])^m/f + b*m*Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_/cos[e_. + f_.*x_]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, m}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.3 (d sin)^n (a+b sec)^m.m",
       "filename": "4.5.1.3 (d sin)^n (a+b sec)^m.m",
            "rhs": "Sin[e + f*x]^ FracPart[m]*(a + b*Csc[e + f*x])^FracPart[m]/(b + a*Sin[e + f*x])^ FracPart[m]* Int[(g*Cos[e + f*x])^p*(b + a*Sin[e + f*x])^m/Sin[e + f*x]^m, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_.*(a_ + b_.*csc[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, m, p}, x] && (EqQ[a^2 - b^2, 0] || IntegersQ[2*m, p])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.3 (d sin)^n (a+b sec)^m.m",
       "filename": "4.5.1.3 (d sin)^n (a+b sec)^m.m",
            "rhs": "Unintegrable[(g*Cos[e + f*x])^p*(a + b*Csc[e + f*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*cos[e_. + f_.*x_])^p_.*(a_ + b_.*csc[e_. + f_.*x_])^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, m, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.3 (d sin)^n (a+b sec)^m.m",
       "filename": "4.5.1.3 (d sin)^n (a+b sec)^m.m",
            "rhs": "Int[(g*Sec[e+f*x])^p*(b+a*Sin[e+f*x])^m/Sin[e+f*x]^m,x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*sec[e_.+f_.*x_])^p_*(a_+b_.*csc[e_.+f_.*x_])^m_.,x_Symbol] ",
        "comment": false,
         "givens": " FreeQ[{a,b,e,f,g,p},x] && Not[IntegerQ[p]] && IntegerQ[m] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.3 (d sin)^n (a+b sec)^m.m",
       "filename": "4.5.1.3 (d sin)^n (a+b sec)^m.m",
            "rhs": "g^IntPart[p]*(g*Sec[e + f*x])^FracPart[p]*Cos[e + f*x]^FracPart[p]* Int[(a + b*Csc[e + f*x])^m/Cos[e + f*x]^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*sec[e_. + f_.*x_])^p_*(a_ + b_.*csc[e_. + f_.*x_])^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, g, m, p}, x] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.4 (d tan)^n (a+b sec)^m.m",
       "filename": "4.5.1.4 (d tan)^n (a+b sec)^m.m",
            "rhs": "1/(a^(m - n - 1)*b^n*d)* Subst[Int[(a - b*x)^((m - 1)/2)*(a + b*x)^((m - 1)/2 + n)/ x^(m + n), x], x, Sin[c + d*x]]",
     "rulenumber": 0,
            "lhs": "Int[cot[c_. + d_.*x_]^m_.*(a_ + b_.*csc[c_. + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.4 (d tan)^n (a+b sec)^m.m",
       "filename": "4.5.1.4 (d tan)^n (a+b sec)^m.m",
            "rhs": "-1/(d*b^(m - 1))* Subst[Int[(-a + b*x)^((m - 1)/2)*(a + b*x)^((m - 1)/2 + n)/x, x], x, Csc[c + d*x]]",
     "rulenumber": 0,
            "lhs": "Int[cot[c_. + d_.*x_]^m_.*(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.4 (d tan)^n (a+b sec)^m.m",
       "filename": "4.5.1.4 (d tan)^n (a+b sec)^m.m",
            "rhs": "-e*(e*Cot[c + d*x])^(m - 1)*(a*m + b*(m - 1)*Csc[c + d*x])/(d*m*(m - 1)) - e^2/m* Int[(e*Cot[c + d*x])^(m - 2)*(a*m + b*(m - 1)*Csc[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cot[c_. + d_.*x_])^m_*(a_ + b_.*csc[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && GtQ[m, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.4 (d tan)^n (a+b sec)^m.m",
       "filename": "4.5.1.4 (d tan)^n (a+b sec)^m.m",
            "rhs": "-(e*Cot[c + d*x])^(m + 1)*(a + b*Csc[c + d*x])/(d* e*(m + 1)) - 1/(e^2*(m + 1))* Int[(e*Cot[c + d*x])^(m + 2)*(a*(m + 1) + b*(m + 2)*Csc[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cot[c_. + d_.*x_])^m_*(a_ + b_.*csc[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.4 (d tan)^n (a+b sec)^m.m",
       "filename": "4.5.1.4 (d tan)^n (a+b sec)^m.m",
            "rhs": "Int[(b + a*Sin[c + d*x])/Cos[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[c_. + d_.*x_])/cot[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.4 (d tan)^n (a+b sec)^m.m",
       "filename": "4.5.1.4 (d tan)^n (a+b sec)^m.m",
            "rhs": "a*Int[(e*Cot[c + d*x])^m, x] + b*Int[(e*Cot[c + d*x])^m*Csc[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cot[c_. + d_.*x_])^m_.*(a_ + b_.*csc[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.4 (d tan)^n (a+b sec)^m.m",
       "filename": "4.5.1.4 (d tan)^n (a+b sec)^m.m",
            "rhs": "-(-1)^((m - 1)/2)/(d*b^(m - 1))* Subst[Int[(b^2 - x^2)^((m - 1)/2)*(a + x)^n/x, x], x, b*Csc[c + d*x]]",
     "rulenumber": 0,
            "lhs": "Int[cot[c_. + d_.*x_]^m_.*(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.4 (d tan)^n (a+b sec)^m.m",
       "filename": "4.5.1.4 (d tan)^n (a+b sec)^m.m",
            "rhs": "Int[ExpandIntegrand[(e*Cot[c + d*x])^m, (a + b*Csc[c + d*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cot[c_. + d_.*x_])^m_*(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.4 (d tan)^n (a+b sec)^m.m",
       "filename": "4.5.1.4 (d tan)^n (a+b sec)^m.m",
            "rhs": "-2*a^(m/2 + n + 1/2)/d* Subst[Int[x^m*(2 + a*x^2)^(m/2 + n - 1/2)/(1 + a*x^2), x], x, Cot[c + d*x]/Sqrt[a + b*Csc[c + d*x]]]",
     "rulenumber": 0,
            "lhs": "Int[cot[c_. + d_.*x_]^m_.*(a_ + b_.*csc[c_. + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m/2] && IntegerQ[n - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.4 (d tan)^n (a+b sec)^m.m",
       "filename": "4.5.1.4 (d tan)^n (a+b sec)^m.m",
            "rhs": "a^(2*n)*e^(-2*n)* Int[(e*Cot[c + d*x])^(m + 2*n)/(-a + b*Csc[c + d*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cot[c_. + d_.*x_])^m_*(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && EqQ[a^2 - b^2, 0] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.4 (d tan)^n (a+b sec)^m.m",
       "filename": "4.5.1.4 (d tan)^n (a+b sec)^m.m",
            "rhs": "-2^(m + n + 1)*(e*Cot[c + d*x])^(m + 1)*(a + b*Csc[c + d*x])^ n/(d*e*(m + 1))*(a/(a + b*Csc[c + d*x]))^(m + n + 1)* AppellF1[(m + 1)/2, m + n, 1, (m + 3)/ 2, -(a - b*Csc[c + d*x])/(a + b*Csc[c + d*x]), (a - b*Csc[c + d*x])/(a + b*Csc[c + d*x])]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cot[c_. + d_.*x_])^m_*(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[a^2 - b^2, 0] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.4 (d tan)^n (a+b sec)^m.m",
       "filename": "4.5.1.4 (d tan)^n (a+b sec)^m.m",
            "rhs": "1/a*Int[Sqrt[e*Cot[c + d*x]], x] - b/a*Int[Sqrt[e*Cot[c + d*x]]/(b + a*Sin[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[e_.*cot[c_. + d_.*x_]]/(a_ + b_.*csc[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.4 (d tan)^n (a+b sec)^m.m",
       "filename": "4.5.1.4 (d tan)^n (a+b sec)^m.m",
            "rhs": "-e^2/b^2* Int[(e*Cot[c + d*x])^(m - 2)*(a - b*Csc[c + d*x]), x] + e^2*(a^2 - b^2)/b^2* Int[(e*Cot[c + d*x])^(m - 2)/(a + b*Csc[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cot[c_. + d_.*x_])^m_/(a_ + b_.*csc[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m - 1/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.4 (d tan)^n (a+b sec)^m.m",
       "filename": "4.5.1.4 (d tan)^n (a+b sec)^m.m",
            "rhs": "1/a*Int[1/Sqrt[e*Cot[c + d*x]], x] - b/a*Int[1/(Sqrt[e*Cot[c + d*x]]*(b + a*Sin[c + d*x])), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[e_.*cot[c_. + d_.*x_]]*(a_ + b_.*csc[c_. + d_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.4 (d tan)^n (a+b sec)^m.m",
       "filename": "4.5.1.4 (d tan)^n (a+b sec)^m.m",
            "rhs": "1/(a^2 - b^2)*Int[(e*Cot[c + d*x])^m*(a - b*Csc[c + d*x]), x] + b^2/(e^2*(a^2 - b^2))* Int[(e*Cot[c + d*x])^(m + 2)/(a + b*Csc[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cot[c_. + d_.*x_])^m_/(a_ + b_.*csc[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2, 0] && ILtQ[m + 1/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.4 (d tan)^n (a+b sec)^m.m",
       "filename": "4.5.1.4 (d tan)^n (a+b sec)^m.m",
            "rhs": "Int[(-1 + Csc[c + d*x]^2)*(a + b*Csc[c + d*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[cot[c_. + d_.*x_]^2*(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.4 (d tan)^n (a+b sec)^m.m",
       "filename": "4.5.1.4 (d tan)^n (a+b sec)^m.m",
            "rhs": "Int[ExpandIntegrand[(a + b*Csc[c + d*x])^ n, (-1 + Csc[c + d*x]^2)^(m/2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[cot[c_. + d_.*x_]^m_*(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m/2, 0] && IntegerQ[n - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.4 (d tan)^n (a+b sec)^m.m",
       "filename": "4.5.1.4 (d tan)^n (a+b sec)^m.m",
            "rhs": "Int[ExpandIntegrand[(a + b*Csc[c + d*x])^ n, (-1 + Sec[c + d*x]^2)^(-m/2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[cot[c_. + d_.*x_]^m_*(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 - b^2, 0] && ILtQ[m/2, 0] && IntegerQ[n - 1/2] && EqQ[m, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.4 (d tan)^n (a+b sec)^m.m",
       "filename": "4.5.1.4 (d tan)^n (a+b sec)^m.m",
            "rhs": "Int[ExpandIntegrand[(e*Cot[c + d*x])^m, (a + b*Csc[c + d*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cot[c_. + d_.*x_])^m_*(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && NeQ[a^2 - b^2, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.4 (d tan)^n (a+b sec)^m.m",
       "filename": "4.5.1.4 (d tan)^n (a+b sec)^m.m",
            "rhs": "Int[Cos[c + d*x]^m*(b + a*Sin[c + d*x])^n/Sin[c + d*x]^(m + n), x]",
     "rulenumber": 0,
            "lhs": "Int[cot[c_. + d_.*x_]^m_.*(a_ + b_.*csc[c_. + d_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[n] && IntegerQ[m] && (IntegerQ[m/2] || LeQ[m, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.4 (d tan)^n (a+b sec)^m.m",
       "filename": "4.5.1.4 (d tan)^n (a+b sec)^m.m",
            "rhs": "Unintegrable[(e*Cot[c + d*x])^m*(a + b*Csc[c + d*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cot[c_. + d_.*x_])^m_.*(a_. + b_.*csc[c_. + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.4 (d tan)^n (a+b sec)^m.m",
       "filename": "4.5.1.4 (d tan)^n (a+b sec)^m.m",
            "rhs": "(e*Cot[c + d*x])^m*Tan[c + d*x]^m* Int[(a + b*Sec[c + d*x])^n/Tan[c + d*x]^m, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cot[c_. + d_.*x_])^m_*(a_ + b_.*sec[c_. + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.4 (d tan)^n (a+b sec)^m.m",
       "filename": "4.5.1.4 (d tan)^n (a+b sec)^m.m",
            "rhs": "(e*Tan[c + d*x]^p)^m/(e*Tan[c + d*x])^(m*p)* Int[(e*Tan[c + d*x])^(m*p)*(a + b*Sec[c + d*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*tan[c_. + d_.*x_]^p_)^m_*(a_ + b_.*sec[c_. + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && Not[IntegerQ[m]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.1.4 (d tan)^n (a+b sec)^m.m",
       "filename": "4.5.1.4 (d tan)^n (a+b sec)^m.m",
            "rhs": "(e*Cot[c + d*x]^p)^m/(e*Cot[c + d*x])^(m*p)* Int[(e*Cot[c + d*x])^(m*p)*(a + b*Csc[c + d*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cot[c_. + d_.*x_]^p_)^m_*(a_ + b_.*csc[c_. + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && Not[IntegerQ[m]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.10 (c+d x)^m (a+b sec)^n.m",
       "filename": "4.5.10 (c+d x)^m (a+b sec)^n.m",
            "rhs": "-2*(c + d*x)^m* ArcTanh[E^(-I*k*Pi)*E^(-I*e + f*fz*x)]/(f*fz*I) - d*m/(f*fz*I)* Int[(c + d*x)^(m - 1)*Log[1 - E^(-I*k*Pi)*E^(-I*e + f*fz*x)], x] + d*m/(f*fz*I)* Int[(c + d*x)^(m - 1)*Log[1 + E^(-I*k*Pi)*E^(-I*e + f*fz*x)], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*csc[e_. + k_.*Pi + f_.*Complex[0, fz_]*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.10 (c+d x)^m (a+b sec)^n.m",
       "filename": "4.5.10 (c+d x)^m (a+b sec)^n.m",
            "rhs": "-2*(c + d*x)^m* ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f - d*m/f* Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x] + d*m/f* Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x))], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*csc[e_. + k_.*Pi + f_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.10 (c+d x)^m (a+b sec)^n.m",
       "filename": "4.5.10 (c+d x)^m (a+b sec)^n.m",
            "rhs": "-2*(c + d*x)^m*ArcTanh[E^(-I*e + f*fz*x)]/(f*fz*I) - d*m/(f*fz*I)* Int[(c + d*x)^(m - 1)*Log[1 - E^(-I*e + f*fz*x)], x] + d*m/(f*fz*I)* Int[(c + d*x)^(m - 1)*Log[1 + E^(-I*e + f*fz*x)], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*csc[e_. + f_.*Complex[0, fz_]*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.10 (c+d x)^m (a+b sec)^n.m",
       "filename": "4.5.10 (c+d x)^m (a+b sec)^n.m",
            "rhs": "-2*(c + d*x)^m*ArcTanh[E^(I*(e + f*x))]/f - d*m/f*Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x] + d*m/f*Int[(c + d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*csc[e_. + f_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.10 (c+d x)^m (a+b sec)^n.m",
       "filename": "4.5.10 (c+d x)^m (a+b sec)^n.m",
            "rhs": "-(c + d*x)^m*Cot[e + f*x]/f + d*m/f*Int[(c + d*x)^(m - 1)*Cot[e + f*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*csc[e_. + f_.*x_]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e, f}, x] && GtQ[m, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.10 (c+d x)^m (a+b sec)^n.m",
       "filename": "4.5.10 (c+d x)^m (a+b sec)^n.m",
            "rhs": "-b^2*(c + d*x)* Cot[e + f*x]*(b*Csc[e + f*x])^(n - 2)/(f*(n - 1)) - b^2*d*(b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2)) + b^2*(n - 2)/(n - 1)*Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)*(b_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.10 (c+d x)^m (a+b sec)^n.m",
       "filename": "4.5.10 (c+d x)^m (a+b sec)^n.m",
            "rhs": "-b^2*(c + d*x)^m* Cot[e + f*x]*(b*Csc[e + f*x])^(n - 2)/(f*(n - 1)) - b^2*d* m*(c + d*x)^(m - 1)*(b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2)) + b^2*(n - 2)/(n - 1)* Int[(c + d*x)^m*(b*Csc[e + f*x])^(n - 2), x] + b^2*d^2*m*(m - 1)/(f^2*(n - 1)*(n - 2))* Int[(c + d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_*(b_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.10 (c+d x)^m (a+b sec)^n.m",
       "filename": "4.5.10 (c+d x)^m (a+b sec)^n.m",
            "rhs": "d*(b*Csc[e + f*x])^n/(f^2*n^2) + (c + d*x)*Cos[e + f*x]*(b*Csc[e + f*x])^(n + 1)/(b*f*n) + (n + 1)/(b^2*n)*Int[(c + d*x)*(b*Csc[e + f*x])^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)*(b_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f}, x] && LtQ[n, -1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.10 (c+d x)^m (a+b sec)^n.m",
       "filename": "4.5.10 (c+d x)^m (a+b sec)^n.m",
            "rhs": "d*m*(c + d*x)^(m - 1)*(b*Csc[e + f*x])^n/(f^2*n^2) + (c + d*x)^m*Cos[e + f*x]*(b*Csc[e + f*x])^(n + 1)/(b*f*n) + (n + 1)/(b^2*n)*Int[(c + d*x)^m*(b*Csc[e + f*x])^(n + 2), x] - d^2*m*(m - 1)/(f^2*n^2)* Int[(c + d*x)^(m - 2)*(b*Csc[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_*(b_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f}, x] && LtQ[n, -1] && GtQ[m, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.10 (c+d x)^m (a+b sec)^n.m",
       "filename": "4.5.10 (c+d x)^m (a+b sec)^n.m",
            "rhs": "(b*Sin[e + f*x])^n*(b*Csc[e + f*x])^n* Int[(c + d*x)^m/(b*Sin[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(b_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f, m, n}, x] && Not[IntegerQ[n]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.10 (c+d x)^m (a+b sec)^n.m",
       "filename": "4.5.10 (c+d x)^m (a+b sec)^n.m",
            "rhs": "Int[ExpandIntegrand[(c + d*x)^m, (a + b*Csc[e + f*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(a_ + b_.*csc[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.10 (c+d x)^m (a+b sec)^n.m",
       "filename": "4.5.10 (c+d x)^m (a+b sec)^n.m",
            "rhs": "Int[ExpandIntegrand[(c + d*x)^m, Sin[e + f*x]^(-n)/(b + a*Sin[e + f*x])^(-n), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(a_ + b_.*csc[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] && IGtQ[m, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.10 (c+d x)^m (a+b sec)^n.m",
       "filename": "4.5.10 (c+d x)^m (a+b sec)^n.m",
            "rhs": "If[MatchQ[f, f1_.*Complex[0, j_]], If[MatchQ[e, e1_. + Pi/2], Unintegrable[(c + d*x)^m*Sech[I*(e - Pi/2) + I*f*x]^n, x], (-I)^n*Unintegrable[(c + d*x)^m*Csch[-I*e - I*f*x]^n, x]], If[MatchQ[e, e1_. + Pi/2], Unintegrable[(c + d*x)^m*Sec[e - Pi/2 + f*x]^n, x], Unintegrable[(c + d*x)^m*Csc[e + f*x]^n, x]]]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*csc[e_. + f_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e, f, m, n}, x] && IntegerQ[n]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.10 (c+d x)^m (a+b sec)^n.m",
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            "rhs": "Unintegrable[(c + d*x)^m*(a + b*Csc[e + f*x])^n, x]",
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            "lhs": "Int[(c_. + d_.*x_)^m_.*(a_. + b_.*csc[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.10 (c+d x)^m (a+b sec)^n.m",
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            "rhs": "Int[ExpandToSum[u, x]^m*(a + b*Sec[ExpandToSum[v, x]])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*(a_. + b_.*Sec[v_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && LinearQ[{u, v}, x] && Not[LinearMatchQ[{u, v}, x]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.10 (c+d x)^m (a+b sec)^n.m",
       "filename": "4.5.10 (c+d x)^m (a+b sec)^n.m",
            "rhs": "Int[ExpandToSum[u, x]^m*(a + b*Csc[ExpandToSum[v, x]])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*(a_. + b_.*Csc[v_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && LinearQ[{u, v}, x] && Not[LinearMatchQ[{u, v}, x]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
       "filename": "4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
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            "lhs": "Int[(a_. + b_.*Sec[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, p}, x] && IGtQ[1/n, 0] && IntegerQ[p]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
       "filename": "4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
            "rhs": "1/n*Subst[Int[x^(1/n - 1)*(a + b*Csc[c + d*x])^p, x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Csc[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, p}, x] && IGtQ[1/n, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
       "filename": "4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
            "rhs": "Unintegrable[(a + b*Sec[c + d*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Sec[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
       "filename": "4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
            "rhs": "Unintegrable[(a + b*Csc[c + d*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Csc[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
       "filename": "4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
            "rhs": "1/Coefficient[u, x, 1]* Subst[Int[(a + b*Sec[c + d*x^n])^p, x], x, u]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Sec[c_. + d_.*u_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x] && NeQ[u, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
       "filename": "4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
            "rhs": "1/Coefficient[u, x, 1]* Subst[Int[(a + b*Csc[c + d*x^n])^p, x], x, u]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Csc[c_. + d_.*u_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x] && LinearQ[u, x] && NeQ[u, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
       "filename": "4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
            "rhs": "Int[(a + b*Sec[ExpandToSum[u, x]])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*Sec[u_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
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            "rhs": "Int[(a + b*Csc[ExpandToSum[u, x]])^p, x]",
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            "lhs": "Int[(a_. + b_.*Csc[u_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
       "filename": "4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
            "rhs": "1/n*Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sec[c + d*x])^p, x], x, x^n]",
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            "lhs": "Int[x_^m_.*(a_. + b_.*Sec[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 1)/n], 0] && IntegerQ[p]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
       "filename": "4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
            "rhs": "1/n*Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Csc[c + d*x])^p, x], x, x^n]",
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            "lhs": "Int[x_^m_.*(a_. + b_.*Csc[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 1)/n], 0] && IntegerQ[p]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
       "filename": "4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
            "rhs": "Unintegrable[x^m*(a + b*Sec[c + d*x^n])^p, x]",
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            "lhs": "Int[x_^m_.*(a_. + b_.*Sec[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p}, x]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
       "filename": "4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
            "rhs": "Unintegrable[x^m*(a + b*Csc[c + d*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*Csc[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p}, x]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
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            "rhs": "e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Sec[c + d*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_*x_)^m_.*(a_. + b_.*Sec[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x]"
  },
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
       "filename": "4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
            "rhs": "e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Csc[c + d*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_*x_)^m_.*(a_. + b_.*Csc[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
       "filename": "4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
            "rhs": "Int[(e*x)^m*(a + b*Sec[ExpandToSum[u, x]])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_*x_)^m_.*(a_. + b_.*Sec[u_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
       "filename": "4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
            "rhs": "Int[(e*x)^m*(a + b*Csc[ExpandToSum[u, x]])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_*x_)^m_.*(a_. + b_.*Csc[u_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
       "filename": "4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
            "rhs": "x^(m - n + 1)*Sec[a + b*x^n]^(p - 1)/(b*n*(p - 1)) - (m - n + 1)/(b*n*(p - 1))* Int[x^(m - n)*Sec[a + b*x^n]^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Sec[a_. + b_.*x_^n_.]^p_*Sin[a_. + b_.*x_^n_.], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m - n, 0] && NeQ[p, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
       "filename": "4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m",
            "rhs": "-x^(m - n + 1)*Csc[a + b*x^n]^(p - 1)/(b*n*(p - 1)) + (m - n + 1)/(b*n*(p - 1))* Int[x^(m - n)*Csc[a + b*x^n]^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Csc[a_. + b_.*x_^n_.]^p_*Cos[a_. + b_.*x_^n_.], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m - n, 0] && NeQ[p, 1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "c^n*Int[ExpandTrig[(1 + d/c*csc[e + f*x])^n, (a + b*csc[e + f*x])^m, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0] && ILtQ[n, 0] && LtQ[m + n, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "(-a*c)^m* Int[Cot[e + f*x]^(2*m)*(c + d*Csc[e + f*x])^(n - m), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && RationalQ[n] && Not[IntegerQ[n] && GtQ[m - n, 0]]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "(-a*c)^(m + 1/2)* Cot[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]])* Int[Cot[e + f*x]^(2*m), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(c_ + d_.*csc[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m + 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "2*a*c*Cot[ e + f*x]*(c + d*Csc[e + f*x])^(n - 1)/(f*(2*n - 1)* Sqrt[a + b*Csc[e + f*x]]) + c*Int[Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && GtQ[n, 1/2]"
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       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "-2*a* Cot[e + f*x]*(c + d*Csc[e + f*x])^ n/(f*(2*n + 1)*Sqrt[a + b*Csc[e + f*x]]) + 1/c*Int[Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])^(n + 1), x]",
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            "lhs": "Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[n, -1/2]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
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            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^(3/2)*(c_ + d_.*csc[e_. + f_.*x_])^ n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[n, -1/2]"
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  {
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            "rhs": "-2*a^2* Cot[e + f*x]*(c + d*Csc[e + f*x])^ n/(f*(2*n + 1)*Sqrt[a + b*Csc[e + f*x]]) + a*Int[Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])^n, x]",
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            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^(3/2)*(c_ + d_.*csc[e_. + f_.*x_])^ n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && Not[LeQ[n, -1/2]]"
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            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^(5/2)*(c_ + d_.*csc[e_. + f_.*x_])^ n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[n, -1/2]"
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            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m - 1/2] && EqQ[m + n, 0]"
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            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]"
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            "rhs": "a*c*x + b*d*Int[Csc[e + f*x]^2, x]",
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            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])*(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0]"
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            "rhs": "a*c*x + (b*c + a*d)*Int[Csc[e + f*x], x] + b*d*Int[Csc[e + f*x]^2, x]",
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            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])*(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[b*c + a*d, 0]"
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            "rhs": "c*Int[Sqrt[a + b*Csc[e + f*x]], x] + d*Int[Sqrt[a + b*Csc[e + f*x]]*Csc[e + f*x], x]",
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            "lhs": "Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol]",
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         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]"
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            "rhs": "a*c*Int[1/Sqrt[a + b*Csc[e + f*x]], x] + Int[Csc[e + f*x]*(b*c + a*d + b*d*Csc[e + f*x])/ Sqrt[a + b*Csc[e + f*x]], x]",
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         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]"
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            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && GtQ[m, 1] && EqQ[a^2 - b^2, 0] && IntegerQ[2*m]"
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            "rhs": "-b*d* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)/(f*m) + 1/m*Int[(a + b*Csc[e + f*x])^(m - 2)* Simp[a^2*c*m + (b^2*d*(m - 1) + 2*a*b*c*m + a^2*d*m)* Csc[e + f*x] + b*(b*c*m + a*d*(2*m - 1))*Csc[e + f*x]^2, x], x]",
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            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && GtQ[m, 1] && NeQ[a^2 - b^2, 0] && IntegerQ[2*m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "c*x/a - (b*c - a*d)/a*Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_ + d_.*csc[e_. + f_.*x_])/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "c/a*Int[Sqrt[a + b*Csc[e + f*x]], x] - (b*c - a*d)/a* Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_ + d_.*csc[e_. + f_.*x_])/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "c*Int[1/Sqrt[a + b*Csc[e + f*x]], x] + d*Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_ + d_.*csc[e_. + f_.*x_])/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "-(b*c - a*d)* Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(b*f*(2*m + 1)) + 1/(a^2*(2*m + 1))* Int[(a + b*Csc[e + f*x])^(m + 1)* Simp[a*c*(2*m + 1) - (b*c - a*d)*(m + 1)*Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && EqQ[a^2 - b^2, 0] && IntegerQ[2*m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "b*(b*c - a*d)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(a* f*(m + 1)*(a^2 - b^2)) + 1/(a*(m + 1)*(a^2 - b^2))*Int[(a + b*Csc[e + f*x])^(m + 1)* Simp[c*(a^2 - b^2)*(m + 1) - (a*(b*c - a*d)*(m + 1))* Csc[e + f*x] + b*(b*c - a*d)*(m + 2)*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && NeQ[a^2 - b^2, 0] && IntegerQ[2*m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "c*Int[(a + b*Csc[e + f*x])^m, x] + d*Int[(a + b*Csc[e + f*x])^m*Csc[e + f*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && Not[IntegerQ[2*m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "1/c*Int[Sqrt[a + b*Csc[e + f*x]], x] - d/c*Int[Csc[e + f*x]*Sqrt[a + b*Csc[e + f*x]]/(c + d*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]/(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "a/c*Int[1/Sqrt[a + b*Csc[e + f*x]], x] + (b*c - a*d)/c* Int[Csc[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]/(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "a/c*Int[Sqrt[a + b*Csc[e + f*x]], x] + (b*c - a*d)/c* Int[Csc[e + f*x]*Sqrt[a + b*Csc[e + f*x]]/(c + d*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^(3/2)/(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "b/d*Int[Sqrt[a+b*Csc[e+f*x]],x] -  (b*c-a*d)/d*Int[Sqrt[a+b*Csc[e+f*x]]/(c+d*Csc[e+f*x]),x]",
     "rulenumber": 0,
            "lhs": "Int[(a_+b_.*csc[e_.+f_.*x_])^(3/2)/(c_+d_.*csc[e_.+f_.*x_]),x_ Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e,f},x] && NeQ[b*c-a*d,0] && NeQ[a^2-b^2,0] &&  NeQ[c^2-d^2,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "1/(c*d)*Int[(a^2*d + b^2*c*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] - (b*c - a*d)^2/(c*d)* Int[Csc[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^(3/2)/(c_ + d_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "1/(c*(b*c - a*d))* Int[(b*c - a*d - b*d*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] + d^2/(c*(b*c - a*d))* Int[Csc[e + f*x]*Sqrt[a + b*Csc[e + f*x]]/(c + d*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*(c_ + d_.*csc[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "1/c*Int[1/Sqrt[a + b*Csc[e + f*x]], x] - d/c*Int[Csc[ e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*(c_ + d_.*csc[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]]/Cot[e + f*x]* Int[Cot[e + f*x], x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*Sqrt[c_ + d_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "c*Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[c + d*Csc[e + f*x]], x] + d*Int[ Csc[e + f*x]*Sqrt[a + b*Csc[e + f*x]]/Sqrt[c + d*Csc[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*Sqrt[c_ + d_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "1/c*Int[Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]], x] - d/c*Int[ Csc[e + f*x]*Sqrt[a + b*Csc[e + f*x]]/Sqrt[c + d*Csc[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]/Sqrt[c_ + d_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "-2*a/f* Subst[Int[1/(1 + a*c*x^2), x], x, Cot[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]* Sqrt[c + d*Csc[e + f*x]])]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]/Sqrt[c_ + d_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "a/c*Int[Sqrt[c + d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x] + (b*c - a*d)/c* Int[Csc[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]* Sqrt[c + d*Csc[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]/Sqrt[c_ + d_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "2*(a + b*Csc[e + f*x])/(c*f*Rt[(a + b)/(c + d), 2]*Cot[e + f*x])* Sqrt[(b*c - a*d)*(1 + Csc[e + f*x])/((c - d)*(a + b*Csc[e + f*x]))]* Sqrt[-(b*c - a*d)*(1 - Csc[e + f*x])/((c + d)*(a + b*Csc[e + f*x]))]* EllipticPi[a*(c + d)/(c*(a + b)), ArcSin[Rt[(a + b)/(c + d), 2]* Sqrt[c + d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]]], (a - b)*(c + d)/((a + b)*(c - d))]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]/Sqrt[c_ + d_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "Cot[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]])* Int[1/Cot[e + f*x], x]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_ + b_.*csc[e_. + f_.*x_]]* Sqrt[c_ + d_.*csc[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "1/a*Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[c + d*Csc[e + f*x]], x] - b/a*Int[ Csc[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_ + b_.*csc[e_. + f_.*x_]]* Sqrt[c_ + d_.*csc[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "1/c*Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[c + d*Csc[e + f*x]], x] - d/c*Int[ Csc[e + f*x]*Sqrt[a + b*Csc[e + f*x]]/(c + d*Csc[e + f*x])^(3/2), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]/(c_ + d_.*csc[e_. + f_.*x_])^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "a^2*Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]* Sqrt[a - b*Csc[e + f*x]])* Subst[ Int[(a + b*x)^(m - 1/2)*(c + d*x)^n/(x*Sqrt[a - b*x]), x], x, Csc[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && IntegerQ[m - 1/2]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "Int[(b + a*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n/ Sin[e + f*x]^(m + n), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IntegerQ[n] && LeQ[-2, m + n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "Sqrt[d + c*Sin[e + f*x]]* Sqrt[a + b*Csc[e + f*x]]/(Sqrt[b + a*Sin[e + f*x]]* Sqrt[c + d*Csc[e + f*x]])* Int[(b + a*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n/ Sin[e + f*x]^(m + n), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && IntegerQ[m + 1/2] && IntegerQ[n + 1/2] && LeQ[-2, m + n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "Sin[e + f*x]^(m + n)*(a + b*Csc[e + f*x])^ m*(c + d*Csc[e + f*x])^ n/((b + a*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n)* Int[(b + a*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n/ Sin[e + f*x]^Simplify[m + n], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n, 0] && Not[IntegerQ[2*m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "Int[ExpandTrig[(a + b*csc[e + f*x])^m, (c + d*csc[e + f*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "Unintegrable[(a + b*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "d^m*Int[(b + a*Cos[e + f*x])^m*(d*Cos[e + f*x])^(n - m), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sec[e_. + f_.*x_])^m_.*(d_./sec[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, n}, x] && Not[IntegerQ[n]] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "d^m*Int[(b + a*Sin[e + f*x])^m*(d*Sin[e + f*x])^(n - m), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*csc[e_. + f_.*x_])^m_.*(d_./csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, n}, x] && Not[IntegerQ[n]] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "c^IntPart[n]*(c*(d*Sec[e + f*x])^p)^ FracPart[n]/(d*Sec[e + f*x])^(p*FracPart[n])* Int[(a + b*Sec[e + f*x])^m*(d*Sec[e + f*x])^(n*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sec[e_. + f_.*x_])^ m_.*(c_.*(d_.*sec[e_. + f_.*x_])^p_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.1 (a+b sec)^m (c+d sec)^n.m",
            "rhs": "c^IntPart[n]*(c*(d*Csc[e + f*x])^p)^ FracPart[n]/(d*Csc[e + f*x])^(p*FracPart[n])* Int[(a + b*Cos[e + f*x])^m*(d*Cos[e + f*x])^(n*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*csc[e_. + f_.*x_])^ m_.*(c_.*(d_.*csc[e_. + f_.*x_])^p_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
            "rhs": "b*Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(c + d*Csc[e + f*x])^n/(a*f*(2*m + 1))",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && NeQ[2*m + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
            "rhs": "b*Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(c + d*Csc[e + f*x])^n/(a*f*(2*m + 1)) + (m + n + 1)/(a*(2*m + 1))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(c + d*Csc[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && ILtQ[m + n + 1, 0] && NeQ[2*m + 1, 0] && Not[LtQ[n, 0]] && Not[IGtQ[n + 1/2, 0] && LtQ[n + 1/2, -(m + n)]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
            "rhs": "a*c*Log[1 + b/a*Csc[e + f*x]]* Cot[e + f*x]/(b*f*Sqrt[a + b*Csc[e + f*x]]* Sqrt[c + d*Csc[e + f*x]])",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]* Sqrt[c_ + d_.*csc[e_. + f_.*x_]]/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
            "rhs": "2*a*c*Cot[ e + f*x]*(a + b*Csc[e + f*x])^ m/(b*f*(2*m + 1)*Sqrt[c + d*Csc[e + f*x]])",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^m_.* Sqrt[c_ + d_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[m, -1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
            "rhs": "2*a*c*Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(c + d*Csc[e + f*x])^(n - 1)/(b*f*(2*m + 1)) - d*(2*n - 1)/(b*(2*m + 1))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(c + d*Csc[e + f*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[n - 1/2, 0] && LtQ[m, -1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
            "rhs": "-d* Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(c + d*Csc[e + f*x])^(n - 1)/(f*(m + n)) + c*(2*n - 1)/(m + n)* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^ m*(c + d*Csc[e + f*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[n - 1/2, 0] && Not[LtQ[m, -1/2]] && Not[IGtQ[m - 1/2, 0] && LtQ[m, n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
            "rhs": "-2*d* Cot[e + f*x]*(c + d*Csc[e + f*x])^(n - 1)/(f*(2*n - 1)* Sqrt[a + b*Csc[e + f*x]]) + 2*c*(2*n - 1)/(2*n - 1)* Int[Csc[e + f*x]*(c + d*Csc[e + f*x])^(n - 1)/ Sqrt[a + b*Csc[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(c_ + d_.*csc[e_. + f_.*x_])^n_./ Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
            "rhs": "2*a*c*Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(c + d*Csc[e + f*x])^(n - 1)/(b*f*(2*m + 1)) - d*(2*n - 1)/(b*(2*m + 1))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(c + d*Csc[e + f*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[n, 0] && LtQ[m, -1/2] && IntegerQ[2*m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
            "rhs": "(-a*c)^m* Int[ExpandTrig[ csc[e + f*x]*cot[e + f*x]^(2*m), (c + d*csc[e + f*x])^(n - m), x], x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegersQ[m, n] && GeQ[n - m, 0] && GtQ[m*n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
            "rhs": "(-a*c)^(m + 1/2)* Cot[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]])* Int[Csc[e + f*x]*Cot[e + f*x]^(2*m), x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m + 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
            "rhs": "b*Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(c + d*Csc[e + f*x])^n/(a*f*(2*m + 1)) + (m + n + 1)/(a*(2*m + 1))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(c + d*Csc[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && (ILtQ[m, 0] && ILtQ[n - 1/2, 0] || ILtQ[m - 1/2, 0] && ILtQ[n - 1/2, 0] && LtQ[m, n])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
            "rhs": "a*c*Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]* Sqrt[c + d*Csc[e + f*x]])* Subst[Int[(a + b*x)^(m - 1/2)*(c + d*x)^(n - 1/2), x], x, Csc[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
            "rhs": "(-a*c)^m* Int[ExpandTrig[(g*csc[e + f*x])^p* cot[e + f*x]^(2*m), (c + d*csc[e + f*x])^(n - m), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*csc[e_. + f_.*x_])^p_.*(a_ + b_.*csc[e_. + f_.*x_])^ m_.*(c_ + d_.*csc[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegersQ[m, n] && GeQ[n - m, 0] && GtQ[m*n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
            "rhs": "(-a*c)^(m + 1/2)* Cot[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]])* Int[(g*Csc[e + f*x])^p*Cot[e + f*x]^(2*m), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*csc[e_. + f_.*x_])^p_.*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m + 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
            "rhs": "a*c*g*Cot[ e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]])* Subst[ Int[(g*x)^(p - 1)*(a + b*x)^(m - 1/2)*(c + d*x)^(n - 1/2), x], x, Csc[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*csc[e_. + f_.*x_])^p_.*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]"
  },
  {
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]"
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       "filename": "4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]"
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            "rhs": "-Sqrt[a + b*Csc[e + f*x]]* Sqrt[c/(c + d*Csc[e + f*x])]/(d*f* Sqrt[c*d*(a + b*Csc[e + f*x])/((b*c + a*d)*(c + d*Csc[e + f*x]))])* EllipticE[ ArcSin[c* Cot[e + f*x]/(c + d*Csc[e + f*x])], -(b*c - a*d)/(b*c + a*d)]",
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         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]"
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
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            "rhs": "g/d*Int[Sqrt[g*Csc[e + f*x]]*Sqrt[a + b*Csc[e + f*x]], x] - c*g/d* Int[Sqrt[g*Csc[e + f*x]]* Sqrt[a + b*Csc[e + f*x]]/(c + d*Csc[e + f*x]), x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]"
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            "rhs": "b/d*Int[(g*Csc[e + f*x])^(3/2)/Sqrt[a + b*Csc[e + f*x]], x] - (b*c - a*d)/d* Int[(g*Csc[e + f*x])^(3/2)/(Sqrt[ a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])), x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]"
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            "rhs": "b/(b*c - a*d)*Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x] - d/(b*c - a*d)* Int[Csc[e + f*x]*Sqrt[a + b*Csc[e + f*x]]/(c + d*Csc[e + f*x]), x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])"
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       "filename": "4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
            "rhs": "-2* Cot[e + f*x]/(f*(c + d)*Sqrt[a + b*Csc[e + f*x]]* Sqrt[-Cot[e + f*x]^2])*Sqrt[(a + b*Csc[e + f*x])/(a + b)]* EllipticPi[2*d/(c + d), ArcSin[Sqrt[1 - Csc[e + f*x]]/Sqrt[2]], 2*b/(a + b)]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
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            "rhs": "-a*g/(b*c - a*d)* Int[Sqrt[g*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x] + c*g/(b*c - a*d)* Int[Sqrt[g*Csc[e + f*x]]* Sqrt[a + b*Csc[e + f*x]]/(c + d*Csc[e + f*x]), x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]"
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            "rhs": "g*Sqrt[g*Csc[e + f*x]]* Sqrt[b + a*Sin[e + f*x]]/Sqrt[a + b*Csc[e + f*x]]* Int[1/(Sqrt[b + a*Sin[e + f*x]]*(d + c*Sin[e + f*x])), x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]"
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            "rhs": "-a/(b*c - a*d)* Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x] + c/(b*c - a*d)* Int[Csc[e + f*x]*Sqrt[a + b*Csc[e + f*x]]/(c + d*Csc[e + f*x]), x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])"
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            "rhs": "1/d*Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x] - c/d*Int[ Csc[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])), x]",
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         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]"
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            "rhs": "g/d*Int[(g*Csc[e + f*x])^(3/2)/Sqrt[a + b*Csc[e + f*x]], x] - c*g/d* Int[(g*Csc[e + f*x])^(3/2)/(Sqrt[ a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])), x]",
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         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
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         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]"
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         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
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            "rhs": "-2*a/(b*f)* Subst[Int[1/(2 + (a*c - b*d)*x^2), x], x, Cot[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]* Sqrt[c + d*Csc[e + f*x]])]",
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         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
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         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
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            "rhs": "(a - b)/(c - d)* Int[Csc[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]* Sqrt[c + d*Csc[e + f*x]]), x] + (b*c - a*d)/(c - d)* Int[Csc[e + f*x]*(1 + Csc[e + f*x])/(Sqrt[ a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])^(3/2)), x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]* Sqrt[a_ + b_.*csc[e_. + f_.*x_]]/(c_ + d_.*csc[e_. + f_.*x_])^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
            "rhs": "a^2*g*Cot[ e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x]])* Subst[ Int[(g*x)^(p - 1)*(a + b*x)^(m - 1/2)*(c + d*x)^n/Sqrt[a - b*x], x], x, Csc[e + f*x]]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*csc[e_. + f_.*x_])^p_.*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && (EqQ[p, 1] || IntegerQ[m - 1/2])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
            "rhs": "1/g^(m + n)* Int[(g*Csc[e + f*x])^(m + n + p)*(b + a*Sin[e + f*x])^ m*(d + c*Sin[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*csc[e_. + f_.*x_])^p_.*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
            "rhs": "(g*Csc[e + f*x])^(m + p)*(c + d*Csc[e + f*x])^ n/(g^m*(d + c*Sin[e + f*x])^n)* Int[(b + a*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*csc[e_. + f_.*x_])^p_.*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + p, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
            "rhs": "(g*Csc[e + f*x])^p*(a + b*Csc[e + f*x])^ m*(c + d*Csc[e + f*x])^ n/((b + a*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n)* Int[(b + a*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*csc[e_. + f_.*x_])^p_.*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + p, 0] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
            "rhs": "Sqrt[d + c*Sin[e + f*x]]* Sqrt[a + b*Csc[e + f*x]]/(Sqrt[b + a*Sin[e + f*x]]* Sqrt[c + d*Csc[e + f*x]])* Int[(b + a*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n/ Sin[e + f*x]^(m + n + p), x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]^p_.*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && IntegerQ[m - 1/2] && IntegerQ[n - 1/2] && IntegerQ[p] && LeQ[-2, m + n + p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
            "rhs": "Int[ExpandTrig[(g*csc[e + f*x])^p*(a + b*csc[e + f*x])^ m*(c + d*csc[e + f*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*csc[e_. + f_.*x_])^p_.*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(c_ + d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[b*c - a*d, 0] && (IntegersQ[m, n] || IntegersQ[m, p] || IntegersQ[n, p])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
            "rhs": "Unintegrable[(g*Csc[e + f*x])^p*(a + b*Csc[e + f*x])^ m*(c + d*Csc[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*csc[e_. + f_.*x_])^p_.*(a_. + b_.*csc[e_. + f_.*x_])^ m_*(c_. + d_.*csc[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
            "rhs": "2*A*(1 + Sec[e + f*x])* Sqrt[(b*c - a*d)*(1 - Sec[e + f*x])/((a + b)*(c + d*Sec[e + f*x]))]/ (f*(b*c - a*d)*Rt[(c + d)/(a + b), 2]*Tan[e + f*x]* Sqrt[-(b*c - a*d)*(1 + Sec[e + f*x])/((a - b)*(c + d*Sec[e + f*x]))])* EllipticE[ ArcSin[Rt[(c + d)/(a + b), 2]* Sqrt[a + b*Sec[e + f*x]]/Sqrt[c + d*Sec[e + f*x]]], (a + b)*(c - d)/((a - b)*(c + d))]",
     "rulenumber": 0,
            "lhs": "Int[sec[e_. + f_.*x_]*(A_ + B_.*sec[e_. + f_.*x_])/(Sqrt[ a_ + b_.*sec[e_. + f_.*x_]]*(c_ + d_.*sec[e_. + f_.*x_])^(3/ 2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A, B]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
       "filename": "4.5.2.2 (g sec)^p (a+b sec)^m (c+d sec)^n.m",
            "rhs": "-2*A*(1 + Csc[e + f*x])* Sqrt[(b*c - a*d)*(1 - Csc[e + f*x])/((a + b)*(c + d*Csc[e + f*x]))]/ (f*(b*c - a*d)*Rt[(c + d)/(a + b), 2]*Cot[e + f*x]* Sqrt[-(b*c - a*d)*(1 + Csc[e + f*x])/((a - b)*(c + d*Csc[e + f*x]))])* EllipticE[ ArcSin[Rt[(c + d)/(a + b), 2]* Sqrt[a + b*Csc[e + f*x]]/Sqrt[c + d*Csc[e + f*x]]], (a + b)*(c - d)/((a - b)*(c + d))]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(A_ + B_.*csc[e_. + f_.*x_])/(Sqrt[ a_ + b_.*csc[e_. + f_.*x_]]*(c_ + d_.*csc[e_. + f_.*x_])^(3/ 2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A, B]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "A*a*Cot[e + f*x]*(d*Csc[e + f*x])^n/(f*n) + 1/(d*n)* Int[(d*Csc[e + f*x])^(n + 1)* Simp[n*(B*a + A*b) + (B*b*n + A*a*(n + 1))*Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && LeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "-b*B*Cot[e + f*x]*(d*Csc[e + f*x])^n/(f*(n + 1)) + 1/(n + 1)* Int[(d*Csc[e + f*x])^n* Simp[A*a*(n + 1) + B*b*n + (A*b + B*a)*(n + 1)*Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])*(d_.*csc[e_. + f_.*x_])^ n_.*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && Not[LeQ[n, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "B/b*Int[Csc[e + f*x], x] + (A*b - a*B)/b* Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(A_ + B_.*csc[e_. + f_.*x_])/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B}, x] && NeQ[A*b - a*B, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "-B*Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(f*(m + 1))",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B, e, f, m}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && EqQ[a*B*m + A*b*(m + 1), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "(A*b - a*B)* Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(a*f*(2*m + 1)) + (a*B*m + A*b*(m + 1))/(a*b*(2*m + 1))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B, e, f}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && NeQ[a*B*m + A*b*(m + 1), 0] && LtQ[m, -1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "-B*Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(f*(m + 1)) + (a*B*m + A*b*(m + 1))/(b*(m + 1))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B, e, f, m}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && NeQ[a*B*m + A*b*(m + 1), 0] && Not[LtQ[m, -1/2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "-B*Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(f*(m + 1)) + 1/(m + 1)* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)* Simp[b*B*m + a*A*(m + 1) + (a*B*m + A*b*(m + 1))*Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B, e, f}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "-(A*b - a*B)* Cot[e + f* x]*(a + b*Csc[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2)) + 1/((m + 1)*(a^2 - b^2))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)* Simp[(a*A - b*B)*(m + 1) - (A*b - a*B)*(m + 2)*Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B, e, f}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "-2*(A*b - a*B)*Rt[a + b*B/A, 2]* Sqrt[b*(1 - Csc[e + f*x])/(a + b)]* Sqrt[-b*(1 + Csc[e + f*x])/(a - b)]/(b^2*f*Cot[e + f*x])* EllipticE[ ArcSin[Sqrt[a + b*Csc[e + f*x]]/ Rt[a + b*B/A, 2]], (a*A + b*B)/(a*A - b*B)]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(A_ + B_.*csc[e_. + f_.*x_])/ Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "(A - B)* Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x] + B*Int[Csc[e + f*x]*(1 + Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(A_ + B_.*csc[e_. + f_.*x_])/ Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && NeQ[A^2 - B^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "2*Sqrt[2]*A*(a + b*Csc[e + f*x])^m*(A - B*Csc[e + f*x])* Sqrt[(A + B*Csc[e + f*x])/A]/(B*f* Cot[e + f*x]*(A*(a + b*Csc[e + f*x])/(a*A + b*B))^m)* AppellF1[1/2, -(1/2), -m, 3/2, (A - B*Csc[e + f*x])/(2*A), (b*(A - B*Csc[e + f*x]))/(A*b + a*B)]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B, e, f}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0] && Not[IntegerQ[2*m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "(A*b - a*B)/b* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m, x] + B/b*Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B, e, f, m}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "-(A*b - a*B)* Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(b*f*(2*m + 1)) + 1/(b^2*(2*m + 1))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)* Simp[A*b*m - a*B*m + b*B*(2*m + 1)*Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]^2*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "a*(A*b - a*B)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(b* f*(m + 1)*(a^2 - b^2)) - 1/(b*(m + 1)*(a^2 - b^2))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)* Simp[b*(A*b - a*B)*(m + 1) - (a*A*b*(m + 2) - B*(a^2 + b^2*(m + 1)))*Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]^2*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "-B* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 2)) + 1/(b*(m + 2))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m* Simp[b*B*(m + 1) + (A*b*(m + 2) - a*B)*Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]^2*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] && Not[LtQ[m, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n/(f*n)",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, m, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && EqQ[a*A*m - b*B*n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "-(A*b - a*B)* Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^n/(b*f*(2*m + 1)) + (a*A*m + b*B*(m + 1))/(a^2*(2*m + 1))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && LeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n/(f*n) - (a*A*m - b*B*n)/(b*d*n)* Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, m, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && Not[LeQ[m, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "-2*b*B* Cot[e + f*x]*(d*Csc[e + f*x])^ n/(f*(2*n + 1)*Sqrt[a + b*Csc[e + f*x]])",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && EqQ[A*b*(2*n + 1) + 2*a*B*n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "A*b^2*Cot[ e + f*x]*(d*Csc[e + f*x])^n/(a*f*n*Sqrt[a + b*Csc[e + f*x]]) + (A*b*(2*n + 1) + 2*a*B*n)/(2*a*d*n)* Int[Sqrt[a + b*Csc[e + f*x]]*(d*Csc[e + f*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && NeQ[A*b*(2*n + 1) + 2*a*B*n, 0] && LtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "-2*b*B* Cot[e + f*x]*(d*Csc[e + f*x])^ n/(f*(2*n + 1)*Sqrt[a + b*Csc[e + f*x]]) + (A*b*(2*n + 1) + 2*a*B*n)/(b*(2*n + 1))* Int[Sqrt[a + b*Csc[e + f*x]]*(d*Csc[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && NeQ[A*b*(2*n + 1) + 2*a*B*n, 0] && Not[LtQ[n, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "a*A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^ n/(f*n) - b/(a*d*n)* Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)* Simp[a*A*(m - n - 1) - b*B*n - (a*B*n + A*b*(m + n))*Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && GtQ[m, 1/2] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "-b*B* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^ n/(f*(m + n)) + 1/(d*(m + n))* Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n* Simp[a*A*d*(m + n) + B*(b*d*n) + (A*b*d*(m + n) + a*B*d*(2*m + n - 1))* Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && GtQ[m, 1/2] && Not[LtQ[n, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "d*(A*b - a*B)* Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^(n - 1)/(a*f*(2*m + 1)) - 1/(a*b*(2*m + 1))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)* Simp[A*(a*d*(n - 1)) - B*(b*d*(n - 1)) - d*(a*B*(m - n + 1) + A*b*(m + n))*Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -1/2] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "-(A*b - a*B)* Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^n/(b*f*(2*m + 1)) - 1/(a^2*(2*m + 1))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n* Simp[b*B*n - a*A*(2*m + n + 1) + (A*b - a*B)*(m + n + 1)*Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -1/2] && Not[GtQ[n, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "-B*d* Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^(n - 1)/(f*(m + n)) + d/(b*(m + n))* Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 1)* Simp[b*B*(n - 1) + (A*b*(m + n) + a*B*m)*Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && GtQ[n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n/(f*n) - 1/(b*d*n)* Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)* Simp[a*A*m - b*B*n - A*b*(m + n + 1)*Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && LtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "(A*b - a*B)/b* Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n, x] + B/b*Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "a^2*A*Cos[e + f*x]*(d*Csc[e + f*x])^(n + 1)/(d*f*n) + 1/(d*n)* Int[(d*Csc[e + f*x])^(n + 1)*(a*(2*A*b + a*B)* n + (2*a*b*B*n + A*(b^2*n + a^2*(n + 1)))*Csc[e + f*x] + b^2*B*n*Csc[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^2*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "a*A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^ n/(f*n) + 1/(d*n)*Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^(n + 1)* Simp[a*(a*B*n - A*b*(m - n - 1)) + (2*a*b*B*n + A*(b^2*n + a^2*(1 + n)))*Csc[e + f*x] + b*(b*B*n + a*A*(m + n))*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && LeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "-b*B* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^ n/(f*(m + n)) + 1/(m + n)*Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^n* Simp[a^2*A*(m + n) + a*b*B*n + (a*(2*A*b + a*B)*(m + n) + b^2*B*(m + n - 1))* Csc[e + f*x] + b*(A*b*(m + n) + a*B*(2*m + n - 1))*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && Not[IGtQ[n, 1] && Not[IntegerQ[m]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "-d*(A*b - a*B)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)/(f*(m + 1)*(a^2 - b^2)) + 1/((m + 1)*(a^2 - b^2))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)* Simp[d*(n - 1)*(A*b - a*B) + d*(a*A - b*B)*(m + 1)*Csc[e + f*x] - d*(A*b - a*B)*(m + n + 1)*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && LtQ[0, n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "-a^2*(A*b - a*B)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(b^2* f*(m + 1)*(a^2 - b^2)) + 1/(b^2*(m + 1)*(a^2 - b^2))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)* Simp[a*b*(A*b - a*B)*(m + 1) - (A*b - a*B)*(a^2 + b^2*(m + 1))* Csc[e + f*x] + b*B*(m + 1)*(a^2 - b^2)*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]^3*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "a*d^2*(A*b - a*B)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 2)/(b*f*(m + 1)*(a^2 - b^2)) - d/(b*(m + 1)*(a^2 - b^2))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 2)* Simp[a*d*(A*b - a*B)*(n - 2) + b*d*(A*b - a*B)*(m + 1)* Csc[e + f*x] - (a*A*b*d*(m + n) - d*B*(a^2*(n - 1) + b^2*(m + 1)))*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "b*(A*b - a*B)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^ n/(a*f*(m + 1)*(a^2 - b^2)) + 1/(a*(m + 1)*(a^2 - b^2))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n* Simp[A*(a^2*(m + 1) - b^2*(m + n + 1)) + a*b*B*n - a*(A*b - a*B)*(m + 1)*Csc[e + f*x] + b*(A*b - a*B)*(m + n + 2)*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && Not[ILtQ[m + 1/2, 0] && ILtQ[n, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "-B*d* Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^(n - 1)/(f*(m + n)) + d/(m + n)* Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n - 1)* Simp[a*B*(n - 1) + (b*B*(m + n - 1) + a*A*(m + n))* Csc[e + f*x] + (a*B*m + A*b*(m + n))*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[0, m, 1] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n/(f*n) - 1/(d*n)*Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)* Simp[A*b*m - a*B*n - (b*B*n + a*A*(n + 1))*Csc[e + f*x] - A*b*(m + n + 1)*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[0, m, 1] && LeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "-B*d^2* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 2)/(b*f*(m + n)) + d^2/(b*(m + n))* Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 2)* Simp[a*B*(n - 2) + B*b*(m + n - 1)*Csc[e + f*x] + (A*b*(m + n) - a*B*(n - 1))* Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[n, 1] && NeQ[m + n, 0] && Not[IGtQ[m, 1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^ n/(a*f*n) + 1/(a*d*n)*Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)* Simp[a*B*n - A*b*(m + n + 1) + A*a*(n + 1)*Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "A/a*Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x] - (A*b - a*B)/(a*d)* Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*csc[e_. + f_.*x_])/(Sqrt[d_.*csc[e_. + f_.*x_]]* Sqrt[a_ + b_.*csc[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "A*Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x] + B/d*Int[(d*Csc[e + f*x])^(3/2)/Sqrt[a + b*Csc[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[d_.*csc[e_. + f_.*x_]]*(A_ + B_.*csc[e_. + f_.*x_])/ Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "B/d*Int[Sqrt[a + b*Csc[e + f*x]]*Sqrt[d*Csc[e + f*x]], x] + A*Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_ + b_.*csc[e_. + f_.*x_]]*(A_ + B_.*csc[e_. + f_.*x_])/ Sqrt[d_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "A/a*Int[(d*Csc[e + f*x])^n, x] - (A*b - a*B)/(a*d)* Int[(d*Csc[e + f*x])^(n + 1)/(a + b*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*csc[e_. + f_.*x_])^ n_*(A_ + B_.*csc[e_. + f_.*x_])/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "Unintegrable[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^ n*(A + B*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_.*(A_ + B_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, m, n}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": " (-a*c)^m*Int[Cot[e+f*x]^(2*m)*(c+d*Csc[e+f*x])^(n-m)*(A+B*Csc[e+f*x])^ p,x]",
     "rulenumber": 0,
            "lhs": "Int[(a_+b_.*csc[e_.+f_.*x_])^m_.*(c_+d_.*csc[e_.+f_.*x_])^n_.*(A_.+ B_.*csc[e_.+f_.*x_])^p_.,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,A,B,n,p},x] && EqQ[b*c+a*d,0] &&  EqQ[a^2-b^2,0] && IntegerQ[m] && Not[IntegerQ[n] && (LtQ[m,0] && GtQ[n,0] || LtQ[0,n,m] ||  LtQ[m,n,0])] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
       "filename": "4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m",
            "rhs": "(-a*c)^m* Int[Cos[e + f*x]^(2*m)*(d + c*Sin[e + f*x])^(n - m)*(B + A*Sin[e + f*x])^p/Sin[e + f*x]^(m + n + p), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(c_ + d_.*csc[e_. + f_.*x_])^ n_.*(A_. + B_.*csc[e_. + f_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegersQ[m, n, p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "1/b^2*Int[(a + b*Csc[e + f*x])^(m + 1)* Simp[b*B - a*C + b*C*Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^ m_.*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B, C, m}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "C/b^2*Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[-a + b*Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A*b^2 + a^2*C, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "A*Cot[e + f*x]*(b*Csc[e + f*x])^m/(f*m)",
     "rulenumber": 0,
            "lhs": "Int[(b_.*csc[e_. + f_.*x_])^m_.*(A_ + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, e, f, A, C, m}, x] && EqQ[C*m + A*(m + 1), 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "Int[(C + A*Sin[e + f*x]^2)/Sin[e + f*x]^(m + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]^m_.*(A_ + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && ILtQ[(m + 1)/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "A*Cot[e + f*x]*(b*Csc[e + f*x])^m/(f*m) + (C*m + A*(m + 1))/(b^2*m)*Int[(b*Csc[e + f*x])^(m + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*csc[e_. + f_.*x_])^m_.*(A_ + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && LeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "-C*Cot[e + f*x]*(b*Csc[e + f*x])^m/(f*(m + 1)) + (C*m + A*(m + 1))/(m + 1)*Int[(b*Csc[e + f*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*csc[e_. + f_.*x_])^m_.*(A_ + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, e, f, A, C, m}, x] && NeQ[C*m + A*(m + 1), 0] && Not[LeQ[m, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "B/b*Int[(b*Csc[e + f*x])^(m + 1), x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*csc[e_. + f_.*x_])^ m_.*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, e, f, A, B, C, m}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "-b*C*Csc[e + f*x]*Cot[e + f*x]/(2*f) + 1/2*Int[ Simp[2*A*a + (2*B*a + b*(2*A + C))*Csc[e + f*x] + 2*(a*C + B*b)*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B, C}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "-b*C*Csc[e + f*x]*Cot[e + f*x]/(2*f) + 1/2*Int[ Simp[2*A*a + b*(2*A + C)*Csc[e + f*x] + 2*a*C*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, C}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "C/b*Int[Csc[e + f*x], x] + 1/b*Int[(A*b + (b*B - a*C)*Csc[e + f*x])/(a + b*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2)/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B, C}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "C/b*Int[Csc[e + f*x], x] + 1/b*Int[(A*b - a*C*Csc[e + f*x])/(a + b*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + C_.*csc[e_. + f_.*x_]^2)/(a_ + b_.*csc[e_. + f_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, C}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "-(a*A - b*B + a*C)* Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(a*f*(2*m + 1)) + 1/(a*b*(2*m + 1))* Int[(a + b*Csc[e + f*x])^(m + 1)* Simp[A*b*(2*m + 1) + (b*B*(m + 1) - a*(A*(m + 1) - C*m))* Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B, C}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "-a*(A + C)* Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(a*f*(2*m + 1)) + 1/(a*b*(2*m + 1))* Int[(a + b*Csc[e + f*x])^(m + 1)* Simp[A*b*(2*m + 1) - a*(A*(m + 1) - C*m)*Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, C}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "-C*Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(f*(m + 1)) + 1/(b*(m + 1))* Int[(a + b*Csc[e + f*x])^m* Simp[A*b*(m + 1) + (a*C*m + b*B*(m + 1))*Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^ m_.*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B, C, m}, x] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1/2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "-C*Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(f*(m + 1)) + 1/(b*(m + 1))* Int[(a + b*Csc[e + f*x])^m* Simp[A*b*(m + 1) + a*C*m*Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1/2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "-C*Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(f*(m + 1)) + 1/(m + 1)*Int[(a + b*Csc[e + f*x])^(m - 1)* Simp[a*A*(m + 1) + ((A*b + a*B)*(m + 1) + b*C*m)* Csc[e + f*x] + (b*B*(m + 1) + a*C*m)*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^ m_.*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && IGtQ[2*m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "-C*Cot[e + f*x]*(a + b*Csc[e + f*x])^m/(f*(m + 1)) + 1/(m + 1)* Int[(a + b*Csc[e + f*x])^(m - 1)* Simp[a*A*(m + 1) + (A*b*(m + 1) + b*C*m)*Csc[e + f*x] + a*C*m*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, C}, x] && NeQ[a^2 - b^2, 0] && IGtQ[2*m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "Int[(A + (B - C)*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] + C*Int[Csc[e + f*x]*(1 + Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2)/ Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "Int[(A - C*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] + C*Int[Csc[e + f*x]*(1 + Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + C_.*csc[e_. + f_.*x_]^2)/Sqrt[a_ + b_.*csc[e_. + f_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, C}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "(A*b^2 - a*b*B + a^2*C)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(a* f*(m + 1)*(a^2 - b^2)) + 1/(a*(m + 1)*(a^2 - b^2))*Int[(a + b*Csc[e + f*x])^(m + 1)* Simp[A*(a^2 - b^2)*(m + 1) - a*(A*b - a*B + b*C)*(m + 1)* Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + 2)* Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "(A*b^2 + a^2*C)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(a* f*(m + 1)*(a^2 - b^2)) + 1/(a*(m + 1)*(a^2 - b^2))*Int[(a + b*Csc[e + f*x])^(m + 1)* Simp[A*(a^2 - b^2)*(m + 1) - a*b*(A + C)*(m + 1)*Csc[e + f*x] + (A*b^2 + a^2*C)*(m + 2)* Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, C}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[2*m] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "1/b*Int[(a + b*Csc[e + f*x])^m*(A*b + (b*B - a*C)*Csc[e + f*x]), x] + C/b*Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && Not[IntegerQ[2*m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "1/b*Int[(a + b*Csc[e + f*x])^m*(A*b - a*C*Csc[e + f*x]), x] + C/b*Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, C, m}, x] && NeQ[a^2 - b^2, 0] && Not[IntegerQ[2*m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "b^2*Int[(b*Cos[e + f*x])^(m - 2)*(C + B*Cos[e + f*x] + A*Cos[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*cos[e_. + f_.*x_])^ m_*(A_. + B_.*sec[e_. + f_.*x_] + C_.*sec[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, e, f, A, B, C, m}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "b^2*Int[(b*Sin[e + f*x])^(m - 2)*(C + B*Sin[e + f*x] + A*Sin[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*sin[e_. + f_.*x_])^ m_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, e, f, A, B, C, m}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "b^2*Int[(b*Cos[e + f*x])^(m - 2)*(C + A*Cos[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*cos[e_. + f_.*x_])^m_*(A_. + C_.*sec[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, e, f, A, C, m}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "b^2*Int[(b*Sin[e + f*x])^(m - 2)*(C + A*Sin[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*sin[e_. + f_.*x_])^m_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, e, f, A, C, m}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "a^IntPart[m]*(a*(b*Sec[e + f*x])^p)^ FracPart[m]/(b*Sec[e + f*x])^(p*FracPart[m])* Int[(b*Sec[e + f*x])^(m*p)*(A + B*Sec[e + f*x] + C*Sec[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*(b_.*sec[e_. + f_.*x_])^p_)^ m_*(A_. + B_.*sec[e_. + f_.*x_] + C_.*sec[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B, C, m, p}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "a^IntPart[m]*(a*(b*Csc[e + f*x])^p)^ FracPart[m]/(b*Csc[e + f*x])^(p*FracPart[m])* Int[(b*Csc[e + f*x])^(m*p)*(A + B*Csc[e + f*x] + C*Csc[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*(b_.*csc[e_. + f_.*x_])^p_)^ m_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B, C, m, p}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "a^IntPart[m]*(a*(b*Sec[e + f*x])^p)^ FracPart[m]/(b*Sec[e + f*x])^(p*FracPart[m])* Int[(b*Sec[e + f*x])^(m*p)*(A + C*Sec[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*(b_.*sec[e_. + f_.*x_])^p_)^ m_*(A_. + C_.*sec[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, C, m, p}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
       "filename": "4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m",
            "rhs": "a^IntPart[m]*(a*(b*Csc[e + f*x])^p)^ FracPart[m]/(b*Csc[e + f*x])^(p*FracPart[m])* Int[(b*Csc[e + f*x])^(m*p)*(A + C*Csc[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*(b_.*csc[e_. + f_.*x_])^p_)^ m_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, C, m, p}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "1/b^2*Int[(a + b*Csc[e + f*x])^(m + 1)*(c + d*Csc[e + f*x])^ n*(b*B - a*C + b*C*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*csc[e_. + f_.*x_])^m_.*(c_. + d_.*csc[e_. + f_.*x_])^ n_.*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "-C/b^2* Int[(a + b*Csc[e + f*x])^(m + 1)*(c + d*Csc[e + f*x])^ n*(a - b*Csc[e + f*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*csc[e_. + f_.*x_])^m_.*(c_. + d_.*csc[e_. + f_.*x_])^ n_.*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A*b^2 + a^2*C, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "A*a*Cot[e + f*x]*(d*Csc[e + f*x])^n/(f*n) + 1/(d*n)* Int[(d*Csc[e + f*x])^(n + 1)* Simp[n*(B*a + A*b) + (n*(a*C + B*b) + A*a*(n + 1))* Csc[e + f*x] + b*C*n*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, C}, x] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "A*a*Cot[e + f*x]*(d*Csc[e + f*x])^n/(f*n) + 1/(d*n)* Int[(d*Csc[e + f*x])^(n + 1)* Simp[A*b*n + a*(C*n + A*(n + 1))*Csc[e + f*x] + b*C*n*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, C}, x] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "-b*C*Csc[e + f*x]* Cot[e + f*x]*(d*Csc[e + f*x])^n/(f*(n + 2)) + 1/(n + 2)* Int[(d*Csc[e + f*x])^n* Simp[A*a*(n + 2) + (B*a*(n + 2) + b*(C*(n + 1) + A*(n + 2)))* Csc[e + f*x] + (a*C + B*b)*(n + 2)*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*csc[e_. + f_.*x_])^ n_.*(a_ + b_.*csc[e_. + f_.*x_])*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, C, n}, x] && Not[LtQ[n, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "-b*C*Csc[e + f*x]* Cot[e + f*x]*(d*Csc[e + f*x])^n/(f*(n + 2)) + 1/(n + 2)* Int[(d*Csc[e + f*x])^n* Simp[A*a*(n + 2) + b*(C*(n + 1) + A*(n + 2))*Csc[e + f*x] + a*C*(n + 2)*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*csc[e_. + f_.*x_])^ n_.*(a_ + b_.*csc[e_. + f_.*x_])*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, C, n}, x] && Not[LtQ[n, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "-(a*A - b*B + a*C)*Cot[e + f*x]* Csc[e + f*x]*(a + b*Csc[e + f*x])^m/(a*f*(2*m + 1)) - 1/(a*b*(2*m + 1))*Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)* Simp[a*B - b*C - 2*A*b*(m + 1) - (b*B*(m + 2) - a*(A*(m + 2) - C*(m - 1)))* Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "-(A + C)*Cot[e + f*x]* Csc[e + f*x]*(a + b*Csc[e + f*x])^m/(f*(2*m + 1)) - 1/(a*b*(2*m + 1))*Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)* Simp[-b*C - 2*A*b*(m + 1) + a*(A*(m + 2) - C*(m - 1))*Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, C}, x] && LtQ[m, -1] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "-(A*b^2 - a*b*B + a^2*C)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(b* f*(m + 1)*(a^2 - b^2)) + 1/(b*(m + 1)*(a^2 - b^2))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)* Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "-(A*b^2 + a^2*C)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(b* f*(m + 1)*(a^2 - b^2)) + 1/(b*(m + 1)*(a^2 - b^2))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)* Simp[a*b*(A + C)*(m + 1) - (A*b^2 + a^2*C + b*(A*b + b*C)*(m + 1))*Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "-C* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 2)) + 1/(b*(m + 2))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m* Simp[b*A*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B, C, m}, x] && Not[LtQ[m, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "-C* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 2)) + 1/(b*(m + 2))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m* Simp[b*A*(m + 2) + b*C*(m + 1) - a*C*Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, C, m}, x] && Not[LtQ[m, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "-(a*A - b*B + a*C)* Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^n/(a*f*(2*m + 1)) - 1/(a*b*(2*m + 1))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n* Simp[a*B*n - b*C*n - A*b*(2*m + n + 1) - (b*B*(m + n + 1) - a*(A*(m + n + 1) - C*(m - n)))*Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, C, n}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "-a*(A + C)* Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^n/(a*f*(2*m + 1)) + 1/(a*b*(2*m + 1))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n* Simp[b*C*n + A*b*(2*m + n + 1) - (a*(A*(m + n + 1) - C*(m - n)))* Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, C, n}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n/(f*n) - 1/(b*d*n)* Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)* Simp[a*A*m - b*B*n - b*(A*(m + n + 1) + C*n)*Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, C, m}, x] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1/2]] && (LtQ[n, -1/2] || EqQ[m + n + 1, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n/(f*n) - 1/(b*d*n)* Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)* Simp[a*A*m - b*(A*(m + n + 1) + C*n)*Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, C, m}, x] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1/2]] && (LtQ[n, -1/2] || EqQ[m + n + 1, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "-C* Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^n/(f*(m + n + 1)) + 1/(b*(m + n + 1))* Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n* Simp[A*b*(m + n + 1) + b*C*n + (a*C*m + b*B*(m + n + 1))*Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, C, m, n}, x] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1/2]] && Not[LtQ[n, -1/2]] && NeQ[m + n + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "-C* Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^n/(f*(m + n + 1)) + 1/(b*(m + n + 1))* Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n* Simp[A*b*(m + n + 1) + b*C*n + a*C*m*Csc[e + f*x], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, C, m, n}, x] && EqQ[a^2 - b^2, 0] && Not[LtQ[m, -1/2]] && Not[LtQ[n, -1/2]] && NeQ[m + n + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "a*(A*b^2 - a*b*B + a^2*C)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(b^2* f*(m + 1)*(a^2 - b^2)) - 1/(b^2*(m + 1)*(a^2 - b^2))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)* Simp[b*(m + 1)*(-a*(b*B - a*C) + A*b^2) + (b*B*(a^2 + b^2*(m + 1)) - a*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))*Csc[e + f*x] - b*C*(m + 1)*(a^2 - b^2)*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]^2*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "a*(A*b^2 + a^2*C)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(b^2* f*(m + 1)*(a^2 - b^2)) - 1/(b^2*(m + 1)*(a^2 - b^2))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)* Simp[b*(m + 1)*(a^2*C + A*b^2) - a*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1)))*Csc[e + f*x] - b*C*(m + 1)*(a^2 - b^2)*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]^2*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "-C*Csc[e + f*x]* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 3)) + 1/(b*(m + 3))*Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m* Simp[a*C + b*(C*(m + 2) + A*(m + 3))* Csc[e + f*x] - (2*a*C - b*B*(m + 3))*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]^2*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && Not[LtQ[m, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "-C*Csc[e + f*x]* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 3)) + 1/(b*(m + 3))* Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m* Simp[a*C + b*(C*(m + 2) + A*(m + 3))*Csc[e + f*x] - 2*a*C*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[csc[e_. + f_.*x_]^2*(a_ + b_.*csc[e_. + f_.*x_])^ m_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, A, C, m}, x] && NeQ[a^2 - b^2, 0] && Not[LtQ[m, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n/(f*n) - 1/(d*n)*Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)* Simp[A*b*m - a*B*n - (b*B*n + a*(C*n + A*(n + 1)))*Csc[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n/(f*n) - 1/(d*n)*Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)* Simp[A*b*m - a*(C*n + A*(n + 1))*Csc[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, C}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "-C* Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^n/(f*(m + n + 1)) + 1/(m + n + 1)*Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n* Simp[a*A*(m + n + 1) + a*C*n + ((A*b + a*B)*(m + n + 1) + b*C*(m + n))* Csc[e + f*x] + (b*B*(m + n + 1) + a*C*m)*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, C, n}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && Not[LeQ[n, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "-C* Cot[e + f*x]*(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^n/(f*(m + n + 1)) + 1/(m + n + 1)*Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n* Simp[a*A*(m + n + 1) + a*C*n + b*(A*(m + n + 1) + C*(m + n))*Csc[e + f*x] + a*C*m*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, C, n}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && Not[LeQ[n, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "-d*(A*b^2 - a*b*B + a^2*C)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)/(b*f*(a^2 - b^2)*(m + 1)) + d/(b*(a^2 - b^2)*(m + 1))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)* Simp[A*b^2*(n - 1) - a*(b*B - a*C)*(n - 1) + b*(a*A - b*B + a*C)*(m + 1)*Csc[e + f*x] - (b*(A*b - a*B)*(m + n + 1) + C*(a^2*n + b^2*(m + 1)))* Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "-d*(A*b^2 + a^2*C)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)/(b*f*(a^2 - b^2)*(m + 1)) + d/(b*(a^2 - b^2)*(m + 1))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)* Simp[A*b^2*(n - 1) + a^2*C*(n - 1) + a*b*(A + C)*(m + 1)* Csc[e + f*x] - (A*b^2*(m + n + 1) + C*(a^2*n + b^2*(m + 1)))* Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "(A*b^2 - a*b*B + a^2*C)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^ n/(a*f*(m + 1)*(a^2 - b^2)) + 1/(a*(m + 1)*(a^2 - b^2))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n* Simp[a*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C)*(m + n + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + n + 2)*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, C, n}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && Not[ILtQ[m + 1/2, 0] && ILtQ[n, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "(A*b^2 + a^2*C)* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^ n/(a*f*(m + 1)*(a^2 - b^2)) + 1/(a*(m + 1)*(a^2 - b^2))* Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n* Simp[a^2*(A + C)*(m + 1) - (A*b^2 + a^2*C)*(m + n + 1) - a*b*(A + C)*(m + 1)* Csc[e + f*x] + (A*b^2 + a^2*C)*(m + n + 2)*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, C, n}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && Not[ILtQ[m + 1/2, 0] && ILtQ[n, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "-C*d* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)/(b*f*(m + n + 1)) + d/(b*(m + n + 1))* Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 1)* Simp[a*C*(n - 1) + (A*b*(m + n + 1) + b*C*(m + n))* Csc[e + f*x] + (b*B*(m + n + 1) - a*C*n)*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 0] (* && Not[IGtQ[m,0] && Not[IntegerQ[ n]]] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "-C*d* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)/(b*f*(m + n + 1)) + d/(b*(m + n + 1))* Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 1)* Simp[a*C*(n - 1) + (A*b*(m + n + 1) + b*C*(m + n))* Csc[e + f*x] - a*C*n*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, C, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 0] (* && Not[IGtQ[m,0] && Not[IntegerQ[n]]] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^ n/(a*f*n) + 1/(a*d*n)*Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)* Simp[a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)*Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^ n/(a*f*n) + 1/(a*d*n)*Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)* Simp[-A*b*(m + n + 1) + a*(A + A*n + C*n)*Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_*(d_.*csc[e_. + f_.*x_])^ n_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "(A*b^2 - a*b*B + a^2*C)/(a^2*d^2)* Int[(d*Csc[e + f*x])^(3/2)/(a + b*Csc[e + f*x]), x] + 1/a^2* Int[(a*A - (A*b - a*B)*Csc[e + f*x])/Sqrt[d*Csc[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2)/(Sqrt[ d_.*csc[e_. + f_.*x_]]*(a_ + b_.*csc[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "(A*b^2 + a^2*C)/(a^2*d^2)* Int[(d*Csc[e + f*x])^(3/2)/(a + b*Csc[e + f*x]), x] + 1/a^2*Int[(a*A - A*b*Csc[e + f*x])/Sqrt[d*Csc[e + f*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + C_.*csc[e_. + f_.*x_]^2)/(Sqrt[ d_.*csc[e_. + f_.*x_]]*(a_ + b_.*csc[e_. + f_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, C}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "C/d^2*Int[(d*Csc[e + f*x])^(3/2)/Sqrt[a + b*Csc[e + f*x]], x] + Int[(A + B*Csc[e + f*x])/(Sqrt[d*Csc[e + f*x]]* Sqrt[a + b*Csc[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2)/(Sqrt[d_.*csc[e_. + f_.*x_]]* Sqrt[a_ + b_.*csc[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "C/d^2*Int[(d*Csc[e + f*x])^(3/2)/Sqrt[a + b*Csc[e + f*x]], x] + A*Int[1/(Sqrt[d*Csc[e + f*x]]*Sqrt[a + b*Csc[e + f*x]]), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + C_.*csc[e_. + f_.*x_]^2)/(Sqrt[d_.*csc[e_. + f_.*x_]]* Sqrt[a_ + b_.*csc[e_. + f_.*x_]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, C}, x] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "Unintegrable[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^ n*(A + B*Csc[e + f*x] + C*Csc[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(d_.*csc[e_. + f_.*x_])^ n_.*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, C, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "Unintegrable[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^ n*(A + C*Csc[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(d_.*csc[e_. + f_.*x_])^ n_.*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, C, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "d^(m + 2)* Int[(b + a*Cos[e + f*x])^ m*(d*Cos[e + f*x])^(n - m - 2)*(C + B*Cos[e + f*x] + A*Cos[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sec[e_. + f_.*x_])^m_.*(d_.*cos[e_. + f_.*x_])^ n_*(A_. + B_.*sec[e_. + f_.*x_] + C_.*sec[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, C, n}, x] && Not[IntegerQ[n]] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "d^(m + 2)* Int[(b + a*Sin[e + f*x])^ m*(d*Sin[e + f*x])^(n - m - 2)*(C + B*Sin[e + f*x] + A*Sin[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(d_.*sin[e_. + f_.*x_])^ n_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, B, C, n}, x] && Not[IntegerQ[n]] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "d^(m + 2)* Int[(b + a*Cos[e + f*x])^ m*(d*Cos[e + f*x])^(n - m - 2)*(C + A*Cos[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sec[e_. + f_.*x_])^m_.*(d_.*cos[e_. + f_.*x_])^ n_*(A_. + C_.*sec[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, C, n}, x] && Not[IntegerQ[n]] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "d^(m + 2)* Int[(b + a*Sin[e + f*x])^ m*(d*Sin[e + f*x])^(n - m - 2)*(C + A*Sin[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(d_.*sin[e_. + f_.*x_])^ n_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, A, C, n}, x] && Not[IntegerQ[n]] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "c^IntPart[n]*(c*(d*Sec[e + f*x])^p)^ FracPart[n]/(d*Sec[e + f*x])^(p*FracPart[n])* Int[(a + b*Sec[e + f*x])^ m*(d*Sec[e + f*x])^(n*p)*(A + B*Sec[e + f*x] + C*Sec[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sec[e_. + f_.*x_])^m_.*(c_.*(d_.*sec[e_. + f_.*x_])^p_)^ n_*(A_. + B_.*sec[e_. + f_.*x_] + C_.*sec[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, m, n, p}, x] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "c^IntPart[n]*(c*(d*Csc[e + f*x])^p)^ FracPart[n]/(d*Csc[e + f*x])^(p*FracPart[n])* Int[(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^(n*p)*(A + B*Csc[e + f*x] + C*Csc[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(c_.*(d_.*csc[e_. + f_.*x_])^p_)^ n_*(A_. + B_.*csc[e_. + f_.*x_] + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, m, n, p}, x] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "c^IntPart[n]*(c*(d*Sec[e + f*x])^p)^ FracPart[n]/(d*Sec[e + f*x])^(p*FracPart[n])* Int[(a + b*Sec[e + f*x])^ m*(d*Sec[e + f*x])^(n*p)*(A + C*Sec[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sec[e_. + f_.*x_])^m_.*(c_.*(d_.*sec[e_. + f_.*x_])^p_)^ n_*(A_. + C_.*sec[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C, m, n, p}, x] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
       "filename": "4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m",
            "rhs": "c^IntPart[n]*(c*(d*Csc[e + f*x])^p)^ FracPart[n]/(d*Csc[e + f*x])^(p*FracPart[n])* Int[(a + b*Csc[e + f*x])^ m*(d*Csc[e + f*x])^(n*p)*(A + C*Csc[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*csc[e_. + f_.*x_])^m_.*(c_.*(d_.*csc[e_. + f_.*x_])^p_)^ n_*(A_. + C_.*csc[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, C, m, n, p}, x] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "b^p*Int[ActivateTrig[u*tan[e + f*x]^(2*p)], x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_ + b_.*sec[e_. + f_.*x_]^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "Int[ActivateTrig[u*(b*tan[e + f*x]^2)^p], x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_ + b_.*sec[e_. + f_.*x_]^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, b*ff/f* Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*sec[e_. + f_.*x_]^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, e, f, p}, x] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "b^IntPart[p]*(b*(c*Sec[e + f*x])^n)^ FracPart[p]/(c*Sec[e + f*x])^(n*FracPart[p])* Int[(c*Sec[e + f*x])^(n*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*(c_.*sec[e_. + f_.*x_])^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, e, f, n, p}, x] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "b/(2*f)*Subst[Int[(-1 + x)^((m - 1)/2)*(b*x)^(p - 1), x], x, Sec[e + f*x]^2]",
     "rulenumber": 0,
            "lhs": "Int[tan[e_. + f_.*x_]^m_.*(b_.*sec[e_. + f_.*x_]^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, e, f, p}, x] && Not[IntegerQ[p]] && IntegerQ[(m - 1)/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Sec[e + f*x], x]}, (b*ff^n)^ IntPart[p]*(b*Sec[e + f*x]^n)^ FracPart[p]/(Sec[e + f*x]/ff)^(n*FracPart[p])* Int[ActivateTrig[u]*(Sec[e + f*x]/ff)^(n*p), x]] /; FreeQ[{b, e, f, n, p}, x] && Not[IntegerQ[p]] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, (d_.*trig_[e + f*x])^m_.",
     "rulenumber": 0,
            "lhs": "Int[u_.*(b_.*sec[e_. + f_.*x_]^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "b^IntPart[p]*(b*(c*Sec[e + f*x])^n)^ FracPart[p]/(c*Sec[e + f*x])^(n*FracPart[p])* Int[ActivateTrig[u]*(c*Sec[e + f*x])^(n*p), x] /; FreeQ[{b, c, e, f, n, p}, x] && Not[IntegerQ[p]] && Not[IntegerQ[n]] && (EqQ[u, 1] || MatchQ[u, (d_.*trig_[e + f*x])^m_.",
     "rulenumber": 0,
            "lhs": "Int[u_.*(b_.*(c_.*sec[e_. + f_.*x_])^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "x/a - b/a*Int[1/(b + a*Cos[e + f*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*sec[e_. + f_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && NeQ[a + b, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[(a + b + b*ff^2*x^2)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sec[e_. + f_.*x_]^2)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, p}, x] && NeQ[a + b, 0] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[(a + b + 2*b*ff^2*x^2 + b*ff^4*x^4)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sec[e_. + f_.*x_]^4)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, p}, x] && IntegerQ[2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[(a + b*(1 + ff^2*x^2)^(n/2))^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*sec[e_. + f_.*x_]^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, p}, x] && IntegerQ[n/2] && IGtQ[p, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "Unintegrable[(a + b*(c*Sec[e + f*x])^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*(c_.*sec[e_. + f_.*x_])^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, e, f, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, ff^(m + 1)/f* Subst[Int[ x^m*ExpandToSum[a + b*(1 + ff^2*x^2)^(n/2), x]^ p/(1 + ff^2*x^2)^(m/2 + 1), x], x, Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[sin[e_. + f_.*x_]^m_*(a_ + b_.*sec[e_. + f_.*x_]^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && IntegerQ[n/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Cos[e + f*x], x]}, -ff/f* Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(b + a*(ff*x)^n)^ p/(ff*x)^(n*p), x], x, Cos[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[sin[e_. + f_.*x_]^m_.*(a_ + b_.*sec[e_. + f_.*x_]^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Cos[e + f*x], x]}, 1/(f*ff^m)* Subst[Int[(-1 + ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p/ x^(m + 1), x], x, Sec[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[sin[e_. + f_.*x_]^m_.*(a_ + b_.*(c_.*sec[e_. + f_.*x_])^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (GtQ[m, 0] || EqQ[n, 2] || EqQ[n, 4])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "Unintegrable[(d*Sin[e + f*x])^m*(a + b*(c*Sec[e + f*x])^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sin[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*sec[e_. + f_.*x_])^n_)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "d^(n*p)*Int[(d*Cos[e + f*x])^(m - n*p)*(b + a*Cos[e + f*x]^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*cos[e_. + f_.*x_])^m_*(a_ + b_.*sec[e_. + f_.*x_]^n_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, n, p}, x] && Not[IntegerQ[m]] && IntegersQ[n, p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "(d*Cos[e + f*x])^FracPart[m]*(Sec[e + f*x]/d)^ FracPart[m]* Int[(Sec[e + f*x]/d)^(-m)*(a + b*(c*Sec[e + f*x])^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*cos[e_. + f_.*x_])^m_*(a_ + b_.*(c_.*sec[e_. + f_.*x_])^n_)^ p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "Module[{ff = FreeFactors[Cos[e + f*x], x]}, -1/(f*ff^(m + n*p - 1))* Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(b + a*(ff*x)^n)^p/ x^(m + n*p), x], x, Cos[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[tan[e_. + f_.*x_]^m_.*(a_ + b_.*sec[e_. + f_.*x_]^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, n}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Sec[e + f*x], x]}, 1/f*Subst[ Int[(-1 + ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p/x, x], x, Sec[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[tan[e_. + f_.*x_]^m_.*(a_ + b_.*(c_.*sec[e_. + f_.*x_])^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (GtQ[m, 0] || EqQ[n, 2] || EqQ[n, 4] || IGtQ[p, 0] || IntegersQ[2*n, p])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, b*ff/f* Subst[Int[(d*ff*x)^m*(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*tan[e_. + f_.*x_])^m_*(b_.*sec[e_. + f_.*x_]^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, d, e, f, m, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[(d*ff*x)^ m*(a + b*(1 + ff^2*x^2)^(n/2))^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*tan[e_. + f_.*x_])^m_*(a_ + b_.*sec[e_. + f_.*x_]^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f, m, p}, x] && IntegerQ[n/2] && (IntegerQ[m/2] || EqQ[n, 2])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "d*(d*Tan[e + f*x])^(m - 1)*(b*(c*Sec[e + f*x])^n)^ p/(f*(p*n + m - 1)) - d^2*(m - 1)/(p*n + m - 1)* Int[(d*Tan[e + f*x])^(m - 2)*(b*(c*Sec[e + f*x])^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*tan[e_. + f_.*x_])^m_*(b_.*(c_.*sec[e_. + f_.*x_])^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f, p, n}, x] && GtQ[m, 1] && NeQ[p*n + m - 1, 0] && IntegersQ[2*p*n, 2*m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "(d*Tan[e + f*x])^(m + 1)*(b*(c*Sec[e + f*x])^n)^ p/(d*f*(m + 1)) - (p*n + m + 1)/(d^2*(m + 1))* Int[(d*Tan[e + f*x])^(m + 2)*(b*(c*Sec[e + f*x])^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*tan[e_. + f_.*x_])^m_*(b_.*(c_.*sec[e_. + f_.*x_])^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f, p, n}, x] && LtQ[m, -1] && NeQ[p*n + m + 1, 0] && IntegersQ[2*p*n, 2*m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "Unintegrable[(d*Tan[e + f*x])^m*(a + b*(c*Sec[e + f*x])^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*tan[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*sec[e_. + f_.*x_])^n_)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "(d*Cot[e + f*x])^FracPart[m]*(Tan[e + f*x]/d)^ FracPart[m]* Int[(Tan[e + f*x]/d)^(-m)*(a + b*(c*Sec[e + f*x])^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*cot[e_. + f_.*x_])^m_*(a_ + b_.*(c_.*sec[e_. + f_.*x_])^n_)^ p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Tan[e + f*x], x]}, ff/f* Subst[Int[(1 + ff^2*x^2)^(m/2 - 1)* ExpandToSum[a + b*(1 + ff^2*x^2)^(n/2), x]^p, x], x, Tan[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[sec[e_. + f_.*x_]^m_*(a_ + b_.*sec[e_. + f_.*x_]^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && IntegerQ[n/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Sin[e + f*x], x]}, ff/f* Subst[Int[ ExpandToSum[b + a*(1 - ff^2*x^2)^(n/2), x]^ p/(1 - ff^2*x^2)^((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[sec[e_. + f_.*x_]^m_.*(a_ + b_.*sec[e_. + f_.*x_]^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "With[{ff = FreeFactors[Sin[e + f*x], x]}, ff/f* Subst[Int[(a + b/(1 - ff^2*x^2)^(n/2))^ p/(1 - ff^2*x^2)^((m + 1)/2), x], x, Sin[e + f*x]/ff]]",
     "rulenumber": 0,
            "lhs": "Int[sec[e_. + f_.*x_]^m_.*(a_ + b_.*sec[e_. + f_.*x_]^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "Int[ExpandTrig[sec[e + f*x]^m*(a + b*sec[e + f*x]^n)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[sec[e_. + f_.*x_]^m_.*(a_ + b_.*sec[e_. + f_.*x_]^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && IntegersQ[m, n, p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "Unintegrable[(d*Sec[e + f*x])^m*(a + b*(c*Sec[e + f*x])^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*sec[e_. + f_.*x_])^m_.*(a_ + b_.*(c_.*sec[e_. + f_.*x_])^n_)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
       "filename": "4.5.7 (d trig)^m (a+b (c sec)^n)^p.m",
            "rhs": "(d*Csc[e + f*x])^FracPart[m]*(Sin[e + f*x]/d)^ FracPart[m]* Int[(Sin[e + f*x]/d)^(-m)*(a + b*(c*Sec[e + f*x])^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*csc[e_. + f_.*x_])^m_*(a_ + b_.*(c_.*sec[e_. + f_.*x_])^n_)^ p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
       "filename": "4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
            "rhs": "1/(4^p*c^p)*Int[(b + 2*c*Sec[d + e*x]^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sec[d_. + e_.*x_]^n_. + c_.*sec[d_. + e_.*x_]^n2_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
       "filename": "4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
            "rhs": "1/(4^p*c^p)*Int[(b + 2*c*Csc[d + e*x]^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*csc[d_. + e_.*x_]^n_. + c_.*csc[d_. + e_.*x_]^n2_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
       "filename": "4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
            "rhs": "(a + b*Sec[d + e*x]^n + c*Sec[d + e*x]^(2*n))^ p/(b + 2*c*Sec[d + e*x]^n)^(2*p)* Int[u*(b + 2*c*Sec[d + e*x]^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*sec[d_. + e_.*x_]^n_. + c_.*sec[d_. + e_.*x_]^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
       "filename": "4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
            "rhs": "(a + b*Csc[d + e*x]^n + c*Csc[d + e*x]^(2*n))^ p/(b + 2*c*Csc[d + e*x]^n)^(2*p)* Int[u*(b + 2*c*Csc[d + e*x]^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*csc[d_. + e_.*x_]^n_. + c_.*csc[d_. + e_.*x_]^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
       "filename": "4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
            "rhs": "Module[{q = Rt[b^2 - 4*a*c, 2]}, 2*c/q*Int[1/(b - q + 2*c*Sec[d + e*x]^n), x] - 2*c/q*Int[1/(b + q + 2*c*Sec[d + e*x]^n), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_. + b_.*sec[d_. + e_.*x_]^n_. + c_.*sec[d_. + e_.*x_]^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
       "filename": "4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
            "rhs": "Module[{q = Rt[b^2 - 4*a*c, 2]}, 2*c/q*Int[1/(b - q + 2*c*Csc[d + e*x]^n), x] - 2*c/q*Int[1/(b + q + 2*c*Csc[d + e*x]^n), x]]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_. + b_.*csc[d_. + e_.*x_]^n_. + c_.*csc[d_. + e_.*x_]^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
       "filename": "4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
            "rhs": "Module[{f = FreeFactors[Cos[d + e*x], x]}, -f/e* Subst[Int[(1 - f^2*x^2)^((m - 1)/2)*(b + a*(f*x)^n)^p/(f*x)^(n*p), x], x, Cos[d + e*x]/f]]",
     "rulenumber": 0,
            "lhs": "Int[sin[d_. + e_.*x_]^ m_.*(a_. + b_.*sec[d_. + e_.*x_]^n_. + c_.*sec[d_. + e_.*x_]^n2_)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[(m - 1)/2] && IntegersQ[n, p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
       "filename": "4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
            "rhs": "Module[{f = FreeFactors[Sin[d + e*x], x]}, f/e*Subst[ Int[(1 - f^2*x^2)^((m - 1)/2)*(b + a*(f*x)^n)^p/(f*x)^(n*p), x], x, Sin[d + e*x]/f]]",
     "rulenumber": 0,
            "lhs": "Int[cos[d_. + e_.*x_]^ m_.*(a_. + b_.*csc[d_. + e_.*x_]^n_. + c_.*csc[d_. + e_.*x_]^n2_)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[(m - 1)/2] && IntegersQ[n, p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
       "filename": "4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
            "rhs": "Module[{f = FreeFactors[Tan[d + e*x], x]}, f^(m + 1)/e* Subst[Int[ x^m*ExpandToSum[a + b*(1 + f^2*x^2)^(n/2) + c*(1 + f^2*x^2)^n, x]^p/(1 + f^2*x^2)^(m/2 + 1), x], x, Tan[d + e*x]/f]]",
     "rulenumber": 0,
            "lhs": "Int[sin[d_. + e_.*x_]^ m_*(a_. + b_.*sec[d_. + e_.*x_]^n_ + c_.*sec[d_. + e_.*x_]^n2_)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && IntegerQ[n/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
       "filename": "4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
            "rhs": "Module[{f = FreeFactors[Cot[d + e*x], x]}, -f^(m + 1)/e* Subst[Int[ x^m*ExpandToSum[a + b*(1 + f^2*x^2)^(n/2) + c*(1 + f^2*x^2)^n, x]^p/(1 + f^2*x^2)^(m/2 + 1), x], x, Cot[d + e*x]/f]]",
     "rulenumber": 0,
            "lhs": "Int[cos[d_. + e_.*x_]^ m_*(a_. + b_.*csc[d_. + e_.*x_]^n_ + c_.*csc[d_. + e_.*x_]^n2_)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && IntegerQ[n/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
       "filename": "4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
            "rhs": "1/(4^p*c^p)*Int[Sec[d + e*x]^m*(b + 2*c*Sec[d + e*x]^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[sec[d_. + e_.*x_]^ m_.*(a_. + b_.*sec[d_. + e_.*x_]^n_. + c_.*sec[d_. + e_.*x_]^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
       "filename": "4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
            "rhs": "1/(4^p*c^p)*Int[Csc[d + e*x]^m*(b + 2*c*Csc[d + e*x]^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[csc[d_. + e_.*x_]^ m_.*(a_. + b_.*csc[d_. + e_.*x_]^n_. + c_.*csc[d_. + e_.*x_]^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
       "filename": "4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
            "rhs": "(a + b*Sec[d + e*x]^n + c*Sec[d + e*x]^(2*n))^ p/(b + 2*c*Sec[d + e*x]^n)^(2*p)* Int[Sec[d + e*x]^m*(b + 2*c*Sec[d + e*x]^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[sec[d_. + e_.*x_]^ m_.*(a_. + b_.*sec[d_. + e_.*x_]^n_. + c_.*sec[d_. + e_.*x_]^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
       "filename": "4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
            "rhs": "(a + b*Csc[d + e*x]^n + c*Csc[d + e*x]^(2*n))^ p/(b + 2*c*Csc[d + e*x]^n)^(2*p)* Int[Csc[d + e*x]^m*(b + 2*c*Csc[d + e*x]^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[csc[d_. + e_.*x_]^ m_.*(a_. + b_.*csc[d_. + e_.*x_]^n_. + c_.*csc[d_. + e_.*x_]^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
       "filename": "4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
            "rhs": "Int[ExpandTrig[ sec[d + e*x]^m*(a + b*sec[d + e*x]^n + c*sec[d + e*x]^(2*n))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[sec[d_. + e_.*x_]^ m_.*(a_. + b_.*sec[d_. + e_.*x_]^n_. + c_.*sec[d_. + e_.*x_]^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegersQ[m, n, p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
       "filename": "4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
            "rhs": "Int[ExpandTrig[ csc[d + e*x]^m*(a + b*csc[d + e*x]^n + c*csc[d + e*x]^(2*n))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[csc[d_. + e_.*x_]^ m_.*(a_. + b_.*csc[d_. + e_.*x_]^n_. + c_.*csc[d_. + e_.*x_]^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegersQ[m, n, p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
       "filename": "4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
            "rhs": "Module[{f = FreeFactors[Cos[d + e*x], x]}, -1/(e*f^(m + n*p - 1))* Subst[Int[(1 - f^2*x^2)^((m - 1)/ 2)*(c + b*(f*x)^n + c*(f*x)^(2*n))^p/x^(m + 2*n*p), x], x, Cos[d + e*x]/f]]",
     "rulenumber": 0,
            "lhs": "Int[tan[d_. + e_.*x_]^ m_.*(a_ + b_.*sec[d_. + e_.*x_]^n_. + c_.*sec[d_. + e_.*x_]^n2_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && IntegerQ[(m - 1)/2] && IntegerQ[n] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
       "filename": "4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
            "rhs": "Module[{f = FreeFactors[Sin[d + e*x], x]}, 1/(e*f^(m + n*p - 1))* Subst[Int[(1 - f^2*x^2)^((m - 1)/ 2)*(c + b*(f*x)^n + c*(f*x)^(2*n))^p/x^(m + 2*n*p), x], x, Sin[d + e*x]/f]]",
     "rulenumber": 0,
            "lhs": "Int[cot[d_. + e_.*x_]^ m_.*(a_ + b_.*csc[d_. + e_.*x_]^n_. + c_.*sec[d_. + e_.*x_]^n2_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && IntegerQ[(m - 1)/2] && IntegerQ[n] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
       "filename": "4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
            "rhs": "Module[{f = FreeFactors[Tan[d + e*x], x]}, f^(m + 1)/e* Subst[Int[ x^m*ExpandToSum[a + b*(1 + f^2*x^2)^(n/2) + c*(1 + f^2*x^2)^n, x]^p/(1 + f^2*x^2), x], x, Tan[d + e*x]/f]]",
     "rulenumber": 0,
            "lhs": "Int[tan[d_. + e_.*x_]^ m_.*(a_ + b_.*sec[d_. + e_.*x_]^n_ + c_.*sec[d_. + e_.*x_]^n2_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && IntegerQ[n/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
       "filename": "4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
            "rhs": "Module[{f = FreeFactors[Cot[d + e*x], x]}, -f^(m + 1)/e* Subst[Int[ x^m*ExpandToSum[a + b*(1 + f^2*x^2)^(n/2) + c*(1 + f^2*x^2)^n, x]^p/(1 + f^2*x^2), x], x, Cot[d + e*x]/f]]",
     "rulenumber": 0,
            "lhs": "Int[cot[d_. + e_.*x_]^ m_.*(a_ + b_.*csc[d_. + e_.*x_]^n_ + c_.*sec[d_. + e_.*x_]^n2_.)^ p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && IntegerQ[n/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
       "filename": "4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
            "rhs": "1/(4^n*c^n)* Int[(A + B*Sec[d + e*x])*(b + 2*c*Sec[d + e*x])^(2*n), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*sec[d_. + e_.*x_])*(a_ + b_.*sec[d_. + e_.*x_] + c_.*sec[d_. + e_.*x_]^2)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
       "filename": "4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
            "rhs": "1/(4^n*c^n)* Int[(A + B*Csc[d + e*x])*(b + 2*c*Csc[d + e*x])^(2*n), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*csc[d_. + e_.*x_])*(a_ + b_.*csc[d_. + e_.*x_] + c_.*csc[d_. + e_.*x_]^2)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
       "filename": "4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
            "rhs": "(a + b*Sec[d + e*x] + c*Sec[d + e*x]^2)^ n/(b + 2*c*Sec[d + e*x])^(2*n)* Int[(A + B*Sec[d + e*x])*(b + 2*c*Sec[d + e*x])^(2*n), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*sec[d_. + e_.*x_])*(a_ + b_.*sec[d_. + e_.*x_] + c_.*sec[d_. + e_.*x_]^2)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
       "filename": "4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
            "rhs": "(a + b*Csc[d + e*x] + c*Csc[d + e*x]^2)^ n/(b + 2*c*Csc[d + e*x])^(2*n)* Int[(A + B*Csc[d + e*x])*(b + 2*c*Csc[d + e*x])^(2*n), x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*csc[d_. + e_.*x_])*(a_ + b_.*csc[d_. + e_.*x_] + c_.*csc[d_. + e_.*x_]^2)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
       "filename": "4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
            "rhs": "Module[{q = Rt[b^2 - 4*a*c, 2]}, (B + (b*B - 2*A*c)/q)*Int[1/(b + q + 2*c*Sec[d + e*x]), x] + (B - (b*B - 2*A*c)/q)*Int[1/(b - q + 2*c*Sec[d + e*x]), x]]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*sec[d_. + e_.*x_])/(a_. + b_.*sec[d_. + e_.*x_] + c_.*sec[d_. + e_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
       "filename": "4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
            "rhs": "Module[{q = Rt[b^2 - 4*a*c, 2]}, (B + (b*B - 2*A*c)/q)*Int[1/(b + q + 2*c*Csc[d + e*x]), x] + (B - (b*B - 2*A*c)/q)*Int[1/(b - q + 2*c*Csc[d + e*x]), x]]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*csc[d_. + e_.*x_])/(a_. + b_.*csc[d_. + e_.*x_] + c_.*csc[d_. + e_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
       "filename": "4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
            "rhs": "Int[ExpandTrig[(A + B*sec[d + e*x])*(a + b*sec[d + e*x] + c*sec[d + e*x]^2)^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*sec[d_. + e_.*x_])*(a_. + b_.*sec[d_. + e_.*x_] + c_.*sec[d_. + e_.*x_]^2)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.5 Secant/4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
       "filename": "4.5.9 trig^m (a+b sec^n+c sec^(2 n))^p.m",
            "rhs": "Int[ExpandTrig[(A + B*csc[d + e*x])*(a + b*csc[d + e*x] + c*csc[d + e*x]^2)^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(A_ + B_.*csc[d_. + e_.*x_])*(a_. + b_.*csc[d_. + e_.*x_] + c_.*csc[d_. + e_.*x_]^2)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.1 Sine normalization rules.m",
       "filename": "4.7.1 Sine normalization rules.m",
            "rhs": "(c*Tan[a + b*x])^ m*(d*Cos[a + b*x])^m/(d*Sin[a + b*x])^m* Int[ActivateTrig[u]*(d*Sin[a + b*x])^(m + n)/(d*Cos[a + b*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*tan[a_. + b_.*x_])^m_.*(d_.*sin[a_. + b_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && KnownSineIntegrandQ[u, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.1 Sine normalization rules.m",
       "filename": "4.7.1 Sine normalization rules.m",
            "rhs": "(c*Tan[a + b*x])^ m*(d*Cos[a + b*x])^m/(d*Sin[a + b*x])^m* Int[ActivateTrig[u]*(d*Sin[a + b*x])^m/(d*Cos[a + b*x])^(m - n), x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*tan[a_. + b_.*x_])^m_.*(d_.*cos[a_. + b_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && KnownSineIntegrandQ[u, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.1 Sine normalization rules.m",
       "filename": "4.7.1 Sine normalization rules.m",
            "rhs": "(c*Cot[a + b*x])^ m*(d*Sin[a + b*x])^m/(d*Cos[a + b*x])^m* Int[ActivateTrig[u]*(d*Cos[a + b*x])^m/(d*Sin[a + b*x])^(m - n), x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*cot[a_. + b_.*x_])^m_.*(d_.*sin[a_. + b_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && KnownSineIntegrandQ[u, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.1 Sine normalization rules.m",
       "filename": "4.7.1 Sine normalization rules.m",
            "rhs": "(c*Cot[a + b*x])^ m*(d*Sin[a + b*x])^m/(d*Cos[a + b*x])^m* Int[ActivateTrig[u]*(d*Cos[a + b*x])^(m + n)/(d*Sin[a + b*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*cot[a_. + b_.*x_])^m_.*(d_.*cos[a_. + b_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && KnownSineIntegrandQ[u, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.1 Sine normalization rules.m",
       "filename": "4.7.1 Sine normalization rules.m",
            "rhs": "(c*Sec[a + b*x])^m*(d*Cos[a + b*x])^m* Int[ActivateTrig[u]*(d*Cos[a + b*x])^(n - m), x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*sec[a_. + b_.*x_])^m_.*(d_.*cos[a_. + b_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && KnownSineIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.1 Sine normalization rules.m",
       "filename": "4.7.1 Sine normalization rules.m",
            "rhs": "(c*Csc[a + b*x])^m*(d*Sin[a + b*x])^m* Int[ActivateTrig[u]*(d*Sin[a + b*x])^(n - m), x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*sec[a_. + b_.*x_])^m_.*(d_.*cos[a_. + b_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && KnownSineIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.1 Sine normalization rules.m",
       "filename": "4.7.1 Sine normalization rules.m",
            "rhs": "(c*Tan[a + b*x])^ m*(c*Cos[a + b*x])^m/(c*Sin[a + b*x])^m* Int[ActivateTrig[u]*(c*Sin[a + b*x])^m/(c*Cos[a + b*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*tan[a_. + b_.*x_])^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m}, x] && Not[IntegerQ[m]] && KnownSineIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.1 Sine normalization rules.m",
       "filename": "4.7.1 Sine normalization rules.m",
            "rhs": "(c*Cot[a + b*x])^ m*(c*Sin[a + b*x])^m/(c*Cos[a + b*x])^m* Int[ActivateTrig[u]*(c*Cos[a + b*x])^m/(c*Sin[a + b*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*cot[a_. + b_.*x_])^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m}, x] && Not[IntegerQ[m]] && KnownSineIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.1 Sine normalization rules.m",
       "filename": "4.7.1 Sine normalization rules.m",
            "rhs": "(c*Sec[a + b*x])^m*(c*Cos[a + b*x])^m* Int[ActivateTrig[u]/(c*Cos[a + b*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*sec[a_. + b_.*x_])^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m}, x] && Not[IntegerQ[m]] && KnownSineIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.1 Sine normalization rules.m",
       "filename": "4.7.1 Sine normalization rules.m",
            "rhs": "(c*Csc[a + b*x])^m*(c*Sin[a + b*x])^m* Int[ActivateTrig[u]/(c*Sin[a + b*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*csc[a_. + b_.*x_])^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m}, x] && Not[IntegerQ[m]] && KnownSineIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.1 Sine normalization rules.m",
       "filename": "4.7.1 Sine normalization rules.m",
            "rhs": "c*Int[ActivateTrig[u]*(c*Sin[a + b*x])^(n - 1)*(B + A*Sin[a + b*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*sin[a_. + b_.*x_])^n_.*(A_ + B_.*csc[a_. + b_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, B, n}, x] && KnownSineIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.1 Sine normalization rules.m",
       "filename": "4.7.1 Sine normalization rules.m",
            "rhs": "c*Int[ActivateTrig[u]*(c*Cos[a + b*x])^(n - 1)*(B + A*Cos[a + b*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*cos[a_. + b_.*x_])^n_.*(A_ + B_.*sec[a_. + b_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, B, n}, x] && KnownSineIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.1 Sine normalization rules.m",
       "filename": "4.7.1 Sine normalization rules.m",
            "rhs": "Int[ActivateTrig[u]*(B + A*Sin[a + b*x])/Sin[a + b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(A_ + B_.*csc[a_. + b_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B}, x] && KnownSineIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.1 Sine normalization rules.m",
       "filename": "4.7.1 Sine normalization rules.m",
            "rhs": "Int[ActivateTrig[u]*(B + A*Cos[a + b*x])/Cos[a + b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(A_ + B_.*sec[a_. + b_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B}, x] && KnownSineIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.1 Sine normalization rules.m",
       "filename": "4.7.1 Sine normalization rules.m",
            "rhs": "c^2*Int[ActivateTrig[ u]*(c*Sin[a + b*x])^(n - 2)*(C + B*Sin[a + b*x] + A*Sin[a + b*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(c_.*sin[a_. + b_.*x_])^ n_.*(A_. + B_.*csc[a_. + b_.*x_] + C_.*csc[a_. + b_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, B, C, n}, x] && KnownSineIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.1 Sine normalization rules.m",
       "filename": "4.7.1 Sine normalization rules.m",
            "rhs": "c^2*Int[ActivateTrig[ u]*(c*Cos[a + b*x])^(n - 2)*(C + B*Cos[a + b*x] + A*Cos[a + b*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(c_.*cos[a_. + b_.*x_])^ n_.*(A_. + B_.*sec[a_. + b_.*x_] + C_.*sec[a_. + b_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, B, C, n}, x] && KnownSineIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.1 Sine normalization rules.m",
       "filename": "4.7.1 Sine normalization rules.m",
            "rhs": "c^2*Int[ActivateTrig[ u]*(c*Sin[a + b*x])^(n - 2)*(C + A*Sin[a + b*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(c_.*sin[a_. + b_.*x_])^n_.*(A_ + C_.*csc[a_. + b_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, C, n}, x] && KnownSineIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.1 Sine normalization rules.m",
       "filename": "4.7.1 Sine normalization rules.m",
            "rhs": "c^2*Int[ActivateTrig[ u]*(c*Cos[a + b*x])^(n - 2)*(C + A*Cos[a + b*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(c_.*cos[a_. + b_.*x_])^n_.*(A_ + C_.*sec[a_. + b_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, C, n}, x] && KnownSineIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.1 Sine normalization rules.m",
       "filename": "4.7.1 Sine normalization rules.m",
            "rhs": "Int[ActivateTrig[u]*(C + B*Sin[a + b*x] + A*Sin[a + b*x]^2)/ Sin[a + b*x]^2, x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(A_. + B_.*csc[a_. + b_.*x_] + C_.*csc[a_. + b_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B, C}, x] && KnownSineIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.1 Sine normalization rules.m",
       "filename": "4.7.1 Sine normalization rules.m",
            "rhs": "Int[ActivateTrig[u]*(C + B*Cos[a + b*x] + A*Cos[a + b*x]^2)/ Cos[a + b*x]^2, x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(A_. + B_.*sec[a_. + b_.*x_] + C_.*sec[a_. + b_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B, C}, x] && KnownSineIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.1 Sine normalization rules.m",
       "filename": "4.7.1 Sine normalization rules.m",
            "rhs": "Int[ActivateTrig[u]*(C + A*Sin[a + b*x]^2)/Sin[a + b*x]^2, x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(A_ + C_.*csc[a_. + b_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, C}, x] && KnownSineIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.1 Sine normalization rules.m",
       "filename": "4.7.1 Sine normalization rules.m",
            "rhs": "Int[ActivateTrig[u]*(C + A*Cos[a + b*x]^2)/Cos[a + b*x]^2, x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(A_ + C_.*sec[a_. + b_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, C}, x] && KnownSineIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.1 Sine normalization rules.m",
       "filename": "4.7.1 Sine normalization rules.m",
            "rhs": "Int[ActivateTrig[u]*(C + A*Sin[a + b*x] + B*Sin[a + b*x]^2)/ Sin[a + b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(A_. + B_.*sin[a_. + b_.*x_] + C_.*csc[a_. + b_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B, C}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.1 Sine normalization rules.m",
       "filename": "4.7.1 Sine normalization rules.m",
            "rhs": "Int[ActivateTrig[u]*(C + A*Cos[a + b*x] + B*Cos[a + b*x]^2)/ Cos[a + b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(A_. + B_.*cos[a_. + b_.*x_] + C_.*sec[a_. + b_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B, C}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.1 Sine normalization rules.m",
       "filename": "4.7.1 Sine normalization rules.m",
            "rhs": "Int[ActivateTrig[u]* Sin[a + b*x]^n*(A + B*Sin[a + b*x] + C*Sin[a + b*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(A_.*sin[a_. + b_.*x_]^n_. + B_.*sin[a_. + b_.*x_]^n1_ + C_.*sin[a_. + b_.*x_]^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B, C, n}, x] && EqQ[n1, n + 1] && EqQ[n2, n + 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.1 Sine normalization rules.m",
       "filename": "4.7.1 Sine normalization rules.m",
            "rhs": "Int[ActivateTrig[u]* Cos[a + b*x]^n*(A + B*Cos[a + b*x] + C*Cos[a + b*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(A_.*cos[a_. + b_.*x_]^n_. + B_.*cos[a_. + b_.*x_]^n1_ + C_.*cos[a_. + b_.*x_]^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B, C, n}, x] && EqQ[n1, n + 1] && EqQ[n2, n + 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.2 Tangent normalization rules.m",
       "filename": "4.7.2 Tangent normalization rules.m",
            "rhs": "(c*Cot[a + b*x])^m*(d*Tan[a + b*x])^m* Int[ActivateTrig[u]*(d*Tan[a + b*x])^(n - m), x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*cot[a_. + b_.*x_])^m_.*(d_.*tan[a_. + b_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && KnownTangentIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.2 Tangent normalization rules.m",
       "filename": "4.7.2 Tangent normalization rules.m",
            "rhs": "(c*Tan[a + b*x])^ m*(d*Cos[a + b*x])^m/(d*Sin[a + b*x])^m* Int[ActivateTrig[u]*(d*Sin[a + b*x])^m/(d*Cos[a + b*x])^(m - n), x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*tan[a_. + b_.*x_])^m_.*(d_.*cos[a_. + b_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && KnownCotangentIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.2 Tangent normalization rules.m",
       "filename": "4.7.2 Tangent normalization rules.m",
            "rhs": "(c*Cot[a + b*x])^m*(c*Tan[a + b*x])^m* Int[ActivateTrig[u]/(c*Tan[a + b*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*cot[a_. + b_.*x_])^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m}, x] && Not[IntegerQ[m]] && KnownTangentIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.2 Tangent normalization rules.m",
       "filename": "4.7.2 Tangent normalization rules.m",
            "rhs": "(c*Cot[a + b*x])^m*(c*Tan[a + b*x])^m* Int[ActivateTrig[u]/(c*Cot[a + b*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*tan[a_. + b_.*x_])^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m}, x] && Not[IntegerQ[m]] && KnownCotangentIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.2 Tangent normalization rules.m",
       "filename": "4.7.2 Tangent normalization rules.m",
            "rhs": "c*Int[ActivateTrig[u]*(c*Tan[a + b*x])^(n - 1)*(B + A*Tan[a + b*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*tan[a_. + b_.*x_])^n_.*(A_ + B_.*cot[a_. + b_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, B, n}, x] && KnownTangentIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.2 Tangent normalization rules.m",
       "filename": "4.7.2 Tangent normalization rules.m",
            "rhs": "c*Int[ActivateTrig[u]*(c*Cot[a + b*x])^(n - 1)*(B + A*Cot[a + b*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*cot[a_. + b_.*x_])^n_.*(A_ + B_.*tan[a_. + b_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, B, n}, x] && KnownCotangentIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.2 Tangent normalization rules.m",
       "filename": "4.7.2 Tangent normalization rules.m",
            "rhs": "Int[ActivateTrig[u]*(B + A*Tan[a + b*x])/Tan[a + b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(A_ + B_.*cot[a_. + b_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B}, x] && KnownTangentIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.2 Tangent normalization rules.m",
       "filename": "4.7.2 Tangent normalization rules.m",
            "rhs": "Int[ActivateTrig[u]*(B + A*Cot[a + b*x])/Cot[a + b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(A_ + B_.*tan[a_. + b_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B}, x] && KnownCotangentIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.2 Tangent normalization rules.m",
       "filename": "4.7.2 Tangent normalization rules.m",
            "rhs": "c^2*Int[ActivateTrig[ u]*(c*Tan[a + b*x])^(n - 2)*(C + B*Tan[a + b*x] + A*Tan[a + b*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(c_.*tan[a_. + b_.*x_])^ n_.*(A_. + B_.*cot[a_. + b_.*x_] + C_.*cot[a_. + b_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, B, C, n}, x] && KnownTangentIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.2 Tangent normalization rules.m",
       "filename": "4.7.2 Tangent normalization rules.m",
            "rhs": "c^2*Int[ActivateTrig[ u]*(c*Cot[a + b*x])^(n - 2)*(C + B*Cot[a + b*x] + A*Cot[a + b*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(c_.*cot[a_. + b_.*x_])^ n_.*(A_. + B_.*tan[a_. + b_.*x_] + C_.*tan[a_. + b_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, B, C, n}, x] && KnownCotangentIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.2 Tangent normalization rules.m",
       "filename": "4.7.2 Tangent normalization rules.m",
            "rhs": "c^2*Int[ActivateTrig[ u]*(c*Tan[a + b*x])^(n - 2)*(C + A*Tan[a + b*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(c_.*tan[a_. + b_.*x_])^n_.*(A_ + C_.*cot[a_. + b_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, C, n}, x] && KnownTangentIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.2 Tangent normalization rules.m",
       "filename": "4.7.2 Tangent normalization rules.m",
            "rhs": "c^2*Int[ActivateTrig[ u]*(c*Cot[a + b*x])^(n - 2)*(C + A*Cot[a + b*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(c_.*cot[a_. + b_.*x_])^n_.*(A_ + C_.*tan[a_. + b_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, C, n}, x] && KnownCotangentIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.2 Tangent normalization rules.m",
       "filename": "4.7.2 Tangent normalization rules.m",
            "rhs": "Int[ActivateTrig[u]*(C + B*Tan[a + b*x] + A*Tan[a + b*x]^2)/ Tan[a + b*x]^2, x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(A_. + B_.*cot[a_. + b_.*x_] + C_.*cot[a_. + b_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B, C}, x] && KnownTangentIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.2 Tangent normalization rules.m",
       "filename": "4.7.2 Tangent normalization rules.m",
            "rhs": "Int[ActivateTrig[u]*(C + B*Cot[a + b*x] + A*Cot[a + b*x]^2)/ Cot[a + b*x]^2, x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(A_. + B_.*tan[a_. + b_.*x_] + C_.*tan[a_. + b_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B, C}, x] && KnownCotangentIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.2 Tangent normalization rules.m",
       "filename": "4.7.2 Tangent normalization rules.m",
            "rhs": "Int[ActivateTrig[u]*(C + A*Tan[a + b*x]^2)/Tan[a + b*x]^2, x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(A_ + C_.*cot[a_. + b_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, C}, x] && KnownTangentIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.2 Tangent normalization rules.m",
       "filename": "4.7.2 Tangent normalization rules.m",
            "rhs": "Int[ActivateTrig[u]*(C + A*Cot[a + b*x]^2)/Cot[a + b*x]^2, x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(A_ + C_.*tan[a_. + b_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, C}, x] && KnownCotangentIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.2 Tangent normalization rules.m",
       "filename": "4.7.2 Tangent normalization rules.m",
            "rhs": "Int[ActivateTrig[u]*(C + A*Tan[a + b*x] + B*Tan[a + b*x]^2)/ Tan[a + b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(A_. + B_.*tan[a_. + b_.*x_] + C_.*cot[a_. + b_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B, C}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.2 Tangent normalization rules.m",
       "filename": "4.7.2 Tangent normalization rules.m",
            "rhs": "Int[ActivateTrig[u]* Tan[a + b*x]^n*(A + B*Tan[a + b*x] + C*Tan[a + b*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(A_.*tan[a_. + b_.*x_]^n_. + B_.*tan[a_. + b_.*x_]^n1_ + C_.*tan[a_. + b_.*x_]^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B, C, n}, x] && EqQ[n1, n + 1] && EqQ[n2, n + 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.2 Tangent normalization rules.m",
       "filename": "4.7.2 Tangent normalization rules.m",
            "rhs": "Int[ActivateTrig[u]* Cot[a + b*x]^n*(A + B*Cot[a + b*x] + C*Cot[a + b*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(A_.*cot[a_. + b_.*x_]^n_. + B_.*cot[a_. + b_.*x_]^n1_ + C_.*cot[a_. + b_.*x_]^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B, C, n}, x] && EqQ[n1, n + 1] && EqQ[n2, n + 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.3 Secant normalization rules.m",
       "filename": "4.7.3 Secant normalization rules.m",
            "rhs": "(c*Sin[a + b*x])^m*(d*Csc[a + b*x])^m* Int[ActivateTrig[u]*(d*Csc[a + b*x])^(n - m), x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*sin[a_. + b_.*x_])^m_.*(d_.*csc[a_. + b_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && KnownSecantIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.3 Secant normalization rules.m",
       "filename": "4.7.3 Secant normalization rules.m",
            "rhs": "(c*Cos[a + b*x])^m*(d*Sec[a + b*x])^m* Int[ActivateTrig[u]*(d*Sec[a + b*x])^(n - m), x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*cos[a_. + b_.*x_])^m_.*(d_.*sec[a_. + b_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && KnownSecantIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.3 Secant normalization rules.m",
       "filename": "4.7.3 Secant normalization rules.m",
            "rhs": "(c*Tan[a + b*x])^ m*(d*Csc[a + b*x])^m/(d*Sec[a + b*x])^m* Int[ActivateTrig[u]*(d*Sec[a + b*x])^(m + n)/(d*Csc[a + b*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*tan[a_. + b_.*x_])^m_.*(d_.*sec[a_. + b_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && KnownSecantIntegrandQ[u, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.3 Secant normalization rules.m",
       "filename": "4.7.3 Secant normalization rules.m",
            "rhs": "(c*Tan[a + b*x])^ m*(d*Csc[a + b*x])^m/(d*Sec[a + b*x])^m* Int[ActivateTrig[u]*(d*Sec[a + b*x])^m/(d*Csc[a + b*x])^(m - n), x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*tan[a_. + b_.*x_])^m_.*(d_.*csc[a_. + b_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && KnownSecantIntegrandQ[u, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.3 Secant normalization rules.m",
       "filename": "4.7.3 Secant normalization rules.m",
            "rhs": "(c*Cot[a + b*x])^ m*(d*Sec[a + b*x])^m/(d*Csc[a + b*x])^m* Int[ActivateTrig[u]*(d*Csc[a + b*x])^m/(d*Sec[a + b*x])^(m - n), x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*cot[a_. + b_.*x_])^m_.*(d_.*sec[a_. + b_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && KnownSecantIntegrandQ[u, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.3 Secant normalization rules.m",
       "filename": "4.7.3 Secant normalization rules.m",
            "rhs": "(c*Cot[a + b*x])^ m*(d*Sec[a + b*x])^m/(d*Csc[a + b*x])^m* Int[ActivateTrig[u]*(d*Csc[a + b*x])^(m + n)/(d*Sec[a + b*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*cot[a_. + b_.*x_])^m_.*(d_.*csc[a_. + b_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && KnownSecantIntegrandQ[u, x] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.3 Secant normalization rules.m",
       "filename": "4.7.3 Secant normalization rules.m",
            "rhs": "(c*Csc[a + b*x])^m*(c*Sin[a + b*x])^m* Int[ActivateTrig[u]/(c*Csc[a + b*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*sin[a_. + b_.*x_])^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m}, x] && Not[IntegerQ[m]] && KnownSecantIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.3 Secant normalization rules.m",
       "filename": "4.7.3 Secant normalization rules.m",
            "rhs": "(c*Cos[a + b*x])^m*(c*Sec[a + b*x])^m* Int[ActivateTrig[u]/(c*Sec[a + b*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*cos[a_. + b_.*x_])^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m}, x] && Not[IntegerQ[m]] && KnownSecantIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.3 Secant normalization rules.m",
       "filename": "4.7.3 Secant normalization rules.m",
            "rhs": "(c*Tan[a + b*x])^ m*(c*Csc[a + b*x])^m/(c*Sec[a + b*x])^m* Int[ActivateTrig[u]*(c*Sec[a + b*x])^m/(c*Csc[a + b*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*tan[a_. + b_.*x_])^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m}, x] && Not[IntegerQ[m]] && KnownSecantIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.3 Secant normalization rules.m",
       "filename": "4.7.3 Secant normalization rules.m",
            "rhs": "(c*Cot[a + b*x])^ m*(c*Sec[a + b*x])^m/(c*Csc[a + b*x])^m* Int[ActivateTrig[u]*(c*Csc[a + b*x])^m/(c*Sec[a + b*x])^m, x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*cot[a_. + b_.*x_])^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m}, x] && Not[IntegerQ[m]] && KnownSecantIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.3 Secant normalization rules.m",
       "filename": "4.7.3 Secant normalization rules.m",
            "rhs": "c*Int[ActivateTrig[u]*(c*Sec[a + b*x])^(n - 1)*(B + A*Sec[a + b*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*sec[a_. + b_.*x_])^n_.*(A_ + B_.*cos[a_. + b_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, B, n}, x] && KnownSecantIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.3 Secant normalization rules.m",
       "filename": "4.7.3 Secant normalization rules.m",
            "rhs": "c*Int[ActivateTrig[u]*(c*Csc[a + b*x])^(n - 1)*(B + A*Csc[a + b*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*csc[a_. + b_.*x_])^n_.*(A_ + B_.*sin[a_. + b_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, B, n}, x] && KnownSecantIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.3 Secant normalization rules.m",
       "filename": "4.7.3 Secant normalization rules.m",
            "rhs": "Int[ActivateTrig[u]*(B + A*Sec[a + b*x])/Sec[a + b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(A_ + B_.*cos[a_. + b_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B}, x] && KnownSecantIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.3 Secant normalization rules.m",
       "filename": "4.7.3 Secant normalization rules.m",
            "rhs": "Int[ActivateTrig[u]*(B + A*Csc[a + b*x])/Csc[a + b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(A_ + B_.*sin[a_. + b_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B}, x] && KnownSecantIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.3 Secant normalization rules.m",
       "filename": "4.7.3 Secant normalization rules.m",
            "rhs": "c^2*Int[ActivateTrig[ u]*(c*Sec[a + b*x])^(n - 2)*(C + B*Sec[a + b*x] + A*Sec[a + b*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(c_.*sec[a_. + b_.*x_])^ n_.*(A_. + B_.*cos[a_. + b_.*x_] + C_.*cos[a_. + b_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, B, C, n}, x] && KnownSecantIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.3 Secant normalization rules.m",
       "filename": "4.7.3 Secant normalization rules.m",
            "rhs": "c^2*Int[ActivateTrig[ u]*(c*Csc[a + b*x])^(n - 2)*(C + B*Csc[a + b*x] + A*Csc[a + b*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(c_.*csc[a_. + b_.*x_])^ n_.*(A_. + B_.*sin[a_. + b_.*x_] + C_.*sin[a_. + b_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, B, C, n}, x] && KnownSecantIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.3 Secant normalization rules.m",
       "filename": "4.7.3 Secant normalization rules.m",
            "rhs": "c^2*Int[ActivateTrig[ u]*(c*Sec[a + b*x])^(n - 2)*(C + A*Sec[a + b*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(c_.*sec[a_. + b_.*x_])^n_.*(A_ + C_.*cos[a_. + b_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, C, n}, x] && KnownSecantIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.3 Secant normalization rules.m",
       "filename": "4.7.3 Secant normalization rules.m",
            "rhs": "c^2*Int[ActivateTrig[ u]*(c*Csc[a + b*x])^(n - 2)*(C + A*Csc[a + b*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(c_.*csc[a_. + b_.*x_])^n_.*(A_ + C_.*sin[a_. + b_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, C, n}, x] && KnownSecantIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.3 Secant normalization rules.m",
       "filename": "4.7.3 Secant normalization rules.m",
            "rhs": "Int[ActivateTrig[u]*(C + B*Sec[a + b*x] + A*Sec[a + b*x]^2)/ Sec[a + b*x]^2, x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(A_. + B_.*cos[a_. + b_.*x_] + C_.*cos[a_. + b_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B, C}, x] && KnownSecantIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.3 Secant normalization rules.m",
       "filename": "4.7.3 Secant normalization rules.m",
            "rhs": "Int[ActivateTrig[u]*(C + B*Csc[a + b*x] + A*Csc[a + b*x]^2)/ Csc[a + b*x]^2, x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(A_. + B_.*sin[a_. + b_.*x_] + C_.*sin[a_. + b_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B, C}, x] && KnownSecantIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.3 Secant normalization rules.m",
       "filename": "4.7.3 Secant normalization rules.m",
            "rhs": "Int[ActivateTrig[u]*(C + A*Sec[a + b*x]^2)/Sec[a + b*x]^2, x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(A_ + C_.*cos[a_. + b_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, C}, x] && KnownSecantIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.3 Secant normalization rules.m",
       "filename": "4.7.3 Secant normalization rules.m",
            "rhs": "Int[ActivateTrig[u]*(C + A*Csc[a + b*x]^2)/Csc[a + b*x]^2, x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(A_ + C_.*sin[a_. + b_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, C}, x] && KnownSecantIntegrandQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.3 Secant normalization rules.m",
       "filename": "4.7.3 Secant normalization rules.m",
            "rhs": "Int[ActivateTrig[u]* Sec[a + b*x]^n*(A + B*Sec[a + b*x] + C*Sec[a + b*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(A_.*sec[a_. + b_.*x_]^n_. + B_.*sec[a_. + b_.*x_]^n1_ + C_.*sec[a_. + b_.*x_]^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B, C, n}, x] && EqQ[n1, n + 1] && EqQ[n2, n + 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.3 Secant normalization rules.m",
       "filename": "4.7.3 Secant normalization rules.m",
            "rhs": "Int[ActivateTrig[u]* Csc[a + b*x]^n*(A + B*Csc[a + b*x] + C*Csc[a + b*x]^2), x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(A_.*csc[a_. + b_.*x_]^n_. + B_.*csc[a_. + b_.*x_]^n1_ + C_.*csc[a_. + b_.*x_]^n2_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B, C, n}, x] && EqQ[n1, n + 1] && EqQ[n2, n + 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "Sin[a - c + (b - d)*x]/(2*(b - d)) - Sin[a + c + (b + d)*x]/(2*(b + d))",
     "rulenumber": 0,
            "lhs": "Int[sin[a_. + b_.*x_]*sin[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "Sin[a - c + (b - d)*x]/(2*(b - d)) + Sin[a + c + (b + d)*x]/(2*(b + d))",
     "rulenumber": 0,
            "lhs": "Int[cos[a_. + b_.*x_]*cos[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "-Cos[a - c + (b - d)*x]/(2*(b - d)) - Cos[a + c + (b + d)*x]/(2*(b + d))",
     "rulenumber": 0,
            "lhs": "Int[sin[a_. + b_.*x_]*cos[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "1/2*Int[(g*Sin[c + d*x])^p, x] + 1/2*Int[Cos[c + d*x]*(g*Sin[c + d*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[cos[a_. + b_.*x_]^2*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && IGtQ[p/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "1/2*Int[(g*Sin[c + d*x])^p, x] - 1/2*Int[Cos[c + d*x]*(g*Sin[c + d*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[sin[a_. + b_.*x_]^2*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && IGtQ[p/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "2^p/e^p*Int[(e*Cos[a + b*x])^(m + p)*Sin[a + b*x]^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cos[a_. + b_.*x_])^m_.*sin[c_. + d_.*x_]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "2^p/f^p*Int[Cos[a + b*x]^p*(f*Sin[a + b*x])^(n + p), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*sin[a_. + b_.*x_])^n_.*sin[c_. + d_.*x_]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, n}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "e^2*(e*Cos[a + b*x])^(m - 2)*(g*Sin[c + d*x])^(p + 1)/(2*b* g*(p + 1))",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cos[a_. + b_.*x_])^m_*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, g, m, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && EqQ[m + p - 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "-e^2*(e*Sin[a + b*x])^(m - 2)*(g*Sin[c + d*x])^(p + 1)/(2*b*g*(p + 1))",
     "rulenumber": 0,
            "lhs": "Int[(e_.*sin[a_. + b_.*x_])^m_*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, g, m, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && EqQ[m + p - 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "-(e*Cos[a + b*x])^ m*(g*Sin[c + d*x])^(p + 1)/(b*g*m)",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cos[a_. + b_.*x_])^m_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, g, m, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && EqQ[m + 2*p + 2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "(e*Sin[a + b*x])^ m*(g*Sin[c + d*x])^(p + 1)/(b*g*m)",
     "rulenumber": 0,
            "lhs": "Int[(e_.*sin[a_. + b_.*x_])^m_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, g, m, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && EqQ[m + 2*p + 2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "e^2*(e*Cos[a + b*x])^(m - 2)*(g*Sin[c + d*x])^(p + 1)/(2*b* g*(p + 1)) + e^4*(m + p - 1)/(4*g^2*(p + 1))* Int[(e*Cos[a + b*x])^(m - 4)*(g*Sin[c + d*x])^(p + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cos[a_. + b_.*x_])^m_*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[m, 2] && LtQ[p, -1] && (GtQ[m, 3] || EqQ[p, -3/2]) && IntegersQ[2*m, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "-e^2*(e*Sin[a + b*x])^(m - 2)*(g*Sin[c + d*x])^(p + 1)/(2*b*g*(p + 1)) + e^4*(m + p - 1)/(4*g^2*(p + 1))* Int[(e*Sin[a + b*x])^(m - 4)*(g*Sin[c + d*x])^(p + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*sin[a_. + b_.*x_])^m_*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[m, 2] && LtQ[p, -1] && (GtQ[m, 3] || EqQ[p, -3/2]) && IntegersQ[2*m, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "(e*Cos[a + b*x])^ m*(g*Sin[c + d*x])^(p + 1)/(2*b*g*(p + 1)) + e^2*(m + 2*p + 2)/(4*g^2*(p + 1))* Int[(e*Cos[a + b*x])^(m - 2)*(g*Sin[c + d*x])^(p + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cos[a_. + b_.*x_])^m_*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[m, 1] && LtQ[p, -1] && NeQ[m + 2*p + 2, 0] && (LtQ[p, -2] || EqQ[m, 2]) && IntegersQ[2*m, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "-(e*Sin[a + b*x])^ m*(g*Sin[c + d*x])^(p + 1)/(2*b*g*(p + 1)) + e^2*(m + 2*p + 2)/(4*g^2*(p + 1))* Int[(e*Sin[a + b*x])^(m - 2)*(g*Sin[c + d*x])^(p + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*sin[a_. + b_.*x_])^m_*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[m, 1] && LtQ[p, -1] && NeQ[m + 2*p + 2, 0] && (LtQ[p, -2] || EqQ[m, 2]) && IntegersQ[2*m, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "e^2*(e*Cos[a + b*x])^(m - 2)*(g*Sin[c + d*x])^(p + 1)/(2*b* g*(m + 2*p)) + e^2*(m + p - 1)/(m + 2*p)* Int[(e*Cos[a + b*x])^(m - 2)*(g*Sin[c + d*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cos[a_. + b_.*x_])^m_*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[m, 1] && NeQ[m + 2*p, 0] && IntegersQ[2*m, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "-e^2*(e*Sin[a + b*x])^(m - 2)*(g*Sin[c + d*x])^(p + 1)/(2*b*g*(m + 2*p)) + e^2*(m + p - 1)/(m + 2*p)* Int[(e*Sin[a + b*x])^(m - 2)*(g*Sin[c + d*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*sin[a_. + b_.*x_])^m_*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[m, 1] && NeQ[m + 2*p, 0] && IntegersQ[2*m, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "-(e*Cos[a + b*x])^ m*(g*Sin[c + d*x])^(p + 1)/(2*b*g*(m + p + 1)) + (m + 2*p + 2)/(e^2*(m + p + 1))* Int[(e*Cos[a + b*x])^(m + 2)*(g*Sin[c + d*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cos[a_. + b_.*x_])^m_*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && LtQ[m, -1] && NeQ[m + 2*p + 2, 0] && NeQ[m + p + 1, 0] && IntegersQ[2*m, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "(e*Sin[a + b*x])^ m*(g*Sin[c + d*x])^(p + 1)/(2*b*g*(m + p + 1)) + (m + 2*p + 2)/(e^2*(m + p + 1))* Int[(e*Sin[a + b*x])^(m + 2)*(g*Sin[c + d*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*sin[a_. + b_.*x_])^m_*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && LtQ[m, -1] && NeQ[m + 2*p + 2, 0] && NeQ[m + p + 1, 0] && IntegersQ[2*m, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "2*Sin[a + b*x]*(g*Sin[c + d*x])^p/(d*(2*p + 1)) + 2*p*g/(2*p + 1)*Int[Sin[a + b*x]*(g*Sin[c + d*x])^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[cos[a_. + b_.*x_]*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[p, 0] && IntegerQ[2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "-2*Cos[a + b*x]*(g*Sin[c + d*x])^p/(d*(2*p + 1)) + 2*p*g/(2*p + 1)*Int[Cos[a + b*x]*(g*Sin[c + d*x])^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[sin[a_. + b_.*x_]*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[p, 0] && IntegerQ[2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "Cos[a + b*x]*(g*Sin[c + d*x])^(p + 1)/(2*b*g*(p + 1)) + (2*p + 3)/(2*g*(p + 1))* Int[Sin[a + b*x]*(g*Sin[c + d*x])^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[cos[a_. + b_.*x_]*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && LtQ[p, -1] && IntegerQ[2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "-Sin[ a + b*x]*(g*Sin[c + d*x])^(p + 1)/(2*b*g*(p + 1)) + (2*p + 3)/(2*g*(p + 1))* Int[Cos[a + b*x]*(g*Sin[c + d*x])^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[sin[a_. + b_.*x_]*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && LtQ[p, -1] && IntegerQ[2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "-ArcSin[Cos[a + b*x] - Sin[a + b*x]]/d + Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[c + d*x]]]/d",
     "rulenumber": 0,
            "lhs": "Int[cos[a_. + b_.*x_]/Sqrt[sin[c_. + d_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "-ArcSin[Cos[a + b*x] - Sin[a + b*x]]/d - Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[c + d*x]]]/d",
     "rulenumber": 0,
            "lhs": "Int[sin[a_. + b_.*x_]/Sqrt[sin[c_. + d_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "2*g*Int[Sin[a + b*x]*(g*Sin[c + d*x])^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*sin[c_. + d_.*x_])^p_/cos[a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && IntegerQ[2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "2*g*Int[Cos[a + b*x]*(g*Sin[c + d*x])^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_.*sin[c_. + d_.*x_])^p_/sin[a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && IntegerQ[2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "  -(e*Cos[a+b*x])^(m+1)*Sin[a+b*x]*(g*Sin[c+d*x])^p/(b*e*(m+p+1)*(Sin[ a+b*x]^2)^((p+1)/2))* Hypergeometric2F1[-(p-1)/2,(m+p+1)/2,(m+p+3)/2,Cos[a+b*x]^2]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cos[a_.+b_.*x_])^m_*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e,g,m,p},x] && EqQ[b*c-a*d,0] && EqQ[d/b,2] &&  Not[IntegerQ[p]] && Not[IntegerQ[m+p]] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "  -Cos[a+b*x]*(f*Sin[a+b*x])^(n+1)*(g*Sin[c+d*x])^p/(b*f*(p+1)*(Sin[a+ b*x]^2)^((n+p+1)/2))* Hypergeometric2F1[-(n+p-1)/2,(p+1)/2,(p+3)/2,Cos[a+b*x]^2]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*sin[a_.+b_.*x_])^n_.*(g_.*sin[c_.+d_.*x_])^p_,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,f,g,n,p},x] && EqQ[b*c-a*d,0] && EqQ[d/b,2] &&  Not[IntegerQ[p]] && Not[IntegerQ[n+p]] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "(g*Sin[c + d*x])^ p/((e*Cos[a + b*x])^p*Sin[a + b*x]^p)* Int[(e*Cos[a + b*x])^(m + p)*Sin[a + b*x]^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cos[a_. + b_.*x_])^m_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, g, m, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "(g*Sin[c + d*x])^ p/(Cos[a + b*x]^p*(f*Sin[a + b*x])^p)* Int[Cos[a + b*x]^p*(f*Sin[a + b*x])^(n + p), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*sin[a_. + b_.*x_])^n_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f, g, n, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "1/4*Int[(g*Sin[c + d*x])^p, x] - 1/4*Int[Cos[c + d*x]^2*(g*Sin[c + d*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[cos[a_. + b_.*x_]^2* sin[a_. + b_.*x_]^2*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && IGtQ[p/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "2^p/(e^p*f^p)* Int[(e*Cos[a + b*x])^(m + p)*(f*Sin[a + b*x])^(n + p), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cos[a_. + b_.*x_])^m_.*(f_.*sin[a_. + b_.*x_])^n_.* sin[c_. + d_.*x_]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "e*(e*Cos[a + b*x])^(m - 1)*(f*Sin[a + b*x])^(n + 1)*(g*Sin[c + d*x])^ p/(b*f*(n + p + 1))",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cos[a_. + b_.*x_])^m_.*(f_.*sin[a_. + b_.*x_])^ n_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && EqQ[m + p + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "-e*(e*Sin[a + b*x])^(m - 1)*(f*Cos[a + b*x])^(n + 1)*(g*Sin[c + d*x])^p/(b*f*(n + p + 1))",
     "rulenumber": 0,
            "lhs": "Int[(e_.*sin[a_. + b_.*x_])^m_*(f_.*cos[a_. + b_.*x_])^ n_*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && EqQ[m + p + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "-(e*Cos[a + b*x])^(m + 1)*(f*Sin[a + b*x])^(n + 1)*(g*Sin[c + d*x])^p/(b*e*f*(m + p + 1))",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cos[a_. + b_.*x_])^m_.*(f_.*sin[a_. + b_.*x_])^ n_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && EqQ[m + n + 2*p + 2, 0] && NeQ[m + p + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "e^2*(e*Cos[a + b*x])^(m - 2)*(f*Sin[a + b*x])^ n*(g*Sin[c + d*x])^(p + 1)/(2*b*g*(n + p + 1)) + e^4*(m + p - 1)/(4*g^2*(n + p + 1))* Int[(e*Cos[a + b*x])^(m - 4)*(f*Sin[a + b*x])^ n*(g*Sin[c + d*x])^(p + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cos[a_. + b_.*x_])^m_*(f_.*sin[a_. + b_.*x_])^ n_*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[m, 3] && LtQ[p, -1] && NeQ[n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "-e^2*(e*Sin[a + b*x])^(m - 2)*(f*Cos[a + b*x])^ n*(g*Sin[c + d*x])^(p + 1)/(2*b*g*(n + p + 1)) + e^4*(m + p - 1)/(4*g^2*(n + p + 1))* Int[(e*Sin[a + b*x])^(m - 4)*(f*Cos[a + b*x])^ n*(g*Sin[c + d*x])^(p + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*sin[a_. + b_.*x_])^m_*(f_.*cos[a_. + b_.*x_])^ n_*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[m, 3] && LtQ[p, -1] && NeQ[n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "(e*Cos[a + b*x])^m*(f*Sin[a + b*x])^ n*(g*Sin[c + d*x])^(p + 1)/(2*b*g*(n + p + 1)) + e^2*(m + n + 2*p + 2)/(4*g^2*(n + p + 1))* Int[(e*Cos[a + b*x])^(m - 2)*(f*Sin[a + b*x])^ n*(g*Sin[c + d*x])^(p + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cos[a_. + b_.*x_])^m_*(f_.*sin[a_. + b_.*x_])^ n_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[m, 1] && LtQ[p, -1] && NeQ[m + n + 2*p + 2, 0] && NeQ[n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p] && (LtQ[p, -2] || EqQ[m, 2] || EqQ[m, 3])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "-(e*Sin[a + b*x])^m*(f*Cos[a + b*x])^ n*(g*Sin[c + d*x])^(p + 1)/(2*b*g*(n + p + 1)) + e^2*(m + n + 2*p + 2)/(4*g^2*(n + p + 1))* Int[(e*Sin[a + b*x])^(m - 2)*(f*Cos[a + b*x])^ n*(g*Sin[c + d*x])^(p + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*sin[a_. + b_.*x_])^m_*(f_.*cos[a_. + b_.*x_])^ n_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[m, 1] && LtQ[p, -1] && NeQ[m + n + 2*p + 2, 0] && NeQ[n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p] && (LtQ[p, -2] || EqQ[m, 2] || EqQ[m, 3])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "e*(e*Cos[a + b*x])^(m - 1)*(f*Sin[a + b*x])^(n + 1)*(g*Sin[c + d*x])^ p/(b*f*(n + p + 1)) + e^2*(m + p - 1)/(f^2*(n + p + 1))* Int[(e*Cos[a + b*x])^(m - 2)*(f*Sin[a + b*x])^(n + 2)*(g* Sin[c + d*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cos[a_. + b_.*x_])^m_*(f_.*sin[a_. + b_.*x_])^ n_*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[m, 1] && LtQ[n, -1] && NeQ[n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "-e*(e*Sin[a + b*x])^(m - 1)*(f*Cos[a + b*x])^(n + 1)*(g*Sin[c + d*x])^p/(b*f*(n + p + 1)) + e^2*(m + p - 1)/(f^2*(n + p + 1))* Int[(e*Sin[a + b*x])^(m - 2)*(f*Cos[a + b*x])^(n + 2)*(g* Sin[c + d*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*sin[a_. + b_.*x_])^m_*(f_.*cos[a_. + b_.*x_])^ n_*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[m, 1] && LtQ[n, -1] && NeQ[n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "e*(e*Cos[a + b*x])^(m - 1)*(f*Sin[a + b*x])^(n + 1)*(g*Sin[c + d*x])^ p/(b*f*(m + n + 2*p)) + e^2*(m + p - 1)/(m + n + 2*p)* Int[(e*Cos[a + b*x])^(m - 2)*(f*Sin[a + b*x])^n*(g*Sin[c + d*x])^ p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cos[a_. + b_.*x_])^m_*(f_.*sin[a_. + b_.*x_])^ n_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[m, 1] && NeQ[m + n + 2*p, 0] && IntegersQ[2*m, 2*n, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "-e*(e*Sin[a + b*x])^(m - 1)*(f*Cos[a + b*x])^(n + 1)*(g*Sin[c + d*x])^p/(b*f*(m + n + 2*p)) + e^2*(m + p - 1)/(m + n + 2*p)* Int[(e*Sin[a + b*x])^(m - 2)*(f*Cos[a + b*x])^n*(g*Sin[c + d*x])^ p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*sin[a_. + b_.*x_])^m_*(f_.*cos[a_. + b_.*x_])^ n_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && GtQ[m, 1] && NeQ[m + n + 2*p, 0] && IntegersQ[2*m, 2*n, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "-f*(e*Cos[a + b*x])^(m + 1)*(f*Sin[a + b*x])^(n - 1)*(g*Sin[c + d*x])^p/(b*e*(m + n + 2*p)) + 2*f*g*(n + p - 1)/(e*(m + n + 2*p))* Int[(e*Cos[a + b*x])^(m + 1)*(f*Sin[a + b*x])^(n - 1)*(g* Sin[c + d*x])^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cos[a_. + b_.*x_])^m_*(f_.*sin[a_. + b_.*x_])^ n_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n + 2*p, 0] && IntegersQ[2*m, 2*n, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "f*(e*Sin[a + b*x])^(m + 1)*(f*Cos[a + b*x])^(n - 1)*(g*Sin[c + d*x])^ p/(b*e*(m + n + 2*p)) + 2*f*g*(n + p - 1)/(e*(m + n + 2*p))* Int[(e*Sin[a + b*x])^(m + 1)*(f*Cos[a + b*x])^(n - 1)*(g* Sin[c + d*x])^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*sin[a_. + b_.*x_])^m_*(f_.*cos[a_. + b_.*x_])^ n_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n + 2*p, 0] && IntegersQ[2*m, 2*n, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "-(e*Cos[a + b*x])^(m + 1)*(f*Sin[a + b*x])^(n + 1)*(g*Sin[c + d*x])^p/(b*e*f*(m + p + 1)) + f*(m + n + 2*p + 2)/(2*e*g*(m + p + 1))* Int[(e*Cos[a + b*x])^(m + 1)*(f*Sin[a + b*x])^(n - 1)*(g* Sin[c + d*x])^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cos[a_. + b_.*x_])^m_*(f_.*sin[a_. + b_.*x_])^ n_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && LtQ[m, -1] && GtQ[n, 0] && LtQ[p, -1] && NeQ[m + n + 2*p + 2, 0] && NeQ[m + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "(e*Sin[a + b*x])^(m + 1)*(f*Cos[a + b*x])^(n + 1)*(g*Sin[c + d*x])^p/(b*e*f*(m + p + 1)) + f*(m + n + 2*p + 2)/(2*e*g*(m + p + 1))* Int[(e*Sin[a + b*x])^(m + 1)*(f*Cos[a + b*x])^(n - 1)*(g* Sin[c + d*x])^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*sin[a_. + b_.*x_])^m_*(f_.*cos[a_. + b_.*x_])^ n_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && LtQ[m, -1] && GtQ[n, 0] && LtQ[p, -1] && NeQ[m + n + 2*p + 2, 0] && NeQ[m + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "-(e*Cos[a + b*x])^(m + 1)*(f*Sin[a + b*x])^(n + 1)*(g*Sin[c + d*x])^p/(b*e*f*(m + p + 1)) + (m + n + 2*p + 2)/(e^2*(m + p + 1))* Int[(e*Cos[a + b*x])^(m + 2)*(f*Sin[a + b*x])^n*(g*Sin[c + d*x])^ p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cos[a_. + b_.*x_])^m_*(f_.*sin[a_. + b_.*x_])^ n_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && LtQ[m, -1] && NeQ[m + n + 2*p + 2, 0] && NeQ[m + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "(e*Sin[a + b*x])^(m + 1)*(f*Cos[a + b*x])^(n + 1)*(g*Sin[c + d*x])^p/(b*e*f*(m + p + 1)) + (m + n + 2*p + 2)/(e^2*(m + p + 1))* Int[(e*Sin[a + b*x])^(m + 2)*(f*Cos[a + b*x])^n*(g*Sin[c + d*x])^ p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*sin[a_. + b_.*x_])^m_*(f_.*cos[a_. + b_.*x_])^ n_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]] && LtQ[m, -1] && NeQ[m + n + 2*p + 2, 0] && NeQ[m + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": " -(e*Cos[a+b*x])^(m+1)*(f*Sin[a+b*x])^(n+1)*(g*Sin[c+d*x])^p/(b*e*f*(m+ p+1)*(Sin[a+b*x]^2)^((n+p+1)/2))* Hypergeometric2F1[-(n+p-1)/2,(m+p+1)/2,(m+p+3)/2,Cos[a+b*x]^2]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cos[a_.+b_.*x_])^m_*(f_.*sin[a_.+b_.*x_])^n_.*(g_.*sin[c_. +d_.*x_])^p_,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e,f,g,m,n,p},x] && EqQ[b*c-a*d,0] && EqQ[d/b,2] &&  Not[IntegerQ[p]] && Not[IntegerQ[m+p]] && Not[IntegerQ[n+p]] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "(g*Sin[c + d*x])^ p/((e*Cos[a + b*x])^p*(f*Sin[a + b*x])^p)* Int[(e*Cos[a + b*x])^(m + p)*(f*Sin[a + b*x])^(n + p), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cos[a_. + b_.*x_])^m_.*(f_.*sin[a_. + b_.*x_])^ n_.*(g_.*sin[c_. + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.4 (c trig)^m (d trig)^n.m",
       "filename": "4.7.4 (c trig)^m (d trig)^n.m",
            "rhs": "-(m + 2)*(e*Cos[a + b*x])^(m + 1)* Cos[(m + 1)*(a + b*x)]/(d*e*(m + 1))",
     "rulenumber": 0,
            "lhs": "Int[(e_.*cos[a_. + b_.*x_])^m_.*sin[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, Abs[m + 2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{v = ActivateTrig[F[c + d*x]]}, a^IntPart[n]*(v/NonfreeFactors[v, x])^(p*IntPart[n])*(a*v^p)^ FracPart[n]/NonfreeFactors[v, x]^(p*FracPart[n])* Int[NonfreeFactors[v, x]^(n*p), x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*F_[c_. + d_.*x_]^p_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n, p}, x] && InertTrigQ[F] && Not[IntegerQ[n]] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{v = ActivateTrig[F[c + d*x]]}, a^IntPart[n]*(a*(b*v)^p)^FracPart[n]/(b*v)^(p*FracPart[n])* Int[(b*v)^(n*p), x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_.*(b_.*F_[c_. + d_.*x_])^p_)^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x] && InertTrigQ[F] && Not[IntegerQ[n]] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, d/(b*c)* Subst[Int[SubstFor[1, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d] /; FunctionOfQ[Sin[c*(a + b*x)]/d, u, x, True]]",
     "rulenumber": 0,
            "lhs": "Int[u_*F_[c_.*(a_. + b_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && (EqQ[F, Cos] || EqQ[F, cos])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -d/(b*c)* Subst[Int[SubstFor[1, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x, True]]",
     "rulenumber": 0,
            "lhs": "Int[u_*F_[c_.*(a_. + b_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && (EqQ[F, Sin] || EqQ[F, sin])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Sinh[c*(a + b*x)], x]}, d/(b*c)* Subst[Int[SubstFor[1, Sinh[c*(a + b*x)]/d, u, x], x], x, Sinh[c*(a + b*x)]/d] /; FunctionOfQ[Sinh[c*(a + b*x)]/d, u, x, True]]",
     "rulenumber": 0,
            "lhs": "Int[u_*Cosh[c_.*(a_. + b_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Cosh[c*(a + b*x)], x]}, d/(b*c)* Subst[Int[SubstFor[1, Cosh[c*(a + b*x)]/d, u, x], x], x, Cosh[c*(a + b*x)]/d] /; FunctionOfQ[Cosh[c*(a + b*x)]/d, u, x, True]]",
     "rulenumber": 0,
            "lhs": "Int[u_*Sinh[c_.*(a_. + b_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, 1/(b*c)* Subst[Int[SubstFor[1/x, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d] /; FunctionOfQ[Sin[c*(a + b*x)]/d, u, x, True]]",
     "rulenumber": 0,
            "lhs": "Int[u_*F_[c_.*(a_. + b_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && (EqQ[F, Cot] || EqQ[F, cot])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -1/(b*c)* Subst[Int[SubstFor[1/x, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x, True]]",
     "rulenumber": 0,
            "lhs": "Int[u_*F_[c_.*(a_. + b_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && (EqQ[F, Tan] || EqQ[F, tan])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Sinh[c*(a + b*x)], x]}, 1/(b*c)* Subst[Int[SubstFor[1/x, Sinh[c*(a + b*x)]/d, u, x], x], x, Sinh[c*(a + b*x)]/d] /; FunctionOfQ[Sinh[c*(a + b*x)]/d, u, x, True]]",
     "rulenumber": 0,
            "lhs": "Int[u_*Coth[c_.*(a_. + b_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Cosh[c*(a + b*x)], x]}, 1/(b*c)* Subst[Int[SubstFor[1/x, Cosh[c*(a + b*x)]/d, u, x], x], x, Cosh[c*(a + b*x)]/d] /; FunctionOfQ[Cosh[c*(a + b*x)]/d, u, x, True]]",
     "rulenumber": 0,
            "lhs": "Int[u_*Tanh[c_.*(a_. + b_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Tan[c*(a + b*x)], x]}, d/(b*c)* Subst[Int[SubstFor[1, Tan[c*(a + b*x)]/d, u, x], x], x, Tan[c*(a + b*x)]/d] /; FunctionOfQ[Tan[c*(a + b*x)]/d, u, x, True]]",
     "rulenumber": 0,
            "lhs": "Int[u_*F_[c_.*(a_. + b_.*x_)]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NonsumQ[u] && (EqQ[F, Sec] || EqQ[F, sec])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Tan[c*(a + b*x)], x]}, d/(b*c)* Subst[Int[SubstFor[1, Tan[c*(a + b*x)]/d, u, x], x], x, Tan[c*(a + b*x)]/d] /; FunctionOfQ[Tan[c*(a + b*x)]/d, u, x, True]]",
     "rulenumber": 0,
            "lhs": "Int[u_/cos[c_.*(a_. + b_.*x_)]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NonsumQ[u]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Cot[c*(a + b*x)], x]}, -d/(b*c)* Subst[Int[SubstFor[1, Cot[c*(a + b*x)]/d, u, x], x], x, Cot[c*(a + b*x)]/d] /; FunctionOfQ[Cot[c*(a + b*x)]/d, u, x, True]]",
     "rulenumber": 0,
            "lhs": "Int[u_*F_[c_.*(a_. + b_.*x_)]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NonsumQ[u] && (EqQ[F, Csc] || EqQ[F, csc])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Cot[c*(a + b*x)], x]}, -d/(b*c)* Subst[Int[SubstFor[1, Cot[c*(a + b*x)]/d, u, x], x], x, Cot[c*(a + b*x)]/d] /; FunctionOfQ[Cot[c*(a + b*x)]/d, u, x, True]]",
     "rulenumber": 0,
            "lhs": "Int[u_/sin[c_.*(a_. + b_.*x_)]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NonsumQ[u]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Tanh[c*(a + b*x)], x]}, d/(b*c)* Subst[Int[SubstFor[1, Tanh[c*(a + b*x)]/d, u, x], x], x, Tanh[c*(a + b*x)]/d] /; FunctionOfQ[Tanh[c*(a + b*x)]/d, u, x, True]]",
     "rulenumber": 0,
            "lhs": "Int[u_*Sech[c_.*(a_. + b_.*x_)]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NonsumQ[u]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Coth[c*(a + b*x)], x]}, -d/(b*c)* Subst[Int[SubstFor[1, Coth[c*(a + b*x)]/d, u, x], x], x, Coth[c*(a + b*x)]/d] /; FunctionOfQ[Coth[c*(a + b*x)]/d, u, x, True]]",
     "rulenumber": 0,
            "lhs": "Int[u_*Csch[c_.*(a_. + b_.*x_)]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && NonsumQ[u]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Tan[c*(a + b*x)], x]}, 1/(b*c*d^(n - 1))* Subst[Int[ SubstFor[1/(x^n*(1 + d^2*x^2)), Tan[c*(a + b*x)]/d, u, x], x], x, Tan[c*(a + b*x)]/d] /; FunctionOfQ[Tan[c*(a + b*x)]/d, u, x, True] && TryPureTanSubst[ActivateTrig[u]*Cot[c*(a + b*x)]^n, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*F_[c_.*(a_. + b_.*x_)]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IntegerQ[n] && (EqQ[F, Cot] || EqQ[F, cot])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Cot[c*(a + b*x)], x]}, -1/(b*c*d^(n - 1))* Subst[Int[ SubstFor[1/(x^n*(1 + d^2*x^2)), Cot[c*(a + b*x)]/d, u, x], x], x, Cot[c*(a + b*x)]/d] /; FunctionOfQ[Cot[c*(a + b*x)]/d, u, x, True] && TryPureTanSubst[ActivateTrig[u]*Tan[c*(a + b*x)]^n, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*F_[c_.*(a_. + b_.*x_)]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IntegerQ[n] && (EqQ[F, Tan] || EqQ[F, tan])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Tanh[c*(a + b*x)], x]}, 1/(b*c*d^(n - 1))* Subst[Int[ SubstFor[1/(x^n*(1 - d^2*x^2)), Tanh[c*(a + b*x)]/d, u, x], x], x, Tanh[c*(a + b*x)]/d] /; FunctionOfQ[Tanh[c*(a + b*x)]/d, u, x, True] && TryPureTanSubst[ActivateTrig[u]*Coth[c*(a + b*x)]^n, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*Coth[c_.*(a_. + b_.*x_)]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Coth[c*(a + b*x)], x]}, 1/(b*c*d^(n - 1))* Subst[Int[ SubstFor[1/(x^n*(1 - d^2*x^2)), Coth[c*(a + b*x)]/d, u, x], x], x, Coth[c*(a + b*x)]/d] /; FunctionOfQ[Coth[c*(a + b*x)]/d, u, x, True] && TryPureTanSubst[ActivateTrig[u]*Tanh[c*(a + b*x)]^n, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*Tanh[c_.*(a_. + b_.*x_)]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{v = FunctionOfTrig[u, x]}, With[{d = FreeFactors[Cot[v], x]}, Dist[-d/Coefficient[v, x, 1], Subst[Int[SubstFor[1/(1 + d^2*x^2), Cot[v]/d, u, x], x], x, Cot[v]/d], x]]",
     "rulenumber": 0,
            "lhs": "Int[u_, x_Symbol] := With[{v = FunctionOfTrig[u, x]}, ShowStep[\"\", \"Int[F[Cot[a+b*x]],x]\", \"-1/b*Subst[Int[F[x]/(1+x^2),x],x,Cot[a+b*x]]\", Hold[ With[{d = FreeFactors[Cot[v], x]}, Dist[-d/Coefficient[v, x, 1], Subst[Int[SubstFor[1/(1 + d^2*x^2), Cot[v]/d, u, x], x], x, Cot[v]/d], x]]]] /; Not[FalseQ[v]] && FunctionOfQ[NonfreeFactors[Cot[v], x], u, x, True] && TryPureTanSubst[ActivateTrig[u], x]] /; SimplifyFlag, Int[u_, x_Symbol]",
        "comment": false,
         "givens": "Not[FalseQ[v]] && FunctionOfQ[NonfreeFactors[Cot[v], x], u, x, True] && TryPureTanSubst[ActivateTrig[u], x]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{v = FunctionOfTrig[u, x]}, With[{d = FreeFactors[Tan[v], x]}, Dist[d/Coefficient[v, x, 1], Subst[Int[SubstFor[1/(1 + d^2*x^2), Tan[v]/d, u, x], x], x, Tan[v]/d], x]]",
     "rulenumber": 0,
            "lhs": "Int[u_, x_Symbol] := With[{v = FunctionOfTrig[u, x]}, ShowStep[\"\", \"Int[F[Tan[a+b*x]],x]\", \"1/b*Subst[Int[F[x]/(1+x^2),x],x,Tan[a+b*x]]\", Hold[ With[{d = FreeFactors[Tan[v], x]}, Dist[d/Coefficient[v, x, 1], Subst[Int[SubstFor[1/(1 + d^2*x^2), Tan[v]/d, u, x], x], x, Tan[v]/d], x]]]] /; Not[FalseQ[v]] && FunctionOfQ[NonfreeFactors[Tan[v], x], u, x, True] && TryPureTanSubst[ActivateTrig[u], x]] /; SimplifyFlag, Int[u_, x_Symbol]",
        "comment": false,
         "givens": "Not[FalseQ[v]] && FunctionOfQ[NonfreeFactors[Tan[v], x], u, x, True] && TryPureTanSubst[ActivateTrig[u], x]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "Int[ExpandTrigReduce[ActivateTrig[F[a + b*x]^p*G[c + d*x]^q], x], x]",
     "rulenumber": 0,
            "lhs": "Int[F_[a_. + b_.*x_]^p_.*G_[c_. + d_.*x_]^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && (EqQ[F, sin] || EqQ[F, cos]) && (EqQ[G, sin] || EqQ[G, cos]) && IGtQ[p, 0] && IGtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "Int[ExpandTrigReduce[ ActivateTrig[F[a + b*x]^p*G[c + d*x]^q*H[e + f*x]^r], x], x]",
     "rulenumber": 0,
            "lhs": "Int[F_[a_. + b_.*x_]^p_.*G_[c_. + d_.*x_]^q_.*H_[e_. + f_.*x_]^r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && (EqQ[F, sin] || EqQ[F, cos]) && (EqQ[G, sin] || EqQ[G, cos]) && (EqQ[H, sin] || EqQ[H, cos]) && IGtQ[p, 0] && IGtQ[q, 0] && IGtQ[r, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, d/(b*c)* Subst[Int[SubstFor[1, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d] /; FunctionOfQ[Sin[c*(a + b*x)]/d, u, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*F_[c_.*(a_. + b_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && (EqQ[F, Cos] || EqQ[F, cos])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -d/(b*c)* Subst[Int[SubstFor[1, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*F_[c_.*(a_. + b_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && (EqQ[F, Sin] || EqQ[F, sin])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Sinh[c*(a + b*x)], x]}, d/(b*c)* Subst[Int[SubstFor[1, Sinh[c*(a + b*x)]/d, u, x], x], x, Sinh[c*(a + b*x)]/d] /; FunctionOfQ[Sinh[c*(a + b*x)]/d, u, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*Cosh[c_.*(a_. + b_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Cosh[c*(a + b*x)], x]}, d/(b*c)* Subst[Int[SubstFor[1, Cosh[c*(a + b*x)]/d, u, x], x], x, Cosh[c*(a + b*x)]/d] /; FunctionOfQ[Cosh[c*(a + b*x)]/d, u, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*Sinh[c_.*(a_. + b_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, 1/(b*c)* Subst[Int[SubstFor[1/x, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d] /; FunctionOfQ[Sin[c*(a + b*x)]/d, u, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*F_[c_.*(a_. + b_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && (EqQ[F, Cot] || EqQ[F, cot])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -1/(b*c)* Subst[Int[SubstFor[1/x, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*F_[c_.*(a_. + b_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && (EqQ[F, Tan] || EqQ[F, tan])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Sinh[c*(a + b*x)], x]}, 1/(b*c)* Subst[Int[SubstFor[1/x, Sinh[c*(a + b*x)]/d, u, x], x], x, Sinh[c*(a + b*x)]/d] /; FunctionOfQ[Sinh[c*(a + b*x)]/d, u, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*Coth[c_.*(a_. + b_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Cosh[c*(a + b*x)], x]}, 1/(b*c)* Subst[Int[SubstFor[1/x, Cosh[c*(a + b*x)]/d, u, x], x], x, Cosh[c*(a + b*x)]/d] /; FunctionOfQ[Cosh[c*(a + b*x)]/d, u, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*Tanh[c_.*(a_. + b_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, d/(b*c)* Subst[Int[ SubstFor[(1 - d^2*x^2)^((n - 1)/2), Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d] /; FunctionOfQ[Sin[c*(a + b*x)]/d, u, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*F_[c_.*(a_. + b_.*x_)]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Cos] || EqQ[F, cos])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, d/(b*c)* Subst[Int[ SubstFor[(1 - d^2*x^2)^((-n - 1)/2), Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d] /; FunctionOfQ[Sin[c*(a + b*x)]/d, u, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*F_[c_.*(a_. + b_.*x_)]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Sec] || EqQ[F, sec])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -d/(b*c)* Subst[Int[ SubstFor[(1 - d^2*x^2)^((n - 1)/2), Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*F_[c_.*(a_. + b_.*x_)]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Sin] || EqQ[F, sin])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -d/(b*c)* Subst[Int[ SubstFor[(1 - d^2*x^2)^((-n - 1)/2), Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*F_[c_.*(a_. + b_.*x_)]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Csc] || EqQ[F, csc])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Sinh[c*(a + b*x)], x]}, d/(b*c)* Subst[Int[ SubstFor[(1 + d^2*x^2)^((n - 1)/2), Sinh[c*(a + b*x)]/d, u, x], x], x, Sinh[c*(a + b*x)]/d] /; FunctionOfQ[Sinh[c*(a + b*x)]/d, u, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*Cosh[c_.*(a_. + b_.*x_)]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Sinh[c*(a + b*x)], x]}, d/(b*c)* Subst[Int[ SubstFor[(1 + d^2*x^2)^((-n - 1)/2), Sinh[c*(a + b*x)]/d, u, x], x], x, Sinh[c*(a + b*x)]/d] /; FunctionOfQ[Sinh[c*(a + b*x)]/d, u, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*Sech[c_.*(a_. + b_.*x_)]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Cosh[c*(a + b*x)], x]}, d/(b*c)* Subst[Int[ SubstFor[(-1 + d^2*x^2)^((n - 1)/2), Cosh[c*(a + b*x)]/d, u, x], x], x, Cosh[c*(a + b*x)]/d] /; FunctionOfQ[Cosh[c*(a + b*x)]/d, u, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*Sinh[c_.*(a_. + b_.*x_)]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Cosh[c*(a + b*x)], x]}, d/(b*c)* Subst[Int[ SubstFor[(-1 + d^2*x^2)^((-n - 1)/2), Cosh[c*(a + b*x)]/d, u, x], x], x, Cosh[c*(a + b*x)]/d] /; FunctionOfQ[Cosh[c*(a + b*x)]/d, u, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*Csch[c_.*(a_. + b_.*x_)]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, 1/(b*c*d^(n - 1))* Subst[Int[ SubstFor[(1 - d^2*x^2)^((n - 1)/2)/x^n, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d] /; FunctionOfQ[Sin[c*(a + b*x)]/d, u, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*F_[c_.*(a_. + b_.*x_)]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Cot] || EqQ[F, cot])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -1/(b*c*d^(n - 1))* Subst[Int[ SubstFor[(1 - d^2*x^2)^((n - 1)/2)/x^n, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*F_[c_.*(a_. + b_.*x_)]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Tan] || EqQ[F, tan])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Sinh[c*(a + b*x)], x]}, 1/(b*c*d^(n - 1))* Subst[Int[ SubstFor[(1 + d^2*x^2)^((n - 1)/2)/x^n, Sinh[c*(a + b*x)]/d, u, x], x], x, Sinh[c*(a + b*x)]/d] /; FunctionOfQ[Sinh[c*(a + b*x)]/d, u, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*Coth[c_.*(a_. + b_.*x_)]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{d = FreeFactors[Cosh[c*(a + b*x)], x]}, 1/(b*c*d^(n - 1))* Subst[Int[ SubstFor[(-1 + d^2*x^2)^((n - 1)/2)/x^n, Cosh[c*(a + b*x)]/d, u, x], x], x, Cosh[c*(a + b*x)]/d] /; FunctionOfQ[Cosh[c*(a + b*x)]/d, u, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*Tanh[c_.*(a_. + b_.*x_)]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{e = FreeFactors[Sin[c*(a + b*x)], x]}, Int[ActivateTrig[u*v], x] + d*Int[ActivateTrig[u]*Cos[c*(a + b*x)]^n, x] /; FunctionOfQ[Sin[c*(a + b*x)]/e, u, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*(v_ + d_.*F_[c_.*(a_. + b_.*x_)]^n_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && Not[FreeQ[v, x]] && IntegerQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Cos] || EqQ[F, cos])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{e = FreeFactors[Cos[c*(a + b*x)], x]}, Int[ActivateTrig[u*v], x] + d*Int[ActivateTrig[u]*Sin[c*(a + b*x)]^n, x] /; FunctionOfQ[Cos[c*(a + b*x)]/e, u, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*(v_ + d_.*F_[c_.*(a_. + b_.*x_)]^n_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && Not[FreeQ[v, x]] && IntegerQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Sin] || EqQ[F, sin])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{v = FunctionOfTrig[u, x]}, With[{d = FreeFactors[Sin[v], x]}, Dist[d/Coefficient[v, x, 1], Subst[Int[SubstFor[1, Sin[v]/d, u/Cos[v], x], x], x, Sin[v]/d], x]]",
     "rulenumber": 0,
            "lhs": "Int[u_, x_Symbol] := With[{v = FunctionOfTrig[u, x]}, ShowStep[\"\", \"Int[F[Sin[a+b*x]]*Cos[a+b*x],x]\", \"Subst[Int[F[x],x],x,Sin[a+b*x]]/b\", Hold[ With[{d = FreeFactors[Sin[v], x]}, Dist[d/Coefficient[v, x, 1], Subst[Int[SubstFor[1, Sin[v]/d, u/Cos[v], x], x], x, Sin[v]/d], x]]]] /; Not[FalseQ[v]] && FunctionOfQ[NonfreeFactors[Sin[v], x], u/Cos[v], x]] /; SimplifyFlag, Int[u_, x_Symbol]",
        "comment": false,
         "givens": "Not[FalseQ[v]] && FunctionOfQ[NonfreeFactors[Sin[v], x], u/Cos[v], x]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{v = FunctionOfTrig[u, x]}, With[{d = FreeFactors[Cos[v], x]}, Dist[-d/Coefficient[v, x, 1], Subst[Int[SubstFor[1, Cos[v]/d, u/Sin[v], x], x], x, Cos[v]/d], x]]",
     "rulenumber": 0,
            "lhs": "Int[u_, x_Symbol] := With[{v = FunctionOfTrig[u, x]}, ShowStep[\"\", \"Int[F[Cos[a+b*x]]*Sin[a+b*x],x]\", \"-Subst[Int[F[x],x],x,Cos[a+b*x]]/b\", Hold[ With[{d = FreeFactors[Cos[v], x]}, Dist[-d/Coefficient[v, x, 1], Subst[Int[SubstFor[1, Cos[v]/d, u/Sin[v], x], x], x, Cos[v]/d], x]]]] /; Not[FalseQ[v]] && FunctionOfQ[NonfreeFactors[Cos[v], x], u/Sin[v], x]] /; SimplifyFlag, Int[u_, x_Symbol]",
        "comment": false,
         "givens": "Not[FalseQ[v]] && FunctionOfQ[NonfreeFactors[Cos[v], x], u/Sin[v], x]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "(a + c)^p*Int[ActivateTrig[u], x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_. + b_.*cos[d_. + e_.*x_]^2 + c_.*sin[d_. + e_.*x_]^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[b - c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "(a + c)^p*Int[ActivateTrig[u], x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_. + b_.*tan[d_. + e_.*x_]^2 + c_.*sec[d_. + e_.*x_]^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[b + c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "(a + c)^p*Int[ActivateTrig[u], x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_. + b_.*cot[d_. + e_.*x_]^2 + c_.*csc[d_. + e_.*x_]^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[b + c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{q = DerivativeDivides[ActivateTrig[y], ActivateTrig[u], x]}, q*Log[RemoveContent[ActivateTrig[y], x]] /; Not[FalseQ[q]]]",
     "rulenumber": 0,
            "lhs": "Int[u_/y_, x_Symbol]",
        "comment": false,
         "givens": "Not[InertTrigFreeQ[u]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{q = DerivativeDivides[ActivateTrig[y*w], ActivateTrig[u], x]}, q*Log[RemoveContent[ActivateTrig[y*w], x]] /; Not[FalseQ[q]]]",
     "rulenumber": 0,
            "lhs": "Int[u_/(y_*w_), x_Symbol]",
        "comment": false,
         "givens": "Not[InertTrigFreeQ[u]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{q = DerivativeDivides[ActivateTrig[y], ActivateTrig[u], x]}, q*ActivateTrig[y^(m + 1)]/(m + 1) /; Not[FalseQ[q]]]",
     "rulenumber": 0,
            "lhs": "Int[u_*y_^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[m, x] && NeQ[m, -1] && Not[InertTrigFreeQ[u]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{q = DerivativeDivides[ActivateTrig[y*z], ActivateTrig[u*z^(n - m)], x]}, q*ActivateTrig[y^(m + 1)*z^(m + 1)]/(m + 1) /; Not[FalseQ[q]]]",
     "rulenumber": 0,
            "lhs": "Int[u_*y_^m_.*z_^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{m, n}, x] && NeQ[m, -1] && Not[InertTrigFreeQ[u]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{v = ActivateTrig[F[c + d*x]]}, a^IntPart[n]*(v/NonfreeFactors[v, x])^(p*IntPart[n])*(a*v^p)^ FracPart[n]/NonfreeFactors[v, x]^(p*FracPart[n])* Int[ActivateTrig[u]*NonfreeFactors[v, x]^(n*p), x]]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_.*F_[c_. + d_.*x_]^p_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n, p}, x] && InertTrigQ[F] && Not[IntegerQ[n]] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{v = ActivateTrig[F[c + d*x]]}, a^IntPart[n]*(a*(b*v)^p)^FracPart[n]/(b*v)^(p*FracPart[n])* Int[ActivateTrig[u]*(b*v)^(n*p), x]]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_.*(b_.*F_[c_. + d_.*x_])^p_)^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x] && InertTrigQ[F] && Not[IntegerQ[n]] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{v = FunctionOfTrig[u, x]}, With[{d = FreeFactors[Tan[v], x]}, Dist[d/Coefficient[v, x, 1], Subst[Int[SubstFor[1/(1 + d^2*x^2), Tan[v]/d, u, x], x], x, Tan[v]/d], x]] /; Not[FalseQ[v]] && FunctionOfQ[NonfreeFactors[Tan[v], x], u, x]] /; InverseFunctionFreeQ[u, x] && Not[ MatchQ[u, v_.*(c_.*tan[w_]^n_.*tan[z_]^n_.)^p_.",
     "rulenumber": 0,
            "lhs": "Int[u_, x_Symbol] := With[{v = FunctionOfTrig[u, x]}, ShowStep[\"\", \"Int[F[Tan[a+b*x]],x]\", \"1/b*Subst[Int[F[x]/(1+x^2),x],x,Tan[a+b*x]]\", Hold[ With[{d = FreeFactors[Tan[v], x]}, Dist[d/Coefficient[v, x, 1], Subst[Int[SubstFor[1/(1 + d^2*x^2), Tan[v]/d, u, x], x], x, Tan[v]/d], x]]]] /; Not[FalseQ[v]] && FunctionOfQ[NonfreeFactors[Tan[v], x], u, x]] /; SimplifyFlag && InverseFunctionFreeQ[u, x] && Not[ MatchQ[u, v_.*(c_.*tan[w_]^n_.*tan[z_]^n_.)^p_. /; FreeQ[{c, p}, x] && IntegerQ[n] && LinearQ[w, x] && EqQ[z, 2*w]]], Int[u_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, p}, x] && IntegerQ[n] && LinearQ[w, x] && EqQ[z, 2*w]]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{w = FunctionOfTrig[u*Sin[v/2]^(2*m)/(c*Tan[v/2])^m, x]}, (c*Sin[v])^m*(c*Tan[v/2])^m/Sin[v/2]^(2*m)* Int[u*Sin[v/2]^(2*m)/(c*Tan[v/2])^m, x] /; Not[FalseQ[w]] && FunctionOfQ[NonfreeFactors[Tan[w], x], u*Sin[v/2]^(2*m)/(c*Tan[v/2])^m, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*sin[v_])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[c, x] && LinearQ[v, x] && IntegerQ[m + 1/2] && Not[SumQ[u]] && InverseFunctionFreeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "Int[ActivateTrig[u]*Sec[c + d*x]^(n*p)*(b + a*Sin[c + d*x]^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_.*tan[c_. + d_.*x_]^n_. + b_.*sec[c_. + d_.*x_]^n_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "Int[ActivateTrig[u]*Csc[c + d*x]^(n*p)*(b + a*Cos[c + d*x]^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_.*cot[c_. + d_.*x_]^n_. + b_.*csc[c_. + d_.*x_]^n_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "Int[ActivateTrig[u*F[c + d*x]^(n*p)*(a + b*F[c + d*x]^(q - p))^n], x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(a_*F_[c_. + d_.*x_]^p_. + b_.*F_[c_. + d_.*x_]^q_.)^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, p, q}, x] && InertTrigQ[F] && IntegerQ[n] && PosQ[q - p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "Int[ActivateTrig[ u*F[d + e*x]^(n*p)*(a + b*F[d + e*x]^(q - p) + c*F[d + e*x]^(r - p))^n], x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(a_*F_[d_. + e_.*x_]^p_. + b_.*F_[d_. + e_.*x_]^q_. + c_.*F_[d_. + e_.*x_]^r_.)^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p, q, r}, x] && InertTrigQ[F] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "Int[ActivateTrig[ u*F[d + e*x]^(n*p)*(b + a*F[d + e*x]^(-p) + c*F[d + e*x]^(q - p))^ n], x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(a_ + b_.*F_[d_. + e_.*x_]^p_. + c_.*F_[d_. + e_.*x_]^q_.)^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p, q}, x] && InertTrigQ[F] && IntegerQ[n] && NegQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "Int[ActivateTrig[u]*(a*E^(-a/b*(c + d*x)))^n, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_.*cos[c_. + d_.*x_] + b_.*sin[c_. + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "Int[TrigSimplify[u], x]",
     "rulenumber": 0,
            "lhs": "Int[u_, x_Symbol]",
        "comment": false,
         "givens": "TrigSimplifyQ[u]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{uu = ActivateTrig[u], vv = ActivateTrig[v]}, a^IntPart[p]*(a*vv)^FracPart[p]/(vv^FracPart[p])* Int[uu*vv^p, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_*v_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, p}, x] && Not[IntegerQ[p]] && Not[InertTrigFreeQ[v]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{uu = ActivateTrig[u], vv = ActivateTrig[v]}, (vv^m)^FracPart[p]/(vv^(m*FracPart[p]))*Int[uu*vv^(m*p), x]]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(v_^m_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{m, p}, x] && Not[IntegerQ[p]] && Not[InertTrigFreeQ[v]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{uu = ActivateTrig[u], vv = ActivateTrig[v], ww = ActivateTrig[w]}, (vv^m*ww^n)^FracPart[p]/(vv^(m*FracPart[p])*ww^(n*FracPart[p]))* Int[uu*vv^(m*p)*ww^(n*p), x]]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(v_^m_.*w_^n_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{m, n, p}, x] && Not[IntegerQ[p]] && (Not[InertTrigFreeQ[v]] || Not[InertTrigFreeQ[w]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{v = ExpandTrig[u, x]}, Int[v, x] /; SumQ[v]]",
     "rulenumber": 0,
            "lhs": "Int[u_, x_Symbol]",
        "comment": false,
         "givens": "Not[InertTrigFreeQ[u]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{w = Block[{$ShowSteps = False, $StepCounter = Null}, Int[SubstFor[ 1/(1 + FreeFactors[Tan[FunctionOfTrig[u, x]/2], x]^2*x^2), Tan[FunctionOfTrig[u, x]/2]/ FreeFactors[Tan[FunctionOfTrig[u, x]/2], x], u, x], x]]}, Module[{v = FunctionOfTrig[u, x], d}, d = FreeFactors[Tan[v/2], x]; Dist[2*d/Coefficient[v, x, 1], Subst[Int[SubstFor[1/(1 + d^2*x^2), Tan[v/2]/d, u, x], x], x, Tan[v/2]/d], x]] /; CalculusFreeQ[w, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_, x_Symbol] := With[{w = Block[{$ShowSteps = False, $StepCounter = Null}, Int[SubstFor[ 1/(1 + FreeFactors[Tan[FunctionOfTrig[u, x]/2], x]^2*x^2), Tan[FunctionOfTrig[u, x]/2]/ FreeFactors[Tan[FunctionOfTrig[u, x]/2], x], u, x], x]]}, ShowStep[\"\", \"Int[F[Sin[a+b*x],Cos[a+b*x]],x]\", \"2/b*Subst[Int[1/(1+x^2)*F[2*x/(1+x^2),(1-x^2)/(1+x^2)],x],x, Tan[(a+b*x)/2]]\", Hold[ Module[{v = FunctionOfTrig[u, x], d}, d = FreeFactors[Tan[v/2], x]; Dist[2*d/Coefficient[v, x, 1], Subst[Int[SubstFor[1/(1 + d^2*x^2), Tan[v/2]/d, u, x], x], x, Tan[v/2]/d], x]]]] /; CalculusFreeQ[w, x]] /; SimplifyFlag && InverseFunctionFreeQ[u, x] && Not[FalseQ[FunctionOfTrig[u, x]]], Int[u_, x_Symbol]",
        "comment": false,
         "givens": "InverseFunctionFreeQ[u, x] && Not[FalseQ[FunctionOfTrig[u, x]]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{v=FunctionOfTrig[u,x]}, With[{d=FreeFactors[Tan[v/2],x]}, Dist[2*d/Coefficient[v,x,1],Subst[Int[SubstFor[1/(1+d^2*x^2),Tan[v/ 2]/d,u,x],x],x,Tan[v/2]/d],x]] /; Not[FalseQ[v]]]",
     "rulenumber": 0,
            "lhs": "Int[u_,x_Symbol] := With[{v=FunctionOfTrig[u,x]}, ShowStep[\"\",\"Int[F[Sin[a+b*x],Cos[a+b*x]],x]\",\"2/b*Subst[Int[1/(1+x^ 2)*F[2*x/(1+x^2),(1-x^2)/(1+x^2)],x],x,Tan[(a+b*x)/2]]\",Hold[ With[{d=FreeFactors[Tan[v/2],x]}, Dist[2*d/Coefficient[v,x,1],Subst[Int[SubstFor[1/(1+d^2*x^2),Tan[v/ 2]/d,u,x],x],x,Tan[v/2]/d],x]]]] /; Not[FalseQ[v]]] /; SimplifyFlag &&  InverseFunctionFreeQ[u,x], Int[u_,x_Symbol]",
        "comment": false,
         "givens": " InverseFunctionFreeQ[u,x]] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.5 Inert trig functions.m",
       "filename": "4.7.5 Inert trig functions.m",
            "rhs": "With[{v = ActivateTrig[u]}, CannotIntegrate[v, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_, x_Symbol]",
        "comment": false,
         "givens": "Not[InertTrigFreeQ[u]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
       "filename": "4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
            "rhs": "(c + d*x)^m*Sin[a + b*x]^(n + 1)/(b*(n + 1)) - d*m/(b*(n + 1))*Int[(c + d*x)^(m - 1)*Sin[a + b*x]^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Sin[a_. + b_.*x_]^n_.*Cos[a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
       "filename": "4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
            "rhs": "-(c + d*x)^m*Cos[a + b*x]^(n + 1)/(b*(n + 1)) + d*m/(b*(n + 1))*Int[(c + d*x)^(m - 1)*Cos[a + b*x]^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Sin[a_. + b_.*x_]*Cos[a_. + b_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
       "filename": "4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
            "rhs": "Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Sin[a_. + b_.*x_]^n_.*Cos[a_. + b_.*x_]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
       "filename": "4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
            "rhs": "-Int[(c + d*x)^m*Sin[a + b*x]^n* Tan[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Sin[a + b*x]^(n - 2)*Tan[a + b*x]^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Sin[a_. + b_.*x_]^n_.*Tan[a_. + b_.*x_]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
       "filename": "4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
            "rhs": "-Int[(c + d*x)^m*Cos[a + b*x]^n* Cot[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Cos[a + b*x]^(n - 2)*Cot[a + b*x]^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Cos[a_. + b_.*x_]^n_.*Cot[a_. + b_.*x_]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
       "filename": "4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
            "rhs": "(c + d*x)^m*Sec[a + b*x]^n/(b*n) - d*m/(b*n)*Int[(c + d*x)^(m - 1)*Sec[a + b*x]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Sec[a_. + b_.*x_]^n_.*Tan[a_. + b_.*x_]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
       "filename": "4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
            "rhs": "-(c + d*x)^m*Csc[a + b*x]^n/(b*n) + d*m/(b*n)*Int[(c + d*x)^(m - 1)*Csc[a + b*x]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Csc[a_. + b_.*x_]^n_.*Cot[a_. + b_.*x_]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
       "filename": "4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
            "rhs": "(c + d*x)^m*Tan[a + b*x]^(n + 1)/(b*(n + 1)) - d*m/(b*(n + 1))*Int[(c + d*x)^(m - 1)*Tan[a + b*x]^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Sec[a_. + b_.*x_]^2*Tan[a_. + b_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
       "filename": "4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
            "rhs": "-(c + d*x)^m*Cot[a + b*x]^(n + 1)/(b*(n + 1)) + d*m/(b*(n + 1))*Int[(c + d*x)^(m - 1)*Cot[a + b*x]^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Csc[a_. + b_.*x_]^2*Cot[a_. + b_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
       "filename": "4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
            "rhs": "-Int[(c + d*x)^m*Sec[a + b*x]*Tan[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Sec[a + b*x]^3*Tan[a + b*x]^(p - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Sec[a_. + b_.*x_]*Tan[a_. + b_.*x_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && IGtQ[p/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
       "filename": "4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
            "rhs": "-Int[(c + d*x)^m*Sec[a + b*x]^n* Tan[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Sec[a + b*x]^(n + 2)*Tan[a + b*x]^(p - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Sec[a_. + b_.*x_]^n_.*Tan[a_. + b_.*x_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
       "filename": "4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
            "rhs": "-Int[(c + d*x)^m*Csc[a + b*x]*Cot[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Csc[a + b*x]^3*Cot[a + b*x]^(p - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Csc[a_. + b_.*x_]*Cot[a_. + b_.*x_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && IGtQ[p/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
       "filename": "4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
            "rhs": "-Int[(c + d*x)^m*Csc[a + b*x]^n* Cot[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Csc[a + b*x]^(n + 2)*Cot[a + b*x]^(p - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Csc[a_. + b_.*x_]^n_.*Cot[a_. + b_.*x_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
       "filename": "4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
            "rhs": "Module[{u = IntHide[Sec[a + b*x]^n*Tan[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - d*m*Int[(c + d*x)^(m - 1)*u, x]]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Sec[a_. + b_.*x_]^n_.*Tan[a_. + b_.*x_]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x] && IGtQ[m, 0] && (IntegerQ[n/2] || IntegerQ[(p - 1)/2])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
       "filename": "4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
            "rhs": "Module[{u = IntHide[Csc[a + b*x]^n*Cot[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - d*m*Int[(c + d*x)^(m - 1)*u, x]]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Csc[a_. + b_.*x_]^n_.*Cot[a_. + b_.*x_]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x] && IGtQ[m, 0] && (IntegerQ[n/2] || IntegerQ[(p - 1)/2])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
       "filename": "4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
            "rhs": "2^n*Int[(c + d*x)^m*Csc[2*a + 2*b*x]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Csc[a_. + b_.*x_]^n_.*Sec[a_. + b_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && IntegerQ[n] && RationalQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
       "filename": "4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
            "rhs": "Module[{u = IntHide[Csc[a + b*x]^n*Sec[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - d*m*Int[(c + d*x)^(m - 1)*u, x]]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Csc[a_. + b_.*x_]^n_.*Sec[a_. + b_.*x_]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n, p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
       "filename": "4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
            "rhs": "Int[ExpandToSum[u, x]^m*F[ExpandToSum[v, x]]^n* G[ExpandToSum[v, x]]^p, x]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*F_[v_]^n_.*G_[w_]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{m, n, p}, x] && TrigQ[F] && TrigQ[G] && EqQ[v, w] && LinearQ[{u, v, w}, x] && Not[LinearMatchQ[{u, v, w}, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
       "filename": "4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
            "rhs": "(e + f*x)^ m*(a + b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1)) - f*m/(b*d*(n + 1))* Int[(e + f*x)^(m - 1)*(a + b*Sin[c + d*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Cos[c_. + d_.*x_]*(a_ + b_.*Sin[c_. + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
       "filename": "4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
            "rhs": "-(e + f*x)^ m*(a + b*Cos[c + d*x])^(n + 1)/(b*d*(n + 1)) + f*m/(b*d*(n + 1))* Int[(e + f*x)^(m - 1)*(a + b*Cos[c + d*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Sin[c_. + d_.*x_]*(a_ + b_.*Cos[c_. + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
       "filename": "4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
            "rhs": "(e + f*x)^ m*(a + b*Tan[c + d*x])^(n + 1)/(b*d*(n + 1)) - f*m/(b*d*(n + 1))* Int[(e + f*x)^(m - 1)*(a + b*Tan[c + d*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Sec[c_. + d_.*x_]^2*(a_ + b_.*Tan[c_. + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
       "filename": "4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
            "rhs": "-(e + f*x)^ m*(a + b*Cot[c + d*x])^(n + 1)/(b*d*(n + 1)) + f*m/(b*d*(n + 1))* Int[(e + f*x)^(m - 1)*(a + b*Cot[c + d*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Csc[c_. + d_.*x_]^2*(a_ + b_.*Cot[c_. + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
       "filename": "4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
            "rhs": "(e + f*x)^ m*(a + b*Sec[c + d*x])^(n + 1)/(b*d*(n + 1)) - f*m/(b*d*(n + 1))* Int[(e + f*x)^(m - 1)*(a + b*Sec[c + d*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Sec[c_. + d_.*x_]* Tan[c_. + d_.*x_]*(a_ + b_.*Sec[c_. + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
       "filename": "4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
            "rhs": "-(e + f*x)^ m*(a + b*Csc[c + d*x])^(n + 1)/(b*d*(n + 1)) + f*m/(b*d*(n + 1))* Int[(e + f*x)^(m - 1)*(a + b*Csc[c + d*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Csc[c_. + d_.*x_]* Cot[c_. + d_.*x_]*(a_ + b_.*Csc[c_. + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
       "filename": "4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
            "rhs": "Int[ExpandTrigReduce[(e + f*x)^m, Sin[a + b*x]^p*Sin[c + d*x]^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Sin[a_. + b_.*x_]^p_.*Sin[c_. + d_.*x_]^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IGtQ[q, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
       "filename": "4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
            "rhs": "Int[ExpandTrigReduce[(e + f*x)^m, Cos[a + b*x]^p*Cos[c + d*x]^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Cos[a_. + b_.*x_]^p_.*Cos[c_. + d_.*x_]^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IGtQ[q, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
       "filename": "4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
            "rhs": "Int[ExpandTrigReduce[(e + f*x)^m, Sin[a + b*x]^p*Cos[c + d*x]^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Sin[a_. + b_.*x_]^p_.*Cos[c_. + d_.*x_]^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] && IGtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
       "filename": "4.7.6 (c+d x)^m trig(a+b x)^n trig(a+b x)^p.m",
            "rhs": "Int[ExpandTrigExpand[(e + f*x)^m*G[c + d*x]^q, F, c + d*x, p, b/d, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*F_[a_. + b_.*x_]^p_.*G_[c_. + d_.*x_]^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && MemberQ[{Sin, Cos}, F] && MemberQ[{Sec, Csc}, G] && IGtQ[p, 0] && IGtQ[q, 0] && EqQ[b*c - a*d, 0] && IGtQ[b/d, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "b*c*Log[F]*F^(c*(a + b*x))*Sin[d + e*x]/(e^2 + b^2*c^2*Log[F]^2) - e*F^(c*(a + b*x))*Cos[d + e*x]/(e^2 + b^2*c^2*Log[F]^2)",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Sin[d_. + e_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 + b^2*c^2*Log[F]^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "b*c*Log[F]*F^(c*(a + b*x))*Cos[d + e*x]/(e^2 + b^2*c^2*Log[F]^2) + e*F^(c*(a + b*x))*Sin[d + e*x]/(e^2 + b^2*c^2*Log[F]^2)",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Cos[d_. + e_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 + b^2*c^2*Log[F]^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "b*c*Log[F]*F^(c*(a + b*x))* Sin[d + e*x]^n/(e^2*n^2 + b^2*c^2*Log[F]^2) - e*n*F^(c*(a + b*x))*Cos[d + e*x]* Sin[d + e*x]^(n - 1)/(e^2*n^2 + b^2*c^2*Log[F]^2) + (n*(n - 1)*e^2)/(e^2*n^2 + b^2*c^2*Log[F]^2)* Int[F^(c*(a + b*x))*Sin[d + e*x]^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Sin[d_. + e_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*n^2 + b^2*c^2*Log[F]^2, 0] && GtQ[n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "b*c*Log[F]*F^(c*(a + b*x))* Cos[d + e*x]^m/(e^2*m^2 + b^2*c^2*Log[F]^2) + e*m*F^(c*(a + b*x))*Sin[d + e*x]* Cos[d + e*x]^(m - 1)/(e^2*m^2 + b^2*c^2*Log[F]^2) + (m*(m - 1)*e^2)/(e^2*m^2 + b^2*c^2*Log[F]^2)* Int[F^(c*(a + b*x))*Cos[d + e*x]^(m - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Cos[d_. + e_.*x_]^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*m^2 + b^2*c^2*Log[F]^2, 0] && GtQ[m, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "-b*c*Log[F]*F^(c*(a + b*x))* Sin[d + e*x]^(n + 2)/(e^2*(n + 1)*(n + 2)) + F^(c*(a + b*x))*Cos[d + e*x]*Sin[d + e*x]^(n + 1)/(e*(n + 1))",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Sin[d_. + e_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, n}, x] && EqQ[e^2*(n + 2)^2 + b^2*c^2*Log[F]^2, 0] && NeQ[n, -1] && NeQ[n, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "-b*c*Log[F]*F^(c*(a + b*x))* Cos[d + e*x]^(n + 2)/(e^2*(n + 1)*(n + 2)) - F^(c*(a + b*x))*Sin[d + e*x]*Cos[d + e*x]^(n + 1)/(e*(n + 1))",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Cos[d_. + e_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, n}, x] && EqQ[e^2*(n + 2)^2 + b^2*c^2*Log[F]^2, 0] && NeQ[n, -1] && NeQ[n, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "-b*c*Log[F]*F^(c*(a + b*x))* Sin[d + e*x]^(n + 2)/(e^2*(n + 1)*(n + 2)) + F^(c*(a + b*x))*Cos[d + e*x]*Sin[d + e*x]^(n + 1)/(e*(n + 1)) + (e^2*(n + 2)^2 + b^2*c^2*Log[F]^2)/(e^2*(n + 1)*(n + 2))* Int[F^(c*(a + b*x))*Sin[d + e*x]^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Sin[d_. + e_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*(n + 2)^2 + b^2*c^2*Log[F]^2, 0] && LtQ[n, -1] && NeQ[n, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "-b*c*Log[F]*F^(c*(a + b*x))* Cos[d + e*x]^(n + 2)/(e^2*(n + 1)*(n + 2)) - F^(c*(a + b*x))*Sin[d + e*x]*Cos[d + e*x]^(n + 1)/(e*(n + 1)) + (e^2*(n + 2)^2 + b^2*c^2*Log[F]^2)/(e^2*(n + 1)*(n + 2))* Int[F^(c*(a + b*x))*Cos[d + e*x]^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Cos[d_. + e_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*(n + 2)^2 + b^2*c^2*Log[F]^2, 0] && LtQ[n, -1] && NeQ[n, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "E^(I*n*(d + e*x))*Sin[d + e*x]^n/(-1 + E^(2*I*(d + e*x)))^n* Int[F^(c*(a + b*x))*(-1 + E^(2*I*(d + e*x)))^n/E^(I*n*(d + e*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Sin[d_. + e_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, n}, x] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "E^(I*n*(d + e*x))*Cos[d + e*x]^n/(1 + E^(2*I*(d + e*x)))^n* Int[F^(c*(a + b*x))*(1 + E^(2*I*(d + e*x)))^n/E^(I*n*(d + e*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Cos[d_. + e_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, n}, x] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "I^n*Int[ExpandIntegrand[ F^(c*(a + b*x))*(1 - E^(2*I*(d + e*x)))^ n/(1 + E^(2*I*(d + e*x)))^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Tan[d_. + e_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "(-I)^n* Int[ExpandIntegrand[ F^(c*(a + b*x))*(1 + E^(2*I*(d + e*x)))^ n/(1 - E^(2*I*(d + e*x)))^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Cot[d_. + e_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "b*c*Log[F]* F^(c*(a + b*x))*(Sec[d + e x]^n/(e^2*n^2 + b^2*c^2*Log[F]^2)) - e*n*F^(c*(a + b*x))* Sec[d + e x]^(n + 1)*(Sin[d + e x]/(e^2*n^2 + b^2*c^2*Log[F]^2)) + e^2*n*((n + 1)/(e^2*n^2 + b^2*c^2*Log[F]^2))* Int[F^(c*(a + b*x))*Sec[d + e x]^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Sec[d_. + e_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*n^2 + b^2*c^2*Log[F]^2, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "b*c*Log[F]* F^(c*(a + b*x))*(Csc[d + e x]^n/(e^2*n^2 + b^2*c^2*Log[F]^2)) + e*n*F^(c*(a + b*x))* Csc[d + e x]^(n + 1)*(Cos[d + e x]/(e^2*n^2 + b^2*c^2*Log[F]^2)) + e^2*n*((n + 1)/(e^2*n^2 + b^2*c^2*Log[F]^2))* Int[F^(c*(a + b*x))*Csc[d + e x]^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Csc[d_. + e_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*n^2 + b^2*c^2*Log[F]^2, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "-b*c*Log[F]*F^(c*(a + b*x))* Sec[d + e x]^(n - 2)/(e^2*(n - 1)*(n - 2)) + F^(c*(a + b*x))*Sec[d + e x]^(n - 1)*Sin[d + e x]/(e*(n - 1))",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Sec[d_. + e_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, n}, x] && EqQ[b^2*c^2*Log[F]^2 + e^2*(n - 2)^2, 0] && NeQ[n, 1] && NeQ[n, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "-b*c*Log[F]*F^(c*(a + b*x))* Csc[d + e x]^(n - 2)/(e^2*(n - 1)*(n - 2)) + F^(c*(a + b*x))*Csc[d + e x]^(n - 1)*Cos[d + e x]/(e*(n - 1))",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Csc[d_. + e_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, n}, x] && EqQ[b^2*c^2*Log[F]^2 + e^2*(n - 2)^2, 0] && NeQ[n, 1] && NeQ[n, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "-b*c*Log[F]*F^(c*(a + b*x))* Sec[d + e x]^(n - 2)/(e^2*(n - 1)*(n - 2)) + F^(c*(a + b*x))*Sec[d + e x]^(n - 1)*Sin[d + e x]/(e*(n - 1)) + (e^2*(n - 2)^2 + b^2*c^2*Log[F]^2)/(e^2*(n - 1)*(n - 2))* Int[F^(c*(a + b*x))*Sec[d + e x]^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Sec[d_. + e_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && NeQ[b^2*c^2*Log[F]^2 + e^2*(n - 2)^2, 0] && GtQ[n, 1] && NeQ[n, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "-b*c*Log[F]*F^(c*(a + b*x))* Csc[d + e x]^(n - 2)/(e^2*(n - 1)*(n - 2)) - F^(c*(a + b*x))*Csc[d + e x]^(n - 1)*Cos[d + e x]/(e*(n - 1)) + (e^2*(n - 2)^2 + b^2*c^2*Log[F]^2)/(e^2*(n - 1)*(n - 2))* Int[F^(c*(a + b*x))*Csc[d + e x]^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Csc[d_. + e_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && NeQ[b^2*c^2*Log[F]^2 + e^2*(n - 2)^2, 0] && GtQ[n, 1] && NeQ[n, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": " 2^n*Int[SimplifyIntegrand[F^(c*(a+b*x))*E^(I*n*(d+e*x))/(1+E^(2*I*(d+ e*x)))^n,x],x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_.+b_.*x_))*Sec[d_.+e_.*x_]^n_.,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F,a,b,c,d,e},x] && IntegerQ[n] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": " (2*I)^n*Int[SimplifyIntegrand[F^(c*(a+b*x))*E^(-I*n*(d+e*x))/(1-E^(-2* I*(d+e*x)))^n,x],x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_.+b_.*x_))*Csc[d_.+e_.*x_]^n_.,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F,a,b,c,d,e},x] && IntegerQ[n] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "2^n*E^(I*k*n*Pi)*E^(I*n*(d + e*x))* F^(c*(a + b*x))/(I*e*n + b*c*Log[F])* Hypergeometric2F1[n, n/2 - I*b*c*Log[F]/(2*e), 1 + n/2 - I*b*c*Log[F]/(2*e), -E^(2*I*k*Pi)*E^(2*I*(d + e*x))]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Sec[d_. + k_.*Pi + e_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[4*k] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "2^n*E^(I*n*(d + e*x))*F^(c*(a + b*x))/(I*e*n + b*c*Log[F])* Hypergeometric2F1[n, n/2 - I*b*c*Log[F]/(2*e), 1 + n/2 - I*b*c*Log[F]/(2*e), -E^(2*I*(d + e*x))]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Sec[d_. + e_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "(-2*I)^n*E^(I*k*n*Pi)* E^(I*n*(d + e*x))*(F^(c*(a + b*x))/(I*e*n + b*c*Log[F]))* Hypergeometric2F1[n, n/2 - I*b*c*Log[F]/(2*e), 1 + n/2 - I*b*c*Log[F]/(2*e), E^(2*I*k*Pi)*E^(2*I*(d + e*x))]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Csc[d_. + k_.*Pi + e_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[4*k] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "(-2*I)^n* E^(I*n*(d + e*x))*(F^(c*(a + b*x))/(I*e*n + b*c*Log[F]))* Hypergeometric2F1[n, n/2 - I*b*c*Log[F]/(2*e), 1 + n/2 - I*b*c*Log[F]/(2*e), E^(2*I*(d + e*x))]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Csc[d_. + e_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "(1 + E^(2*I*(d + e*x)))^n* Sec[d + e*x]^n/E^(I*n*(d + e*x))* Int[SimplifyIntegrand[ F^(c*(a + b*x))*E^(I*n*(d + e*x))/(1 + E^(2*I*(d + e*x)))^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Sec[d_. + e_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "(1 - E^(-2*I*(d + e*x)))^n* Csc[d + e*x]^n/E^(-I*n*(d + e*x))* Int[SimplifyIntegrand[ F^(c*(a + b*x))*E^(-I*n*(d + e*x))/(1 - E^(-2*I*(d + e*x)))^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Csc[d_. + e_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "2^n*f^n*Int[F^(c*(a + b*x))*Cos[d/2 + e*x/2 - f*Pi/(4*g)]^(2*n), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*(f_ + g_.*Sin[d_. + e_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f^2 - g^2, 0] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "2^n*f^n*Int[F^(c*(a + b*x))*Cos[d/2 + e*x/2]^(2*n), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*(f_ + g_.*Cos[d_. + e_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f - g, 0] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "2^n*f^n*Int[F^(c*(a + b*x))*Sin[d/2 + e*x/2]^(2*n), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*(f_ + g_.*Cos[d_. + e_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f + g, 0] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "g^n*Int[F^(c*(a + b*x))*Tan[f*Pi/(4*g) - d/2 - e*x/2]^m, x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))* Cos[d_. + e_.*x_]^m_.*(f_ + g_.*Sin[d_. + e_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f^2 - g^2, 0] && IntegersQ[m, n] && EqQ[m + n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "f^n*Int[F^(c*(a + b*x))*Tan[d/2 + e*x/2]^m, x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))* Sin[d_. + e_.*x_]^m_.*(f_ + g_.*Cos[d_. + e_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f - g, 0] && IntegersQ[m, n] && EqQ[m + n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "f^n*Int[F^(c*(a + b*x))*Cot[d/2 + e*x/2]^m, x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))* Sin[d_. + e_.*x_]^m_.*(f_ + g_.*Cos[d_. + e_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f + g, 0] && IntegersQ[m, n] && EqQ[m + n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "2*i*Int[F^(c*(a + b*x))*(Cos[d + e*x]/(f + g*Sin[d + e*x])), x] + Int[F^(c*(a + b*x))*((h - i*Cos[d + e*x])/(f + g*Sin[d + e*x])), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*(h_ + i_.*Cos[d_. + e_.*x_])/(f_ + g_.*Sin[d_. + e_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g, h, i}, x] && EqQ[f^2 - g^2, 0] && EqQ[h^2 - i^2, 0] && EqQ[g*h - f*i, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "2*i*Int[F^(c*(a + b*x))*(Sin[d + e*x]/(f + g*Cos[d + e*x])), x] + Int[F^(c*(a + b*x))*((h - i*Sin[d + e*x])/(f + g*Cos[d + e*x])), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*(h_ + i_.*Sin[d_. + e_.*x_])/(f_ + g_.*Cos[d_. + e_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g, h, i}, x] && EqQ[f^2 - g^2, 0] && EqQ[h^2 - i^2, 0] && EqQ[g*h + f*i, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "Int[F^(c*ExpandToSum[u, x])*G[ExpandToSum[v, x]]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*u_)*G_[v_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, c, n}, x] && TrigQ[G] && LinearQ[{u, v}, x] && Not[LinearMatchQ[{u, v}, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "Module[{u = IntHide[F^(c*(a + b*x))*Sin[d + e*x]^n, x]}, Dist[(f*x)^m, u, x] - f*m*Int[(f*x)^(m - 1)*u, x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*F_^(c_.*(a_. + b_.*x_))*Sin[d_. + e_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f}, x] && IGtQ[n, 0] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "Module[{u = IntHide[F^(c*(a + b*x))*Cos[d + e*x]^n, x]}, Dist[(f*x)^m, u, x] - f*m*Int[(f*x)^(m - 1)*u, x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*F_^(c_.*(a_. + b_.*x_))*Cos[d_. + e_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f}, x] && IGtQ[n, 0] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "(f*x)^(m + 1)/(f*(m + 1))*F^(c*(a + b*x))* Sin[d + e*x] - e/(f*(m + 1))* Int[(f*x)^(m + 1)*F^(c*(a + b*x))*Cos[d + e*x], x] - b*c*Log[F]/(f*(m + 1))* Int[(f*x)^(m + 1)*F^(c*(a + b*x))*Sin[d + e*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*F_^(c_.*(a_. + b_.*x_))*Sin[d_. + e_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, m}, x] && (LtQ[m, -1] || SumSimplerQ[m, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "(f*x)^(m + 1)/(f*(m + 1))*F^(c*(a + b*x))* Cos[d + e*x] + e/(f*(m + 1))* Int[(f*x)^(m + 1)*F^(c*(a + b*x))*Sin[d + e*x], x] - b*c*Log[F]/(f*(m + 1))* Int[(f*x)^(m + 1)*F^(c*(a + b*x))*Cos[d + e*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*F_^(c_.*(a_. + b_.*x_))*Cos[d_. + e_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, m}, x] && (LtQ[m, -1] || SumSimplerQ[m, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": " I^n/2^n*Int[ExpandIntegrand[(f*x)^m*F^(c*(a+b*x)),(E^(-I*(d+e*x))-E^( I*(d+e*x)))^n,x],x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*F_^(c_.*(a_.+b_.*x_))*Sin[d_.+e_.*x_]^n_.,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{F,a,b,c,d,e,f},x] && IGtQ[n,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": " 1/2^n*Int[ExpandIntegrand[(f*x)^m*F^(c*(a+b*x)),(E^(-I*(d+e*x))+E^(I*( d+e*x)))^n,x],x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*F_^(c_.*(a_.+b_.*x_))*Cos[d_.+e_.*x_]^n_.,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{F,a,b,c,d,e,f},x] && IGtQ[n,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "Int[ExpandTrigReduce[F^(c*(a + b*x)), Sin[d + e*x]^m*Cos[f + g*x]^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Sin[d_. + e_.*x_]^m_.* Cos[f_. + g_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "Int[ExpandTrigReduce[x^p*F^(c*(a + b*x)), Sin[d + e*x]^m*Cos[f + g*x]^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^p_.*F_^(c_.*(a_. + b_.*x_))*Sin[d_. + e_.*x_]^m_.* Cos[f_. + g_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "Int[ExpandTrigToExp[F^(c*(a + b*x)), G[d + e*x]^m*H[d + e*x]^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*G_[d_. + e_.*x_]^m_.*H_[d_. + e_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && IGtQ[m, 0] && IGtQ[n, 0] && TrigQ[G] && TrigQ[H]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "Int[ExpandTrigToExp[F^u, Sin[v]^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[F_^u_*Sin[v_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[F, x] && (LinearQ[u, x] || PolyQ[u, x, 2]) && (LinearQ[v, x] || PolyQ[v, x, 2]) && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "Int[ExpandTrigToExp[F^u, Cos[v]^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[F_^u_*Cos[v_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[F, x] && (LinearQ[u, x] || PolyQ[u, x, 2]) && (LinearQ[v, x] || PolyQ[v, x, 2]) && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
       "filename": "4.7.7 F^(c (a+b x)) trig(d+e x)^n.m",
            "rhs": "Int[ExpandTrigToExp[F^u, Sin[v]^m*Cos[v]^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[F_^u_*Sin[v_]^m_.*Cos[v_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[F, x] && (LinearQ[u, x] || PolyQ[u, x, 2]) && (LinearQ[v, x] || PolyQ[v, x, 2]) && IGtQ[m, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "x*Sin[d*(a + b*Log[c*x^n])]/(b^2*d^2*n^2 + 1) - b*d*n*x*Cos[d*(a + b*Log[c*x^n])]/(b^2*d^2*n^2 + 1)",
     "rulenumber": 0,
            "lhs": "Int[Sin[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && NeQ[b^2*d^2*n^2 + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "x*Cos[d*(a + b*Log[c*x^n])]/(b^2*d^2*n^2 + 1) + b*d*n*x*Sin[d*(a + b*Log[c*x^n])]/(b^2*d^2*n^2 + 1)",
     "rulenumber": 0,
            "lhs": "Int[Cos[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && NeQ[b^2*d^2*n^2 + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "x*Sin[d*(a + b*Log[c*x^n])]^p/(b^2*d^2*n^2*p^2 + 1) - b*d*n*p*x*Cos[d*(a + b*Log[c*x^n])]* Sin[d*(a + b*Log[c*x^n])]^(p - 1)/(b^2*d^2*n^2*p^2 + 1) + b^2*d^2*n^2*p*(p - 1)/(b^2*d^2*n^2*p^2 + 1)* Int[Sin[d*(a + b*Log[c*x^n])]^(p - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[Sin[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "x*Cos[d*(a + b*Log[c*x^n])]^p/(b^2*d^2*n^2*p^2 + 1) + b*d*n*p*x*Cos[d*(a + b*Log[c*x^n])]^(p - 1)* Sin[d*(a + b*Log[c*x^n])]/(b^2*d^2*n^2*p^2 + 1) + b^2*d^2*n^2*p*(p - 1)/(b^2*d^2*n^2*p^2 + 1)* Int[Cos[d*(a + b*Log[c*x^n])]^(p - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[Cos[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "1/(2^p*b^p*d^p*p^p)* Int[ExpandIntegrand[(E^(a*b*d^2*p)*x^(-1/p) - E^(-a*b*d^2*p)*x^(1/p))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Sin[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x] && IGtQ[p, 0] && EqQ[b^2*d^2*p^2 + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "1/2^p*Int[ ExpandIntegrand[(E^(a*b*d^2*p)*x^(-1/p) + E^(-a*b*d^2*p)*x^(1/p))^ p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Cos[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x] && IGtQ[p, 0] && EqQ[b^2*d^2*p^2 + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": " 1/((-2*I)^p*E^(I*a*d*p))*Int[(1-E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I*b*d*p) ,x]",
     "rulenumber": 0,
            "lhs": "Int[Sin[d_.*(a_.+b_.*Log[x_])]^p_.,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,d},x] && IntegerQ[p] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": " 1/(2^p*E^(I*a*d*p))*Int[(1+E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I*b*d*p),x]",
     "rulenumber": 0,
            "lhs": "Int[Cos[d_.*(a_.+b_.*Log[x_])]^p_.,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,d},x] && IntegerQ[p] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "Sin[d*(a + b*Log[x])]^p*x^(I*b*d*p)/(1 - E^(2*I*a*d)*x^(2*I*b*d))^p* Int[(1 - E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I*b*d*p), x]",
     "rulenumber": 0,
            "lhs": "Int[Sin[d_.*(a_. + b_.*Log[x_])]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, p}, x] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "Cos[d*(a + b*Log[x])]^p*x^(I*b*d*p)/(1 + E^(2*I*a*d)*x^(2*I*b*d))^p* Int[(1 + E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I*b*d*p), x]",
     "rulenumber": 0,
            "lhs": "Int[Cos[d_.*(a_. + b_.*Log[x_])]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, p}, x] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "x/(n*(c*x^n)^(1/n))* Subst[Int[x^(1/n - 1)*Sin[d*(a + b*Log[x])]^p, x], x, c*x^n]",
     "rulenumber": 0,
            "lhs": "Int[Sin[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "x/(n*(c*x^n)^(1/n))* Subst[Int[x^(1/n - 1)*Cos[d*(a + b*Log[x])]^p, x], x, c*x^n]",
     "rulenumber": 0,
            "lhs": "Int[Cos[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "(m + 1)*(e*x)^(m + 1)* Sin[d*(a + b*Log[c*x^n])]/(b^2*d^2*e*n^2 + e*(m + 1)^2) - b*d*n*(e*x)^(m + 1)* Cos[d*(a + b*Log[c*x^n])]/(b^2*d^2*e*n^2 + e*(m + 1)^2)",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Sin[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b^2*d^2*n^2 + (m + 1)^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "(m + 1)*(e*x)^(m + 1)* Cos[d*(a + b*Log[c*x^n])]/(b^2*d^2*e*n^2 + e*(m + 1)^2) + b*d*n*(e*x)^(m + 1)* Sin[d*(a + b*Log[c*x^n])]/(b^2*d^2*e*n^2 + e*(m + 1)^2)",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Cos[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b^2*d^2*n^2 + (m + 1)^2, 0]"
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       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "(m + 1)*(e*x)^(m + 1)* Sin[d*(a + b*Log[c*x^n])]^p/(b^2*d^2*e*n^2*p^2 + e*(m + 1)^2) - b*d*n*p*(e*x)^(m + 1)*Cos[d*(a + b*Log[c*x^n])]* Sin[d*(a + b*Log[c*x^n])]^(p - 1)/(b^2*d^2*e*n^2*p^2 + e*(m + 1)^2) + b^2*d^2*n^2*p*(p - 1)/(b^2*d^2*n^2*p^2 + (m + 1)^2)* Int[(e*x)^m*Sin[d*(a + b*Log[c*x^n])]^(p - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Sin[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 + (m + 1)^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "(m + 1)*(e*x)^(m + 1)* Cos[d*(a + b*Log[c*x^n])]^p/(b^2*d^2*e*n^2*p^2 + e*(m + 1)^2) + b*d*n*p*(e*x)^(m + 1)*Sin[d*(a + b*Log[c*x^n])]* Cos[d*(a + b*Log[c*x^n])]^(p - 1)/(b^2*d^2*e*n^2*p^2 + e*(m + 1)^2) + b^2*d^2*n^2*p*(p - 1)/(b^2*d^2*n^2*p^2 + (m + 1)^2)* Int[(e*x)^m*Cos[d*(a + b*Log[c*x^n])]^(p - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Cos[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 + (m + 1)^2, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "(m + 1)^p/(2^p*b^p*d^p*p^p)* Int[ ExpandIntegrand[(e*x)^ m*(E^(a*b*d^2*p/(m + 1))*x^(-(m + 1)/p) - E^(-a*b*d^2*p/(m + 1))*x^((m + 1)/p))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Sin[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, m}, x] && IGtQ[p, 0] && EqQ[b^2*d^2*p^2 + (m + 1)^2, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "1/2^p*Int[ ExpandIntegrand[(e*x)^ m*(E^(a*b*d^2*p/(m + 1))*x^(-(m + 1)/p) + E^(-a*b*d^2*p/(m + 1))*x^((m + 1)/p))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Cos[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, m}, x] && IGtQ[p, 0] && EqQ[b^2*d^2*p^2 + (m + 1)^2, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": " 1/((-2*I)^p*E^(I*a*d*p))*Int[(e*x)^m*(1-E^(2*I*a*d)*x^(2*I*b*d))^p/x^( I*b*d*p),x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Sin[d_.*(a_.+b_.*Log[x_])]^p_.,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,d,e,m},x] && IntegerQ[p] *)"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": " 1/(2^p*E^(I*a*d*p))*Int[(e*x)^m*(1+E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I*b* d*p),x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Cos[d_.*(a_.+b_.*Log[x_])]^p_.,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,d,e,m},x] && IntegerQ[p] *)"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "Sin[d*(a + b*Log[x])]^p*x^(I*b*d*p)/(1 - E^(2*I*a*d)*x^(2*I*b*d))^p* Int[(e*x)^m*(1 - E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I*b*d*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Sin[d_.*(a_. + b_.*Log[x_])]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, m, p}, x] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "Cos[d*(a + b*Log[x])]^p*x^(I*b*d*p)/(1 + E^(2*I*a*d)*x^(2*I*b*d))^p* Int[(e*x)^m*(1 + E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I*b*d*p), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Cos[d_.*(a_. + b_.*Log[x_])]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, m, p}, x] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "(e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n))* Subst[Int[x^((m + 1)/n - 1)*Sin[d*(a + b*Log[x])]^p, x], x, c*x^n]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Sin[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "(e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n))* Subst[Int[x^((m + 1)/n - 1)*Cos[d*(a + b*Log[x])]^p, x], x, c*x^n]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Cos[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "I*E^(-I*a*d)*(c*x^n)^(-I*b*d)/(2*x^(-I*b*d*n))* Int[x^(-I*b*d*n)*(h*(e + f*Log[g*x^m]))^q, x] - I*E^(I*a*d)*(c*x^n)^(I*b*d)/(2*x^(I*b*d*n))* Int[x^(I*b*d*n)*(h*(e + f*Log[g*x^m]))^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(h_.*(e_. + f_.*Log[g_.*x_^m_.]))^q_.* Sin[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n, q}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "E^(-I*a*d)*(c*x^n)^(-I*b*d)/(2*x^(-I*b*d*n))* Int[x^(-I*b*d*n)*(h*(e + f*Log[g*x^m]))^q, x] + E^(I*a*d)*(c*x^n)^(I*b*d)/(2*x^(I*b*d*n))* Int[x^(I*b*d*n)*(h*(e + f*Log[g*x^m]))^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(h_.*(e_. + f_.*Log[g_.*x_^m_.]))^q_.* Cos[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n, q}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "I*E^(-I*a*d)*(i*x)^r*(c*x^n)^(-I*b*d)/(2*x^(r - I*b*d*n))* Int[x^(r - I*b*d*n)*(h*(e + f*Log[g*x^m]))^q, x] - I*E^(I*a*d)*(i*x)^r*(c*x^n)^(I*b*d)/(2*x^(r + I*b*d*n))* Int[x^(r + I*b*d*n)*(h*(e + f*Log[g*x^m]))^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(i_.*x_)^r_.*(h_.*(e_. + f_.*Log[g_.*x_^m_.]))^q_.* Sin[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, i, m, n, q, r}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "E^(-I*a*d)*(i*x)^r*(c*x^n)^(-I*b*d)/(2*x^(r - I*b*d*n))* Int[x^(r - I*b*d*n)*(h*(e + f*Log[g*x^m]))^q, x] + E^(I*a*d)*(i*x)^r*(c*x^n)^(I*b*d)/(2*x^(r + I*b*d*n))* Int[x^(r + I*b*d*n)*(h*(e + f*Log[g*x^m]))^q, x]",
     "rulenumber": 0,
            "lhs": "Int[(i_.*x_)^r_.*(h_.*(e_. + f_.*Log[g_.*x_^m_.]))^q_.* Cos[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, i, m, n, q, r}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "2^p*E^(I*a*d*p)*Int[x^(I*b*d*p)/(1 + E^(2*I*a*d)*x^(2*I*b*d))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[Sec[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x] && IntegerQ[p]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "(-2*I)^p*E^(I*a*d*p)* Int[x^(I*b*d*p)/(1 - E^(2*I*a*d)*x^(2*I*b*d))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[Csc[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x] && IntegerQ[p]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "Sec[d*(a + b*Log[x])]^p*(1 + E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I*b*d*p)* Int[x^(I*b*d*p)/(1 + E^(2*I*a*d)*x^(2*I*b*d))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[Sec[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, p}, x] && Not[IntegerQ[p]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "Csc[d*(a + b*Log[x])]^p*(1 - E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I*b*d*p)* Int[x^(I*b*d*p)/(1 - E^(2*I*a*d)*x^(2*I*b*d))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[Csc[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, p}, x] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "x/(n*(c*x^n)^(1/n))* Subst[Int[x^(1/n - 1)*Sec[d*(a + b*Log[x])]^p, x], x, c*x^n]",
     "rulenumber": 0,
            "lhs": "Int[Sec[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "x/(n*(c*x^n)^(1/n))* Subst[Int[x^(1/n - 1)*Csc[d*(a + b*Log[x])]^p, x], x, c*x^n]",
     "rulenumber": 0,
            "lhs": "Int[Csc[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "2^p*E^(I*a*d*p)* Int[(e*x)^m*x^(I*b*d*p)/(1 + E^(2*I*a*d)*x^(2*I*b*d))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Sec[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, m}, x] && IntegerQ[p]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "(-2*I)^p*E^(I*a*d*p)* Int[(e*x)^m*x^(I*b*d*p)/(1 - E^(2*I*a*d)*x^(2*I*b*d))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Csc[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, m}, x] && IntegerQ[p]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "Sec[d*(a + b*Log[x])]^p*(1 + E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I*b*d*p)* Int[(e*x)^m*x^(I*b*d*p)/(1 + E^(2*I*a*d)*x^(2*I*b*d))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Sec[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, m, p}, x] && Not[IntegerQ[p]]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "Csc[d*(a + b*Log[x])]^p*(1 - E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I*b*d*p)* Int[(e*x)^m*x^(I*b*d*p)/(1 - E^(2*I*a*d)*x^(2*I*b*d))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Csc[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, m, p}, x] && Not[IntegerQ[p]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "(e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n))* Subst[Int[x^((m + 1)/n - 1)*Sec[d*(a + b*Log[x])]^p, x], x, c*x^n]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Sec[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "(e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n))* Subst[Int[x^((m + 1)/n - 1)*Csc[d*(a + b*Log[x])]^p, x], x, c*x^n]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Csc[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "-Cos[a*x*Log[b*x]]/a - Int[Sin[a*x*Log[b*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[Sin[a_.*x_*Log[b_.*x_]]*Log[b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "Sin[a*x*Log[b*x]]/a - Int[Cos[a*x*Log[b*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[Cos[a_.*x_*Log[b_.*x_]]*Log[b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "-Cos[a*x^n*Log[b*x]]/(a*n) - 1/n*Int[x^m*Sin[a*x^n*Log[b*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Sin[a_.*x_^n_.*Log[b_.*x_]]*Log[b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && EqQ[m, n - 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.8 u trig(a+b log(c x^n))^p.m",
       "filename": "4.7.8 u trig(a+b log(c x^n))^p.m",
            "rhs": "Sin[a*x^n*Log[b*x]]/(a*n) - 1/n*Int[x^m*Cos[a*x^n*Log[b*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Cos[a_.*x_^n_.*Log[b_.*x_]]*Log[b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && EqQ[m, n - 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/b*Int[(e + f*x)^m*Sin[c + d*x]^(n - 1), x] - a/b*Int[(e + f*x)^m*Sin[c + d*x]^(n - 1)/(a + b*Sin[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Sin[c_. + d_.*x_]^n_./(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/b*Int[(e + f*x)^m*Cos[c + d*x]^(n - 1), x] - a/b*Int[(e + f*x)^m*Cos[c + d*x]^(n - 1)/(a + b*Cos[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Cos[c_. + d_.*x_]^n_./(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "-I*(e + f*x)^(m + 1)/(b*f*(m + 1)) + 2*Int[(e + f*x)^m*E^(I*(c + d*x))/(a - I*b*E^(I*(c + d*x))), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Cos[c_. + d_.*x_]/(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "I*(e + f*x)^(m + 1)/(b*f*(m + 1)) - 2*I*Int[(e + f*x)^m*E^(I*(c + d*x))/(a + b*E^(I*(c + d*x))), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Sin[c_. + d_.*x_]/(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "-I*(e + f*x)^(m + 1)/(b*f*(m + 1)) + Int[(e + f*x)^m* E^(I*(c + d*x))/(a - Rt[a^2 - b^2, 2] - I*b*E^(I*(c + d*x))), x] + Int[(e + f*x)^m* E^(I*(c + d*x))/(a + Rt[a^2 - b^2, 2] - I*b*E^(I*(c + d*x))), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Cos[c_. + d_.*x_]/(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && PosQ[a^2 - b^2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "I*(e + f*x)^(m + 1)/(b*f*(m + 1)) - I*Int[(e + f*x)^m* E^(I*(c + d*x))/(a - Rt[a^2 - b^2, 2] + b*E^(I*(c + d*x))), x] - I*Int[(e + f*x)^m* E^(I*(c + d*x))/(a + Rt[a^2 - b^2, 2] + b*E^(I*(c + d*x))), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Sin[c_. + d_.*x_]/(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && PosQ[a^2 - b^2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "-I*(e + f*x)^(m + 1)/(b*f*(m + 1)) + I*Int[(e + f*x)^m* E^(I*(c + d*x))/(I*a - Rt[-a^2 + b^2, 2] + b*E^(I*(c + d*x))), x] + I*Int[(e + f*x)^m* E^(I*(c + d*x))/(I*a + Rt[-a^2 + b^2, 2] + b*E^(I*(c + d*x))), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Cos[c_. + d_.*x_]/(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NegQ[a^2 - b^2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "I*(e + f*x)^(m + 1)/(b*f*(m + 1)) + Int[(e + f*x)^m* E^(I*(c + d*x))/(I*a - Rt[-a^2 + b^2, 2] + I*b*E^(I*(c + d*x))), x] + Int[(e + f*x)^m* E^(I*(c + d*x))/(I*a + Rt[-a^2 + b^2, 2] + I*b*E^(I*(c + d*x))), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Sin[c_. + d_.*x_]/(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NegQ[a^2 - b^2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/a*Int[(e + f*x)^m*Cos[c + d*x]^(n - 2), x] - 1/b*Int[(e + f*x)^m*Cos[c + d*x]^(n - 2)*Sin[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Cos[c_. + d_.*x_]^n_/(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 1] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/a*Int[(e + f*x)^m*Sin[c + d*x]^(n - 2), x] - 1/b*Int[(e + f*x)^m*Sin[c + d*x]^(n - 2)*Cos[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Sin[c_. + d_.*x_]^n_/(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 1] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "a/b^2*Int[(e + f*x)^m*Cos[c + d*x]^(n - 2), x] - 1/b*Int[(e + f*x)^m*Cos[c + d*x]^(n - 2)*Sin[c + d*x], x] - (a^2 - b^2)/b^2* Int[(e + f*x)^m*Cos[c + d*x]^(n - 2)/(a + b*Sin[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Cos[c_. + d_.*x_]^n_/(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "a/b^2*Int[(e + f*x)^m*Sin[c + d*x]^(n - 2), x] - 1/b*Int[(e + f*x)^m*Sin[c + d*x]^(n - 2)*Cos[c + d*x], x] - (a^2 - b^2)/b^2* Int[(e + f*x)^m*Sin[c + d*x]^(n - 2)/(a + b*Cos[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Sin[c_. + d_.*x_]^n_/(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/b*Int[(e + f*x)^m*Sec[c + d*x]*Tan[c + d*x]^(n - 1), x] - a/b*Int[(e + f*x)^m*Sec[c + d*x]* Tan[c + d*x]^(n - 1)/(a + b*Sin[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Tan[c_. + d_.*x_]^n_./(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/b*Int[(e + f*x)^m*Csc[c + d*x]*Cot[c + d*x]^(n - 1), x] - a/b*Int[(e + f*x)^m*Csc[c + d*x]* Cot[c + d*x]^(n - 1)/(a + b*Cos[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Cot[c_. + d_.*x_]^n_./(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/a*Int[(e + f*x)^m*Cot[c + d*x]^n, x] - b/a*Int[(e + f*x)^m*Cos[c + d*x]* Cot[c + d*x]^(n - 1)/(a + b*Sin[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Cot[c_. + d_.*x_]^n_./(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/a*Int[(e + f*x)^m*Tan[c + d*x]^n, x] - b/a*Int[(e + f*x)^m*Sin[c + d*x]* Tan[c + d*x]^(n - 1)/(a + b*Cos[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Tan[c_. + d_.*x_]^n_./(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/a*Int[(e + f*x)^m*Sec[c + d*x]^(n + 2), x] - 1/b*Int[(e + f*x)^m*Sec[c + d*x]^(n + 1)*Tan[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Sec[c_. + d_.*x_]^n_./(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/a*Int[(e + f*x)^m*Csc[c + d*x]^(n + 2), x] - 1/b*Int[(e + f*x)^m*Csc[c + d*x]^(n + 1)*Cot[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Csc[c_. + d_.*x_]^n_./(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "-b^2/(a^2 - b^2)* Int[(e + f*x)^m*Sec[c + d*x]^(n - 2)/(a + b*Sin[c + d*x]), x] + 1/(a^2 - b^2)* Int[(e + f*x)^m*Sec[c + d*x]^n*(a - b*Sin[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Sec[c_. + d_.*x_]^n_./(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 - b^2, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "-b^2/(a^2 - b^2)* Int[(e + f*x)^m*Csc[c + d*x]^(n - 2)/(a + b*Cos[c + d*x]), x] + 1/(a^2 - b^2)* Int[(e + f*x)^m*Csc[c + d*x]^n*(a - b*Cos[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Csc[c_. + d_.*x_]^n_./(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 - b^2, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/a*Int[(e + f*x)^m*Csc[c + d*x]^n, x] - b/a*Int[(e + f*x)^m*Csc[c + d*x]^(n - 1)/(a + b*Sin[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Csc[c_. + d_.*x_]^n_./(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/a*Int[(e + f*x)^m*Sec[c + d*x]^n, x] - b/a*Int[(e + f*x)^m*Sec[c + d*x]^(n - 1)/(a + b*Cos[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Sec[c_. + d_.*x_]^n_./(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Unintegrable[(e + f*x)^m*F[c + d*x]^n/(a + b*Sin[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* F_[c_. + d_.*x_]^n_./(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && TrigQ[F]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Unintegrable[(e + f*x)^m*F[c + d*x]^n/(a + b*Cos[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* F_[c_. + d_.*x_]^n_./(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && TrigQ[F]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/b*Int[(e + f*x)^m*Cos[c + d*x]^p*Sin[c + d*x]^(n - 1), x] - a/b*Int[(e + f*x)^m*Cos[c + d*x]^p* Sin[c + d*x]^(n - 1)/(a + b*Sin[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Cos[c_. + d_.*x_]^p_.* Sin[c_. + d_.*x_]^n_./(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/b*Int[(e + f*x)^m*Sin[c + d*x]^p*Cos[c + d*x]^(n - 1), x] - a/b*Int[(e + f*x)^m*Sin[c + d*x]^p* Cos[c + d*x]^(n - 1)/(a + b*Cos[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Sin[c_. + d_.*x_]^p_.* Cos[c_. + d_.*x_]^n_./(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/b*Int[(e + f*x)^m*Cos[c + d*x]^(p - 1)*Tan[c + d*x]^(n - 1), x] - a/b*Int[(e + f*x)^m*Cos[c + d*x]^(p - 1)* Tan[c + d*x]^(n - 1)/(a + b*Sin[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Cos[c_. + d_.*x_]^p_.* Tan[c_. + d_.*x_]^n_./(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/b*Int[(e + f*x)^m*Sin[c + d*x]^(p - 1)*Cot[c + d*x]^(n - 1), x] - a/b*Int[(e + f*x)^m*Sin[c + d*x]^(p - 1)* Cot[c + d*x]^(n - 1)/(a + b*Cos[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Sin[c_. + d_.*x_]^p_.* Cot[c_. + d_.*x_]^n_./(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/a*Int[(e + f*x)^m*Cos[c + d*x]^p*Cot[c + d*x]^n, x] - b/a*Int[(e + f*x)^m*Cos[c + d*x]^(p + 1)* Cot[c + d*x]^(n - 1)/(a + b*Sin[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Cos[c_. + d_.*x_]^p_.* Cot[c_. + d_.*x_]^n_./(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/a*Int[(e + f*x)^m*Sin[c + d*x]^p*Tan[c + d*x]^n, x] - b/a*Int[(e + f*x)^m*Sin[c + d*x]^(p + 1)* Tan[c + d*x]^(n - 1)/(a + b*Cos[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Sin[c_. + d_.*x_]^p_.* Tan[c_. + d_.*x_]^n_./(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/a*Int[(e + f*x)^m*Cos[c + d*x]^p*Csc[c + d*x]^n, x] - b/a*Int[(e + f*x)^m*Cos[c + d*x]^p* Csc[c + d*x]^(n - 1)/(a + b*Sin[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Cos[c_. + d_.*x_]^p_.* Csc[c_. + d_.*x_]^n_./(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/a*Int[(e + f*x)^m*Sin[c + d*x]^p*Sec[c + d*x]^n, x] - b/a*Int[(e + f*x)^m*Sin[c + d*x]^p* Sec[c + d*x]^(n - 1)/(a + b*Cos[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Sin[c_. + d_.*x_]^p_.* Sec[c_. + d_.*x_]^n_./(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Unintegrable[(e + f*x)^m*Cos[c + d*x]^p* F[c + d*x]^n/(a + b*Sin[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Cos[c_. + d_.*x_]^p_.* F_[c_. + d_.*x_]^n_./(a_ + b_.*Sin[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && TrigQ[F]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Unintegrable[(e + f*x)^m*Sin[c + d*x]^p* F[c + d*x]^n/(a + b*Cos[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Sin[c_. + d_.*x_]^p_.* F_[c_. + d_.*x_]^n_./(a_ + b_.*Cos[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && TrigQ[F]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Int[(e + f*x)^m*Cos[c + d*x]*F[c + d*x]^n/(b + a*Cos[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* F_[c_. + d_.*x_]^n_./(a_ + b_.*Sec[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && TrigQ[F] && IntegersQ[m, n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Int[(e + f*x)^m*Sin[c + d*x]*F[c + d*x]^n/(b + a*Sin[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* F_[c_. + d_.*x_]^n_./(a_ + b_.*Csc[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && TrigQ[F] && IntegersQ[m, n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Int[(e + f*x)^m*Cos[c + d*x]*F[c + d*x]^n* G[c + d*x]^p/(b + a*Cos[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*F_[c_. + d_.*x_]^n_.* G_[c_. + d_.*x_]^p_./(a_ + b_.*Sec[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && TrigQ[F] && TrigQ[G] && IntegersQ[m, n, p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Int[(e + f*x)^m*Sin[c + d*x]*F[c + d*x]^n* G[c + d*x]^p/(b + a*Sin[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*F_[c_. + d_.*x_]^n_.* G_[c_. + d_.*x_]^p_./(a_ + b_.*Csc[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && TrigQ[F] && TrigQ[G] && IntegersQ[m, n, p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/2^(p + q)* Int[ExpandIntegrand[(I/E^(I*(c + d*x)) - I*E^(I*(c + d*x)))^ q, (I/E^(I*(a + b*x)) - I*E^(I*(a + b*x)))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Sin[a_. + b_.*x_]^p_.*Sin[c_. + d_.*x_]^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, q}, x] && IGtQ[p, 0] && Not[IntegerQ[q]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/2^(p + q)* Int[ExpandIntegrand[(E^(-I*(c + d*x)) + E^(I*(c + d*x)))^ q, (E^(-I*(a + b*x)) + E^(I*(a + b*x)))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Cos[a_. + b_.*x_]^p_.*Cos[c_. + d_.*x_]^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, q}, x] && IGtQ[p, 0] && Not[IntegerQ[q]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/2^(p + q)* Int[ExpandIntegrand[(E^(-I*(c + d*x)) + E^(I*(c + d*x)))^ q, (I/E^(I*(a + b*x)) - I*E^(I*(a + b*x)))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Sin[a_. + b_.*x_]^p_.*Cos[c_. + d_.*x_]^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, q}, x] && IGtQ[p, 0] && Not[IntegerQ[q]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/2^(p + q)* Int[ExpandIntegrand[(I/E^(I*(c + d*x)) - I*E^(I*(c + d*x)))^ q, (E^(-I*(a + b*x)) + E^(I*(a + b*x)))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Cos[a_. + b_.*x_]^p_.*Sin[c_. + d_.*x_]^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, q}, x] && IGtQ[p, 0] && Not[IntegerQ[q]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Int[E^(-I*(a + b*x))/2 - E^(I*(a + b*x))/2 - E^(-I*(a + b*x))/(1 + E^(2*I*(c + d*x))) + E^(I*(a + b*x))/(1 + E^(2*I*(c + d*x))), x]",
     "rulenumber": 0,
            "lhs": "Int[Sin[a_. + b_.*x_]*Tan[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Int[I*E^(-I*(a + b*x))/2 + I*E^(I*(a + b*x))/2 - I*E^(-I*(a + b*x))/(1 - E^(2*I*(c + d*x))) - I*E^(I*(a + b*x))/(1 - E^(2*I*(c + d*x))), x]",
     "rulenumber": 0,
            "lhs": "Int[Cos[a_. + b_.*x_]*Cot[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Int[-E^(-I*(a + b*x))/2 + E^(I*(a + b*x))/2 + E^(-I*(a + b*x))/(1 - E^(2*I*(c + d*x))) - E^(I*(a + b*x))/(1 - E^(2*I*(c + d*x))), x]",
     "rulenumber": 0,
            "lhs": "Int[Sin[a_. + b_.*x_]*Cot[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Int[-I*E^(-I*(a + b*x))/2 - I*E^(I*(a + b*x))/2 + I*E^(-I*(a + b*x))/(1 + E^(2*I*(c + d*x))) + I*E^(I*(a + b*x))/(1 + E^(2*I*(c + d*x))), x]",
     "rulenumber": 0,
            "lhs": "Int[Cos[a_. + b_.*x_]*Tan[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "-1/d*Subst[Int[Sin[a*x]^n/x^2, x], x, 1/(c + d*x)]",
     "rulenumber": 0,
            "lhs": "Int[Sin[a_./(c_. + d_.*x_)]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "-1/d*Subst[Int[Cos[a*x]^n/x^2, x], x, 1/(c + d*x)]",
     "rulenumber": 0,
            "lhs": "Int[Cos[a_./(c_. + d_.*x_)]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "-1/d* Subst[Int[Sin[b*e/d - e*(b*c - a*d)*x/d]^n/x^2, x], x, 1/(c + d*x)]",
     "rulenumber": 0,
            "lhs": "Int[Sin[e_.*(a_. + b_.*x_)/(c_. + d_.*x_)]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "-1/d* Subst[Int[Cos[b*e/d - e*(b*c - a*d)*x/d]^n/x^2, x], x, 1/(c + d*x)]",
     "rulenumber": 0,
            "lhs": "Int[Cos[e_.*(a_. + b_.*x_)/(c_. + d_.*x_)]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Module[{lst = QuotientOfLinearsParts[u, x]}, Int[Sin[(lst[[1]] + lst[[2]]*x)/(lst[[3]] + lst[[4]]*x)]^n, x]]",
     "rulenumber": 0,
            "lhs": "Int[Sin[u_]^n_., x_Symbol]",
        "comment": false,
         "givens": "IGtQ[n, 0] && QuotientOfLinearsQ[u, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Module[{lst = QuotientOfLinearsParts[u, x]}, Int[Cos[(lst[[1]] + lst[[2]]*x)/(lst[[3]] + lst[[4]]*x)]^n, x]]",
     "rulenumber": 0,
            "lhs": "Int[Cos[u_]^n_., x_Symbol]",
        "comment": false,
         "givens": "IGtQ[n, 0] && QuotientOfLinearsQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Int[u*Sin[v]^(p + q), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*Sin[v_]^p_.*Sin[w_]^q_., x_Symbol]",
        "comment": false,
         "givens": "EqQ[w, v]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Int[u*Cos[v]^(p + q), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*Cos[v_]^p_.*Cos[w_]^q_., x_Symbol]",
        "comment": false,
         "givens": "EqQ[w, v]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Int[ExpandTrigReduce[Sin[v]^p*Sin[w]^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Sin[v_]^p_.*Sin[w_]^q_., x_Symbol]",
        "comment": false,
         "givens": "(PolynomialQ[v, x] && PolynomialQ[w, x] || BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x]) && IGtQ[p, 0] && IGtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Int[ExpandTrigReduce[Cos[v]^p*Cos[w]^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Cos[v_]^p_.*Cos[w_]^q_., x_Symbol]",
        "comment": false,
         "givens": "(PolynomialQ[v, x] && PolynomialQ[w, x] || BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x]) && IGtQ[p, 0] && IGtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Int[ExpandTrigReduce[x^m, Sin[v]^p*Sin[w]^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Sin[v_]^p_.*Sin[w_]^q_., x_Symbol]",
        "comment": false,
         "givens": "IGtQ[m, 0] && IGtQ[p, 0] && IGtQ[q, 0] && (PolynomialQ[v, x] && PolynomialQ[w, x] || BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Int[ExpandTrigReduce[x^m, Cos[v]^p*Cos[w]^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Cos[v_]^p_.*Cos[w_]^q_., x_Symbol]",
        "comment": false,
         "givens": "IGtQ[m, 0] && IGtQ[p, 0] && IGtQ[q, 0] && (PolynomialQ[v, x] && PolynomialQ[w, x] || BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/2^p*Int[u*Sin[2*v]^p, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*Sin[v_]^p_.*Cos[w_]^p_., x_Symbol]",
        "comment": false,
         "givens": "EqQ[w, v] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Int[ExpandTrigReduce[Sin[v]^p*Cos[w]^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Sin[v_]^p_.*Cos[w_]^q_., x_Symbol]",
        "comment": false,
         "givens": "IGtQ[p, 0] && IGtQ[q, 0] && (PolynomialQ[v, x] && PolynomialQ[w, x] || BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Int[ExpandTrigReduce[x^m, Sin[v]^p*Cos[w]^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Sin[v_]^p_.*Cos[w_]^q_., x_Symbol]",
        "comment": false,
         "givens": "IGtQ[m, 0] && IGtQ[p, 0] && IGtQ[q, 0] && (PolynomialQ[v, x] && PolynomialQ[w, x] || BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "-Int[Cos[v]*Tan[w]^(n - 1), x] + Cos[v - w]*Int[Sec[w]*Tan[w]^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[Sin[v_]*Tan[w_]^n_., x_Symbol]",
        "comment": false,
         "givens": "GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "-Int[Sin[v]*Cot[w]^(n - 1), x] + Cos[v - w]*Int[Csc[w]*Cot[w]^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[Cos[v_]*Cot[w_]^n_., x_Symbol]",
        "comment": false,
         "givens": "GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Int[Cos[v]*Cot[w]^(n - 1), x] + Sin[v - w]*Int[Csc[w]*Cot[w]^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[Sin[v_]*Cot[w_]^n_., x_Symbol]",
        "comment": false,
         "givens": "GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Int[Sin[v]*Tan[w]^(n - 1), x] - Sin[v - w]*Int[Sec[w]*Tan[w]^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[Cos[v_]*Tan[w_]^n_., x_Symbol]",
        "comment": false,
         "givens": "GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Cos[v - w]*Int[Tan[w]*Sec[w]^(n - 1), x] + Sin[v - w]*Int[Sec[w]^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[Sin[v_]*Sec[w_]^n_., x_Symbol]",
        "comment": false,
         "givens": "GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Cos[v - w]*Int[Cot[w]*Csc[w]^(n - 1), x] - Sin[v - w]*Int[Csc[w]^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[Cos[v_]*Csc[w_]^n_., x_Symbol]",
        "comment": false,
         "givens": "GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Sin[v - w]*Int[Cot[w]*Csc[w]^(n - 1), x] + Cos[v - w]*Int[Csc[w]^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[Sin[v_]*Csc[w_]^n_., x_Symbol]",
        "comment": false,
         "givens": "GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "-Sin[v - w]*Int[Tan[w]*Sec[w]^(n - 1), x] + Cos[v - w]*Int[Sec[w]^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[Cos[v_]*Sec[w_]^n_., x_Symbol]",
        "comment": false,
         "givens": "GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Int[(e + f*x)^m*(a + b*Sin[2*c + 2*d*x]/2)^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_ + b_.*Sin[c_. + d_.*x_]*Cos[c_. + d_.*x_])^ n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/2^n*Int[x^m*(2*a + b - b*Cos[2*c + 2*d*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*Sin[c_. + d_.*x_]^2)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a + b, 0] && IGtQ[m, 0] && ILtQ[n, 0] && (EqQ[n, -1] || EqQ[m, 1] && EqQ[n, -2])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/2^n*Int[x^m*(2*a + b + b*Cos[2*c + 2*d*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*Cos[c_. + d_.*x_]^2)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a + b, 0] && IGtQ[m, 0] && ILtQ[n, 0] && (EqQ[n, -1] || EqQ[m, 1] && EqQ[n, -2])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "2*Int[(f + g*x)^m/(2*a + b + c + (b - c)*Cos[2*d + 2*e*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^ m_./(a_. + b_.*Cos[d_. + e_.*x_]^2 + c_.*Sin[d_. + e_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && NeQ[a + b, 0] && NeQ[a + c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "2*Int[(f + g*x)^m/(b + c + (b - c)*Cos[2*d + 2*e*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^m_.* Sec[d_. + e_.*x_]^2/(b_ + c_.*Tan[d_. + e_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f, g}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "2*Int[(f + g*x)^m/(2*a + b + c + (b - c)*Cos[2*d + 2*e*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^m_.* Sec[d_. + e_.*x_]^2/(b_. + a_.*Sec[d_. + e_.*x_]^2 + c_.*Tan[d_. + e_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && NeQ[a + b, 0] && NeQ[a + c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "2*Int[(f + g*x)^m/(b + c + (b - c)*Cos[2*d + 2*e*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^m_.* Csc[d_. + e_.*x_]^2/(c_ + b_.*Cot[d_. + e_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f, g}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "2*Int[(f + g*x)^m/(2*a + b + c + (b - c)*Cos[2*d + 2*e*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^m_.* Csc[d_. + e_.*x_]^2/(c_. + b_.*Cot[d_. + e_.*x_]^2 + a_.*Csc[d_. + e_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && NeQ[a + b, 0] && NeQ[a + c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "-B*(e + f*x)* Cos[c + d*x]/(a*d*(a + b*Sin[c + d*x])) + B*f/(a*d)*Int[Cos[c + d*x]/(a + b*Sin[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)*(A_ + B_.*Sin[c_. + d_.*x_])/(a_ + b_.*Sin[c_. + d_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && EqQ[a*A - b*B, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "B*(e + f*x)*Sin[c + d*x]/(a*d*(a + b*Cos[c + d*x])) - B*f/(a*d)*Int[Sin[c + d*x]/(a + b*Cos[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)*(A_ + B_.*Cos[c_. + d_.*x_])/(a_ + b_.*Cos[c_. + d_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && EqQ[a*A - b*B, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "x/(a*d*Sin[a*x]*(c*Sin[a*x] + d*x*Cos[a*x])) + 1/d^2*Int[1/Sin[a*x]^2, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^2/(c_.*Sin[a_.*x_] + d_.*x_*Cos[a_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d}, x] && EqQ[a*c + d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "-x/(a*d*Cos[a*x]*(c*Cos[a*x] + d*x*Sin[a*x])) + 1/d^2*Int[1/Cos[a*x]^2, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^2/(c_.*Cos[a_.*x_] + d_.*x_*Sin[a_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d}, x] && EqQ[a*c - d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/(d^2*x) + Sin[a*x]/(a*d*x*(d*x*Cos[a*x] + c*Sin[a*x]))",
     "rulenumber": 0,
            "lhs": "Int[Sin[a_.*x_]^2/(c_.*Sin[a_.*x_] + d_.*x_*Cos[a_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d}, x] && EqQ[a*c + d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/(d^2*x) - Cos[a*x]/(a*d*x*(d*x*Sin[a*x] + c*Cos[a*x]))",
     "rulenumber": 0,
            "lhs": "Int[Cos[a_.*x_]^2/(c_.*Cos[a_.*x_] + d_.*x_*Sin[a_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d}, x] && EqQ[a*c - d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "b*(b*x)^(m - 1)* Sin[a*x]^(n - 1)/(a*d*(c*Sin[a*x] + d*x*Cos[a*x])) - b^2*(n - 1)/d^2*Int[(b*x)^(m - 2)*Sin[a*x]^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*x_)^m_* Sin[a_.*x_]^n_/(c_.*Sin[a_.*x_] + d_.*x_*Cos[a_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && EqQ[a*c + d, 0] && EqQ[m, 2 - n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "-b*(b*x)^(m - 1)* Cos[a*x]^(n - 1)/(a*d*(c*Cos[a*x] + d*x*Sin[a*x])) - b^2*(n - 1)/d^2*Int[(b*x)^(m - 2)*Cos[a*x]^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*x_)^m_* Cos[a_.*x_]^n_/(c_.*Cos[a_.*x_] + d_.*x_*Sin[a_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && EqQ[a*c - d, 0] && EqQ[m, 2 - n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "b*(b*x)^(m - 1)* Csc[a*x]^(n + 1)/(a*d*(c*Sin[a*x] + d*x*Cos[a*x])) + b^2*(n + 1)/d^2*Int[(b*x)^(m - 2)*Csc[a*x]^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*x_)^m_.* Csc[a_.*x_]^n_./(c_.*Sin[a_.*x_] + d_.*x_*Cos[a_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && EqQ[a*c + d, 0] && EqQ[m, n + 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "-b*(b*x)^(m - 1)* Sec[a*x]^(n + 1)/(a*d*(c*Cos[a*x] + d*x*Sin[a*x])) + b^2*(n + 1)/d^2*Int[(b*x)^(m - 2)*Sec[a*x]^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(b_.*x_)^m_.* Sec[a_.*x_]^n_./(c_.*Cos[a_.*x_] + d_.*x_*Sin[a_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && EqQ[a*c - d, 0] && EqQ[m, n + 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "a^m*c^m*Int[(g + h*x)^p* Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)^p_.*(a_ + b_.*Sin[e_. + f_.*x_])^ m_.*(c_ + d_.*Sin[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && IGtQ[n - m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "a^m*c^m*Int[(g + h*x)^p* Sin[e + f*x]^(2*m)*(c + d*Cos[e + f*x])^(n - m), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)^p_.*(a_ + b_.*Cos[e_. + f_.*x_])^ m_.*(c_ + d_.*Cos[e_. + f_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && IGtQ[n - m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "a^IntPart[m]* c^IntPart[m]*(a + b*Sin[e + f*x])^ FracPart[m]*(c + d*Sin[e + f*x])^FracPart[m]/ Cos[e + f*x]^(2*FracPart[m])* Int[(g + h*x)^p* Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)^p_.*(a_ + b_.*Sin[e_. + f_.*x_])^ m_*(c_ + d_.*Sin[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[p] && IntegerQ[2*m] && IGeQ[n - m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "a^IntPart[m]* c^IntPart[m]*(a + b*Cos[e + f*x])^ FracPart[m]*(c + d*Cos[e + f*x])^FracPart[m]/ Sin[e + f*x]^(2*FracPart[m])* Int[(g + h*x)^p* Sin[e + f*x]^(2*m)*(c + d*Cos[e + f*x])^(n - m), x]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*x_)^p_.*(a_ + b_.*Cos[e_. + f_.*x_])^ m_*(c_ + d_.*Cos[e_. + f_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[p] && IntegerQ[2*m] && IGeQ[n - m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Int[(a*Cos[v] + b*Sin[v])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[Sec[v_]^m_.*(a_ + b_.*Tan[v_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IntegerQ[(m - 1)/2] && EqQ[m + n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Int[(b*Cos[v] + a*Sin[v])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[Csc[v_]^m_.*(a_ + b_.*Cot[v_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IntegerQ[(m - 1)/2] && EqQ[m + n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Int[ExpandTrigReduce[u, Sin[a + b*x]^m*Sin[c + d*x]^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*Sin[a_. + b_.*x_]^m_.*Sin[c_. + d_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Int[ExpandTrigReduce[u, Cos[a + b*x]^m*Cos[c + d*x]^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*Cos[a_. + b_.*x_]^m_.*Cos[c_. + d_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "-Csc[(b*c - a*d)/d]*Int[Tan[a + b*x], x] + Csc[(b*c - a*d)/b]*Int[Tan[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[Sec[a_. + b_.*x_]*Sec[c_ + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Csc[(b*c - a*d)/b]*Int[Cot[a + b*x], x] - Csc[(b*c - a*d)/d]*Int[Cot[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[Csc[a_. + b_.*x_]*Csc[c_ + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "-b*x/d + b/d*Cos[(b*c - a*d)/d]*Int[Sec[a + b*x]*Sec[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[Tan[a_. + b_.*x_]*Tan[c_ + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "-b*x/d + Cos[(b*c - a*d)/d]*Int[Csc[a + b*x]*Csc[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[Cot[a_. + b_.*x_]*Cot[c_ + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "Int[u*(a*E^(-a/b*v))^n, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_.*Cos[v_] + b_.*Sin[v_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n}, x] && EqQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "I/2*Int[E^(-I*d*(a + b*Log[c*x^n])^2), x] - I/2*Int[E^(I*d*(a + b*Log[c*x^n])^2), x]",
     "rulenumber": 0,
            "lhs": "Int[Sin[d_.*(a_. + b_.*Log[c_.*x_^n_.])^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/2*Int[E^(-I*d*(a + b*Log[c*x^n])^2), x] + 1/2*Int[E^(I*d*(a + b*Log[c*x^n])^2), x]",
     "rulenumber": 0,
            "lhs": "Int[Cos[d_.*(a_. + b_.*Log[c_.*x_^n_.])^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "I/2*Int[(e*x)^m*E^(-I*d*(a + b*Log[c*x^n])^2), x] - I/2*Int[(e*x)^m*E^(I*d*(a + b*Log[c*x^n])^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Sin[d_.*(a_. + b_.*Log[c_.*x_^n_.])^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/4 Trig functions/4.7 Miscellaneous/4.7.9 Active trig functions.m",
       "filename": "4.7.9 Active trig functions.m",
            "rhs": "1/2*Int[(e*x)^m*E^(-I*d*(a + b*Log[c*x^n])^2), x] + 1/2*Int[(e*x)^m*E^(I*d*(a + b*Log[c*x^n])^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Cos[d_.*(a_. + b_.*Log[c_.*x_^n_.])^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.1 (a+b arcsin(c x))^n.m",
       "filename": "5.1.1 (a+b arcsin(c x))^n.m",
            "rhs": "x*(a + b*ArcSin[c*x])^n - b*c*n*Int[x*(a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.1 (a+b arcsin(c x))^n.m",
       "filename": "5.1.1 (a+b arcsin(c x))^n.m",
            "rhs": "x*(a + b*ArcCos[c*x])^n + b*c*n*Int[x*(a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.1 (a+b arcsin(c x))^n.m",
       "filename": "5.1.1 (a+b arcsin(c x))^n.m",
            "rhs": "Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1)) + c/(b*(n + 1))* Int[x*(a + b*ArcSin[c*x])^(n + 1)/Sqrt[1 - c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSin[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.1 (a+b arcsin(c x))^n.m",
       "filename": "5.1.1 (a+b arcsin(c x))^n.m",
            "rhs": "-Sqrt[ 1 - c^2*x^2]*(a + b*ArcCos[c*x])^(n + 1)/(b*c*(n + 1)) - c/(b*(n + 1))* Int[x*(a + b*ArcCos[c*x])^(n + 1)/Sqrt[1 - c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCos[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.1 (a+b arcsin(c x))^n.m",
       "filename": "5.1.1 (a+b arcsin(c x))^n.m",
            "rhs": "1/(b*c)*Subst[Int[x^n*Cos[a/b - x/b], x], x, a + b*ArcSin[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSin[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.1 (a+b arcsin(c x))^n.m",
       "filename": "5.1.1 (a+b arcsin(c x))^n.m",
            "rhs": "1/(b*c)*Subst[Int[x^n*Sin[a/b - x/b], x], x, a + b*ArcCos[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCos[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.2 (d x)^m (a+b arcsin(c x))^n.m",
       "filename": "5.1.2 (d x)^m (a+b arcsin(c x))^n.m",
            "rhs": "Subst[Int[(a + b*x)^n/Tan[x], x], x, ArcSin[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSin[c_.*x_])^n_./x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.2 (d x)^m (a+b arcsin(c x))^n.m",
       "filename": "5.1.2 (d x)^m (a+b arcsin(c x))^n.m",
            "rhs": "-Subst[Int[(a + b*x)^n/Cot[x], x], x, ArcCos[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCos[c_.*x_])^n_./x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.2 (d x)^m (a+b arcsin(c x))^n.m",
       "filename": "5.1.2 (d x)^m (a+b arcsin(c x))^n.m",
            "rhs": "(d*x)^(m + 1)*(a + b*ArcSin[c*x])^n/(d*(m + 1)) - b*c*n/(d*(m + 1))* Int[(d*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.2 (d x)^m (a+b arcsin(c x))^n.m",
       "filename": "5.1.2 (d x)^m (a+b arcsin(c x))^n.m",
            "rhs": "(d*x)^(m + 1)*(a + b*ArcCos[c*x])^n/(d*(m + 1)) + b*c*n/(d*(m + 1))* Int[(d*x)^(m + 1)*(a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.2 (d x)^m (a+b arcsin(c x))^n.m",
       "filename": "5.1.2 (d x)^m (a+b arcsin(c x))^n.m",
            "rhs": "x^(m + 1)*(a + b*ArcSin[c*x])^n/(m + 1) - b*c*n/(m + 1)* Int[x^(m + 1)*(a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*ArcSin[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.2 (d x)^m (a+b arcsin(c x))^n.m",
       "filename": "5.1.2 (d x)^m (a+b arcsin(c x))^n.m",
            "rhs": "x^(m + 1)*(a + b*ArcCos[c*x])^n/(m + 1) + b*c*n/(m + 1)* Int[x^(m + 1)*(a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*ArcCos[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.2 (d x)^m (a+b arcsin(c x))^n.m",
       "filename": "5.1.2 (d x)^m (a+b arcsin(c x))^n.m",
            "rhs": "x^m*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1)) - 1/(b*c^(m + 1)*(n + 1))* Subst[Int[ ExpandTrigReduce[(a + b*x)^(n + 1), Sin[x]^(m - 1)*(m - (m + 1)*Sin[x]^2), x], x], x, ArcSin[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*ArcSin[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.2 (d x)^m (a+b arcsin(c x))^n.m",
       "filename": "5.1.2 (d x)^m (a+b arcsin(c x))^n.m",
            "rhs": "-x^m* Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^(n + 1)/(b*c*(n + 1)) - 1/(b*c^(m + 1)*(n + 1))* Subst[Int[ ExpandTrigReduce[(a + b*x)^(n + 1), Cos[x]^(m - 1)*(m - (m + 1)*Cos[x]^2), x], x], x, ArcCos[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*ArcCos[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.2 (d x)^m (a+b arcsin(c x))^n.m",
       "filename": "5.1.2 (d x)^m (a+b arcsin(c x))^n.m",
            "rhs": "x^m*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1)) - m/(b*c*(n + 1))* Int[x^(m - 1)*(a + b*ArcSin[c*x])^(n + 1)/Sqrt[1 - c^2*x^2], x] + c*(m + 1)/(b*(n + 1))* Int[x^(m + 1)*(a + b*ArcSin[c*x])^(n + 1)/Sqrt[1 - c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*ArcSin[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.2 (d x)^m (a+b arcsin(c x))^n.m",
       "filename": "5.1.2 (d x)^m (a+b arcsin(c x))^n.m",
            "rhs": "-x^m* Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^(n + 1)/(b*c*(n + 1)) + m/(b*c*(n + 1))* Int[x^(m - 1)*(a + b*ArcCos[c*x])^(n + 1)/Sqrt[1 - c^2*x^2], x] - c*(m + 1)/(b*(n + 1))* Int[x^(m + 1)*(a + b*ArcCos[c*x])^(n + 1)/Sqrt[1 - c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*ArcCos[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.2 (d x)^m (a+b arcsin(c x))^n.m",
       "filename": "5.1.2 (d x)^m (a+b arcsin(c x))^n.m",
            "rhs": "1/c^(m + 1)* Subst[Int[(a + b*x)^n*Sin[x]^m*Cos[x], x], x, ArcSin[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*ArcSin[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.2 (d x)^m (a+b arcsin(c x))^n.m",
       "filename": "5.1.2 (d x)^m (a+b arcsin(c x))^n.m",
            "rhs": "-1/c^(m + 1)* Subst[Int[(a + b*x)^n*Cos[x]^m*Sin[x], x], x, ArcCos[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*ArcCos[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.2 (d x)^m (a+b arcsin(c x))^n.m",
       "filename": "5.1.2 (d x)^m (a+b arcsin(c x))^n.m",
            "rhs": "Unintegrable[(d*x)^m*(a + b*ArcSin[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.2 (d x)^m (a+b arcsin(c x))^n.m",
       "filename": "5.1.2 (d x)^m (a+b arcsin(c x))^n.m",
            "rhs": "Unintegrable[(d*x)^m*(a + b*ArcCos[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": " 1/(c*Sqrt[d])*Subst[Int[(a+b*x)^n,x],x,ArcSin[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_.+b_.*ArcSin[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e,0] && GtQ[d,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": " -1/(c*Sqrt[d])*Subst[Int[(a+b*x)^n,x],x,ArcCos[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_.+b_.*ArcCos[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e,n},x] && EqQ[c^2*d+e,0] && GtQ[d,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "Log[a + b*ArcSin[c*x]]/(b*c*Sqrt[d])",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcSin[c_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "-Log[a + b*ArcCos[c*x]]/(b*c*Sqrt[d])",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcCos[c_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(a + b*ArcSin[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1))",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSin[c_.*x_])^n_./Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && NeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "-(a + b*ArcCos[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1))",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCos[c_.*x_])^n_./Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && NeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]* Int[(a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSin[c_.*x_])^n_./Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && Not[GtQ[d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]* Int[(a + b*ArcCos[c*x])^n/Sqrt[1 - c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCos[c_.*x_])^n_./Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && Not[GtQ[d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[a + b*ArcCos[c*x], u, x] + b*c*Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": " x*(d+e*x^2)^p*(a+b*ArcSin[c*x])^n/(2*p+1) + 2*d*p/(2*p+1)*Int[(d+e*x^2)^(p-1)*(a+b*ArcSin[c*x])^n,x] - b*c*n*d^p/(2*p+1)*Int[x*(1-c^2*x^2)^(p-1/2)*(a+b*ArcSin[c*x])^(n-1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e,0] && GtQ[n,0] && GtQ[p,0] &&  (IntegerQ[p] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": " x*(d+e*x^2)^p*(a+b*ArcCos[c*x])^n/(2*p+1) + 2*d*p/(2*p+1)*Int[(d+e*x^2)^(p-1)*(a+b*ArcCos[c*x])^n,x] + b*c*n*d^p/(2*p+1)*Int[x*(1-c^2*x^2)^(p-1/2)*(a+b*ArcCos[c*x])^(n-1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e,0] && GtQ[n,0] && GtQ[p,0] &&  (IntegerQ[p] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "x*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n/2 - b*c*n*Sqrt[d + e*x^2]/(2*Sqrt[1 - c^2*x^2])* Int[x*(a + b*ArcSin[c*x])^(n - 1), x] + Sqrt[d + e*x^2]/(2*Sqrt[1 - c^2*x^2])* Int[(a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "x*Sqrt[d + e*x^2]*(a + b*ArcCos[c*x])^n/2 + b*c*n*Sqrt[d + e*x^2]/(2*Sqrt[1 - c^2*x^2])* Int[x*(a + b*ArcCos[c*x])^(n - 1), x] + Sqrt[d + e*x^2]/(2*Sqrt[1 - c^2*x^2])* Int[(a + b*ArcCos[c*x])^n/Sqrt[1 - c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "x*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n/(2*p + 1) + 2*d*p/(2*p + 1)* Int[(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x] - b*c*n* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/((2*p + 1)*(1 - c^2*x^2)^FracPart[p])* Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "x*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n/(2*p + 1) + 2*d*p/(2*p + 1)* Int[(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n, x] + b*c*n* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/((2*p + 1)*(1 - c^2*x^2)^FracPart[p])* Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "x*(a + b*ArcSin[c*x])^n/(d*Sqrt[d + e*x^2]) - b*c*n/Sqrt[d]*Int[x*(a + b*ArcSin[c*x])^(n - 1)/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSin[c_.*x_])^n_./(d_ + e_.*x_^2)^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "x*(a + b*ArcCos[c*x])^n/(d*Sqrt[d + e*x^2]) + b*c*n/Sqrt[d]*Int[x*(a + b*ArcCos[c*x])^(n - 1)/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCos[c_.*x_])^n_./(d_ + e_.*x_^2)^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "x*(a + b*ArcSin[c*x])^n/(d*Sqrt[d + e*x^2]) - b*c*n*Sqrt[1 - c^2*x^2]/(d*Sqrt[d + e*x^2])* Int[x*(a + b*ArcSin[c*x])^(n - 1)/(1 - c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSin[c_.*x_])^n_./(d_ + e_.*x_^2)^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "x*(a + b*ArcCos[c*x])^n/(d*Sqrt[d + e*x^2]) + b*c*n*Sqrt[1 - c^2*x^2]/(d*Sqrt[d + e*x^2])* Int[x*(a + b*ArcCos[c*x])^(n - 1)/(1 - c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCos[c_.*x_])^n_./(d_ + e_.*x_^2)^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": " -x*(d+e*x^2)^(p+1)*(a+b*ArcSin[c*x])^n/(2*d*(p+1)) + (2*p+3)/(2*d*(p+1))*Int[(d+e*x^2)^(p+1)*(a+b*ArcSin[c*x])^n,x] + b*c*n*d^p/(2*(p+1))*Int[x*(1-c^2*x^2)^(p+1/2)*(a+b*ArcSin[c*x])^(n- 1),x]",
     "rulenumber": 0,
            "lhs": "Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e,0] && GtQ[n,0] &&  LtQ[p,-1] && NeQ[p,-3/2] && (IntegerQ[p] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": " -x*(d+e*x^2)^(p+1)*(a+b*ArcCos[c*x])^n/(2*d*(p+1)) + (2*p+3)/(2*d*(p+1))*Int[(d+e*x^2)^(p+1)*(a+b*ArcCos[c*x])^n,x] - b*c*n*d^p/(2*(p+1))*Int[x*(1-c^2*x^2)^(p+1/2)*(a+b*ArcCos[c*x])^(n- 1),x]",
     "rulenumber": 0,
            "lhs": "Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e},x] && EqQ[c^2*d+e,0] && GtQ[n,0] &&  LtQ[p,-1] && NeQ[p,-3/2] && (IntegerQ[p] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "-x*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^ n/(2*d*(p + 1)) + (2*p + 3)/(2*d*(p + 1))* Int[(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x] + b*c*n* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/(2*(p + 1)*(1 - c^2*x^2)^FracPart[p])* Int[x*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "-x*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^ n/(2*d*(p + 1)) + (2*p + 3)/(2*d*(p + 1))* Int[(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n, x] - b*c*n* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/(2*(p + 1)*(1 - c^2*x^2)^FracPart[p])* Int[x*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "1/(c*d)*Subst[Int[(a + b*x)^n*Sec[x], x], x, ArcSin[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSin[c_.*x_])^n_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "-1/(c*d)* Subst[Int[(a + b*x)^n*Csc[x], x], x, ArcCos[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCos[c_.*x_])^n_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": " d^p*(1-c^2*x^2)^(p+1/2)*(a+b*ArcSin[c*x])^(n+1)/(b*c*(n+1)) + c*d^p*(2*p+1)/(b*(n+1))*Int[x*(1-c^2*x^2)^(p-1/2)*(a+b*ArcSin[c*x])^ (n+1),x]",
     "rulenumber": 0,
            "lhs": "Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,p},x] && EqQ[c^2*d+e,0] && LtQ[n,-1] &&  (IntegerQ[p] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": " -d^p*(1-c^2*x^2)^(p+1/2)*(a+b*ArcCos[c*x])^(n+1)/(b*c*(n+1)) - c*d^p*(2*p+1)/(b*(n+1))*Int[x*(1-c^2*x^2)^(p-1/2)*(a+b*ArcCos[c*x])^ (n+1),x]",
     "rulenumber": 0,
            "lhs": "Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,p},x] && EqQ[c^2*d+e,0] && LtQ[n,-1] &&  (IntegerQ[p] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "Sqrt[1 - c^2*x^2]*(d + e*x^2)^ p*(a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1)) + c*(2*p + 1)* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/(b*(n + 1)*(1 - c^2*x^2)^FracPart[p])* Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "-Sqrt[1 - c^2*x^2]*(d + e*x^2)^ p*(a + b*ArcCos[c*x])^(n + 1)/(b*c*(n + 1)) - c*(2*p + 1)* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/(b*(n + 1)*(1 - c^2*x^2)^FracPart[p])* Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "d^p/c*Subst[Int[(a + b*x)^n*Cos[x]^(2*p + 1), x], x, ArcSin[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p, 0] && (IntegerQ[p] || GtQ[d, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "-d^p/c* Subst[Int[(a + b*x)^n*Sin[x]^(2*p + 1), x], x, ArcCos[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p, 0] && (IntegerQ[p] || GtQ[d, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "d^(p - 1/2)*Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]* Int[(1 - c^2*x^2)^p*(a + b*ArcSin[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p, 0] && Not[IntegerQ[p] || GtQ[d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "d^(p - 1/2)*Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]* Int[(1 - c^2*x^2)^p*(a + b*ArcCos[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p, 0] && Not[IntegerQ[p] || GtQ[d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[a + b*ArcCos[c*x], u, x] + b*c*Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n, (d + e*x^2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || IGtQ[n, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcCos[c*x])^n, (d + e*x^2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || IGtQ[n, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "Unintegrable[(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "Unintegrable[(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(-d^2*g/e)^q* Int[(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^p_*(f_ + g_.*x_)^q_*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0] && GtQ[d, 0] && LtQ[g/e, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(-d^2*g/e)^q* Int[(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcCos[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^p_*(f_ + g_.*x_)^q_*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0] && GtQ[d, 0] && LtQ[g/e, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(d + e*x)^q*(f + g*x)^q/(1 - c^2*x^2)^q* Int[(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^p_*(f_ + g_.*x_)^q_*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.3 (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(d + e*x)^q*(f + g*x)^q/(1 - c^2*x^2)^q* Int[(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcCos[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^p_*(f_ + g_.*x_)^q_*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "-1/e* Subst[Int[(a + b*x)^n*Tan[x], x], x, ArcSin[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_*(a_. + b_.*ArcSin[c_.*x_])^n_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "1/e*Subst[Int[(a + b*x)^n*Cot[x], x], x, ArcCos[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_*(a_. + b_.*ArcCos[c_.*x_])^n_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "  (d+e*x^2)^(p+1)*(a+b*ArcSin[c*x])^n/(2*e*(p+1)) + b*n*d^p/(2*c*(p+1))*Int[(1-c^2*x^2)^(p+1/2)*(a+b*ArcSin[c*x])^(n-1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,p},x] && EqQ[c^2*d+e,0] && GtQ[n,0] &&  NeQ[p,-1] && (IntegerQ[p] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "  (d+e*x^2)^(p+1)*(a+b*ArcCos[c*x])^n/(2*e*(p+1)) - b*n*d^p/(2*c*(p+1))*Int[(1-c^2*x^2)^(p+1/2)*(a+b*ArcCos[c*x])^(n-1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,p},x] && EqQ[c^2*d+e,0] && GtQ[n,0] &&  NeQ[p,-1] && (IntegerQ[p] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^ n/(2*e*(p + 1)) + b*n*d^ IntPart[p]*(d + e*x^2)^ FracPart[p]/(2*c*(p + 1)*(1 - c^2*x^2)^FracPart[p])* Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^ n/(2*e*(p + 1)) - b*n*d^ IntPart[p]*(d + e*x^2)^ FracPart[p]/(2*c*(p + 1)*(1 - c^2*x^2)^FracPart[p])* Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "1/d*Subst[Int[(a + b*x)^n/(Cos[x]*Sin[x]), x], x, ArcSin[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSin[c_.*x_])^n_./(x_*(d_ + e_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "-1/d* Subst[Int[(a + b*x)^n/(Cos[x]*Sin[x]), x], x, ArcCos[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCos[c_.*x_])^n_./(x_*(d_ + e_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": " (f*x)^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcSin[c*x])^n/(d*f*(m+1)) - b*c*n*d^p/(f*(m+1))*Int[(f*x)^(m+1)*(1-c^2*x^2)^(p+1/2)*(a+b*ArcSin[ c*x])^(n-1),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,m,p},x] && EqQ[c^2*d+e,0] &&  GtQ[n,0] && EqQ[m+2*p+3,0] && NeQ[m,-1] && (IntegerQ[p] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": " (f*x)^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcCos[c*x])^n/(d*f*(m+1)) + b*c*n*d^p/(f*(m+1))*Int[(f*x)^(m+1)*(1-c^2*x^2)^(p+1/2)*(a+b*ArcCos[ c*x])^(n-1),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,m,p},x] && EqQ[c^2*d+e,0] &&  GtQ[n,0] && EqQ[m+2*p+3,0] && NeQ[m,-1] && (IntegerQ[p] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n/(d*f*(m + 1)) - b*c*n* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/(f*(m + 1)*(1 - c^2*x^2)^FracPart[p])* Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n/(d*f*(m + 1)) + b*c*n* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/(f*(m + 1)*(1 - c^2*x^2)^FracPart[p])* Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(d + e*x^2)^p*(a + b*ArcSin[c*x])/(2*p) - b*c*d^p/(2*p)*Int[(1 - c^2*x^2)^(p - 1/2), x] + d*Int[(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])/x, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(d + e*x^2)^p*(a + b*ArcCos[c*x])/(2*p) + b*c*d^p/(2*p)*Int[(1 - c^2*x^2)^(p - 1/2), x] + d*Int[(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])/x, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*(d + e*x^2)^ p*(a + b*ArcSin[c*x])/(f*(m + 1)) - b*c*d^p/(f*(m + 1))* Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2), x] - 2*e*p/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && ILtQ[(m + 1)/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*(d + e*x^2)^ p*(a + b*ArcCos[c*x])/(f*(m + 1)) + b*c*d^p/(f*(m + 1))* Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2), x] - 2*e*p/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && ILtQ[(m + 1)/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCos[c*x], u, x] + b*c*Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = IntHide[x^m*(1 - c^2*x^2)^p, x]}, Dist[d^p*(a + b*ArcSin[c*x]), u, x] - b*c*d^p*Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[ p - 1/2] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0]) && NeQ[p, -1/2] && GtQ[d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = IntHide[x^m*(1 - c^2*x^2)^p, x]}, Dist[d^p*(a + b*ArcCos[c*x]), u, x] + b*c*d^p*Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[ p - 1/2] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0]) && NeQ[p, -1/2] && GtQ[d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = IntHide[x^m*(1 - c^2*x^2)^p, x]}, (a + b*ArcSin[c*x])*Int[x^m*(d + e*x^2)^p, x] - b*c*d^(p - 1/2)*Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]* Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p + 1/2, 0] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = IntHide[x^m*(1 - c^2*x^2)^p, x]}, (a + b*ArcCos[c*x])*Int[x^m*(d + e*x^2)^p, x] + b*c*d^(p - 1/2)*Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]* Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p + 1/2, 0] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(f*x)^(m+1)*(d+e*x^2)^p*(a+b*ArcSin[c*x])^n/(f*(m+1)) - 2*e*p/(f^2*(m+1))*Int[(f*x)^(m+2)*(d+e*x^2)^(p-1)*(a+b*ArcSin[c*x])^ n,x] - b*c*n*d^p/(f*(m+1))*Int[(f*x)^(m+1)*(1-c^2*x^2)^(p-1/2)*(a+b*ArcSin[ c*x])^(n-1),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e,0] &&  GtQ[n,0] && GtQ[p,0] && LtQ[m,-1] && (IntegerQ[p] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(f*x)^(m+1)*(d+e*x^2)^p*(a+b*ArcCos[c*x])^n/(f*(m+1)) - 2*e*p/(f^2*(m+1))*Int[(f*x)^(m+2)*(d+e*x^2)^(p-1)*(a+b*ArcCos[c*x])^ n,x] + b*c*n*d^p/(f*(m+1))*Int[(f*x)^(m+1)*(1-c^2*x^2)^(p-1/2)*(a+b*ArcCos[ c*x])^(n-1),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e,0] &&  GtQ[n,0] && GtQ[p,0] && LtQ[m,-1] && (IntegerQ[p] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(f*x)^(m + 1)* Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n/(f*(m + 1)) - b*c*n*Sqrt[d + e*x^2]/(f*(m + 1)*Sqrt[1 - c^2*x^2])* Int[(f*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1), x] + c^2*Sqrt[d + e*x^2]/(f^2*(m + 1)*Sqrt[1 - c^2*x^2])* Int[(f*x)^(m + 2)*(a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(f*x)^(m + 1)* Sqrt[d + e*x^2]*(a + b*ArcCos[c*x])^n/(f*(m + 1)) + b*c*n*Sqrt[d + e*x^2]/(f*(m + 1)*Sqrt[1 - c^2*x^2])* Int[(f*x)^(m + 1)*(a + b*ArcCos[c*x])^(n - 1), x] + c^2*Sqrt[d + e*x^2]/(f^2*(m + 1)*Sqrt[1 - c^2*x^2])* Int[(f*x)^(m + 2)*(a + b*ArcCos[c*x])^n/Sqrt[1 - c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*(d + e*x^2)^ p*(a + b*ArcSin[c*x])^n/(f*(m + 1)) - 2*e*p/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x] - b*c*n* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/(f*(m + 1)*(1 - c^2*x^2)^FracPart[p])* Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*(d + e*x^2)^ p*(a + b*ArcCos[c*x])^n/(f*(m + 1)) - 2*e*p/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n, x] + b*c*n* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/(f*(m + 1)*(1 - c^2*x^2)^FracPart[p])* Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": " (f*x)^(m+1)*(d+e*x^2)^p*(a+b*ArcSin[c*x])^n/(f*(m+2*p+1)) + 2*d*p/(m+2*p+1)*Int[(f*x)^m*(d+e*x^2)^(p-1)*(a+b*ArcSin[c*x])^n,x] - b*c*n*d^p/(f*(m+2*p+1))*Int[(f*x)^(m+1)*(1-c^2*x^2)^(p-1/2)*(a+b* ArcSin[c*x])^(n-1),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,m},x] && EqQ[c^2*d+e,0] &&  GtQ[n,0] && GtQ[p,0] && Not[LtQ[m,-1]] && (IntegerQ[p] || GtQ[d,0]) && (RationalQ[m] || EqQ[n,1]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": " (f*x)^(m+1)*(d+e*x^2)^p*(a+b*ArcCos[c*x])^n/(f*(m+2*p+1)) + 2*d*p/(m+2*p+1)*Int[(f*x)^m*(d+e*x^2)^(p-1)*(a+b*ArcCos[c*x])^n,x] + b*c*n*d^p/(f*(m+2*p+1))*Int[(f*x)^(m+1)*(1-c^2*x^2)^(p-1/2)*(a+b* ArcCos[c*x])^(n-1),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,m},x] && EqQ[c^2*d+e,0] &&  GtQ[n,0] && GtQ[p,0] && Not[LtQ[m,-1]] && (IntegerQ[p] || GtQ[d,0]) && (RationalQ[m] || EqQ[n,1]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(f*x)^(m + 1)* Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n/(f*(m + 2)) - b*c*n*Sqrt[d + e*x^2]/(f*(m + 2)*Sqrt[1 - c^2*x^2])* Int[(f*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1), x] + Sqrt[d + e*x^2]/((m + 2)*Sqrt[1 - c^2*x^2])* Int[(f*x)^m*(a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && Not[LtQ[m, -1]] && (RationalQ[m] || EqQ[n, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(f*x)^(m + 1)* Sqrt[d + e*x^2]*(a + b*ArcCos[c*x])^n/(f*(m + 2)) + b*c*n*Sqrt[d + e*x^2]/(f*(m + 2)*Sqrt[1 - c^2*x^2])* Int[(f*x)^(m + 1)*(a + b*ArcCos[c*x])^(n - 1), x] + Sqrt[d + e*x^2]/((m + 2)*Sqrt[1 - c^2*x^2])* Int[(f*x)^m*(a + b*ArcCos[c*x])^n/Sqrt[1 - c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && Not[LtQ[m, -1]] && (RationalQ[m] || EqQ[n, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*(d + e*x^2)^ p*(a + b*ArcSin[c*x])^n/(f*(m + 2*p + 1)) + 2*d*p/(m + 2*p + 1)* Int[(f*x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x] - b*c*n* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/(f*(m + 2*p + 1)*(1 - c^2*x^2)^FracPart[p])* Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && Not[LtQ[m, -1]] && (RationalQ[m] || EqQ[n, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*(d + e*x^2)^ p*(a + b*ArcCos[c*x])^n/(f*(m + 2*p + 1)) + 2*d*p/(m + 2*p + 1)* Int[(f*x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n, x] + b*c*n* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/(f*(m + 2*p + 1)*(1 - c^2*x^2)^FracPart[p])* Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && Not[LtQ[m, -1]] && (RationalQ[m] || EqQ[n, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": " (f*x)^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcSin[c*x])^n/(d*f*(m+1)) + c^2*(m+2*p+3)/(f^2*(m+1))*Int[(f*x)^(m+2)*(d+e*x^2)^p*(a+b*ArcSin[c* x])^n,x] - b*c*n*d^p/(f*(m+1))*Int[(f*x)^(m+1)*(1-c^2*x^2)^(p+1/2)*(a+b*ArcSin[ c*x])^(n-1),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,p},x] && EqQ[c^2*d+e,0] &&  GtQ[n,0] && LtQ[m,-1] && IntegerQ[m] && (IntegerQ[p] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": " (f*x)^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcCos[c*x])^n/(d*f*(m+1)) + c^2*(m+2*p+3)/(f^2*(m+1))*Int[(f*x)^(m+2)*(d+e*x^2)^p*(a+b*ArcCos[c* x])^n,x] + b*c*n*d^p/(f*(m+1))*Int[(f*x)^(m+1)*(1-c^2*x^2)^(p+1/2)*(a+b*ArcCos[ c*x])^(n-1),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,p},x] && EqQ[c^2*d+e,0] &&  GtQ[n,0] && LtQ[m,-1] && IntegerQ[m] && (IntegerQ[p] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n/(d*f*(m + 1)) + c^2*(m + 2*p + 3)/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x] - b*c*n* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/(f*(m + 1)*(1 - c^2*x^2)^FracPart[p])* Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[m, -1] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n/(d*f*(m + 1)) + c^2*(m + 2*p + 3)/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x] + b*c*n* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/(f*(m + 1)*(1 - c^2*x^2)^FracPart[p])* Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[m, -1] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": " f*(f*x)^(m-1)*(d+e*x^2)^(p+1)*(a+b*ArcSin[c*x])^n/(2*e*(p+1)) - f^2*(m-1)/(2*e*(p+1))*Int[(f*x)^(m-2)*(d+e*x^2)^(p+1)*(a+b*ArcSin[c* x])^n,x] + b*f*n*d^p/(2*c*(p+1))*Int[(f*x)^(m-1)*(1-c^2*x^2)^(p+1/2)*(a+b* ArcSin[c*x])^(n-1),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e,0] &&  GtQ[n,0] && LtQ[p,-1] && GtQ[m,1] && (IntegerQ[p] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": " f*(f*x)^(m-1)*(d+e*x^2)^(p+1)*(a+b*ArcCos[c*x])^n/(2*e*(p+1)) - f^2*(m-1)/(2*e*(p+1))*Int[(f*x)^(m-2)*(d+e*x^2)^(p+1)*(a+b*ArcCos[c* x])^n,x] - b*f*n*d^p/(2*c*(p+1))*Int[(f*x)^(m-1)*(1-c^2*x^2)^(p+1/2)*(a+b* ArcCos[c*x])^(n-1),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e,0] &&  GtQ[n,0] && LtQ[p,-1] && GtQ[m,1] && (IntegerQ[p] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^ n/(2*e*(p + 1)) - f^2*(m - 1)/(2*e*(p + 1))* Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x] + b*f*n* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/(2*c*(p + 1)*(1 - c^2*x^2)^FracPart[p])* Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && GtQ[m, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^ n/(2*e*(p + 1)) - f^2*(m - 1)/(2*e*(p + 1))* Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n, x] - b*f*n* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/(2*c*(p + 1)*(1 - c^2*x^2)^FracPart[p])* Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && GtQ[m, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": " -(f*x)^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcSin[c*x])^n/(2*d*f*(p+1)) + (m+2*p+3)/(2*d*(p+1))*Int[(f*x)^m*(d+e*x^2)^(p+1)*(a+b*ArcSin[c*x])^ n,x] + b*c*n*d^p/(2*f*(p+1))*Int[(f*x)^(m+1)*(1-c^2*x^2)^(p+1/2)*(a+b* ArcSin[c*x])^(n-1),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,m},x] && EqQ[c^2*d+e,0] &&  GtQ[n,0] && LtQ[p,-1] && Not[GtQ[m,1]] && (IntegerQ[p] || GtQ[d,0]) &&   (IntegerQ[m] || IntegerQ[p] || EqQ[n,1]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": " -(f*x)^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcCos[c*x])^n/(2*d*f*(p+1)) + (m+2*p+3)/(2*d*(p+1))*Int[(f*x)^m*(d+e*x^2)^(p+1)*(a+b*ArcCos[c*x])^ n,x] - b*c*n*d^p/(2*f*(p+1))*Int[(f*x)^(m+1)*(1-c^2*x^2)^(p+1/2)*(a+b* ArcCos[c*x])^(n-1),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,m},x] && EqQ[c^2*d+e,0] &&  GtQ[n,0] && LtQ[p,-1] && Not[GtQ[m,1]] && (IntegerQ[p] || GtQ[d,0]) &&   (IntegerQ[m] || IntegerQ[p] || EqQ[n,1]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "-(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n/(2*d*f*(p + 1)) + (m + 2*p + 3)/(2*d*(p + 1))* Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x] + b*c*n* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/(2*f*(p + 1)*(1 - c^2*x^2)^FracPart[p])* Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && Not[GtQ[m, 1]] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "-(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n/(2*d*f*(p + 1)) + (m + 2*p + 3)/(2*d*(p + 1))* Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n, x] - b*c*n* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/(2*f*(p + 1)*(1 - c^2*x^2)^FracPart[p])* Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && Not[GtQ[m, 1]] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "f*(f*x)^(m-1)*Sqrt[d+e*x^2]*(a+b*ArcSin[c*x])^n/(e*m) + b*f*n/(c*m*Sqrt[d])*Int[(f*x)^(m-1)*(a+b*ArcSin[c*x])^(n-1),x] + f^2*(m-1)/(c^2*m)*Int[((f*x)^(m-2)*(a+b*ArcSin[c*x])^n)/Sqrt[d+e*x^ 2],x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_.+b_.*ArcSin[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e,0] && GtQ[n,0] &&  GtQ[m,1] && GtQ[d,0] && IntegerQ[m] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "f*(f*x)^(m-1)*Sqrt[d+e*x^2]*(a+b*ArcCos[c*x])^n/(e*m) - b*f*n*Sqrt[1-c^2*x^2]/(c*m*Sqrt[d+e*x^2])*Int[(f*x)^(m-1)*(a+b* ArcCos[c*x])^(n-1),x] + f^2*(m-1)/(c^2*m)*Int[((f*x)^(m-2)*(a+b*ArcCos[c*x])^n)/Sqrt[d+e*x^ 2],x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_.+b_.*ArcCos[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e,0] && GtQ[n,0] &&  GtQ[m,1] && GtQ[d,0] && IntegerQ[m] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n/(e*m) + b*f*n*Sqrt[1 - c^2*x^2]/(c*m*Sqrt[d + e*x^2])* Int[(f*x)^(m - 1)*(a + b*ArcSin[c*x])^(n - 1), x] + f^2*(m - 1)/(c^2*m)* Int[((f*x)^(m - 2)*(a + b*ArcSin[c*x])^n)/Sqrt[d + e*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcSin[c_.*x_])^n_./Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcCos[c*x])^n/(e*m) - b*f*n*Sqrt[1 - c^2*x^2]/(c*m*Sqrt[d + e*x^2])* Int[(f*x)^(m - 1)*(a + b*ArcCos[c*x])^(n - 1), x] + f^2*(m - 1)/(c^2*m)* Int[((f*x)^(m - 2)*(a + b*ArcCos[c*x])^n)/Sqrt[d + e*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcCos[c_.*x_])^n_./Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "1/(c^(m + 1)*Sqrt[d])* Subst[Int[(a + b*x)^n*Sin[x]^m, x], x, ArcSin[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*(a_. + b_.*ArcSin[c_.*x_])^n_./Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && IGtQ[n, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "-1/(c^(m + 1)*Sqrt[d])* Subst[Int[(a + b*x)^n*Cos[x]^m, x], x, ArcCos[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*(a_. + b_.*ArcCos[c_.*x_])^n_./Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && IGtQ[n, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*(a + b*ArcSin[c*x])* Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2]/(Sqrt[d]*f*(m + 1)) - b*c*(f*x)^(m + 2)* HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2]/(Sqrt[d]*f^2*(m + 1)*(m + 2))",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcSin[c_.*x_])/Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*(a + b*ArcCos[c*x])* Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2]/(Sqrt[d]*f*(m + 1)) + b*c*(f*x)^(m + 2)* HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2]/(Sqrt[d]*f^2*(m + 1)*(m + 2))",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcCos[c_.*x_])/Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]* Int[(f*x)^m*(a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcSin[c_.*x_])^n_./Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && Not[GtQ[d, 0]] && (IntegerQ[m] || EqQ[n, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]* Int[(f*x)^m*(a + b*ArcCos[c*x])^n/Sqrt[1 - c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcCos[c_.*x_])^n_./Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && Not[GtQ[d, 0]] && (IntegerQ[m] || EqQ[n, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": " f*(f*x)^(m-1)*(d+e*x^2)^(p+1)*(a+b*ArcSin[c*x])^n/(e*(m+2*p+1)) + f^2*(m-1)/(c^2*(m+2*p+1))*Int[(f*x)^(m-2)*(d+e*x^2)^p*(a+b*ArcSin[c* x])^n,x] + b*f*n*d^p/(c*(m+2*p+1))*Int[(f*x)^(m-1)*(1-c^2*x^2)^(p+1/2)*(a+b* ArcSin[c*x])^(n-1),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,p},x] && EqQ[c^2*d+e,0] &&  GtQ[n,0] && GtQ[m,1] && NeQ[m+2*p+1,0] && (IntegerQ[p] || GtQ[d,0]) &&  IntegerQ[m] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": " f*(f*x)^(m-1)*(d+e*x^2)^(p+1)*(a+b*ArcCos[c*x])^n/(e*(m+2*p+1)) + f^2*(m-1)/(c^2*(m+2*p+1))*Int[(f*x)^(m-2)*(d+e*x^2)^p*(a+b*ArcCos[c* x])^n,x] - b*f*n*d^p/(c*(m+2*p+1))*Int[(f*x)^(m-1)*(1-c^2*x^2)^(p+1/2)*(a+b* ArcCos[c*x])^(n-1),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,p},x] && EqQ[c^2*d+e,0] &&  GtQ[n,0] && GtQ[m,1] && NeQ[m+2*p+1,0] && (IntegerQ[p] || GtQ[d,0]) &&  IntegerQ[m] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^ n/(e*(m + 2*p + 1)) + f^2*(m - 1)/(c^2*(m + 2*p + 1))* Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x] + b*f*n* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/(c*(m + 2*p + 1)*(1 - c^2*x^2)^FracPart[p])* Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^ n/(e*(m + 2*p + 1)) + f^2*(m - 1)/(c^2*(m + 2*p + 1))* Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x] - b*f*n* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/(c*(m + 2*p + 1)*(1 - c^2*x^2)^FracPart[p])* Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": " (f*x)^m*Sqrt[1-c^2*x^2]*(d+e*x^2)^p*(a+b*ArcSin[c*x])^(n+1)/(b*c*(n+1) ) - f*m*d^p/(b*c*(n+1))*Int[(f*x)^(m-1)*(1-c^2*x^2)^(p-1/2)*(a+b*ArcSin[ c*x])^(n+1),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,m,p},x] && EqQ[c^2*d+e,0] &&  LtQ[n,-1] && EqQ[m+2*p+1,0] && (IntegerQ[p] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": " -(f*x)^m*Sqrt[1-c^2*x^2]*(d+e*x^2)^p*(a+b*ArcCos[c*x])^(n+1)/(b*c*(n+ 1)) + f*m*d^p/(b*c*(n+1))*Int[(f*x)^(m-1)*(1-c^2*x^2)^(p-1/2)*(a+b*ArcCos[ c*x])^(n+1),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,m,p},x] && EqQ[c^2*d+e,0] &&  LtQ[n,-1] && EqQ[m+2*p+1,0] && (IntegerQ[p] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(f*x)^m* Sqrt[1 - c^2*x^2]*(d + e*x^2)^ p*(a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1)) - f*m*d^ IntPart[p]*(d + e*x^2)^ FracPart[p]/(b*c*(n + 1)*(1 - c^2*x^2)^FracPart[p])* Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && EqQ[m + 2*p + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "-(f*x)^m* Sqrt[1 - c^2*x^2]*(d + e*x^2)^ p*(a + b*ArcCos[c*x])^(n + 1)/(b*c*(n + 1)) + f*m*d^ IntPart[p]*(d + e*x^2)^ FracPart[p]/(b*c*(n + 1)*(1 - c^2*x^2)^FracPart[p])* Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && EqQ[m + 2*p + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(f*x)^ m*(a + b*ArcSin[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1)) - f*m/(b*c*Sqrt[d]*(n + 1))* Int[(f*x)^(m - 1)*(a + b*ArcSin[c*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(a_. + b_.*ArcSin[c_.*x_])^n_/Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && GtQ[d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "-(f*x)^ m*(a + b*ArcCos[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1)) + f*m/(b*c*Sqrt[d]*(n + 1))* Int[(f*x)^(m - 1)*(a + b*ArcCos[c*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(a_. + b_.*ArcCos[c_.*x_])^n_/Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && GtQ[d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": " (f*x)^m*Sqrt[1-c^2*x^2]*(d+e*x^2)^p*(a+b*ArcSin[c*x])^(n+1)/(b*c*(n+1) ) - f*m*d^p/(b*c*(n+1))*Int[(f*x)^(m-1)*(1-c^2*x^2)^(p-1/2)*(a+b*ArcSin[ c*x])^(n+1),x] + c*(m+2*p+1)*d^p/(b*f*(n+1))*Int[(f*x)^(m+1)*(1-c^2*x^2)^(p-1/2)*(a+ b*ArcSin[c*x])^(n+1),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSin[c_.*x_])^n_,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e,0] &&  LtQ[n,-1] && IGtQ[m,-3] && IGtQ[2*p,0] && (IntegerQ[p] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": " -(f*x)^m*Sqrt[1-c^2*x^2]*(d+e*x^2)^p*(a+b*ArcCos[c*x])^(n+1)/(b*c*(n+ 1)) + f*m*d^p/(b*c*(n+1))*Int[(f*x)^(m-1)*(1-c^2*x^2)^(p-1/2)*(a+b*ArcCos[ c*x])^(n+1),x] - c*(m+2*p+1)*d^p/(b*f*(n+1))*Int[(f*x)^(m+1)*(1-c^2*x^2)^(p-1/2)*(a+ b*ArcCos[c*x])^(n+1),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCos[c_.*x_])^n_,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f},x] && EqQ[c^2*d+e,0] &&  LtQ[n,-1] && IGtQ[m,-3] && IGtQ[2*p,0] && (IntegerQ[p] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(f*x)^m* Sqrt[1 - c^2*x^2]*(d + e*x^2)^ p*(a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1)) - f*m*d^ IntPart[p]*(d + e*x^2)^ FracPart[p]/(b*c*(n + 1)*(1 - c^2*x^2)^FracPart[p])* Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n + 1), x] + c*(m + 2*p + 1)* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/(b*f*(n + 1)*(1 - c^2*x^2)^FracPart[p])* Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IGtQ[m, -3] && IGtQ[2*p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "-(f*x)^m* Sqrt[1 - c^2*x^2]*(d + e*x^2)^ p*(a + b*ArcCos[c*x])^(n + 1)/(b*c*(n + 1)) + f*m*d^ IntPart[p]*(d + e*x^2)^ FracPart[p]/(b*c*(n + 1)*(1 - c^2*x^2)^FracPart[p])* Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n + 1), x] - c*(m + 2*p + 1)* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/(b*f*(n + 1)*(1 - c^2*x^2)^FracPart[p])* Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IGtQ[m, -3] && IGtQ[2*p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "d^p/c^(m + 1)* Subst[Int[(a + b*x)^n*Sin[x]^m*Cos[x]^(2*p + 1), x], x, ArcSin[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "-d^p/c^(m + 1)* Subst[Int[(a + b*x)^n*Cos[x]^m*Sin[x]^(2*p + 1), x], x, ArcCos[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "d^IntPart[p]*(d + e*x^2)^FracPart[p]/(1 - c^2*x^2)^FracPart[p]* Int[x^m*(1 - c^2*x^2)^p*(a + b*ArcSin[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && Not[(IntegerQ[p] || GtQ[d, 0])]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "d^IntPart[p]*(d + e*x^2)^FracPart[p]/(1 - c^2*x^2)^FracPart[p]* Int[x^m*(1 - c^2*x^2)^p*(a + b*ArcCos[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && Not[(IntegerQ[p] || GtQ[d, 0])]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n/ Sqrt[d + e*x^2], (f*x)^m*(d + e*x^2)^(p + 1/2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && IGtQ[p + 1/2, 0] && Not[IGtQ[(m + 1)/2, 0]] && (EqQ[m, -1] || EqQ[m, -2])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcCos[c*x])^n/ Sqrt[d + e*x^2], (f*x)^m*(d + e*x^2)^(p + 1/2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && IGtQ[p + 1/2, 0] && Not[IGtQ[(m + 1)/2, 0]] && (EqQ[m, -1] || EqQ[m, -2])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])/(2* e*(p + 1)) - b*c/(2*e*(p + 1))*Int[(d + e*x^2)^(p + 1)/Sqrt[1 - c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && NeQ[c^2*d + e, 0] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])/(2* e*(p + 1)) + b*c/(2*e*(p + 1))*Int[(d + e*x^2)^(p + 1)/Sqrt[1 - c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && NeQ[c^2*d + e, 0] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCos[c*x], u, x] + b*c*Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n, (f*x)^m*(d + e*x^2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcCos[c*x])^n, (f*x)^m*(d + e*x^2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "Unintegrable[(f*x)^m*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "Unintegrable[(f*x)^m*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(-d^2*g/e)^q* Int[(h*x)^m*(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^ n, x]",
     "rulenumber": 0,
            "lhs": "Int[(h_.*x_)^m_.*(d_ + e_.*x_)^p_*(f_ + g_.*x_)^ q_*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0] && GtQ[d, 0] && LtQ[g/e, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(-d^2*g/e)^q* Int[(h*x)^m*(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcCos[c*x])^ n, x]",
     "rulenumber": 0,
            "lhs": "Int[(h_.*x_)^m_.*(d_ + e_.*x_)^p_*(f_ + g_.*x_)^ q_*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0] && GtQ[d, 0] && LtQ[g/e, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(-d^2*g/e)^IntPart[q]*(d + e*x)^ FracPart[q]*(f + g*x)^FracPart[q]/(1 - c^2*x^2)^FracPart[q]* Int[(h*x)^m*(d + e*x)^(p - q)*(1 - c^2*x^2)^ q*(a + b*ArcSin[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(h_.*x_)^m_.*(d_ + e_.*x_)^p_*(f_ + g_.*x_)^ q_*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
       "filename": "5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m",
            "rhs": "(-d^2*g/e)^IntPart[q]*(d + e*x)^ FracPart[q]*(f + g*x)^FracPart[q]/(1 - c^2*x^2)^FracPart[q]* Int[(h*x)^m*(d + e*x)^(p - q)*(1 - c^2*x^2)^ q*(a + b*ArcCos[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(h_.*x_)^m_.*(d_ + e_.*x_)^p_*(f_ + g_.*x_)^ q_*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "Subst[Int[(a + b*x)^n*Cos[x]/(c*d + e*Sin[x]), x], x, ArcSin[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSin[c_.*x_])^n_./(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "-Subst[Int[(a + b*x)^n*Sin[x]/(c*d + e*Cos[x]), x], x, ArcCos[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCos[c_.*x_])^n_./(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "(d + e*x)^(m + 1)*(a + b*ArcSin[c*x])^ n/(e*(m + 1)) - b*c*n/(e*(m + 1))* Int[(d + e*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1)/ Sqrt[1 - c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_.*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "(d + e*x)^(m + 1)*(a + b*ArcCos[c*x])^ n/(e*(m + 1)) + b*c*n/(e*(m + 1))* Int[(d + e*x)^(m + 1)*(a + b*ArcCos[c*x])^(n - 1)/ Sqrt[1 - c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_.*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x)^m*(a + b*ArcSin[c*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_.*(a_. + b_.*ArcSin[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[m, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x)^m*(a + b*ArcCos[c*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_.*(a_. + b_.*ArcCos[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[m, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "1/c^(m + 1)* Subst[Int[(a + b*x)^n*Cos[x]*(c*d + e*Sin[x])^m, x], x, ArcSin[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*(a_. + b_.*ArcSin[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "-1/c^(m + 1)* Subst[Int[(a + b*x)^n*Sin[x]*(c*d + e*Cos[x])^m, x], x, ArcCos[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*(a_. + b_.*ArcCos[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = IntHide[ExpandExpression[Px, x], x]}, Dist[a + b*ArcSin[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_. + b_.*ArcSin[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && PolynomialQ[Px, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = IntHide[ExpandExpression[Px, x], x]}, Dist[a + b*ArcCos[c*x], u, x] + b*c*Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_. + b_.*ArcCos[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && PolynomialQ[Px, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": " With[{u=IntHide[Px,x]}, Dist[(a+b*ArcSin[c*x])^n,u,x] -  b*c*n*Int[SimplifyIntegrand[u*(a+b*ArcSin[c*x])^(n-1)/Sqrt[1-c^2*x^2], x],x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_.+b_.*ArcSin[c_.*x_])^n_.,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c},x] && PolynomialQ[Px,x] && IGtQ[n,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": " With[{u=IntHide[Px,x]}, Dist[(a+b*ArcCos[c*x])^n,u,x] +  b*c*n*Int[SimplifyIntegrand[u*(a+b*ArcCos[c*x])^(n-1)/Sqrt[1-c^2*x^2], x],x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_.+b_.*ArcCos[c_.*x_])^n_.,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c},x] && PolynomialQ[Px,x] && IGtQ[n,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[Px*(a + b*ArcSin[c*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_. + b_.*ArcSin[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && PolynomialQ[Px, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[Px*(a + b*ArcCos[c*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_. + b_.*ArcCos[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && PolynomialQ[Px, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = IntHide[Px*(d + e*x)^m, x]}, Dist[a + b*ArcSin[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_. + e_.*x_)^m_.*(a_. + b_.*ArcSin[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && PolynomialQ[Px, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = IntHide[Px*(d + e*x)^m, x]}, Dist[a + b*ArcCos[c*x], u, x] + b*c*Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_. + e_.*x_)^m_.*(a_. + b_.*ArcCos[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && PolynomialQ[Px, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = IntHide[(f + g*x)^p*(d + e*x)^m, x]}, Dist[(a + b*ArcSin[c*x])^n, u, x] - b*c*n*Int[ SimplifyIntegrand[ u*(a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^p_.*(d_ + e_.*x_)^m_*(a_. + b_.*ArcSin[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[m, 0] && LtQ[m + p + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = IntHide[(f + g*x)^p*(d + e*x)^m, x]}, Dist[(a + b*ArcCos[c*x])^n, u, x] + b*c*n*Int[ SimplifyIntegrand[ u*(a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^p_.*(d_ + e_.*x_)^m_*(a_. + b_.*ArcCos[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[m, 0] && LtQ[m + p + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = IntHide[(f + g*x + h*x^2)^p/(d + e*x)^2, x]}, Dist[(a + b*ArcSin[c*x])^n, u, x] - b*c*n*Int[ SimplifyIntegrand[ u*(a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_ + h_.*x_^2)^ p_.*(a_. + b_.*ArcSin[c_.*x_])^n_/(d_ + e_.*x_)^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h}, x] && IGtQ[n, 0] && IGtQ[p, 0] && EqQ[e*g - 2*d*h, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = IntHide[(f + g*x + h*x^2)^p/(d + e*x)^2, x]}, Dist[(a + b*ArcCos[c*x])^n, u, x] + b*c*n*Int[ SimplifyIntegrand[ u*(a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_ + h_.*x_^2)^ p_.*(a_. + b_.*ArcCos[c_.*x_])^n_/(d_ + e_.*x_)^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h}, x] && IGtQ[n, 0] && IGtQ[p, 0] && EqQ[e*g - 2*d*h, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[Px*(d + e*x)^m*(a + b*ArcSin[c*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_ + e_.*x_)^m_.*(a_. + b_.*ArcSin[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && PolynomialQ[Px, x] && IGtQ[n, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[Px*(d + e*x)^m*(a + b*ArcCos[c*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_ + e_.*x_)^m_.*(a_. + b_.*ArcCos[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && PolynomialQ[Px, x] && IGtQ[n, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = IntHide[(f + g*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - b*c*Int[Dist[1/Sqrt[1 - c^2*x^2], u, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && (LtQ[m, -2*p - 1] || GtQ[m, 3])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = IntHide[(f + g*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCos[c*x], u, x] + b*c*Int[Dist[1/Sqrt[1 - c^2*x^2], u, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && (LtQ[m, -2*p - 1] || GtQ[m, 3])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, (f + g*x)^m, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_])^ n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n, 0] && (m == 1 || p > 0 || n == 1 && p > -1 || m == 2 && p < -2)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, (f + g*x)^m, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_])^ n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n, 0] && (m == 1 || p > 0 || n == 1 && p > -1 || m == 2 && p < -2)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "(f + g*x)^ m*(d + e*x^2)*(a + b*ArcSin[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1)) - 1/(b*c*Sqrt[d]*(n + 1))* Int[(d*g*m + 2*e*f*x + e*g*(m + 2)*x^2)*(f + g*x)^(m - 1)*(a + b*ArcSin[c*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_* Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && ILtQ[m, 0] && GtQ[d, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "-(f + g*x)^m*(d + e*x^2)*(a + b*ArcCos[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1)) + 1/(b*c*Sqrt[d]*(n + 1))* Int[(d*g*m + 2*e*f*x + e*g*(m + 2)*x^2)*(f + g*x)^(m - 1)*(a + b*ArcCos[c*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_* Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && ILtQ[m, 0] && GtQ[d, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[ Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n, (f + g*x)^ m*(d + e*x^2)^(p - 1/2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_])^ n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IGtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[ Sqrt[d + e*x^2]*(a + b*ArcCos[c*x])^n, (f + g*x)^ m*(d + e*x^2)^(p - 1/2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_])^ n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IGtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "(f + g*x)^ m*(d + e*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n + 1)/(b*c* Sqrt[d]*(n + 1)) - 1/(b*c*Sqrt[d]*(n + 1))* Int[ ExpandIntegrand[(f + g*x)^(m - 1)*(a + b*ArcSin[c*x])^(n + 1), (d*g*m + e*f*(2*p + 1)*x + e*g*(m + 2*p + 1)*x^2)*(d + e*x^2)^(p - 1/2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_])^ n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && ILtQ[m, 0] && IGtQ[p - 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "-(f + g*x)^ m*(d + e*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n + 1)/(b*c* Sqrt[d]*(n + 1)) + 1/(b*c*Sqrt[d]*(n + 1))* Int[ ExpandIntegrand[(f + g*x)^(m - 1)*(a + b*ArcCos[c*x])^(n + 1), (d*g*m + e*f*(2*p + 1)*x + e*g*(m + 2*p + 1)*x^2)*(d + e*x^2)^(p - 1/2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_])^ n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && ILtQ[m, 0] && IGtQ[p - 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "(f + g*x)^ m*(a + b*ArcSin[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1)) - g*m/(b*c*Sqrt[d]*(n + 1))* Int[(f + g*x)^(m - 1)*(a + b*ArcSin[c*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(a_. + b_.*ArcSin[c_.*x_])^n_/ Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && GtQ[d, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "-(f + g*x)^ m*(a + b*ArcCos[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1)) + g*m/(b*c*Sqrt[d]*(n + 1))* Int[(f + g*x)^(m - 1)*(a + b*ArcCos[c*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(a_. + b_.*ArcCos[c_.*x_])^n_/ Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && GtQ[d, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "1/(c^(m + 1)*Sqrt[d])* Subst[Int[(a + b*x)^n*(c*f + g*Sin[x])^m, x], x, ArcSin[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(a_. + b_.*ArcSin[c_.*x_])^n_./ Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && GtQ[d, 0] && (GtQ[m, 0] || IGtQ[n, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "-1/(c^(m + 1)*Sqrt[d])* Subst[Int[(a + b*x)^n*(c*f + g*Cos[x])^m, x], x, ArcCos[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(a_. + b_.*ArcCos[c_.*x_])^n_./ Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && GtQ[d, 0] && (GtQ[m, 0] || IGtQ[n, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n/ Sqrt[d + e*x^2], (f + g*x)^m*(d + e*x^2)^(p + 1/2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_])^ n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcCos[c*x])^n/ Sqrt[d + e*x^2], (f + g*x)^m*(d + e*x^2)^(p + 1/2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_])^ n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "d^IntPart[p]*(d + e*x^2)^FracPart[p]/(1 - c^2*x^2)^FracPart[p]* Int[(f + g*x)^m*(1 - c^2*x^2)^p*(a + b*ArcSin[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_])^ n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p - 1/2] && Not[GtQ[d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "d^IntPart[p]*(d + e*x^2)^FracPart[p]/(1 - c^2*x^2)^FracPart[p]* Int[(f + g*x)^m*(1 - c^2*x^2)^p*(a + b*ArcCos[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_])^ n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p - 1/2] && Not[GtQ[d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "Log[h*(f + g*x)^m]*(a + b*ArcSin[c*x])^(n + 1)/(b*c* Sqrt[d]*(n + 1)) - g*m/(b*c*Sqrt[d]*(n + 1))* Int[(a + b*ArcSin[c*x])^(n + 1)/(f + g*x), x]",
     "rulenumber": 0,
            "lhs": "Int[Log[h_.*(f_. + g_.*x_)^m_.]*(a_. + b_.*ArcSin[c_.*x_])^n_./ Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "-Log[ h*(f + g*x)^m]*(a + b*ArcCos[c*x])^(n + 1)/(b*c* Sqrt[d]*(n + 1)) + g*m/(b*c*Sqrt[d]*(n + 1))* Int[(a + b*ArcCos[c*x])^(n + 1)/(f + g*x), x]",
     "rulenumber": 0,
            "lhs": "Int[Log[h_.*(f_. + g_.*x_)^m_.]*(a_. + b_.*ArcCos[c_.*x_])^n_./ Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "d^IntPart[p]*(d + e*x^2)^FracPart[p]/(1 - c^2*x^2)^FracPart[p]* Int[Log[h*(f + g*x)^m]*(1 - c^2*x^2)^p*(a + b*ArcSin[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[Log[h_.*(f_. + g_.*x_)^m_.]*(d_ + e_.*x_^2)^ p_*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && Not[GtQ[d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "d^IntPart[p]*(d + e*x^2)^FracPart[p]/(1 - c^2*x^2)^FracPart[p]* Int[Log[h*(f + g*x)^m]*(1 - c^2*x^2)^p*(a + b*ArcCos[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[Log[h_.*(f_. + g_.*x_)^m_.]*(d_ + e_.*x_^2)^ p_*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && Not[GtQ[d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = IntHide[(d + e*x)^m*(f + g*x)^m, x]}, Dist[a + b*ArcSin[c*x], u, x] - b*c*Int[Dist[1/Sqrt[1 - c^2*x^2], u, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(f_ + g_.*x_)^m_*(a_. + b_.*ArcSin[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && ILtQ[m + 1/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = IntHide[(d + e*x)^m*(f + g*x)^m, x]}, Dist[a + b*ArcCos[c*x], u, x] + b*c*Int[Dist[1/Sqrt[1 - c^2*x^2], u, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(f_ + g_.*x_)^m_*(a_. + b_.*ArcCos[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && ILtQ[m + 1/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^m*(a + b*ArcSin[c*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_.*(f_ + g_.*x_)^m_.*(a_. + b_.*ArcSin[c_.*x_])^ n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n}, x] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^m*(a + b*ArcCos[c*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_.*(f_ + g_.*x_)^m_.*(a_. + b_.*ArcCos[c_.*x_])^ n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n}, x] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "With[{v = IntHide[u, x]}, Dist[a + b*ArcSin[c*x], v, x] - b*c*Int[SimplifyIntegrand[v/Sqrt[1 - c^2*x^2], x], x] /; InverseFunctionFreeQ[v, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*(a_. + b_.*ArcSin[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "With[{v = IntHide[u, x]}, Dist[a + b*ArcCos[c*x], v, x] + b*c*Int[SimplifyIntegrand[v/Sqrt[1 - c^2*x^2], x], x] /; InverseFunctionFreeQ[v, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*(a_. + b_.*ArcCos[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = ExpandIntegrand[Px*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && PolynomialQ[Px, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = ExpandIntegrand[Px*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && PolynomialQ[Px, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = ExpandIntegrand[Px*(f + g*(d + e*x^2)^p)^m*(a + b*ArcSin[c*x])^n, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[Px_.*(f_ + g_.*(d_ + e_.*x_^2)^p_)^m_.*(a_. + b_.*ArcSin[c_.*x_])^ n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && PolynomialQ[Px, x] && EqQ[c^2*d + e, 0] && IGtQ[p + 1/2, 0] && IntegersQ[m, n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = ExpandIntegrand[Px*(f + g*(d + e*x^2)^p)^m*(a + b*ArcCos[c*x])^n, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[Px_.*(f_ + g_.*(d_ + e_.*x_^2)^p_)^m_.*(a_. + b_.*ArcCos[c_.*x_])^ n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && PolynomialQ[Px, x] && EqQ[c^2*d + e, 0] && IGtQ[p + 1/2, 0] && IntegersQ[m, n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = ExpandIntegrand[ArcSin[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[RFx_*ArcSin[c_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[c, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = ExpandIntegrand[ArcCos[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[RFx_*ArcCos[c_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[c, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[RFx*(a + b*ArcSin[c*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[RFx_*(a_ + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[RFx*(a + b*ArcCos[c*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[RFx_*(a_ + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = ExpandIntegrand[(d + e*x^2)^p*ArcSin[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[RFx_*(d_ + e_.*x_^2)^p_*ArcSin[c_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "With[{u = ExpandIntegrand[(d + e*x^2)^p*ArcCos[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[RFx_*(d_ + e_.*x_^2)^p_*ArcCos[c_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x^2)^p, RFx*(a + b*ArcSin[c*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[RFx_*(d_ + e_.*x_^2)^p_*(a_ + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x^2)^p, RFx*(a + b*ArcCos[c*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[RFx_*(d_ + e_.*x_^2)^p_*(a_ + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "Unintegrable[u*(a + b*ArcSin[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_. + b_.*ArcSin[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.5 u (a+b arcsin(c x))^n.m",
       "filename": "5.1.5 u (a+b arcsin(c x))^n.m",
            "rhs": "Unintegrable[u*(a + b*ArcCos[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_. + b_.*ArcCos[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "1/d*Subst[Int[(a + b*ArcSin[x])^n, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSin[c_ + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "1/d*Subst[Int[(a + b*ArcCos[x])^n, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCos[c_ + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "1/d*Subst[Int[((d*e - c*f)/d + f*x/d)^m*(a + b*ArcSin[x])^n, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcSin[c_ + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "1/d*Subst[Int[((d*e - c*f)/d + f*x/d)^m*(a + b*ArcCos[x])^n, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCos[c_ + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "1/d*Subst[Int[(-C/d^2 + C/d^2*x^2)^p*(a + b*ArcSin[x])^n, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*x_ + C_.*x_^2)^p_.*(a_. + b_.*ArcSin[c_ + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, A, B, C, n, p}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "1/d*Subst[Int[(-C/d^2 + C/d^2*x^2)^p*(a + b*ArcCos[x])^n, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*x_ + C_.*x_^2)^p_.*(a_. + b_.*ArcCos[c_ + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, A, B, C, n, p}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "1/d*Subst[ Int[((d*e - c*f)/d + f*x/d)^m*(-C/d^2 + C/d^2*x^2)^ p*(a + b*ArcSin[x])^n, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(A_. + B_.*x_ + C_.*x_^2)^ p_.*(a_. + b_.*ArcSin[c_ + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, m, n, p}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "1/d*Subst[ Int[((d*e - c*f)/d + f*x/d)^m*(-C/d^2 + C/d^2*x^2)^ p*(a + b*ArcCos[x])^n, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(A_. + B_.*x_ + C_.*x_^2)^ p_.*(a_. + b_.*ArcCos[c_ + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, m, n, p}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "x*Sqrt[a + b*ArcSin[c + d*x^2]] - Sqrt[Pi]*x*(Cos[a/(2*b)] + c*Sin[a/(2*b)])* FresnelC[Sqrt[c/(Pi*b)]*Sqrt[a + b*ArcSin[c + d*x^2]]]/ (Sqrt[ c/b]*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2])) + Sqrt[Pi]*x*(Cos[a/(2*b)] - c*Sin[a/(2*b)])* FresnelS[Sqrt[c/(Pi*b)]*Sqrt[a + b*ArcSin[c + d*x^2]]]/ (Sqrt[ c/b]*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2]))",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_. + b_.*ArcSin[c_ + d_.*x_^2]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "-2*Sqrt[a + b*ArcCos[1 + d*x^2]]* Sin[ArcCos[1 + d*x^2]/2]^2/(d*x) - 2*Sqrt[Pi]*Sin[a/(2*b)]*Sin[ArcCos[1 + d*x^2]/2]* FresnelC[ Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[1 + d*x^2]]]/(Sqrt[1/b]*d* x) + 2*Sqrt[Pi]*Cos[a/(2*b)]*Sin[ArcCos[1 + d*x^2]/2]* FresnelS[ Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[1 + d*x^2]]]/(Sqrt[1/b]*d*x)",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_. + b_.*ArcCos[1 + d_.*x_^2]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "2*Sqrt[a + b*ArcCos[-1 + d*x^2]]* Cos[(1/2)*ArcCos[-1 + d*x^2]]^2/(d*x) - 2*Sqrt[Pi]*Cos[a/(2*b)]*Cos[ArcCos[-1 + d*x^2]/2]* FresnelC[ Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[-1 + d*x^2]]]/(Sqrt[1/b]*d*x) - 2*Sqrt[Pi]*Sin[a/(2*b)]*Cos[ArcCos[-1 + d*x^2]/2]* FresnelS[ Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[-1 + d*x^2]]]/(Sqrt[1/b]*d*x)",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_. + b_.*ArcCos[-1 + d_.*x_^2]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "x*(a + b*ArcSin[c + d*x^2])^n + 2*b*n* Sqrt[-2*c*d*x^2 - d^2*x^4]*(a + b*ArcSin[c + d*x^2])^(n - 1)/(d*x) - 4*b^2*n*(n - 1)*Int[(a + b*ArcSin[c + d*x^2])^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSin[c_ + d_.*x_^2])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1] && GtQ[n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "x*(a + b*ArcCos[c + d*x^2])^n - 2*b*n* Sqrt[-2*c*d*x^2 - d^2*x^4]*(a + b*ArcCos[c + d*x^2])^(n - 1)/(d*x) - 4*b^2*n*(n - 1)*Int[(a + b*ArcCos[c + d*x^2])^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCos[c_ + d_.*x_^2])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1] && GtQ[n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "-x*(c*Cos[a/(2*b)] - Sin[a/(2*b)])* CosIntegral[(c/(2*b))*(a + b*ArcSin[c + d*x^2])]/ (2* b*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2])) - x*(c*Cos[a/(2*b)] + Sin[a/(2*b)])* SinIntegral[(c/(2*b))*(a + b*ArcSin[c + d*x^2])]/ (2* b*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2]))",
     "rulenumber": 0,
            "lhs": "Int[1/(a_. + b_.*ArcSin[c_ + d_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "x*Cos[a/(2*b)]* CosIntegral[(a + b*ArcCos[1 + d*x^2])/(2*b)]/(Sqrt[2]*b* Sqrt[-d*x^2]) + x*Sin[a/(2*b)]* SinIntegral[(a + b*ArcCos[1 + d*x^2])/(2*b)]/(Sqrt[2]*b* Sqrt[-d*x^2])",
     "rulenumber": 0,
            "lhs": "Int[1/(a_. + b_.*ArcCos[1 + d_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "x*Sin[a/(2*b)]* CosIntegral[(a + b*ArcCos[-1 + d*x^2])/(2*b)]/(Sqrt[2]*b* Sqrt[d*x^2]) - x*Cos[a/(2*b)]* SinIntegral[(a + b*ArcCos[-1 + d*x^2])/(2*b)]/(Sqrt[2]*b* Sqrt[d*x^2])",
     "rulenumber": 0,
            "lhs": "Int[1/(a_. + b_.*ArcCos[-1 + d_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "-Sqrt[Pi]*x*(Cos[a/(2*b)] - c*Sin[a/(2*b)])* FresnelC[1/(Sqrt[b*c]*Sqrt[Pi])*Sqrt[a + b*ArcSin[c + d*x^2]]]/ (Sqrt[ b*c]*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2])) - Sqrt[Pi]*x*(Cos[a/(2*b)] + c*Sin[a/(2*b)])* FresnelS[(1/(Sqrt[b*c]*Sqrt[Pi]))*Sqrt[a + b*ArcSin[c + d*x^2]]]/ (Sqrt[ b*c]*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2]))",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_. + b_.*ArcSin[c_ + d_.*x_^2]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "-2*Sqrt[Pi/b]*Cos[a/(2*b)]*Sin[ArcCos[1 + d*x^2]/2]* FresnelC[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[1 + d*x^2]]]/(d*x) - 2*Sqrt[Pi/b]*Sin[a/(2*b)]*Sin[ArcCos[1 + d*x^2]/2]* FresnelS[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[1 + d*x^2]]]/(d*x)",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_. + b_.*ArcCos[1 + d_.*x_^2]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "2*Sqrt[Pi/b]*Sin[a/(2*b)]*Cos[ArcCos[-1 + d*x^2]/2]* FresnelC[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[-1 + d*x^2]]]/(d*x) - 2*Sqrt[Pi/b]*Cos[a/(2*b)]*Cos[ArcCos[-1 + d*x^2]/2]* FresnelS[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[-1 + d*x^2]]]/(d*x)",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_. + b_.*ArcCos[-1 + d_.*x_^2]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "-Sqrt[-2*c*d*x^2 - d^2*x^4]/(b*d*x* Sqrt[a + b*ArcSin[c + d*x^2]]) - (c/b)^(3/2)*Sqrt[Pi]*x*(Cos[a/(2*b)] + c*Sin[a/(2*b)])* FresnelC[Sqrt[c/(Pi*b)]*Sqrt[a + b*ArcSin[c + d*x^2]]]/ (Cos[(1/2)*ArcSin[c + d*x^2]] - c*Sin[ArcSin[c + d*x^2]/2]) + (c/b)^(3/2)*Sqrt[Pi]*x*(Cos[a/(2*b)] - c*Sin[a/(2*b)])* FresnelS[Sqrt[c/(Pi*b)]*Sqrt[a + b*ArcSin[c + d*x^2]]]/ (Cos[(1/2)*ArcSin[c + d*x^2]] - c*Sin[ArcSin[c + d*x^2]/2])",
     "rulenumber": 0,
            "lhs": "Int[1/(a_. + b_.*ArcSin[c_ + d_.*x_^2])^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "Sqrt[-2*d*x^2 - d^2*x^4]/(b*d*x*Sqrt[a + b*ArcCos[1 + d*x^2]]) - 2*(1/b)^(3/2)*Sqrt[Pi]*Sin[a/(2*b)]*Sin[ArcCos[1 + d*x^2]/2]* FresnelC[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[1 + d*x^2]]]/(d*x) + 2*(1/b)^(3/2)*Sqrt[Pi]*Cos[a/(2*b)]*Sin[ArcCos[1 + d*x^2]/2]* FresnelS[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[1 + d*x^2]]]/(d*x)",
     "rulenumber": 0,
            "lhs": "Int[1/(a_. + b_.*ArcCos[1 + d_.*x_^2])^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "Sqrt[2*d*x^2 - d^2*x^4]/(b*d*x*Sqrt[a + b*ArcCos[-1 + d*x^2]]) - 2*(1/b)^(3/2)*Sqrt[Pi]*Cos[a/(2*b)]*Cos[ArcCos[-1 + d*x^2]/2]* FresnelC[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[-1 + d*x^2]]]/(d*x) - 2*(1/b)^(3/2)*Sqrt[Pi]*Sin[a/(2*b)]*Cos[ArcCos[-1 + d*x^2]/2]* FresnelS[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[-1 + d*x^2]]]/(d*x)",
     "rulenumber": 0,
            "lhs": "Int[1/(a_. + b_.*ArcCos[-1 + d_.*x_^2])^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "-Sqrt[-2*c*d*x^2 - d^2*x^4]/(2*b*d* x*(a + b*ArcSin[c + d*x^2])) - x*(Cos[a/(2*b)] + c*Sin[a/(2*b)])* CosIntegral[(c/(2*b))*(a + b*ArcSin[c + d*x^2])]/ (4* b^2*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2])) + x*(Cos[a/(2*b)] - c*Sin[a/(2*b)])* SinIntegral[(c/(2*b))*(a + b*ArcSin[c + d*x^2])]/ (4* b^2*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2]))",
     "rulenumber": 0,
            "lhs": "Int[1/(a_. + b_.*ArcSin[c_ + d_.*x_^2])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "Sqrt[-2*d*x^2 - d^2*x^4]/(2*b*d*x*(a + b*ArcCos[1 + d*x^2])) + x*Sin[a/(2*b)]* CosIntegral[(a + b*ArcCos[1 + d*x^2])/(2*b)]/(2*Sqrt[2]*b^2* Sqrt[(-d)*x^2]) - x*Cos[a/(2*b)]* SinIntegral[(a + b*ArcCos[1 + d*x^2])/(2*b)]/(2*Sqrt[2]*b^2* Sqrt[(-d)*x^2])",
     "rulenumber": 0,
            "lhs": "Int[1/(a_. + b_.*ArcCos[1 + d_.*x_^2])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "Sqrt[2*d*x^2 - d^2*x^4]/(2*b*d*x*(a + b*ArcCos[-1 + d*x^2])) - x*Cos[a/(2*b)]* CosIntegral[(a + b*ArcCos[-1 + d*x^2])/(2*b)]/(2*Sqrt[2]*b^2* Sqrt[d*x^2]) - x*Sin[a/(2*b)]* SinIntegral[(a + b*ArcCos[-1 + d*x^2])/(2*b)]/(2*Sqrt[2]*b^2* Sqrt[d*x^2])",
     "rulenumber": 0,
            "lhs": "Int[1/(a_. + b_.*ArcCos[-1 + d_.*x_^2])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "x*(a + b*ArcSin[c + d*x^2])^(n + 2)/(4*b^2*(n + 1)*(n + 2)) + Sqrt[-2*c*d*x^2 - d^2*x^4]*(a + b*ArcSin[c + d*x^2])^(n + 1)/(2*b*d*(n + 1)*x) - 1/(4*b^2*(n + 1)*(n + 2))* Int[(a + b*ArcSin[c + d*x^2])^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSin[c_ + d_.*x_^2])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1] && LtQ[n, -1] && NeQ[n, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "x*(a + b*ArcCos[c + d*x^2])^(n + 2)/(4*b^2*(n + 1)*(n + 2)) - Sqrt[-2*c*d*x^2 - d^2*x^4]*(a + b*ArcCos[c + d*x^2])^(n + 1)/(2*b*d*(n + 1)*x) - 1/(4*b^2*(n + 1)*(n + 2))* Int[(a + b*ArcCos[c + d*x^2])^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCos[c_ + d_.*x_^2])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1] && LtQ[n, -1] && NeQ[n, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "1/p*Subst[Int[x^n*Cot[x], x], x, ArcSin[a*x^p]]",
     "rulenumber": 0,
            "lhs": "Int[ArcSin[a_.*x_^p_]^n_./x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, p}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "-1/p*Subst[Int[x^n*Tan[x], x], x, ArcCos[a*x^p]]",
     "rulenumber": 0,
            "lhs": "Int[ArcCos[a_.*x_^p_]^n_./x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, p}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "Int[u*ArcCsc[a/c + b*x^n/c]^m, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*ArcSin[c_./(a_. + b_.*x_^n_.)]^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, m}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "Int[u*ArcSec[a/c + b*x^n/c]^m, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*ArcCos[c_./(a_. + b_.*x_^n_.)]^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, m}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "Sqrt[-b*x^2]/(b*x)* Subst[Int[ArcSin[x]^n/Sqrt[1 - x^2], x], x, Sqrt[1 + b*x^2]]",
     "rulenumber": 0,
            "lhs": "Int[ArcSin[Sqrt[1 + b_.*x_^2]]^n_./Sqrt[1 + b_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "Sqrt[-b*x^2]/(b*x)* Subst[Int[ArcCos[x]^n/Sqrt[1 - x^2], x], x, Sqrt[1 + b*x^2]]",
     "rulenumber": 0,
            "lhs": "Int[ArcCos[Sqrt[1 + b_.*x_^2]]^n_./Sqrt[1 + b_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "1/b*Subst[ Int[ReplaceAll[u, x -> -a/b + Sin[x]/b]*f^(c*x^n)*Cos[x], x], x, ArcSin[a + b*x]]",
     "rulenumber": 0,
            "lhs": "Int[u_.*f_^(c_.*ArcSin[a_. + b_.*x_]^n_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, f}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "-1/b* Subst[Int[ReplaceAll[u, x -> -a/b + Cos[x]/b]*f^(c*x^n)*Sin[x], x], x, ArcCos[a + b*x]]",
     "rulenumber": 0,
            "lhs": "Int[u_.*f_^(c_.*ArcCos[a_. + b_.*x_]^n_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, f}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "x*ArcSin[a*x^2 + b*Sqrt[c + d*x^2]] - x*Sqrt[b^2*d + a^2*x^2 + 2*a*b*Sqrt[c + d*x^2]]/ Sqrt[(-x^2)*(b^2*d + a^2*x^2 + 2*a*b*Sqrt[c + d*x^2])]* Int[ x*(b*d + 2*a*Sqrt[c + d*x^2])/(Sqrt[c + d*x^2]* Sqrt[b^2*d + a^2*x^2 + 2*a*b*Sqrt[c + d*x^2]]), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcSin[a_.*x_^2 + b_.*Sqrt[c_ + d_.*x_^2]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[b^2*c, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "x*ArcCos[a*x^2 + b*Sqrt[c + d*x^2]] + x*Sqrt[b^2*d + a^2*x^2 + 2*a*b*Sqrt[c + d*x^2]]/ Sqrt[(-x^2)*(b^2*d + a^2*x^2 + 2*a*b*Sqrt[c + d*x^2])]* Int[ x*(b*d + 2*a*Sqrt[c + d*x^2])/(Sqrt[c + d*x^2]* Sqrt[b^2*d + a^2*x^2 + 2*a*b*Sqrt[c + d*x^2]]), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCos[a_.*x_^2 + b_.*Sqrt[c_ + d_.*x_^2]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[b^2*c, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "x*ArcSin[u] - Int[SimplifyIntegrand[x*D[u, x]/Sqrt[1 - u^2], x], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcSin[u_], x_Symbol]",
        "comment": false,
         "givens": "InverseFunctionFreeQ[u, x] && Not[FunctionOfExponentialQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "x*ArcCos[u] + Int[SimplifyIntegrand[x*D[u, x]/Sqrt[1 - u^2], x], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCos[u_], x_Symbol]",
        "comment": false,
         "givens": "InverseFunctionFreeQ[u, x] && Not[FunctionOfExponentialQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "(c + d*x)^(m + 1)*(a + b*ArcSin[u])/(d*(m + 1)) - b/(d*(m + 1))* Int[SimplifyIntegrand[(c + d*x)^(m + 1)*D[u, x]/Sqrt[1 - u^2], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(a_. + b_.*ArcSin[u_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && Not[FunctionOfQ[(c + d*x)^(m + 1), u, x]] && Not[FunctionOfExponentialQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "(c + d*x)^(m + 1)*(a + b*ArcCos[u])/(d*(m + 1)) + b/(d*(m + 1))* Int[SimplifyIntegrand[(c + d*x)^(m + 1)*D[u, x]/Sqrt[1 - u^2], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(a_. + b_.*ArcCos[u_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && Not[FunctionOfQ[(c + d*x)^(m + 1), u, x]] && Not[FunctionOfExponentialQ[u, x]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "With[{w = IntHide[v, x]}, Dist[(a + b*ArcSin[u]), w, x] - b*Int[SimplifyIntegrand[w*D[u, x]/Sqrt[1 - u^2], x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && Not[MatchQ[v, (c_. + d_.*x)^m_.",
     "rulenumber": 0,
            "lhs": "Int[v_*(a_. + b_.*ArcSin[u_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, m}, x]]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.1 Inverse sine/5.1.6 Miscellaneous inverse sine.m",
       "filename": "5.1.6 Miscellaneous inverse sine.m",
            "rhs": "With[{w = IntHide[v, x]}, Dist[(a + b*ArcCos[u]), w, x] + b*Int[SimplifyIntegrand[w*D[u, x]/Sqrt[1 - u^2], x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && Not[MatchQ[v, (c_. + d_.*x)^m_.",
     "rulenumber": 0,
            "lhs": "Int[v_*(a_. + b_.*ArcCos[u_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, m}, x]]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "x*(a + b*ArcTan[c*x])^p - b*c*p*Int[x*(a + b*ArcTan[c*x])^(p - 1)/(1 + c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "x*(a + b*ArcCot[c*x])^p + b*c*p*Int[x*(a + b*ArcCot[c*x])^(p - 1)/(1 + c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "a*Log[x] + I*b/2*Int[Log[1 - I*c*x]/x, x] - I*b/2*Int[Log[1 + I*c*x]/x, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTan[c_.*x_])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "a*Log[x] + I*b/2*Int[Log[1 - I/(c*x)]/x, x] - I*b/2*Int[Log[1 + I/(c*x)]/x, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCot[c_.*x_])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "2*(a + b*ArcTan[c*x])^p*ArcTanh[1 - 2/(1 + I*c*x)] - 2*b*c*p* Int[(a + b*ArcTan[c*x])^(p - 1)* ArcTanh[1 - 2/(1 + I*c*x)]/(1 + c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTan[c_.*x_])^p_/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IGtQ[p, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "2*(a + b*ArcCot[c*x])^p*ArcCoth[1 - 2/(1 + I*c*x)] + 2*b*c*p* Int[(a + b*ArcCot[c*x])^(p - 1)* ArcCoth[1 - 2/(1 + I*c*x)]/(1 + c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCot[c_.*x_])^p_/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IGtQ[p, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(d*x)^(m + 1)*(a + b*ArcTan[c*x])^p/(d*(m + 1)) - b*c*p/(d*(m + 1))* Int[(d*x)^(m + 1)*(a + b*ArcTan[c*x])^(p - 1)/(1 + c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(d*x)^(m + 1)*(a + b*ArcCot[c*x])^p/(d*(m + 1)) + b*c*p/(d*(m + 1))* Int[(d*x)^(m + 1)*(a + b*ArcCot[c*x])^(p - 1)/(1 + c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-(a + b*ArcTan[c*x])^p*Log[2/(1 + e*x/d)]/e + b*c*p/e* Int[(a + b*ArcTan[c*x])^(p - 1)*Log[2/(1 + e*x/d)]/(1 + c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTan[c_.*x_])^p_./(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-(a + b*ArcCot[c*x])^p*Log[2/(1 + e*x/d)]/e - b*c*p/e* Int[(a + b*ArcCot[c*x])^(p - 1)*Log[2/(1 + e*x/d)]/(1 + c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCot[c_.*x_])^p_./(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)]/e + b*c/e*Int[Log[2/(1 - I*c*x)]/(1 + c^2*x^2), x] + (a + b*ArcTan[c*x])* Log[2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/e - b*c/e* Int[Log[2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/(1 + c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTan[c_.*x_])/(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-(a + b*ArcCot[c*x])*Log[2/(1 - I*c*x)]/e - b*c/e*Int[Log[2/(1 - I*c*x)]/(1 + c^2*x^2), x] + (a + b*ArcCot[c*x])* Log[2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/e + b*c/e* Int[Log[2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/(1 + c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCot[c_.*x_])/(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)]/e + I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)]/e - b^2*PolyLog[3, 1 - 2/(1 - I*c*x)]/(2*e) + (a + b*ArcTan[c*x])^2* Log[2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/e - I*b*(a + b*ArcTan[c*x])* PolyLog[2, 1 - 2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/e + b^2*PolyLog[3, 1 - 2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/(2*e)",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTan[c_.*x_])^2/(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-(a + b*ArcCot[c*x])^2*Log[2/(1 - I*c*x)]/e - I*b*(a + b*ArcCot[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)]/e - b^2*PolyLog[3, 1 - 2/(1 - I*c*x)]/(2*e) + (a + b*ArcCot[c*x])^2* Log[2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/e + I*b*(a + b*ArcCot[c*x])* PolyLog[2, 1 - 2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/e + b^2*PolyLog[3, 1 - 2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/(2*e)",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCot[c_.*x_])^2/(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-(a + b*ArcTan[c*x])^3*Log[2/(1 - I*c*x)]/e + 3*I*b*(a + b*ArcTan[c*x])^2*PolyLog[2, 1 - 2/(1 - I*c*x)]/(2*e) - 3*b^2*(a + b*ArcTan[c*x])*PolyLog[3, 1 - 2/(1 - I*c*x)]/(2*e) - 3*I*b^3*PolyLog[4, 1 - 2/(1 - I*c*x)]/(4*e) + (a + b*ArcTan[c*x])^3* Log[2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/e - 3*I*b*(a + b*ArcTan[c*x])^2* PolyLog[2, 1 - 2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/(2*e) + 3*b^2*(a + b*ArcTan[c*x])* PolyLog[3, 1 - 2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/(2*e) + 3*I*b^3* PolyLog[4, 1 - 2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/(4*e)",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTan[c_.*x_])^3/(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-(a + b*ArcCot[c*x])^3*Log[2/(1 - I*c*x)]/e - 3*I*b*(a + b*ArcCot[c*x])^2*PolyLog[2, 1 - 2/(1 - I*c*x)]/(2*e) - 3*b^2*(a + b*ArcCot[c*x])*PolyLog[3, 1 - 2/(1 - I*c*x)]/(2*e) + 3*I*b^3*PolyLog[4, 1 - 2/(1 - I*c*x)]/(4*e) + (a + b*ArcCot[c*x])^3* Log[2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/e + 3*I*b*(a + b*ArcCot[c*x])^2* PolyLog[2, 1 - 2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/(2*e) + 3*b^2*(a + b*ArcCot[c*x])* PolyLog[3, 1 - 2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/(2*e) - 3*I*b^3* PolyLog[4, 1 - 2*c*(d + e*x)/((c*d + I*e)*(1 - I*c*x))]/(4*e)",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCot[c_.*x_])^3/(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(d + e*x)^(q + 1)*(a + b*ArcTan[c*x])/(e*(q + 1)) - b*c/(e*(q + 1))*Int[(d + e*x)^(q + 1)/(1 + c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^q_.*(a_. + b_.*ArcTan[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(d + e*x)^(q + 1)*(a + b*ArcCot[c*x])/(e*(q + 1)) + b*c/(e*(q + 1))*Int[(d + e*x)^(q + 1)/(1 + c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^q_.*(a_. + b_.*ArcCot[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(d + e*x)^(q + 1)*(a + b*ArcTan[c*x])^ p/(e*(q + 1)) - b*c*p/(e*(q + 1))* Int[ExpandIntegrand[(a + b*ArcTan[c*x])^(p - 1), (d + e*x)^(q + 1)/(1 + c^2*x^2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^q_.*(a_. + b_.*ArcTan[c_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(d + e*x)^(q + 1)*(a + b*ArcCot[c*x])^ p/(e*(q + 1)) + b*c*p/(e*(q + 1))* Int[ExpandIntegrand[(a + b*ArcCot[c*x])^(p - 1), (d + e*x)^(q + 1)/(1 + c^2*x^2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^q_.*(a_. + b_.*ArcCot[c_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "f/e*Int[(f*x)^(m - 1)*(a + b*ArcTan[c*x])^p, x] - d*f/e*Int[(f*x)^(m - 1)*(a + b*ArcTan[c*x])^p/(d + e*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(a_. + b_.*ArcTan[c_.*x_])^p_./(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0] && GtQ[m, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "f/e*Int[(f*x)^(m - 1)*(a + b*ArcCot[c*x])^p, x] - d*f/e*Int[(f*x)^(m - 1)*(a + b*ArcCot[c*x])^p/(d + e*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(a_. + b_.*ArcCot[c_.*x_])^p_./(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(a + b*ArcTan[c*x])^p*Log[2 - 2/(1 + e*x/d)]/d - b*c*p/d* Int[(a + b*ArcTan[c*x])^(p - 1)* Log[2 - 2/(1 + e*x/d)]/(1 + c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTan[c_.*x_])^p_./(x_*(d_ + e_.*x_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(a + b*ArcCot[c*x])^p*Log[2 - 2/(1 + e*x/d)]/d + b*c*p/d* Int[(a + b*ArcCot[c*x])^(p - 1)* Log[2 - 2/(1 + e*x/d)]/(1 + c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCot[c_.*x_])^p_./(x_*(d_ + e_.*x_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "1/d*Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x] - e/(d*f)*Int[(f*x)^(m + 1)*(a + b*ArcTan[c*x])^p/(d + e*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcTan[c_.*x_])^p_./(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "1/d*Int[(f*x)^m*(a + b*ArcCot[c*x])^p, x] - e/(d*f)*Int[(f*x)^(m + 1)*(a + b*ArcCot[c*x])^p/(d + e*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcCot[c_.*x_])^p_./(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[(a + b*ArcTan[c*x]), u] - b*c*Int[SimplifyIntegrand[u/(1 + c^2*x^2), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_)^q_.*(a_. + b_.*ArcTan[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[q, -1] && IntegerQ[ 2*m] && (IGtQ[m, 0] && IGtQ[q, 0] || ILtQ[m + q + 1, 0] && LtQ[m*q, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[(a + b*ArcCot[c*x]), u] + b*c*Int[SimplifyIntegrand[u/(1 + c^2*x^2), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_)^q_.*(a_. + b_.*ArcCot[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[q, -1] && IntegerQ[ 2*m] && (IGtQ[m, 0] && IGtQ[q, 0] || ILtQ[m + q + 1, 0] && LtQ[m*q, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[(a + b*ArcTan[c*x])^p, u] - b*c*p*Int[ ExpandIntegrand[(a + b*ArcTan[c*x])^(p - 1), u/(1 + c^2*x^2), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_)^q_*(a_. + b_.*ArcTan[c_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, q}, x] && IGtQ[p, 1] && EqQ[c^2*d^2 + e^2, 0] && IntegersQ[m, q] && NeQ[m, -1] && NeQ[q, -1] && ILtQ[m + q + 1, 0] && LtQ[m*q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[(a + b*ArcCot[c*x])^p, u] + b*c*p*Int[ ExpandIntegrand[(a + b*ArcCot[c*x])^(p - 1), u/(1 + c^2*x^2), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_)^q_*(a_. + b_.*ArcCot[c_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, q}, x] && IGtQ[p, 1] && EqQ[c^2*d^2 + e^2, 0] && IntegersQ[m, q] && NeQ[m, -1] && NeQ[q, -1] && ILtQ[m + q + 1, 0] && LtQ[m*q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcTan[c*x])^p, (f*x)^m*(d + e*x)^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_)^q_.*(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcCot[c*x])^p, (f*x)^m*(d + e*x)^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_)^q_.*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-b*(d + e*x^2)^q/(2*c*q*(2*q + 1)) + x*(d + e*x^2)^q*(a + b*ArcTan[c*x])/(2*q + 1) + 2*d*q/(2*q + 1)*Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTan[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "b*(d + e*x^2)^q/(2*c*q*(2*q + 1)) + x*(d + e*x^2)^q*(a + b*ArcCot[c*x])/(2*q + 1) + 2*d*q/(2*q + 1)*Int[(d + e*x^2)^(q - 1)*(a + b*ArcCot[c*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCot[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-b* p*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1)/(2*c*q*(2*q + 1)) + x*(d + e*x^2)^q*(a + b*ArcTan[c*x])^p/(2*q + 1) + 2*d*q/(2*q + 1)* Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x] + b^2*d*p*(p - 1)/(2*q*(2*q + 1))* Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^(p - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTan[c_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && GtQ[p, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "b*p*(d + e*x^2)^q*(a + b*ArcCot[c*x])^(p - 1)/(2*c*q*(2*q + 1)) + x*(d + e*x^2)^q*(a + b*ArcCot[c*x])^p/(2*q + 1) + 2*d*q/(2*q + 1)* Int[(d + e*x^2)^(q - 1)*(a + b*ArcCot[c*x])^p, x] + b^2*d*p*(p - 1)/(2*q*(2*q + 1))* Int[(d + e*x^2)^(q - 1)*(a + b*ArcCot[c*x])^(p - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCot[c_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && GtQ[p, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": " 1/(c*d)*Subst[Int[(a+b*x)^p,x],x,ArcTan[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_.+b_.*ArcTan[c_.*x_])^p_./(d_+e_.*x_^2),x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e,p},x] && EqQ[e,c^2*d] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": " -1/(c*d)*Subst[Int[(a+b*x)^p,x],x,ArcCot[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_.+b_.*ArcCot[c_.*x_])^p_./(d_+e_.*x_^2),x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e,p},x] && EqQ[e,c^2*d] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "Log[RemoveContent[a + b*ArcTan[c*x], x]]/(b*c*d)",
     "rulenumber": 0,
            "lhs": "Int[1/((d_ + e_.*x_^2)*(a_. + b_.*ArcTan[c_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-Log[RemoveContent[a + b*ArcCot[c*x], x]]/(b*c*d)",
     "rulenumber": 0,
            "lhs": "Int[1/((d_ + e_.*x_^2)*(a_. + b_.*ArcCot[c_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1))",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTan[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-(a + b*ArcCot[c*x])^(p + 1)/(b*c*d*(p + 1))",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCot[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-2*I*(a + b*ArcTan[c*x])* ArcTan[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]]/(c*Sqrt[d]) + I*b*PolyLog[ 2, -I*Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]]/(c*Sqrt[d]) - I*b*PolyLog[2, I*Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]]/(c*Sqrt[d])",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTan[c_.*x_])/Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-2*I*(a + b*ArcCot[c*x])* ArcTan[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]]/(c*Sqrt[d]) - I*b*PolyLog[ 2, -I*Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]]/(c*Sqrt[d]) + I*b*PolyLog[2, I*Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]]/(c*Sqrt[d])",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCot[c_.*x_])/Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "1/(c*Sqrt[d])*Subst[Int[(a + b*x)^p*Sec[x], x], x, ArcTan[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTan[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && GtQ[d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-x*Sqrt[1 + 1/(c^2*x^2)]/Sqrt[d + e*x^2]* Subst[Int[(a + b*x)^p*Csc[x], x], x, ArcCot[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCot[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && GtQ[d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]* Int[(a + b*ArcTan[c*x])^p/Sqrt[1 + c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTan[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && Not[GtQ[d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]* Int[(a + b*ArcCot[c*x])^p/Sqrt[1 + c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCot[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && Not[GtQ[d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "x*(a + b*ArcTan[c*x])^p/(2*d*(d + e*x^2)) + (a + b*ArcTan[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)) - b*c*p/2*Int[x*(a + b*ArcTan[c*x])^(p - 1)/(d + e*x^2)^2, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTan[c_.*x_])^p_./(d_ + e_.*x_^2)^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "x*(a + b*ArcCot[c*x])^p/(2*d*(d + e*x^2)) - (a + b*ArcCot[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)) + b*c*p/2*Int[x*(a + b*ArcCot[c*x])^(p - 1)/(d + e*x^2)^2, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCot[c_.*x_])^p_./(d_ + e_.*x_^2)^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "b/(c*d*Sqrt[d + e*x^2]) + x*(a + b*ArcTan[c*x])/(d*Sqrt[d + e*x^2])",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTan[c_.*x_])/(d_ + e_.*x_^2)^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-b/(c*d*Sqrt[d + e*x^2]) + x*(a + b*ArcCot[c*x])/(d*Sqrt[d + e*x^2])",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCot[c_.*x_])/(d_ + e_.*x_^2)^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "b*(d + e*x^2)^(q + 1)/(4*c*d*(q + 1)^2) - x*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])/(2*d*(q + 1)) + (2*q + 3)/(2*d*(q + 1))* Int[(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTan[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && NeQ[q, -3/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-b*(d + e*x^2)^(q + 1)/(4*c*d*(q + 1)^2) - x*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])/(2*d*(q + 1)) + (2*q + 3)/(2*d*(q + 1))* Int[(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCot[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && NeQ[q, -3/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "b*p*(a + b*ArcTan[c*x])^(p - 1)/(c*d*Sqrt[d + e*x^2]) + x*(a + b*ArcTan[c*x])^p/(d*Sqrt[d + e*x^2]) - b^2*p*(p - 1)* Int[(a + b*ArcTan[c*x])^(p - 2)/(d + e*x^2)^(3/2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTan[c_.*x_])^p_/(d_ + e_.*x_^2)^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-b* p*(a + b*ArcCot[c*x])^(p - 1)/(c*d*Sqrt[d + e*x^2]) + x*(a + b*ArcCot[c*x])^p/(d*Sqrt[d + e*x^2]) - b^2*p*(p - 1)* Int[(a + b*ArcCot[c*x])^(p - 2)/(d + e*x^2)^(3/2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCot[c_.*x_])^p_/(d_ + e_.*x_^2)^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "b*p*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^(p - 1)/(4*c* d*(q + 1)^2) - x*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p/(2*d*(q + 1)) + (2*q + 3)/(2*d*(q + 1))* Int[(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x] - b^2*p*(p - 1)/(4*(q + 1)^2)* Int[(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTan[c_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-b* p*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^(p - 1)/(4*c* d*(q + 1)^2) - x*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^p/(2*d*(q + 1)) + (2*q + 3)/(2*d*(q + 1))* Int[(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^p, x] - b^2*p*(p - 1)/(4*(q + 1)^2)* Int[(d + e*x^2)^q*(a + b*ArcCot[c*x])^(p - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCot[c_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^(p + 1)/(b* c*d*(p + 1)) - 2*c*(q + 1)/(b*(p + 1))* Int[x*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTan[c_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^(p + 1)/(b* c*d*(p + 1)) + 2*c*(q + 1)/(b*(p + 1))* Int[x*(d + e*x^2)^q*(a + b*ArcCot[c*x])^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCot[c_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "d^q/c*Subst[Int[(a + b*x)^p/Cos[x]^(2*(q + 1)), x], x, ArcTan[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && ILtQ[2*(q + 1), 0] && (IntegerQ[q] || GtQ[d, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "d^(q + 1/2)*Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]* Int[(1 + c^2*x^2)^q*(a + b*ArcTan[c*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && ILtQ[2*(q + 1), 0] && Not[IntegerQ[q] || GtQ[d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-d^q/c* Subst[Int[(a + b*x)^p/Sin[x]^(2*(q + 1)), x], x, ArcCot[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && ILtQ[2*(q + 1), 0] && IntegerQ[q]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-d^(q + 1/2)*x* Sqrt[(1 + c^2*x^2)/(c^2*x^2)]/Sqrt[d + e*x^2]* Subst[Int[(a + b*x)^p/Sin[x]^(2*(q + 1)), x], x, ArcCot[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && ILtQ[2*(q + 1), 0] && Not[IntegerQ[q]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "I/2*Int[Log[1 - I*c*x]/(d + e*x^2), x] - I/2*Int[Log[1 + I*c*x]/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTan[c_.*x_]/(d_. + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "I/2*Int[Log[1 - I/(c*x)]/(d + e*x^2), x] - I/2*Int[Log[1 + I/(c*x)]/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCot[c_.*x_]/(d_. + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "a*Int[1/(d + e*x^2), x] + b*Int[ArcTan[c*x]/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*ArcTan[c_.*x_])/(d_. + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "a*Int[1/(d + e*x^2), x] + b*Int[ArcCot[c*x]/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*ArcCot[c_.*x_])/(d_. + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "With[{u = IntHide[(d + e*x^2)^q, x]}, Dist[a + b*ArcTan[c*x], u, x] - b*c*Int[u/(1 + c^2*x^2), x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_^2)^q_.*(a_. + b_.*ArcTan[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && (IntegerQ[q] || ILtQ[q + 1/2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "With[{u = IntHide[(d + e*x^2)^q, x]}, Dist[a + b*ArcCot[c*x], u, x] + b*c*Int[u/(1 + c^2*x^2), x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_^2)^q_.*(a_. + b_.*ArcCot[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && (IntegerQ[q] || ILtQ[q + 1/2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcTan[c*x])^p, (d + e*x^2)^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IntegerQ[q] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcCot[c*x])^p, (d + e*x^2)^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IntegerQ[q] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "f^2/e*Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x])^p, x] - d*f^2/e*Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x])^p/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcTan[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "f^2/e*Int[(f*x)^(m - 2)*(a + b*ArcCot[c*x])^p, x] - d*f^2/e*Int[(f*x)^(m - 2)*(a + b*ArcCot[c*x])^p/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcCot[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "1/d*Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x] - e/(d*f^2)* Int[(f*x)^(m + 2)*(a + b*ArcTan[c*x])^p/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcTan[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]"
  },
  {
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       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "1/d*Int[(f*x)^m*(a + b*ArcCot[c*x])^p, x] - e/(d*f^2)* Int[(f*x)^(m + 2)*(a + b*ArcCot[c*x])^p/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcCot[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-I*(a + b*ArcTan[c*x])^(p + 1)/(b*e*(p + 1)) - 1/(c*d)*Int[(a + b*ArcTan[c*x])^p/(I - c*x), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(a_. + b_.*ArcTan[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "I*(a + b*ArcCot[c*x])^(p + 1)/(b*e*(p + 1)) - 1/(c*d)*Int[(a + b*ArcCot[c*x])^p/(I - c*x), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(a_. + b_.*ArcCot[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "x*(a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1)) - 1/(b*c*d*(p + 1))*Int[(a + b*ArcTan[c*x])^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(a_. + b_.*ArcTan[c_.*x_])^p_/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && Not[IGtQ[p, 0]] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-x*(a + b*ArcCot[c*x])^(p + 1)/(b*c*d*(p + 1)) + 1/(b*c*d*(p + 1))*Int[(a + b*ArcCot[c*x])^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(a_. + b_.*ArcCot[c_.*x_])^p_/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && Not[IGtQ[p, 0]] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-I*(a + b*ArcTan[c*x])^(p + 1)/(b*d*(p + 1)) + I/d*Int[(a + b*ArcTan[c*x])^p/(x*(I + c*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTan[c_.*x_])^p_./(x_*(d_ + e_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "I*(a + b*ArcCot[c*x])^(p + 1)/(b*d*(p + 1)) + I/d*Int[(a + b*ArcCot[c*x])^p/(x*(I + c*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCot[c_.*x_])^p_./(x_*(d_ + e_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(f*x)^ m*(a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1)) - f*m/(b*c*d*(p + 1))* Int[(f*x)^(m - 1)*(a + b*ArcTan[c*x])^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcTan[c_.*x_])^p_/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-(f*x)^ m*(a + b*ArcCot[c*x])^(p + 1)/(b*c*d*(p + 1)) + f*m/(b*c*d*(p + 1))* Int[(f*x)^(m - 1)*(a + b*ArcCot[c*x])^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcCot[c_.*x_])^p_/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcTan[c*x]), x^m/(d + e*x^2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*ArcTan[c_.*x_])/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] && Not[EqQ[m, 1] && NeQ[a, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcCot[c*x]), x^m/(d + e*x^2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*ArcCot[c_.*x_])/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] && Not[EqQ[m, 1] && NeQ[a, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^ p/(2*e*(q + 1)) - b*p/(2*c*(q + 1))* Int[(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^ p/(2*e*(q + 1)) + b*p/(2*c*(q + 1))* Int[(d + e*x^2)^q*(a + b*ArcCot[c*x])^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "x*(a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1)*(d + e*x^2)) - (1 - c^2*x^2)*(a + b*ArcTan[c*x])^(p + 2)/(b^2* e*(p + 1)*(p + 2)*(d + e*x^2)) - 4/(b^2*(p + 1)*(p + 2))* Int[x*(a + b*ArcTan[c*x])^(p + 2)/(d + e*x^2)^2, x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(a_. + b_.*ArcTan[c_.*x_])^p_/(d_ + e_.*x_^2)^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[p, -1] && NeQ[p, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-x*(a + b*ArcCot[c*x])^(p + 1)/(b*c* d*(p + 1)*(d + e*x^2)) - (1 - c^2*x^2)*(a + b*ArcCot[c*x])^(p + 2)/(b^2* e*(p + 1)*(p + 2)*(d + e*x^2)) - 4/(b^2*(p + 1)*(p + 2))* Int[x*(a + b*ArcCot[c*x])^(p + 2)/(d + e*x^2)^2, x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(a_. + b_.*ArcCot[c_.*x_])^p_/(d_ + e_.*x_^2)^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[p, -1] && NeQ[p, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-b*(d + e*x^2)^(q + 1)/(4*c^3*d*(q + 1)^2) + x*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])/(2*c^2*d*(q + 1)) - 1/(2*c^2*d*(q + 1))* Int[(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^2*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTan[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && NeQ[q, -5/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "b*(d + e*x^2)^(q + 1)/(4*c^3*d*(q + 1)^2) + x*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])/(2*c^2*d*(q + 1)) - 1/(2*c^2*d*(q + 1))* Int[(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^2*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCot[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && NeQ[q, -5/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(a + b*ArcTan[c*x])^(p + 1)/(2*b*c^3* d^2*(p + 1)) - x*(a + b*ArcTan[c*x])^p/(2*c^2*d*(d + e*x^2)) + b*p/(2*c)*Int[x*(a + b*ArcTan[c*x])^(p - 1)/(d + e*x^2)^2, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^2*(a_. + b_.*ArcTan[c_.*x_])^p_./(d_ + e_.*x_^2)^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-(a + b*ArcCot[c*x])^(p + 1)/(2*b*c^3*d^2*(p + 1)) - x*(a + b*ArcCot[c*x])^p/(2*c^2*d*(d + e*x^2)) - b*p/(2*c)*Int[x*(a + b*ArcCot[c*x])^(p - 1)/(d + e*x^2)^2, x]",
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            "lhs": "Int[x_^2*(a_. + b_.*ArcCot[c_.*x_])^p_./(d_ + e_.*x_^2)^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "b*(f*x)^m*(d + e*x^2)^(q + 1)/(c*d*m^2) - f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])/(c^2*d* m) + f^2*(m - 1)/(c^2*d*m)* Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]), x]",
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            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTan[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-b*(f*x)^m*(d + e*x^2)^(q + 1)/(c*d*m^2) - f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])/(c^2*d* m) + f^2*(m - 1)/(c^2*d*m)* Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x]), x]",
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            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCot[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "b*p*(f*x)^ m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^(p - 1)/(c*d*m^2) - f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^ p/(c^2*d*m) - b^2*p*(p - 1)/m^2* Int[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 2), x] + f^2*(m - 1)/(c^2*d*m)* Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTan[c_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1] && GtQ[p, 1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-b* p*(f*x)^m*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^(p - 1)/(c*d* m^2) - f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^ p/(c^2*d*m) - b^2*p*(p - 1)/m^2* Int[(f*x)^m*(d + e*x^2)^q*(a + b*ArcCot[c*x])^(p - 2), x] + f^2*(m - 1)/(c^2*d*m)* Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCot[c_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1] && GtQ[p, 1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(f*x)^ m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^(p + 1)/(b*c* d*(p + 1)) - f*m/(b*c*(p + 1))* Int[(f*x)^(m - 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x]",
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            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTan[c_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 2, 0] && LtQ[p, -1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-(f*x)^ m*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^(p + 1)/(b*c* d*(p + 1)) + f*m/(b*c*(p + 1))* Int[(f*x)^(m - 1)*(d + e*x^2)^q*(a + b*ArcCot[c*x])^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCot[c_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 2, 0] && LtQ[p, -1]"
  },
  {
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       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p/(d*f*(m + 1)) - b*c*p/(f*(m + 1))* Int[(f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^p/(d*f*(m + 1)) + b*c*p/(f*(m + 1))* Int[(f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcCot[c*x])^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(f*x)^(m + 1)* Sqrt[d + e*x^2]*(a + b*ArcTan[c*x])/(f*(m + 2)) - b*c*d/(f*(m + 2))*Int[(f*x)^(m + 1)/Sqrt[d + e*x^2], x] + d/(m + 2)*Int[(f*x)^m*(a + b*ArcTan[c*x])/Sqrt[d + e*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcTan[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && NeQ[m, -2]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(f*x)^(m + 1)* Sqrt[d + e*x^2]*(a + b*ArcCot[c*x])/(f*(m + 2)) + b*c*d/(f*(m + 2))*Int[(f*x)^(m + 1)/Sqrt[d + e*x^2], x] + d/(m + 2)*Int[(f*x)^m*(a + b*ArcCot[c*x])/Sqrt[d + e*x^2], x]",
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            "lhs": "Int[(f_.*x_)^m_*Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcCot[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && NeQ[m, -2]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTan[c*x])^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && IGtQ[q, 1] && (EqQ[p, 1] || IntegerQ[m])"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcCot[c*x])^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && IGtQ[q, 1] && (EqQ[p, 1] || IntegerQ[m])"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "d*Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x] + c^2*d/f^2* Int[(f*x)^(m + 2)*(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || EqQ[p, 1] && IntegerQ[q])"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "d*Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcCot[c*x])^p, x] + c^2*d/f^2* Int[(f*x)^(m + 2)*(d + e*x^2)^(q - 1)*(a + b*ArcCot[c*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || EqQ[p, 1] && IntegerQ[q])"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcTan[c*x])^p/(c^2*d*m) - b*f*p/(c*m)* Int[(f*x)^(m - 1)*(a + b*ArcTan[c*x])^(p - 1)/Sqrt[d + e*x^2], x] - f^2*(m - 1)/(c^2*m)* Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x])^p/Sqrt[d + e*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcTan[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && GtQ[m, 1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcCot[c*x])^p/(c^2*d*m) + b*f*p/(c*m)* Int[(f*x)^(m - 1)*(a + b*ArcCot[c*x])^(p - 1)/Sqrt[d + e*x^2], x] - f^2*(m - 1)/(c^2*m)* Int[(f*x)^(m - 2)*(a + b*ArcCot[c*x])^p/Sqrt[d + e*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcCot[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && GtQ[m, 1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-2/Sqrt[d]*(a + b*ArcTan[c*x])* ArcTanh[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]] + I*b/Sqrt[d]*PolyLog[2, -Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]] - I*b/Sqrt[d]*PolyLog[2, Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTan[c_.*x_])/(x_*Sqrt[d_ + e_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-2/Sqrt[d]*(a + b*ArcCot[c*x])* ArcTanh[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]] - I*b/Sqrt[d]*PolyLog[2, -Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]] + I*b/Sqrt[d]*PolyLog[2, Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]]",
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            "lhs": "Int[(a_. + b_.*ArcCot[c_.*x_])/(x_*Sqrt[d_ + e_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "1/Sqrt[d]*Subst[Int[(a + b*x)^p*Csc[x], x], x, ArcTan[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTan[c_.*x_])^p_/(x_*Sqrt[d_ + e_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && GtQ[d, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-c*x*Sqrt[1 + 1/(c^2*x^2)]/Sqrt[d + e*x^2]* Subst[Int[(a + b*x)^p*Sec[x], x], x, ArcCot[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCot[c_.*x_])^p_/(x_*Sqrt[d_ + e_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && GtQ[d, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]* Int[(a + b*ArcTan[c*x])^p/(x*Sqrt[1 + c^2*x^2]), x]",
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            "lhs": "Int[(a_. + b_.*ArcTan[c_.*x_])^p_./(x_*Sqrt[d_ + e_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && Not[GtQ[d, 0]]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]* Int[(a + b*ArcCot[c*x])^p/(x*Sqrt[1 + c^2*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCot[c_.*x_])^p_./(x_*Sqrt[d_ + e_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && Not[GtQ[d, 0]]"
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            "rhs": "-Sqrt[d + e*x^2]*(a + b*ArcTan[c*x])^p/(d*x) + b*c*p*Int[(a + b*ArcTan[c*x])^(p - 1)/(x*Sqrt[d + e*x^2]), x]",
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            "lhs": "Int[(a_. + b_.*ArcTan[c_.*x_])^p_./(x_^2*Sqrt[d_ + e_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
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            "lhs": "Int[(a_. + b_.*ArcCot[c_.*x_])^p_./(x_^2*Sqrt[d_ + e_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]"
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            "rhs": "(f*x)^(m + 1)* Sqrt[d + e*x^2]*(a + b*ArcTan[c*x])^p/(d*f*(m + 1)) - b*c*p/(f*(m + 1))* Int[(f*x)^(m + 1)*(a + b*ArcTan[c*x])^(p - 1)/Sqrt[d + e*x^2], x] - c^2*(m + 2)/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(a + b*ArcTan[c*x])^p/Sqrt[d + e*x^2], x]",
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            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcTan[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && LtQ[m, -1] && NeQ[m, -2]"
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            "rhs": "(f*x)^(m + 1)* Sqrt[d + e*x^2]*(a + b*ArcCot[c*x])^p/(d*f*(m + 1)) + b*c*p/(f*(m + 1))* Int[(f*x)^(m + 1)*(a + b*ArcCot[c*x])^(p - 1)/Sqrt[d + e*x^2], x] - c^2*(m + 2)/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(a + b*ArcCot[c*x])^p/Sqrt[d + e*x^2], x]",
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            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcCot[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && LtQ[m, -1] && NeQ[m, -2]"
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2*q] && LtQ[q, -1] && IGtQ[m, 1] && NeQ[p, -1]"
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            "rhs": "1/e*Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^p, x] - d/e*Int[x^(m - 2)*(d + e*x^2)^q*(a + b*ArcCot[c*x])^p, x]",
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            "lhs": "Int[x_^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2*q] && LtQ[q, -1] && IGtQ[m, 1] && NeQ[p, -1]"
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            "rhs": "1/d*Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x] - e/d*Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^p, x]",
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            "lhs": "Int[x_^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2*q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]"
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            "rhs": "1/d*Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^p, x] - e/d*Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcCot[c*x])^p, x]",
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            "lhs": "Int[x_^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2*q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1)) - m/(b*c*(p + 1))* Int[x^(m - 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x] - c*(m + 2*q + 2)/(b*(p + 1))* Int[x^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && EqQ[e, c^2*d] && IntegerQ[m] && LtQ[q, -1] && LtQ[p, -1] && NeQ[m + 2*q + 2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-x^ m*(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])^(p + 1)/(b*c* d*(p + 1)) + m/(b*c*(p + 1))* Int[x^(m - 1)*(d + e*x^2)^q*(a + b*ArcCot[c*x])^(p + 1), x] + c*(m + 2*q + 2)/(b*(p + 1))* Int[x^(m + 1)*(d + e*x^2)^q*(a + b*ArcCot[c*x])^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && EqQ[e, c^2*d] && IntegerQ[m] && LtQ[q, -1] && LtQ[p, -1] && NeQ[m + 2*q + 2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "d^q/c^(m + 1)* Subst[Int[(a + b*x)^p*Sin[x]^m/Cos[x]^(m + 2*(q + 1)), x], x, ArcTan[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && (IntegerQ[q] || GtQ[d, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "d^(q + 1/2)*Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]* Int[x^m*(1 + c^2*x^2)^q*(a + b*ArcTan[c*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && Not[IntegerQ[q] || GtQ[d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-d^q/c^(m + 1)* Subst[Int[(a + b*x)^p*Cos[x]^m/Sin[x]^(m + 2*(q + 1)), x], x, ArcCot[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && IntegerQ[q]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-d^(q + 1/2)*x* Sqrt[(1 + c^2*x^2)/(c^2*x^2)]/(c^m*Sqrt[d + e*x^2])* Subst[Int[(a + b*x)^p*Cos[x]^m/Sin[x]^(m + 2*(q + 1)), x], x, ArcCot[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && Not[IntegerQ[q]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])/(2* e*(q + 1)) - b*c/(2*e*(q + 1))*Int[(d + e*x^2)^(q + 1)/(1 + c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d_. + e_.*x_^2)^q_.*(a_. + b_.*ArcTan[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x])/(2* e*(q + 1)) + b*c/(2*e*(q + 1))*Int[(d + e*x^2)^(q + 1)/(1 + c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d_. + e_.*x_^2)^q_.*(a_. + b_.*ArcCot[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "With[{u = IntHide[(f*x)^m*(d + e*x^2)^q, x]}, Dist[a + b*ArcTan[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/(1 + c^2*x^2), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_^2)^q_.*(a_. + b_.*ArcTan[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, q}, x] && ( IGtQ[q, 0] && Not[ILtQ[(m - 1)/2, 0] && GtQ[m + 2*q + 3, 0]] || IGtQ[(m + 1)/2, 0] && Not[ILtQ[q, 0] && GtQ[m + 2*q + 3, 0]] || ILtQ[(m + 2*q + 1)/2, 0] && Not[ILtQ[(m - 1)/2, 0]] )"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "With[{u = IntHide[(f*x)^m*(d + e*x^2)^q, x]}, Dist[a + b*ArcCot[c*x], u, x] + b*c*Int[SimplifyIntegrand[u/(1 + c^2*x^2), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_^2)^q_.*(a_. + b_.*ArcCot[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, q}, x] && ( IGtQ[q, 0] && Not[ILtQ[(m - 1)/2, 0] && GtQ[m + 2*q + 3, 0]] || IGtQ[(m + 1)/2, 0] && Not[ILtQ[q, 0] && GtQ[m + 2*q + 3, 0]] || ILtQ[(m + 2*q + 1)/2, 0] && Not[ILtQ[(m - 1)/2, 0]] )"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "1/(4*d^2*Rt[-e/d, 2])* Int[(a + b*ArcTan[c*x])^p/(1 - Rt[-e/d, 2]*x)^2, x] - 1/(4*d^2*Rt[-e/d, 2])* Int[(a + b*ArcTan[c*x])^p/(1 + Rt[-e/d, 2]*x)^2, x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(a_. + b_.*ArcTan[c_.*x_])^p_./(d_ + e_.*x_^2)^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "1/(4*d^2*Rt[-e/d, 2])* Int[(a + b*ArcCot[c*x])^p/(1 - Rt[-e/d, 2]*x)^2, x] - 1/(4*d^2*Rt[-e/d, 2])* Int[(a + b*ArcCot[c*x])^p/(1 + Rt[-e/d, 2]*x)^2, x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(a_. + b_.*ArcCot[c_.*x_])^p_./(d_ + e_.*x_^2)^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "With[{u = ExpandIntegrand[(a + b*ArcTan[c*x])^p, (f*x)^m*(d + e*x^2)^q, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && IntegerQ[q] && IGtQ[p, 0] && (EqQ[p, 1] && GtQ[q, 0] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "With[{u = ExpandIntegrand[(a + b*ArcCot[c*x])^p, (f*x)^m*(d + e*x^2)^q, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && IntegerQ[q] && IGtQ[p, 0] && (EqQ[p, 1] && GtQ[q, 0] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "a*Int[(f*x)^m*(d + e*x^2)^q, x] + b*Int[(f*x)^m*(d + e*x^2)^q*ArcTan[c*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_ + b_.*ArcTan[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, q}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "a*Int[(f*x)^m*(d + e*x^2)^q, x] + b*Int[(f*x)^m*(d + e*x^2)^q*ArcCot[c*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_ + b_.*ArcCot[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, q}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcTan[c*x])^p/(d + e*x^2), (f + g*x)^m, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(a_. + b_.*ArcTan[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcCot[c*x])^p/(d + e*x^2), (f + g*x)^m, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(a_. + b_.*ArcCot[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "1/2*Int[Log[1 + u]*(a + b*ArcTan[c*x])^p/(d + e*x^2), x] - 1/2*Int[Log[1 - u]*(a + b*ArcTan[c*x])^p/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTanh[u_]*(a_. + b_.*ArcTan[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - 2*I/(I + c*x))^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "1/2*Int[Log[ SimplifyIntegrand[1 + 1/u, x]]*(a + b*ArcCot[c*x])^ p/(d + e*x^2), x] - 1/2*Int[ Log[SimplifyIntegrand[1 - 1/u, x]]*(a + b*ArcCot[c*x])^ p/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCoth[u_]*(a_. + b_.*ArcCot[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - 2*I/(I + c*x))^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "1/2*Int[Log[1 + u]*(a + b*ArcTan[c*x])^p/(d + e*x^2), x] - 1/2*Int[Log[1 - u]*(a + b*ArcTan[c*x])^p/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTanh[u_]*(a_. + b_.*ArcTan[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - 2*I/(I - c*x))^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "1/2*Int[Log[ SimplifyIntegrand[1 + 1/u, x]]*(a + b*ArcCot[c*x])^ p/(d + e*x^2), x] - 1/2*Int[ Log[SimplifyIntegrand[1 - 1/u, x]]*(a + b*ArcCot[c*x])^ p/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCoth[u_]*(a_. + b_.*ArcCot[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - 2*I/(I - c*x))^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(a + b*ArcTan[c*x])^(p + 1)* Log[f + g*x]/(b*c*d*(p + 1)) - g/(b*c*d*(p + 1))*Int[(a + b*ArcTan[c*x])^(p + 1)/(f + g*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTan[c_.*x_])^p_.*Log[f_ + g_.*x_]/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[c^2*f^2 + g^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(a + b*ArcCot[c*x])^(p + 1)* Log[f + g*x]/(b*c*d*(p + 1)) - g/(b*c*d*(p + 1))*Int[(a + b*ArcCot[c*x])^(p + 1)/(f + g*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCot[c_.*x_])^p_.*Log[f_ + g_.*x_]/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[c^2*f^2 + g^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "I*(a + b*ArcTan[c*x])^p*PolyLog[2, 1 - u]/(2*c*d) - b*p*I/2* Int[(a + b*ArcTan[c*x])^(p - 1)*PolyLog[2, 1 - u]/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTan[c_.*x_])^p_.*Log[u_]/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[(1 - u)^2 - (1 - 2*I/(I + c*x))^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "I*(a + b*ArcCot[c*x])^p*PolyLog[2, 1 - u]/(2*c*d) + b*p*I/2* Int[(a + b*ArcCot[c*x])^(p - 1)*PolyLog[2, 1 - u]/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCot[c_.*x_])^p_.*Log[u_]/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[(1 - u)^2 - (1 - 2*I/(I + c*x))^2, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-I*(a + b*ArcTan[c*x])^p* PolyLog[2, 1 - u]/(2*c*d) + b*p*I/2* Int[(a + b*ArcTan[c*x])^(p - 1)*PolyLog[2, 1 - u]/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTan[c_.*x_])^p_.*Log[u_]/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[(1 - u)^2 - (1 - 2*I/(I - c*x))^2, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-I*(a + b*ArcCot[c*x])^p* PolyLog[2, 1 - u]/(2*c*d) - b*p*I/2* Int[(a + b*ArcCot[c*x])^(p - 1)*PolyLog[2, 1 - u]/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCot[c_.*x_])^p_.*Log[u_]/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[(1 - u)^2 - (1 - 2*I/(I - c*x))^2, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-I*(a + b*ArcTan[c*x])^p* PolyLog[k + 1, u]/(2*c*d) + b*p*I/2* Int[(a + b*ArcTan[c*x])^(p - 1)*PolyLog[k + 1, u]/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTan[c_.*x_])^p_.*PolyLog[k_, u_]/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - 2*I/(I + c*x))^2, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-I*(a + b*ArcCot[c*x])^p* PolyLog[k + 1, u]/(2*c*d) - b*p*I/2* Int[(a + b*ArcCot[c*x])^(p - 1)*PolyLog[k + 1, u]/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCot[c_.*x_])^p_.*PolyLog[k_, u_]/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - 2*I/(I + c*x))^2, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "I*(a + b*ArcTan[c*x])^p*PolyLog[k + 1, u]/(2*c*d) - b*p*I/2* Int[(a + b*ArcTan[c*x])^(p - 1)*PolyLog[k + 1, u]/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTan[c_.*x_])^p_.*PolyLog[k_, u_]/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - 2*I/(I - c*x))^2, 0]"
  },
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "I*(a + b*ArcCot[c*x])^p*PolyLog[k + 1, u]/(2*c*d) + b*p*I/2* Int[(a + b*ArcCot[c*x])^(p - 1)*PolyLog[k + 1, u]/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCot[c_.*x_])^p_.*PolyLog[k_, u_]/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - 2*I/(I - c*x))^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(-Log[a + b*ArcCot[c*x]] + Log[a + b*ArcTan[c*x]])/(b*c* d*(2*a + b*ArcCot[c*x] + b*ArcTan[c*x]))",
     "rulenumber": 0,
            "lhs": "Int[1/((d_ + e_.*x_^2)*(a_. + b_.*ArcCot[c_.*x_])*(a_. + b_.*ArcTan[c_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "-(a + b*ArcCot[c*x])^(q + 1)*(a + b*ArcTan[c*x])^ p/(b*c*d*(q + 1)) + p/(q + 1)* Int[(a + b*ArcCot[c*x])^(q + 1)*(a + b*ArcTan[c*x])^(p - 1)/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCot[c_.*x_])^ q_.*(a_. + b_.*ArcTan[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && IGeQ[q, p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(a + b*ArcTan[c*x])^(q + 1)*(a + b*ArcCot[c*x])^ p/(b*c*d*(q + 1)) + p/(q + 1)* Int[(a + b*ArcTan[c*x])^(q + 1)*(a + b*ArcCot[c*x])^(p - 1)/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTan[c_.*x_])^ q_.*(a_. + b_.*ArcCot[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && IGeQ[q, p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "I/2*Int[Log[1 - I*a*x]/(c + d*x^n), x] - I/2*Int[Log[1 + I*a*x]/(c + d*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTan[a_.*x_]/(c_ + d_.*x_^n_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d}, x] && IntegerQ[n] && Not[EqQ[n, 2] && EqQ[d, a^2*c]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "I/2*Int[Log[1 - I/(a*x)]/(c + d*x^n), x] - I/2*Int[Log[1 + I/(a*x)]/(c + d*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCot[a_.*x_]/(c_ + d_.*x_^n_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d}, x] && IntegerQ[n] && Not[EqQ[n, 2] && EqQ[d, a^2*c]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "I/2*Int[Log[d*x^m]*Log[1 - I*c*x^n]/x, x] - I/2*Int[Log[d*x^m]*Log[1 + I*c*x^n]/x, x]",
     "rulenumber": 0,
            "lhs": "Int[Log[d_.*x_^m_.]*ArcTan[c_.*x_^n_.]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, m, n}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "I/2*Int[Log[d*x^m]*Log[1 - I/(c*x^n)]/x, x] - I/2*Int[Log[d*x^m]*Log[1 + I/(c*x^n)]/x, x]",
     "rulenumber": 0,
            "lhs": "Int[Log[d_.*x_^m_.]*ArcCot[c_.*x_^n_.]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, m, n}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "a*Int[Log[d*x^m]/x, x] + b*Int[(Log[d*x^m]*ArcTan[c*x^n])/x, x]",
     "rulenumber": 0,
            "lhs": "Int[Log[d_.*x_^m_.]*(a_ + b_.*ArcTan[c_.*x_^n_.])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "a*Int[Log[d*x^m]/x, x] + b*Int[(Log[d*x^m]*ArcCot[c*x^n])/x, x]",
     "rulenumber": 0,
            "lhs": "Int[Log[d_.*x_^m_.]*(a_ + b_.*ArcCot[c_.*x_^n_.])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "x*(d + e*Log[f + g*x^2])*(a + b*ArcTan[c*x]) - 2*e*g*Int[x^2*(a + b*ArcTan[c*x])/(f + g*x^2), x] - b*c*Int[x*(d + e*Log[f + g*x^2])/(1 + c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcTan[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "x*(d + e*Log[f + g*x^2])*(a + b*ArcCot[c*x]) - 2*e*g*Int[x^2*(a + b*ArcCot[c*x])/(f + g*x^2), x] + b*c*Int[x*(d + e*Log[f + g*x^2])/(1 + c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcCot[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(Log[f + g*x^2] - Log[1 - I*c*x] - Log[1 + I*c*x])* Int[ArcTan[c*x]/x, x] + I/2*Int[Log[1 - I*c*x]^2/x, x] - I/2*Int[Log[1 + I*c*x]^2/x, x]",
     "rulenumber": 0,
            "lhs": "Int[Log[f_. + g_.*x_^2]*ArcTan[c_.*x_]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, f, g}, x] && EqQ[g, c^2*f]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(Log[f + g*x^2] - Log[c^2*x^2] - Log[1 - I/(c*x)] - Log[1 + I/(c*x)])*Int[ArcCot[c*x]/x, x] + Int[Log[c^2*x^2]*ArcCot[c*x]/x, x] + I/2*Int[Log[1 - I/(c*x)]^2/x, x] - I/2*Int[Log[1 + I/(c*x)]^2/x, x]",
     "rulenumber": 0,
            "lhs": "Int[Log[f_. + g_.*x_^2]*ArcCot[c_.*x_]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, f, g}, x] && EqQ[g, c^2*f]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "a*Int[Log[f + g*x^2]/x, x] + b*Int[Log[f + g*x^2]*ArcTan[c*x]/x, x]",
     "rulenumber": 0,
            "lhs": "Int[Log[f_. + g_.*x_^2]*(a_ + b_.*ArcTan[c_.*x_])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, f, g}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "a*Int[Log[f + g*x^2]/x, x] + b*Int[Log[f + g*x^2]*ArcCot[c*x]/x, x]",
     "rulenumber": 0,
            "lhs": "Int[Log[f_. + g_.*x_^2]*(a_ + b_.*ArcCot[c_.*x_])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, f, g}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "d*Int[(a + b*ArcTan[c*x])/x, x] + e*Int[Log[f + g*x^2]*(a + b*ArcTan[c*x])/x, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcTan[c_.*x_])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "d*Int[(a + b*ArcCot[c*x])/x, x] + e*Int[Log[f + g*x^2]*(a + b*ArcCot[c*x])/x, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcCot[c_.*x_])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "x^(m + 1)*(d + e*Log[f + g*x^2])*(a + b*ArcTan[c*x])/(m + 1) - 2*e*g/(m + 1)*Int[x^(m + 2)*(a + b*ArcTan[c*x])/(f + g*x^2), x] - b*c/(m + 1)* Int[x^(m + 1)*(d + e*Log[f + g*x^2])/(1 + c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcTan[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && ILtQ[m/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "x^(m + 1)*(d + e*Log[f + g*x^2])*(a + b*ArcCot[c*x])/(m + 1) - 2*e*g/(m + 1)*Int[x^(m + 2)*(a + b*ArcCot[c*x])/(f + g*x^2), x] + b*c/(m + 1)* Int[x^(m + 1)*(d + e*Log[f + g*x^2])/(1 + c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcCot[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && ILtQ[m/2, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "With[{u = IntHide[x^m*(d + e*Log[f + g*x^2]), x]}, Dist[a + b*ArcTan[c*x], u, x] - b*c*Int[ExpandIntegrand[u/(1 + c^2*x^2), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcTan[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[(m + 1)/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "With[{u = IntHide[x^m*(d + e*Log[f + g*x^2]), x]}, Dist[a + b*ArcCot[c*x], u, x] + b*c*Int[ExpandIntegrand[u/(1 + c^2*x^2), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcCot[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[(m + 1)/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "With[{u = IntHide[x^m*(a + b*ArcTan[c*x]), x]}, Dist[d + e*Log[f + g*x^2], u, x] - 2*e*g*Int[ExpandIntegrand[x*u/(f + g*x^2), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcTan[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && IntegerQ[m] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "With[{u = IntHide[x^m*(a + b*ArcCot[c*x]), x]}, Dist[d + e*Log[f + g*x^2], u, x] - 2*e*g*Int[ExpandIntegrand[x*u/(f + g*x^2), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcCot[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && IntegerQ[m] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(f + g*x^2)*(d + e*Log[f + g*x^2])*(a + b*ArcTan[c*x])^2/(2*g) - e*x^2*(a + b*ArcTan[c*x])^2/2 - b/c*Int[(d + e*Log[f + g*x^2])*(a + b*ArcTan[c*x]), x] + b*c*e*Int[x^2*(a + b*ArcTan[c*x])/(1 + c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d_. + e_.*Log[f_ + g_.*x_^2])*(a_. + b_.*ArcTan[c_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[g, c^2*f]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(f + g*x^2)*(d + e*Log[f + g*x^2])*(a + b*ArcCot[c*x])^2/(2*g) - e*x^2*(a + b*ArcCot[c*x])^2/2 + b/c*Int[(d + e*Log[f + g*x^2])*(a + b*ArcCot[c*x]), x] - b*c*e*Int[x^2*(a + b*ArcCot[c*x])/(1 + c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d_. + e_.*Log[f_ + g_.*x_^2])*(a_. + b_.*ArcCot[c_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[g, c^2*f]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "Unintegrable[u*(a + b*ArcTan[c*x])^p, x] /; FreeQ[{a, b, c, p}, x] && (EqQ[u, 1] || MatchQ[u, (d_. + e_.*x)^q_. /; FreeQ[{d, e, q}, x]] || MatchQ[ u, (f_.*x)^m_.*(d_. + e_.*x)^q_. /; FreeQ[{d, e, f, m, q}, x]] || MatchQ[u, (d_. + e_.*x^2)^q_. /; FreeQ[{d, e, q}, x]] || MatchQ[ u, (f_.*x)^m_.*(d_. + e_.*x^2)^q_.",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_. + b_.*ArcTan[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{d, e, f, m, q}, x]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "Unintegrable[u*(a + b*ArcCot[c*x])^p, x] /; FreeQ[{a, b, c, p}, x] && (EqQ[u, 1] || MatchQ[u, (d_. + e_.*x)^q_. /; FreeQ[{d, e, q}, x]] || MatchQ[ u, (f_.*x)^m_.*(d_. + e_.*x)^q_. /; FreeQ[{d, e, f, m, q}, x]] || MatchQ[u, (d_. + e_.*x^2)^q_. /; FreeQ[{d, e, q}, x]] || MatchQ[ u, (f_.*x)^m_.*(d_. + e_.*x^2)^q_.",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_. + b_.*ArcCot[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{d, e, f, m, q}, x]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "x*ArcTan[c*x^n] - c*n*Int[x^n/(1 + c^2*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTan[c_.*x_^n_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "x*ArcCot[c*x^n] + c*n*Int[x^n/(1 + c^2*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCot[c_.*x_^n_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(a + (I*b*Log[1 - I*c*x^n])/ 2 - (I*b*Log[1 + I*c*x^n])/2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTan[c_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(a + (I*b*Log[1 - I*x^(-n)/c])/ 2 - (I*b*Log[1 + I*x^(-n)/c])/2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCot[c_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "1/n*Subst[Int[(a + b*ArcTan[c*x])^p/x, x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTan[c_.*x_^n_])^p_./x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "1/n*Subst[Int[(a + b*ArcCot[c*x])^p/x, x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCot[c_.*x_^n_])^p_./x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(d*x)^(m + 1)*(a + b*ArcTan[c*x^n])/(d*(m + 1)) - b*c*n/(d*(m + 1))* Int[x^(n - 1)*(d*x)^(m + 1)/(1 + c^2*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_. + b_.*ArcTan[c_.*x_^n_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "(d*x)^(m + 1)*(a + b*ArcCot[c*x^n])/(d*(m + 1)) + b*c*n/(d*(m + 1))* Int[x^(n - 1)*(d*x)^(m + 1)/(1 + c^2*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_. + b_.*ArcCot[c_.*x_^n_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(d*x)^ m*(a + (I*b*Log[1 - I*c*x^n])/2 - (I*b*Log[1 + I*c*x^n])/2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_. + b_.*ArcTan[c_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p, 0] && IntegerQ[m] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(d*x)^ m*(a + (I*b*Log[1 - I*x^(-n)/c])/2 - (I*b*Log[1 + I*x^(-n)/c])/ 2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_. + b_.*ArcCot[c_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p, 0] && IntegerQ[m] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "Unintegrable[u*(a + b*ArcTan[c*x^n])^p, x] /; FreeQ[{a, b, c, n, p}, x] && (EqQ[u, 1] || MatchQ[u, (d_.*x)^m_.",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_. + b_.*ArcTan[c_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{d, m}, x]]) "
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.1 u (a+b arctan(c x^n))^p.m",
       "filename": "5.3.1 u (a+b arctan(c x^n))^p.m",
            "rhs": "Unintegrable[u*(a + b*ArcCot[c*x^n])^p, x] /; FreeQ[{a, b, c, n, p}, x] && (EqQ[u, 1] || MatchQ[u, (d_.*x)^m_.",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_. + b_.*ArcCot[c_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{d, m}, x]]) "
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.2 u (a+b arctan(c+d x))^p.m",
       "filename": "5.3.2 u (a+b arctan(c+d x))^p.m",
            "rhs": "1/d*Subst[Int[(a + b*ArcTan[x])^p, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTan[c_ + d_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.2 u (a+b arctan(c+d x))^p.m",
       "filename": "5.3.2 u (a+b arctan(c+d x))^p.m",
            "rhs": "1/d*Subst[Int[(a + b*ArcCot[x])^p, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCot[c_ + d_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.2 u (a+b arctan(c+d x))^p.m",
       "filename": "5.3.2 u (a+b arctan(c+d x))^p.m",
            "rhs": "Unintegrable[(a + b*ArcTan[c + d*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTan[c_ + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, p}, x] && Not[IGtQ[p, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.2 u (a+b arctan(c+d x))^p.m",
       "filename": "5.3.2 u (a+b arctan(c+d x))^p.m",
            "rhs": "Unintegrable[(a + b*ArcCot[c + d*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCot[c_ + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, p}, x] && Not[IGtQ[p, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.2 u (a+b arctan(c+d x))^p.m",
       "filename": "5.3.2 u (a+b arctan(c+d x))^p.m",
            "rhs": "1/d*Subst[Int[(f*x/d)^m*(a + b*ArcTan[x])^p, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcTan[c_ + d_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.2 u (a+b arctan(c+d x))^p.m",
       "filename": "5.3.2 u (a+b arctan(c+d x))^p.m",
            "rhs": "1/d*Subst[Int[(f*x/d)^m*(a + b*ArcCot[x])^p, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCot[c_ + d_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.2 u (a+b arctan(c+d x))^p.m",
       "filename": "5.3.2 u (a+b arctan(c+d x))^p.m",
            "rhs": "(e + f*x)^(m + 1)*(a + b*ArcTan[c + d*x])^ p/(f*(m + 1)) - b*d*p/(f*(m + 1))* Int[(e + f*x)^(m + 1)*(a + b*ArcTan[c + d*x])^(p - 1)/(1 + (c + d*x)^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_*(a_. + b_.*ArcTan[c_ + d_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && ILtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.2 u (a+b arctan(c+d x))^p.m",
       "filename": "5.3.2 u (a+b arctan(c+d x))^p.m",
            "rhs": "(e + f*x)^(m + 1)*(a + b*ArcCot[c + d*x])^ p/(f*(m + 1)) + b*d*p/(f*(m + 1))* Int[(e + f*x)^(m + 1)*(a + b*ArcCot[c + d*x])^(p - 1)/(1 + (c + d*x)^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_*(a_. + b_.*ArcCot[c_ + d_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && ILtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.2 u (a+b arctan(c+d x))^p.m",
       "filename": "5.3.2 u (a+b arctan(c+d x))^p.m",
            "rhs": "1/d*Subst[Int[((d*e - c*f)/d + f*x/d)^m*(a + b*ArcTan[x])^p, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcTan[c_ + d_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.2 u (a+b arctan(c+d x))^p.m",
       "filename": "5.3.2 u (a+b arctan(c+d x))^p.m",
            "rhs": "1/d*Subst[Int[((d*e - c*f)/d + f*x/d)^m*(a + b*ArcCot[x])^p, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCot[c_ + d_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.2 u (a+b arctan(c+d x))^p.m",
       "filename": "5.3.2 u (a+b arctan(c+d x))^p.m",
            "rhs": "Unintegrable[(e + f*x)^m*(a + b*ArcTan[c + d*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcTan[c_ + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p}, x] && Not[IGtQ[p, 0]]"
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            "lhs": "Int[x_^2*(c_ + d_.*x_^2)^p_*E^(n_.*ArcTan[a_.*x_]), x_Symbol]",
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         "givens": "FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && LtQ[p, -1] && Not[IntegerQ[I*n]] && NeQ[n^2 + 4*(p + 1)^2, 0] && IntegerQ[2*p]"
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            "rhs": "c^p*Int[x^m*(1 + a^2*x^2)^(p - I*n/2)*(1 - I*a*x)^(I*n), x]",
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         "givens": "FreeQ[{a, c, d, m, p}, x] && EqQ[d, a^2*c] && (IntegerQ[p] || GtQ[c, 0]) && IntegerQ[(I*n + 1)/2] && Not[IntegerQ[p - I*n/2]]"
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         "givens": "FreeQ[{a, c, d, m, n, p}, x] && EqQ[d, a^2*c] && (IntegerQ[p] || GtQ[c, 0])"
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            "rhs": "c^(I*n/2)*Int[x^m*(c + d*x^2)^(p - I*n/2)*(1 - I*a*x)^(I*n), x]",
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            "lhs": "Int[x_^m_.*(c_ + d_.*x_^2)^p_*E^(n_*ArcTan[a_.*x_]), x_Symbol]",
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         "givens": "FreeQ[{a, c, d, m, p}, x] && EqQ[d, a^2*c] && Not[IntegerQ[p] || GtQ[c, 0]] && IGtQ[I*n/2, 0]"
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            "rhs": "c^IntPart[p]*(c + d*x^2)^FracPart[p]/(1 + a^2*x^2)^FracPart[p]* Int[x^m*(1 + a^2*x^2)^p*E^(n*ArcTan[a*x]), x]",
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            "rhs": "c^IntPart[p]*(c + d*x^2)^ FracPart[p]/((1 - I*a*x)^FracPart[p]*(1 + I*a*x)^FracPart[p])* Int[u*(1 - I*a*x)^(p + I*n/2)*(1 + I*a*x)^(p - I*n/2), x]",
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         "givens": "FreeQ[{a, c, d, p}, x] && EqQ[c - a^2*d, 0] && Not[IntegerQ[p]] && IntegerQ[I*n/2] && GtQ[c, 0]"
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            "rhs": "x^(2*p)*(c + d/x^2)^p/((1 - I*a*x)^p*(1 + I*a*x)^p)* Int[u/x^(2*p)*(1 - I*a*x)^p*(1 + I*a*x)^p*E^(n*ArcTan[a*x]), x]",
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            "rhs": "x^(2*p)*(c + d/x^2)^p/(1 + a^2*x^2)^p* Int[u/x^(2*p)*(1 + a^2*x^2)^p*E^(n*ArcTan[a*x]), x]",
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         "givens": "FreeQ[{a, c, d, n, p}, x] && EqQ[c - a^2*d, 0] && Not[IntegerQ[p]] && Not[IntegerQ[I*n/2]]"
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            "lhs": "Int[E^(n_.*ArcTan[c_.*(a_ + b_.*x_)]), x_Symbol]",
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         "givens": "FreeQ[{a, b, c, n}, x]"
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         "givens": "FreeQ[{a, b, c}, x] && ILtQ[m, 0] && LtQ[-1, I*n, 1]"
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         "givens": "FreeQ[{a, b, c, d, e, m, n}, x]"
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            "rhs": "(c/(1 + a^2))^p* Int[u*(1 - I*a - I*b*x)^(p + I*n/2)*(1 + I*a + I*b*x)^(p - I*n/2), x]",
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            "lhs": "Int[u_.*(c_ + d_.*x_ + e_.*x_^2)^p_.*E^(n_.*ArcTan[a_ + b_.*x_]), x_Symbol]",
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         "givens": "FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, 2*a*e] && EqQ[b^2*c - e (1 + a^2), 0] && (IntegerQ[p] || GtQ[c/(1 + a^2), 0])"
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         "givens": "FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, 2*a*e] && EqQ[b^2*c - e (1 + a^2), 0] && Not[IntegerQ[p] || GtQ[c/(1 + a^2), 0]]"
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         "givens": "FreeQ[{a, b, c, n}, x]"
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         "givens": "FreeQ[{a, c, d, n}, x] && EqQ[a^2*c^2 + d^2, 0] && Not[IntegerQ[I*n/2]] && IntegerQ[p]"
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         "givens": "FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 + a^2*d^2, 0] && Not[IntegerQ[I*n/2]] && (IntegerQ[p] || GtQ[c, 0])"
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         "givens": "FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 + a^2*d^2, 0] && Not[IntegerQ[I*n/2]] && Not[IntegerQ[p] || GtQ[c, 0]]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.3 Exponentials of inverse tangent.m",
       "filename": "5.3.3 Exponentials of inverse tangent.m",
            "rhs": "-c^p*x^m*(1/x)^m* Subst[Int[(1 + d*x/c)^ p*(1 - I*x/a)^(I*n/2)/(x^(m + 2)*(1 + I*x/a)^(I*n/2)), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*(c_ + d_./x_)^p_.*E^(n_.*ArcCot[a_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, m, n, p}, x] && EqQ[c^2 + a^2*d^2, 0] && Not[IntegerQ[I*n/2]] && (IntegerQ[p] || GtQ[c, 0]) && Not[IntegerQ[m]]"
  },
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.3 Exponentials of inverse tangent.m",
       "filename": "5.3.3 Exponentials of inverse tangent.m",
            "rhs": "(c + d/x)^p/(1 + d/(c*x))^p* Int[u*(1 + d/(c*x))^p*E^(n*ArcCot[a*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(c_ + d_./x_)^p_*E^(n_.*ArcCot[a_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 + a^2*d^2, 0] && Not[IntegerQ[I*n/2]] && Not[IntegerQ[p] || GtQ[c, 0]]"
  },
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.3 Exponentials of inverse tangent.m",
       "filename": "5.3.3 Exponentials of inverse tangent.m",
            "rhs": "-E^(n*ArcCot[a*x])/(a*c*n)",
     "rulenumber": 0,
            "lhs": "Int[E^(n_.*ArcCot[a_.*x_])/(c_ + d_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.3 Exponentials of inverse tangent.m",
       "filename": "5.3.3 Exponentials of inverse tangent.m",
            "rhs": "-(n - a*x)* E^(n*ArcCot[a*x])/(a*c*(n^2 + 1)*Sqrt[c + d*x^2])",
     "rulenumber": 0,
            "lhs": "Int[E^(n_.*ArcCot[a_.*x_])/(c_ + d_.*x_^2)^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && Not[IntegerQ[(I*n - 1)/2]]"
  },
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.3 Exponentials of inverse tangent.m",
       "filename": "5.3.3 Exponentials of inverse tangent.m",
            "rhs": "-(n + 2*a*(p + 1)*x)*(c + d*x^2)^(p + 1)* E^(n*ArcCot[a*x])/(a*c*(n^2 + 4*(p + 1)^2)) + 2*(p + 1)*(2*p + 3)/(c*(n^2 + 4*(p + 1)^2))* Int[(c + d*x^2)^(p + 1)*E^(n*ArcCot[a*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_ + d_.*x_^2)^p_*E^(n_.*ArcCot[a_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && LtQ[p, -1] && NeQ[p, -3/2] && NeQ[n^2 + 4*(p + 1)^2, 0] && Not[IntegerQ[p] && IntegerQ[I*n/2]] && Not[Not[IntegerQ[p]] && IntegerQ[(I*n - 1)/2]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.3 Exponentials of inverse tangent.m",
       "filename": "5.3.3 Exponentials of inverse tangent.m",
            "rhs": "-(1 + a*n*x)* E^(n*ArcCot[a*x])/(a^2*c*(n^2 + 1)*Sqrt[c + d*x^2])",
     "rulenumber": 0,
            "lhs": "Int[x_*E^(n_.*ArcCot[a_.*x_])/(c_ + d_.*x_^2)^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && Not[IntegerQ[(I*n - 1)/2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.3 Exponentials of inverse tangent.m",
       "filename": "5.3.3 Exponentials of inverse tangent.m",
            "rhs": "(2*(p + 1) - a*n*x)*(c + d*x^2)^(p + 1)* E^(n*ArcCot[a*x])/(a^2*c*(n^2 + 4*(p + 1)^2)) + n*(2*p + 3)/(a*c*(n^2 + 4*(p + 1)^2))* Int[(c + d*x^2)^(p + 1)*E^(n*ArcCot[a*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(c_ + d_.*x_^2)^p_*E^(n_.*ArcCot[a_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && LeQ[p, -1] && NeQ[p, -3/2] && NeQ[n^2 + 4*(p + 1)^2, 0] && Not[IntegerQ[p] && IntegerQ[I*n/2]] && Not[Not[IntegerQ[p]] && IntegerQ[(I*n - 1)/2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.3 Exponentials of inverse tangent.m",
       "filename": "5.3.3 Exponentials of inverse tangent.m",
            "rhs": "(n + 2*(p + 1)*a*x)*(c + d*x^2)^(p + 1)* E^(n*ArcCot[a*x])/(a^3*c*n^2*(n^2 + 1))",
     "rulenumber": 0,
            "lhs": "Int[x_^2*(c_ + d_.*x_^2)^p_.*E^(n_.*ArcCot[a_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && EqQ[n^2 - 2*(p + 1), 0] && NeQ[n^2 + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.3 Exponentials of inverse tangent.m",
       "filename": "5.3.3 Exponentials of inverse tangent.m",
            "rhs": "(n + 2*(p + 1)*a*x)*(c + d*x^2)^(p + 1)* E^(n*ArcCot[a*x])/(a^3*c*(n^2 + 4*(p + 1)^2)) + (n^2 - 2*(p + 1))/(a^2*c*(n^2 + 4*(p + 1)^2))* Int[(c + d*x^2)^(p + 1)*E^(n*ArcCot[a*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^2*(c_ + d_.*x_^2)^p_*E^(n_.*ArcCot[a_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && LeQ[p, -1] && NeQ[n^2 - 2*(p + 1), 0] && NeQ[n^2 + 4*(p + 1)^2, 0] && Not[IntegerQ[p] && IntegerQ[I*n/2]] && Not[Not[IntegerQ[p]] && IntegerQ[(I*n - 1)/2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.3 Exponentials of inverse tangent.m",
       "filename": "5.3.3 Exponentials of inverse tangent.m",
            "rhs": "-c^p/a^(m + 1)* Subst[Int[E^(n*x)*Cot[x]^(m + 2*(p + 1))/Cos[x]^(2*(p + 1)), x], x, ArcCot[a*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(c_ + d_.*x_^2)^p_*E^(n_.*ArcCot[a_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && IntegerQ[m] && LeQ[3, m, -2 (p + 1)] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.3 Exponentials of inverse tangent.m",
       "filename": "5.3.3 Exponentials of inverse tangent.m",
            "rhs": "d^p*Int[u*x^(2*p)*(1 + 1/(a^2*x^2))^p*E^(n*ArcCot[a*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(c_ + d_.*x_^2)^p_.*E^(n_.*ArcCot[a_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c] && Not[IntegerQ[I*n/2]] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.3 Exponentials of inverse tangent.m",
       "filename": "5.3.3 Exponentials of inverse tangent.m",
            "rhs": "(c + d*x^2)^p/(x^(2*p)*(1 + 1/(a^2*x^2))^p)* Int[u*x^(2*p)*(1 + 1/(a^2*x^2))^p*E^(n*ArcCot[a*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(c_ + d_.*x_^2)^p_*E^(n_.*ArcCot[a_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n, p}, x] && EqQ[d, a^2*c] && Not[IntegerQ[I*n/2]] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.3 Exponentials of inverse tangent.m",
       "filename": "5.3.3 Exponentials of inverse tangent.m",
            "rhs": "c^p/(I*a)^(2*p)* Int[u/x^(2*p)*(-1 + I*a*x)^(p - I*n/2)*(1 + I*a*x)^(p + I*n/2), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(c_ + d_./x_^2)^p_.*E^(n_.*ArcCot[a_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n, p}, x] && EqQ[c, a^2*d] && Not[IntegerQ[I*n/2]] && (IntegerQ[p] || GtQ[c, 0]) && IntegersQ[2*p, p + I*n/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.3 Exponentials of inverse tangent.m",
       "filename": "5.3.3 Exponentials of inverse tangent.m",
            "rhs": "-c^p* Subst[Int[(1 - I*x/a)^(p + I*n/2)*(1 + I*x/a)^(p - I*n/2)/x^2, x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(c_ + d_./x_^2)^p_.*E^(n_.*ArcCot[a_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n, p}, x] && EqQ[c, a^2*d] && Not[IntegerQ[I*n/2]] && (IntegerQ[p] || GtQ[c, 0]) && Not[IntegerQ[2*p] && IntegerQ[p + I*n/2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.3 Exponentials of inverse tangent.m",
       "filename": "5.3.3 Exponentials of inverse tangent.m",
            "rhs": "-c^p* Subst[Int[(1 - I*x/a)^(p + I*n/2)*(1 + I*x/a)^(p - I*n/2)/ x^(m + 2), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(c_ + d_./x_^2)^p_.*E^(n_.*ArcCot[a_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n, p}, x] && EqQ[c, a^2*d] && Not[IntegerQ[I*n/2]] && (IntegerQ[p] || GtQ[c, 0]) && Not[IntegerQ[2*p] && IntegerQ[p + I*n/2]] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.3 Exponentials of inverse tangent.m",
       "filename": "5.3.3 Exponentials of inverse tangent.m",
            "rhs": "-c^p*x^m*(1/x)^m* Subst[Int[(1 - I*x/a)^(p + I*n/2)*(1 + I*x/a)^(p - I*n/2)/ x^(m + 2), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*(c_ + d_./x_^2)^p_.*E^(n_.*ArcCot[a_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, m, n, p}, x] && EqQ[c, a^2*d] && Not[IntegerQ[I*n/2]] && (IntegerQ[p] || GtQ[c, 0]) && Not[IntegerQ[2*p] && IntegerQ[p + I*n/2]] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.3 Exponentials of inverse tangent.m",
       "filename": "5.3.3 Exponentials of inverse tangent.m",
            "rhs": "(c + d/x^2)^p/(1 + 1/(a^2*x^2))^p* Int[u*(1 + 1/(a^2*x^2))^p*E^(n*ArcCot[a*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(c_ + d_./x_^2)^p_*E^(n_.*ArcCot[a_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n, p}, x] && EqQ[c, a^2*d] && Not[IntegerQ[I*n/2]] && Not[IntegerQ[p] || GtQ[c, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.3 Exponentials of inverse tangent.m",
       "filename": "5.3.3 Exponentials of inverse tangent.m",
            "rhs": "(-1)^(I*n/2)*Int[u*E^(-n*ArcTan[c*(a + b*x)]), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*E^(n_*ArcCot[c_.*(a_ + b_.*x_)]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IntegerQ[I*n/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.3 Exponentials of inverse tangent.m",
       "filename": "5.3.3 Exponentials of inverse tangent.m",
            "rhs": "(I*c*(a + b*x))^(I* n/2)*(1 + 1/(I*c*(a + b*x)))^(I*n/2)/(1 + I*a*c + I*b*c*x)^(I* n/2)* Int[(1 + I*a*c + I*b*c*x)^(I*n/2)/(-1 + I*a*c + I*b*c*x)^(I* n/2), x]",
     "rulenumber": 0,
            "lhs": "Int[E^(n_.*ArcCot[c_.*(a_ + b_.*x_)]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && Not[IntegerQ[I*n/2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.3 Exponentials of inverse tangent.m",
       "filename": "5.3.3 Exponentials of inverse tangent.m",
            "rhs": "4/(I^m*n*b^(m + 1)*c^(m + 1))* Subst[ Int[x^(2/(I*n))*(1 + I*a*c + (1 - I*a*c)*x^(2/(I*n)))^ m/(-1 + x^(2/(I*n)))^(m + 2), x], x, (1 + 1/(I*c*(a + b*x)))^(I*n/2)/(1 - 1/(I*c*(a + b*x)))^(I* n/2)]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*E^(n_*ArcCoth[c_.*(a_ + b_.*x_)]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && ILtQ[m, 0] && LtQ[-1, I*n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.3 Exponentials of inverse tangent.m",
       "filename": "5.3.3 Exponentials of inverse tangent.m",
            "rhs": "(I*c*(a + b*x))^(I* n/2)*(1 + 1/(I*c*(a + b*x)))^(I*n/2)/(1 + I*a*c + I*b*c*x)^(I* n/2)* Int[(d + e*x)^ m*(1 + I*a*c + I*b*c*x)^(I*n/2)/(-1 + I*a*c + I*b*c*x)^(I*n/2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*E^(n_.*ArcCoth[c_.*(a_ + b_.*x_)]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && Not[IntegerQ[I*n/2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.3 Exponentials of inverse tangent.m",
       "filename": "5.3.3 Exponentials of inverse tangent.m",
            "rhs": "(c/(1 + a^2))^ p*((I*a + I*b*x)/(1 + I*a + I*b*x))^(I* n/2)*((1 + I*a + I*b*x)/(I*a + I*b*x))^(I*n/2)* ((1 - I*a - I*b*x)^(I*n/2)/(-1 + I*a + I*b*x)^(I*n/2))* Int[ u*(1 - I*a - I*b*x)^(p - I*n/2)*(1 + I*a + I*b*x)^(p + I*n/2), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(c_ + d_.*x_ + e_.*x_^2)^p_.*E^(n_.*ArcCot[a_ + b_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p}, x] && Not[IntegerQ[I*n/2]] && EqQ[b*d - 2*a*e, 0] && EqQ[b^2*c - e (1 + a^2), 0] && (IntegerQ[p] || GtQ[c/(1 + a^2), 0])"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.3 Exponentials of inverse tangent.m",
       "filename": "5.3.3 Exponentials of inverse tangent.m",
            "rhs": "(c + d*x + e*x^2)^p/(1 + a^2 + 2*a*b*x + b^2*x^2)^p* Int[u*(1 + a^2 + 2*a*b*x + b^2*x^2)^p*E^(n*ArcCot[a*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(c_ + d_.*x_ + e_.*x_^2)^p_.*E^(n_.*ArcCot[a_ + b_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p}, x] && Not[IntegerQ[I*n/2]] && EqQ[b*d - 2*a*e, 0] && EqQ[b^2*c - e (1 + a^2), 0] && Not[IntegerQ[p] || GtQ[c/(1 + a^2), 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.3 Exponentials of inverse tangent.m",
       "filename": "5.3.3 Exponentials of inverse tangent.m",
            "rhs": "Int[u*E^(n*ArcTan[a/c + b*x/c]), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*E^(n_.*ArcCot[c_./(a_. + b_.*x_)]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "x*ArcTan[a + b*x^n] - b*n*Int[x^n/(1 + a^2 + 2*a*b*x^n + b^2*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTan[a_ + b_.*x_^n_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "x*ArcCot[a + b*x^n] + b*n*Int[x^n/(1 + a^2 + 2*a*b*x^n + b^2*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCot[a_ + b_.*x_^n_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "I/2*Int[Log[1 - I*a - I*b*x^n]/x, x] - I/2*Int[Log[1 + I*a + I*b*x^n]/x, x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTan[a_. + b_.*x_^n_]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "I/2*Int[Log[1 - I/(a + b*x^n)]/x, x] - I/2*Int[Log[1 + I/(a + b*x^n)]/x, x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCot[a_. + b_.*x_^n_]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "x^(m + 1)*ArcTan[a + b*x^n]/(m + 1) - b*n/(m + 1)* Int[x^(m + n)/(1 + a^2 + 2*a*b*x^n + b^2*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*ArcTan[a_ + b_.*x_^n_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && RationalQ[m, n] && m + 1 != 0 && m + 1 != n"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "x^(m + 1)*ArcCot[a + b*x^n]/(m + 1) + b*n/(m + 1)* Int[x^(m + n)/(1 + a^2 + 2*a*b*x^n + b^2*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*ArcCot[a_ + b_.*x_^n_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && RationalQ[m, n] && m + 1 != 0 && m + 1 != n"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "I/2*Int[Log[1 - I*a - I*b*f^(c + d*x)], x] - I/2*Int[Log[1 + I*a + I*b*f^(c + d*x)], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTan[a_. + b_.*f_^(c_. + d_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "I/2*Int[Log[1 - I/(a + b*f^(c + d*x))], x] - I/2*Int[Log[1 + I/(a + b*f^(c + d*x))], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCot[a_. + b_.*f_^(c_. + d_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f}, x] "
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "I/2*Int[x^m*Log[1 - I*a - I*b*f^(c + d*x)], x] - I/2*Int[x^m*Log[1 + I*a + I*b*f^(c + d*x)], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*ArcTan[a_. + b_.*f_^(c_. + d_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f}, x] && IntegerQ[m] && m > 0"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "I/2*Int[x^m*Log[1 - I/(a + b*f^(c + d*x))], x] - I/2*Int[x^m*Log[1 + I/(a + b*f^(c + d*x))], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*ArcCot[a_. + b_.*f_^(c_. + d_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f}, x] && IntegerQ[m] && m > 0"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "Int[u*ArcCot[a/c + b*x^n/c]^m, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*ArcTan[c_./(a_. + b_.*x_^n_.)]^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, m}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "Int[u*ArcTan[a/c + b*x^n/c]^m, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*ArcCot[c_./(a_. + b_.*x_^n_.)]^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, m}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "x*ArcTan[(c*x)/Sqrt[a + b*x^2]] - c*Int[x/Sqrt[a + b*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTan[c_.*x_/Sqrt[a_. + b_.*x_^2]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[b + c^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "x*ArcCot[(c*x)/Sqrt[a + b*x^2]] + c*Int[x/Sqrt[a + b*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCot[c_.*x_/Sqrt[a_. + b_.*x_^2]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[b + c^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "ArcTan[c*x/Sqrt[a + b*x^2]]*Log[x] - c*Int[Log[x]/Sqrt[a + b*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTan[c_.*x_/Sqrt[a_. + b_.*x_^2]]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[b + c^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "ArcCot[c*x/Sqrt[a + b*x^2]]*Log[x] + c*Int[Log[x]/Sqrt[a + b*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCot[c_.*x_/Sqrt[a_. + b_.*x_^2]]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[b + c^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "(d*x)^(m + 1)* ArcTan[(c*x)/Sqrt[a + b*x^2]]/(d*(m + 1)) - c/(d*(m + 1))*Int[(d*x)^(m + 1)/Sqrt[a + b*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*ArcTan[c_.*x_/Sqrt[a_. + b_.*x_^2]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && EqQ[b + c^2, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "(d*x)^(m + 1)* ArcCot[(c*x)/Sqrt[a + b*x^2]]/(d*(m + 1)) + c/(d*(m + 1))*Int[(d*x)^(m + 1)/Sqrt[a + b*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*ArcCot[c_.*x_/Sqrt[a_. + b_.*x_^2]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && EqQ[b + c^2, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "1/c*Log[ArcTan[c*x/Sqrt[a + b*x^2]]]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_. + b_.*x_^2]*ArcTan[c_.*x_/Sqrt[a_. + b_.*x_^2]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[b + c^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "-1/c*Log[ArcCot[c*x/Sqrt[a + b*x^2]]]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_. + b_.*x_^2]*ArcCot[c_.*x_/Sqrt[a_. + b_.*x_^2]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[b + c^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "ArcTan[c*x/Sqrt[a + b*x^2]]^(m + 1)/(c*(m + 1))",
     "rulenumber": 0,
            "lhs": "Int[ArcTan[c_.*x_/Sqrt[a_. + b_.*x_^2]]^m_./Sqrt[a_. + b_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m}, x] && EqQ[b + c^2, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "-ArcCot[c*x/Sqrt[a + b*x^2]]^(m + 1)/(c*(m + 1))",
     "rulenumber": 0,
            "lhs": "Int[ArcCot[c_.*x_/Sqrt[a_. + b_.*x_^2]]^m_./Sqrt[a_. + b_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m}, x] && EqQ[b + c^2, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "Sqrt[a + b*x^2]/Sqrt[d + e*x^2]* Int[ArcTan[c*x/Sqrt[a + b*x^2]]^m/Sqrt[a + b*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTan[c_.*x_/Sqrt[a_. + b_.*x_^2]]^m_./Sqrt[d_. + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && EqQ[b + c^2, 0] && EqQ[b*d - a*e, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "Sqrt[a + b*x^2]/Sqrt[d + e*x^2]* Int[ArcCot[c*x/Sqrt[a + b*x^2]]^m/Sqrt[a + b*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCot[c_.*x_/Sqrt[a_. + b_.*x_^2]]^m_./Sqrt[d_. + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && EqQ[b + c^2, 0] && EqQ[b*d - a*e, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "Pi*s/4*Int[u, x] + 1/2*Int[u*ArcTan[v], x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*ArcTan[v_ + s_.*Sqrt[w_]], x_Symbol]",
        "comment": false,
         "givens": "EqQ[s^2, 1] && EqQ[w, v^2 + 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "Pi*s/4*Int[u, x] - 1/2*Int[u*ArcTan[v], x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*ArcCot[v_ + s_.*Sqrt[w_]], x_Symbol]",
        "comment": false,
         "givens": "EqQ[s^2, 1] && EqQ[w, v^2 + 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "With[{tmp = InverseFunctionOfLinear[u, x]}, (-Discriminant[v, x]/(4*Coefficient[v, x, 2]))^n/ Coefficient[tmp[[1]], x, 1]* Subst[ Int[SimplifyIntegrand[ SubstForInverseFunction[u, tmp, x]*Sec[x]^(2*(n + 1)), x], x], x, tmp] /; Not[FalseQ[tmp]] && EqQ[Head[tmp], ArcTan] && EqQ[Discriminant[v, x]*tmp[[1]]^2 + D[v, x]^2, 0]] /; QuadraticQ[v, x] && ILtQ[n, 0] && NegQ[Discriminant[v, x]] && MatchQ[u, r_.*f_^w_",
     "rulenumber": 0,
            "lhs": "Int[u_*v_^n_., x_Symbol] := With[{tmp = InverseFunctionOfLinear[u, x]}, ShowStep[\"\", \"Int[f[x,ArcTan[a+b*x]]/(1+(a+b*x)^2),x]\", \"Subst[Int[f[-a/b+Tan[x]/b,x],x],x,ArcTan[a+b*x]]/b\", Hold[ (-Discriminant[v, x]/(4*Coefficient[v, x, 2]))^n/ Coefficient[tmp[[1]], x, 1]* Subst[ Int[SimplifyIntegrand[ SubstForInverseFunction[u, tmp, x]*Sec[x]^(2*(n + 1)), x], x], x, tmp]]] /; Not[FalseQ[tmp]] && EqQ[Head[tmp], ArcTan] && EqQ[Discriminant[v, x]*tmp[[1]]^2 + D[v, x]^2, 0]] /; SimplifyFlag && QuadraticQ[v, x] && ILtQ[n, 0] && NegQ[Discriminant[v, x]] && MatchQ[u, r_.*f_^w_ /; FreeQ[f, x]], Int[u_*v_^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[f, x]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "With[{tmp = InverseFunctionOfLinear[u, x]}, -(-Discriminant[v, x]/(4*Coefficient[v, x, 2]))^n/ Coefficient[tmp[[1]], x, 1]* Subst[ Int[SimplifyIntegrand[ SubstForInverseFunction[u, tmp, x]*Csc[x]^(2*(n + 1)), x], x], x, tmp] /; Not[FalseQ[tmp]] && EqQ[Head[tmp], ArcCot] && EqQ[Discriminant[v, x]*tmp[[1]]^2 + D[v, x]^2, 0]] /; QuadraticQ[v, x] && ILtQ[n, 0] && NegQ[Discriminant[v, x]] && MatchQ[u, r_.*f_^w_",
     "rulenumber": 0,
            "lhs": "Int[u_*v_^n_., x_Symbol] := With[{tmp = InverseFunctionOfLinear[u, x]}, ShowStep[\"\", \"Int[f[x,ArcCot[a+b*x]]/(1+(a+b*x)^2),x]\", \"-Subst[Int[f[-a/b+Cot[x]/b,x],x],x,ArcCot[a+b*x]]/b\", Hold[ -(-Discriminant[v, x]/(4*Coefficient[v, x, 2]))^n/ Coefficient[tmp[[1]], x, 1]* Subst[ Int[SimplifyIntegrand[ SubstForInverseFunction[u, tmp, x]*Csc[x]^(2*(n + 1)), x], x], x, tmp]]] /; Not[FalseQ[tmp]] && EqQ[Head[tmp], ArcCot] && EqQ[Discriminant[v, x]*tmp[[1]]^2 + D[v, x]^2, 0]] /; SimplifyFlag && QuadraticQ[v, x] && ILtQ[n, 0] && NegQ[Discriminant[v, x]] && MatchQ[u, r_.*f_^w_ /; FreeQ[f, x]], Int[u_*v_^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[f, x]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "x*ArcTan[c + d*Tan[a + b*x]] - I*b*Int[x/(c + I*d + c*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTan[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[(c + I*d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "x*ArcCot[c + d*Tan[a + b*x]] + I*b*Int[x/(c + I*d + c*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCot[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[(c + I*d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "x*ArcTan[c + d*Cot[a + b*x]] - I*b*Int[x/(c - I*d - c*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTan[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[(c - I*d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "x*ArcCot[c + d*Cot[a + b*x]] + I*b*Int[x/(c - I*d - c*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCot[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[(c - I*d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "x*ArcTan[c + d*Tan[a + b*x]] - b*(1 + I*c + d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 + I*c - d + (1 + I*c + d)*E^(2*I*a + 2*I*b*x)), x] + b*(1 - I*c - d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 - I*c + d + (1 - I*c - d)*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTan[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[(c + I*d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "x*ArcCot[c + d*Tan[a + b*x]] + b*(1 + I*c + d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 + I*c - d + (1 + I*c + d)*E^(2*I*a + 2*I*b*x)), x] - b*(1 - I*c - d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 - I*c + d + (1 - I*c - d)*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCot[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[(c + I*d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "x*ArcTan[c + d*Cot[a + b*x]] + b*(1 + I*c - d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 + I*c + d - (1 + I*c - d)*E^(2*I*a + 2*I*b*x)), x] - b*(1 - I*c + d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 - I*c - d - (1 - I*c + d)*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTan[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[(c + I*d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "x*ArcCot[c + d*Cot[a + b*x]] - b*(1 + I*c - d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 + I*c + d - (1 + I*c - d)*E^(2*I*a + 2*I*b*x)), x] + b*(1 - I*c + d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 - I*c - d - (1 - I*c + d)*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCot[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[(c - I*d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcTan[c + d*Tan[a + b*x]]/(f*(m + 1)) - I*b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c + I*d + c*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcTan[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c + I*d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcCot[c + d*Tan[a + b*x]]/(f*(m + 1)) + I*b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c + I*d + c*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcCot[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c + I*d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcTan[c + d*Cot[a + b*x]]/(f*(m + 1)) - I*b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c - I*d - c*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcTan[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - I*d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcCot[c + d*Cot[a + b*x]]/(f*(m + 1)) + I*b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c - I*d - c*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcCot[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - I*d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcTan[c + d*Tan[a + b*x]]/(f*(m + 1)) - b*(1 + I*c + d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 + I*c - d + (1 + I*c + d)*E^(2*I*a + 2*I*b*x)), x] + b*(1 - I*c - d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 - I*c + d + (1 - I*c - d)*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcTan[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c + I*d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcCot[c + d*Tan[a + b*x]]/(f*(m + 1)) + b*(1 + I*c + d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 + I*c - d + (1 + I*c + d)*E^(2*I*a + 2*I*b*x)), x] - b*(1 - I*c - d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 - I*c + d + (1 - I*c - d)*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcCot[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c + I*d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcTan[c + d*Cot[a + b*x]]/(f*(m + 1)) + b*(1 + I*c - d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 + I*c + d - (1 + I*c - d)*E^(2*I*a + 2*I*b*x)), x] - b*(1 - I*c + d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 - I*c - d - (1 - I*c + d)*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcTan[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - I*d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcCot[c + d*Cot[a + b*x]]/(f*(m + 1)) - b*(1 + I*c - d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 + I*c + d - (1 + I*c - d)*E^(2*I*a + 2*I*b*x)), x] + b*(1 - I*c + d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 - I*c - d - (1 - I*c + d)*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcCot[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - I*d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "x*ArcTan[Tanh[a + b*x]] - b*Int[x*Sech[2*a + 2*b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTan[Tanh[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "x*ArcCot[Tanh[a + b*x]] + b*Int[x*Sech[2*a + 2*b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCot[Tanh[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "x*ArcTan[Coth[a + b*x]] + b*Int[x*Sech[2*a + 2*b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTan[Coth[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "x*ArcCot[Coth[a + b*x]] - b*Int[x*Sech[2*a + 2*b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCot[Coth[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcTan[Tanh[a + b*x]]/(f*(m + 1)) - b/(f*(m + 1))*Int[(e + f*x)^(m + 1)*Sech[2*a + 2*b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcTan[Tanh[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcCot[Tanh[a + b*x]]/(f*(m + 1)) + b/(f*(m + 1))*Int[(e + f*x)^(m + 1)*Sech[2*a + 2*b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcCot[Tanh[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcTan[Coth[a + b*x]]/(f*(m + 1)) + b/(f*(m + 1))*Int[(e + f*x)^(m + 1)*Sech[2*a + 2*b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcTan[Coth[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcCot[Coth[a + b*x]]/(f*(m + 1)) - b/(f*(m + 1))*Int[(e + f*x)^(m + 1)*Sech[2*a + 2*b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcCot[Coth[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "x*ArcTan[c + d*Tanh[a + b*x]] - b*Int[x/(c - d + c*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTan[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[(c - d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "x*ArcCot[c + d*Tanh[a + b*x]] + b*Int[x/(c - d + c*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCot[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[(c - d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "x*ArcTan[c + d*Coth[a + b*x]] - b*Int[x/(c - d - c*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTan[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[(c - d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "x*ArcCot[c + d*Coth[a + b*x]] + b*Int[x/(c - d - c*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCot[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[(c - d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "x*ArcTan[c + d*Tanh[a + b*x]] + I*b*(I - c - d)* Int[x*E^(2*a + 2*b*x)/(I - c + d + (I - c - d)*E^(2*a + 2*b*x)), x] - I*b*(I + c + d)* Int[x*E^(2*a + 2*b*x)/(I + c - d + (I + c + d)*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTan[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[(c - d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "x*ArcCot[c + d*Tanh[a + b*x]] - I*b*(I - c - d)* Int[x*E^(2*a + 2*b*x)/(I - c + d + (I - c - d)*E^(2*a + 2*b*x)), x] + I*b*(I + c + d)* Int[x*E^(2*a + 2*b*x)/(I + c - d + (I + c + d)*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCot[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[(c - d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "x*ArcTan[c + d*Coth[a + b*x]] - I*b*(I - c - d)* Int[x*E^(2*a + 2*b*x)/(I - c + d - (I - c - d)*E^(2*a + 2*b*x)), x] + I*b*(I + c + d)* Int[x*E^(2*a + 2*b*x)/(I + c - d - (I + c + d)*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTan[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[(c - d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "x*ArcCot[c + d*Coth[a + b*x]] + I*b*(I - c - d)* Int[x*E^(2*a + 2*b*x)/(I - c + d - (I - c - d)*E^(2*a + 2*b*x)), x] - I*b*(I + c + d)* Int[x*E^(2*a + 2*b*x)/(I + c - d - (I + c + d)*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCot[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[(c - d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcTan[c + d*Tanh[a + b*x]]/(f*(m + 1)) - b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c - d + c*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcTan[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcCot[c + d*Tanh[a + b*x]]/(f*(m + 1)) + b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c - d + c*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcCot[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcTan[c + d*Coth[a + b*x]]/(f*(m + 1)) - b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c - d - c*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcTan[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcCot[c + d*Coth[a + b*x]]/(f*(m + 1)) + b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c - d - c*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcCot[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcTan[c + d*Tanh[a + b*x]]/(f*(m + 1)) + I*b*(I - c - d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(I - c + d + (I - c - d)*E^(2*a + 2*b*x)), x] - I*b*(I + c + d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(I + c - d + (I + c + d)*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcTan[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcCot[c + d*Tanh[a + b*x]]/(f*(m + 1)) - I*b*(I - c - d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(I - c + d + (I - c - d)*E^(2*a + 2*b*x)), x] + I*b*(I + c + d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(I + c - d + (I + c + d)*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcCot[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcTan[c + d*Coth[a + b*x]]/(f*(m + 1)) - I*b*(I - c - d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(I - c + d - (I - c - d)*E^(2*a + 2*b*x)), x] + I*b*(I + c + d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(I + c - d - (I + c + d)*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcTan[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcCot[c + d*Coth[a + b*x]]/(f*(m + 1)) + I*b*(I - c - d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(I - c + d - (I - c - d)*E^(2*a + 2*b*x)), x] - I*b*(I + c + d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(I + c - d - (I + c + d)*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcCot[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - d)^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "x*ArcTan[u] - Int[SimplifyIntegrand[x*D[u, x]/(1 + u^2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTan[u_], x_Symbol]",
        "comment": false,
         "givens": "InverseFunctionFreeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "x*ArcCot[u] + Int[SimplifyIntegrand[x*D[u, x]/(1 + u^2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCot[u_], x_Symbol]",
        "comment": false,
         "givens": "InverseFunctionFreeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "(c + d*x)^(m + 1)*(a + b*ArcTan[u])/(d*(m + 1)) - b/(d*(m + 1))* Int[SimplifyIntegrand[(c + d*x)^(m + 1)*D[u, x]/(1 + u^2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(a_. + b_.*ArcTan[u_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && Not[FunctionOfQ[(c + d*x)^(m + 1), u, x]] && FalseQ[PowerVariableExpn[u, m + 1, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "(c + d*x)^(m + 1)*(a + b*ArcCot[u])/(d*(m + 1)) + b/(d*(m + 1))* Int[SimplifyIntegrand[(c + d*x)^(m + 1)*D[u, x]/(1 + u^2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(a_. + b_.*ArcCot[u_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && Not[FunctionOfQ[(c + d*x)^(m + 1), u, x]] && FalseQ[PowerVariableExpn[u, m + 1, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "With[{w = IntHide[v, x]}, Dist[(a + b*ArcTan[u]), w, x] - b*Int[SimplifyIntegrand[w*D[u, x]/(1 + u^2), x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && Not[MatchQ[v, (c_. + d_.*x)^m_.",
     "rulenumber": 0,
            "lhs": "Int[v_*(a_. + b_.*ArcTan[u_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, m}, x]]] && FalseQ[FunctionOfLinear[v*(a + b*ArcTan[u]), x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "With[{w = IntHide[v, x]}, Dist[(a + b*ArcCot[u]), w, x] + b*Int[SimplifyIntegrand[w*D[u, x]/(1 + u^2), x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && Not[MatchQ[v, (c_. + d_.*x)^m_.",
     "rulenumber": 0,
            "lhs": "Int[v_*(a_. + b_.*ArcCot[u_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, m}, x]]] && FalseQ[FunctionOfLinear[v*(a + b*ArcCot[u]), x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "I/2*Int[Log[1 - I*v]*Log[w]/(a + b*x), x] - I/2*Int[Log[1 + I*v]*Log[w]/(a + b*x), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTan[v_]*Log[w_]/(a_. + b_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && LinearQ[v, x] && LinearQ[w, x] && EqQ[Simplify[D[v/(a + b*x), x]], 0] && EqQ[Simplify[D[w/(a + b*x), x]], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "x*ArcTan[v]*Log[w] - Int[SimplifyIntegrand[x*Log[w]*D[v, x]/(1 + v^2), x], x] - Int[SimplifyIntegrand[x*ArcTan[v]*D[w, x]/w, x], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTan[v_]*Log[w_], x_Symbol]",
        "comment": false,
         "givens": "InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[w, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "x*ArcCot[v]*Log[w] + Int[SimplifyIntegrand[x*Log[w]*D[v, x]/(1 + v^2), x], x] - Int[SimplifyIntegrand[x*ArcCot[v]*D[w, x]/w, x], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCot[v_]*Log[w_], x_Symbol]",
        "comment": false,
         "givens": "InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[w, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "With[{z = IntHide[u, x]}, Dist[ArcTan[v]*Log[w], z, x] - Int[SimplifyIntegrand[z*Log[w]*D[v, x]/(1 + v^2), x], x] - Int[SimplifyIntegrand[z*ArcTan[v]*D[w, x]/w, x], x] /; InverseFunctionFreeQ[z, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*ArcTan[v_]*Log[w_], x_Symbol]",
        "comment": false,
         "givens": "InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[w, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 Miscellaneous inverse tangent.m",
       "filename": "5.3.4 Miscellaneous inverse tangent.m",
            "rhs": "With[{z = IntHide[u, x]}, Dist[ArcCot[v]*Log[w], z, x] + Int[SimplifyIntegrand[z*Log[w]*D[v, x]/(1 + v^2), x], x] - Int[SimplifyIntegrand[z*ArcCot[v]*D[w, x]/w, x], x] /; InverseFunctionFreeQ[z, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*ArcCot[v_]*Log[w_], x_Symbol]",
        "comment": false,
         "givens": "InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[w, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "x*ArcSec[c*x] - 1/c*Int[1/(x*Sqrt[1 - 1/(c^2*x^2)]), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcSec[c_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[c, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "x*ArcCsc[c*x] + 1/c*Int[1/(x*Sqrt[1 - 1/(c^2*x^2)]), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCsc[c_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[c, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "1/c*Subst[Int[(a + b*x)^n*Sec[x]*Tan[x], x], x, ArcSec[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSec[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "-1/c* Subst[Int[(a + b*x)^n*Csc[x]*Cot[x], x], x, ArcCsc[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCsc[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "-Subst[Int[(a + b*ArcCos[x/c])/x, x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSec[c_.*x_])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "-Subst[Int[(a + b*ArcSin[x/c])/x, x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCsc[c_.*x_])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "(d*x)^(m + 1)*(a + b*ArcSec[c*x])/(d*(m + 1)) - b*d/(c*(m + 1))*Int[(d*x)^(m - 1)/Sqrt[1 - 1/(c^2*x^2)], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_. + b_.*ArcSec[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "(d*x)^(m + 1)*(a + b*ArcCsc[c*x])/(d*(m + 1)) + b*d/(c*(m + 1))*Int[(d*x)^(m - 1)/Sqrt[1 - 1/(c^2*x^2)], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_. + b_.*ArcCsc[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "1/c^(m + 1)* Subst[Int[(a + b*x)^n*Sec[x]^(m + 1)*Tan[x], x], x, ArcSec[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*ArcSec[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (GtQ[n, 0] || LtQ[m, -1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "-1/c^(m + 1)* Subst[Int[(a + b*x)^n*Csc[x]^(m + 1)*Cot[x], x], x, ArcCsc[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*ArcCsc[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (GtQ[n, 0] || LtQ[m, -1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "(a + b*ArcSec[c*x])* Log[1 + (e - Sqrt[-c^2*d^2 + e^2])*E^(I*ArcSec[c*x])/(c*d)]/e + (a + b*ArcSec[c*x])* Log[1 + (e + Sqrt[-c^2*d^2 + e^2])*E^(I*ArcSec[c*x])/(c*d)]/e - (a + b*ArcSec[c*x])*Log[1 + E^(2*I*ArcSec[c*x])]/e - b/(c*e)* Int[Log[1 + (e - Sqrt[-c^2*d^2 + e^2])* E^(I*ArcSec[c*x])/(c*d)]/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x] - b/(c*e)* Int[Log[1 + (e + Sqrt[-c^2*d^2 + e^2])* E^(I*ArcSec[c*x])/(c*d)]/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x] + b/(c*e)* Int[Log[1 + E^(2*I*ArcSec[c*x])]/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSec[c_.*x_])/(d_. + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "(a + b*ArcCsc[c*x])* Log[1 - I*(e - Sqrt[-c^2*d^2 + e^2])*E^(I*ArcCsc[c*x])/(c*d)]/ e + (a + b*ArcCsc[c*x])* Log[1 - I*(e + Sqrt[-c^2*d^2 + e^2])*E^(I*ArcCsc[c*x])/(c*d)]/ e - (a + b*ArcCsc[c*x])*Log[1 - E^(2*I*ArcCsc[c*x])]/e + b/(c*e)* Int[Log[1 - I*(e - Sqrt[-c^2*d^2 + e^2])*E^(I*ArcCsc[c*x])/(c*d)]/(x^2* Sqrt[1 - 1/(c^2*x^2)]), x] + b/(c*e)* Int[Log[1 - I*(e + Sqrt[-c^2*d^2 + e^2])*E^(I*ArcCsc[c*x])/(c*d)]/(x^2* Sqrt[1 - 1/(c^2*x^2)]), x] - b/(c*e)* Int[Log[1 - E^(2*I*ArcCsc[c*x])]/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCsc[c_.*x_])/(d_. + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "(d + e*x)^(m + 1)*(a + b*ArcSec[c*x])/(e*(m + 1)) - b/(c*e*(m + 1))* Int[(d + e*x)^(m + 1)/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*(a_. + b_.*ArcSec[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "(d + e*x)^(m + 1)*(a + b*ArcCsc[c*x])/(e*(m + 1)) + b/(c*e*(m + 1))* Int[(d + e*x)^(m + 1)/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*(a_. + b_.*ArcCsc[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[(a + b*ArcSec[c*x]), u, x] - b*c*x/Sqrt[c^2*x^2]* Int[SimplifyIntegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcSec[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[(a + b*ArcCsc[c*x]), u, x] + b*c*x/Sqrt[c^2*x^2]* Int[SimplifyIntegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcCsc[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "-Subst[ Int[(e + d*x^2)^p*(a + b*ArcCos[x/c])^n/x^(2*(p + 1)), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcSec[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "-Subst[ Int[(e + d*x^2)^p*(a + b*ArcSin[x/c])^n/x^(2*(p + 1)), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcCsc[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "-Sqrt[x^2]/x* Subst[Int[(e + d*x^2)^p*(a + b*ArcCos[x/c])^n/x^(2*(p + 1)), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcSec[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p + 1/2] && GtQ[e, 0] && LtQ[d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "-Sqrt[x^2]/x* Subst[Int[(e + d*x^2)^p*(a + b*ArcSin[x/c])^n/x^(2*(p + 1)), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcCsc[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p + 1/2] && GtQ[e, 0] && LtQ[d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "-Sqrt[d + e*x^2]/(x*Sqrt[e + d/x^2])* Subst[Int[(e + d*x^2)^p*(a + b*ArcCos[x/c])^n/x^(2*(p + 1)), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcSec[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p + 1/2] && Not[GtQ[e, 0] && LtQ[d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "-Sqrt[d + e*x^2]/(x*Sqrt[e + d/x^2])* Subst[Int[(e + d*x^2)^p*(a + b*ArcSin[x/c])^n/x^(2*(p + 1)), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcCsc[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p + 1/2] && Not[GtQ[e, 0] && LtQ[d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "(d + e*x^2)^(p + 1)*(a + b*ArcSec[c*x])/(2* e*(p + 1)) - b*c*x/(2*e*(p + 1)*Sqrt[c^2*x^2])* Int[(d + e*x^2)^(p + 1)/(x*Sqrt[c^2*x^2 - 1]), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcSec[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "(d + e*x^2)^(p + 1)*(a + b*ArcCsc[c*x])/(2* e*(p + 1)) + b*c*x/(2*e*(p + 1)*Sqrt[c^2*x^2])* Int[(d + e*x^2)^(p + 1)/(x*Sqrt[c^2*x^2 - 1]), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcCsc[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[(a + b*ArcSec[c*x]), u, x] - b*c*x/Sqrt[c^2*x^2]* Int[SimplifyIntegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcSec[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p}, x] && ( IGtQ[p, 0] && Not[ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0]] || IGtQ[(m + 1)/2, 0] && Not[ILtQ[p, 0] && GtQ[m + 2*p + 3, 0]] || ILtQ[(m + 2*p + 1)/2, 0] && Not[ILtQ[(m - 1)/2, 0]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[(a + b*ArcCsc[c*x]), u, x] + b*c*x/Sqrt[c^2*x^2]* Int[SimplifyIntegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcCsc[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p}, x] && ( IGtQ[p, 0] && Not[ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0]] || IGtQ[(m + 1)/2, 0] && Not[ILtQ[p, 0] && GtQ[m + 2*p + 3, 0]] || ILtQ[(m + 2*p + 1)/2, 0] && Not[ILtQ[(m - 1)/2, 0]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "-Subst[ Int[(e + d*x^2)^p*(a + b*ArcCos[x/c])^n/x^(m + 2*(p + 1)), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcSec[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && IntegerQ[m] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "-Subst[ Int[(e + d*x^2)^p*(a + b*ArcSin[x/c])^n/x^(m + 2*(p + 1)), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcCsc[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && IntegerQ[m] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "-Sqrt[x^2]/x* Subst[Int[(e + d*x^2)^p*(a + b*ArcCos[x/c])^n/x^(m + 2*(p + 1)), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcSec[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p + 1/2] && GtQ[e, 0] && LtQ[d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "-Sqrt[x^2]/x* Subst[Int[(e + d*x^2)^p*(a + b*ArcSin[x/c])^n/x^(m + 2*(p + 1)), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcCsc[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p + 1/2] && GtQ[e, 0] && LtQ[d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "-Sqrt[d + e*x^2]/(x*Sqrt[e + d/x^2])* Subst[Int[(e + d*x^2)^p*(a + b*ArcCos[x/c])^n/x^(m + 2*(p + 1)), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcSec[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p + 1/2] && Not[GtQ[e, 0] && LtQ[d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "-Sqrt[d + e*x^2]/(x*Sqrt[e + d/x^2])* Subst[Int[(e + d*x^2)^p*(a + b*ArcSin[x/c])^n/x^(m + 2*(p + 1)), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcCsc[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p + 1/2] && Not[GtQ[e, 0] && LtQ[d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "With[{v = IntHide[u, x]}, Dist[(a + b*ArcSec[c*x]), v, x] - b/c* Int[SimplifyIntegrand[v/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x] /; InverseFunctionFreeQ[v, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*(a_. + b_.*ArcSec[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "With[{v = IntHide[u, x]}, Dist[(a + b*ArcCsc[c*x]), v, x] + b/c* Int[SimplifyIntegrand[v/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x] /; InverseFunctionFreeQ[v, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*(a_. + b_.*ArcCsc[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "Unintegrable[u*(a + b*ArcSec[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_. + b_.*ArcSec[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m",
       "filename": "5.5.1 u (a+b arcsec(c x))^n.m",
            "rhs": "Unintegrable[u*(a + b*ArcCsc[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_. + b_.*ArcCsc[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.2 Miscellaneous inverse secant.m",
       "filename": "5.5.2 Miscellaneous inverse secant.m",
            "rhs": "(c + d*x)*ArcSec[c + d*x]/d - Int[1/((c + d*x)*Sqrt[1 - 1/(c + d*x)^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcSec[c_ + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.2 Miscellaneous inverse secant.m",
       "filename": "5.5.2 Miscellaneous inverse secant.m",
            "rhs": "(c + d*x)*ArcCsc[c + d*x]/d + Int[1/((c + d*x)*Sqrt[1 - 1/(c + d*x)^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCsc[c_ + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.2 Miscellaneous inverse secant.m",
       "filename": "5.5.2 Miscellaneous inverse secant.m",
            "rhs": "1/d*Subst[Int[(a + b*ArcSec[x])^p, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSec[c_ + d_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.2 Miscellaneous inverse secant.m",
       "filename": "5.5.2 Miscellaneous inverse secant.m",
            "rhs": "1/d*Subst[Int[(a + b*ArcCsc[x])^p, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCsc[c_ + d_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.2 Miscellaneous inverse secant.m",
       "filename": "5.5.2 Miscellaneous inverse secant.m",
            "rhs": "Unintegrable[(a + b*ArcSec[c + d*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSec[c_ + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, p}, x] && Not[IGtQ[p, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.2 Miscellaneous inverse secant.m",
       "filename": "5.5.2 Miscellaneous inverse secant.m",
            "rhs": "Unintegrable[(a + b*ArcCsc[c + d*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCsc[c_ + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, p}, x] && Not[IGtQ[p, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.2 Miscellaneous inverse secant.m",
       "filename": "5.5.2 Miscellaneous inverse secant.m",
            "rhs": "1/d*Subst[Int[(f*x/d)^m*(a + b*ArcSec[x])^p, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcSec[c_ + d_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.2 Miscellaneous inverse secant.m",
       "filename": "5.5.2 Miscellaneous inverse secant.m",
            "rhs": "1/d*Subst[Int[(f*x/d)^m*(a + b*ArcCsc[x])^p, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCsc[c_ + d_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.2 Miscellaneous inverse secant.m",
       "filename": "5.5.2 Miscellaneous inverse secant.m",
            "rhs": "1/d^(m + 1)* Subst[Int[(a + b*x)^p*Sec[x]*Tan[x]*(d*e - c*f + f*Sec[x])^m, x], x, ArcSec[c + d*x]]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcSec[c_ + d_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.2 Miscellaneous inverse secant.m",
       "filename": "5.5.2 Miscellaneous inverse secant.m",
            "rhs": "-1/d^(m + 1)* Subst[Int[(a + b*x)^p*Csc[x]*Cot[x]*(d*e - c*f + f*Csc[x])^m, x], x, ArcCsc[c + d*x]]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCsc[c_ + d_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.2 Miscellaneous inverse secant.m",
       "filename": "5.5.2 Miscellaneous inverse secant.m",
            "rhs": "1/d*Subst[Int[((d*e - c*f)/d + f*x/d)^m*(a + b*ArcSec[x])^p, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcSec[c_ + d_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.2 Miscellaneous inverse secant.m",
       "filename": "5.5.2 Miscellaneous inverse secant.m",
            "rhs": "1/d*Subst[Int[((d*e - c*f)/d + f*x/d)^m*(a + b*ArcCsc[x])^p, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCsc[c_ + d_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.2 Miscellaneous inverse secant.m",
       "filename": "5.5.2 Miscellaneous inverse secant.m",
            "rhs": "Unintegrable[(e + f*x)^m*(a + b*ArcSec[c + d*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcSec[c_ + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p}, x] && Not[IGtQ[p, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.2 Miscellaneous inverse secant.m",
       "filename": "5.5.2 Miscellaneous inverse secant.m",
            "rhs": "Unintegrable[(e + f*x)^m*(a + b*ArcCsc[c + d*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCsc[c_ + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p}, x] && Not[IGtQ[p, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.2 Miscellaneous inverse secant.m",
       "filename": "5.5.2 Miscellaneous inverse secant.m",
            "rhs": "Int[u*ArcCos[a/c + b*x^n/c]^m, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*ArcSec[c_./(a_. + b_.*x_^n_.)]^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, m}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.2 Miscellaneous inverse secant.m",
       "filename": "5.5.2 Miscellaneous inverse secant.m",
            "rhs": "Int[u*ArcSin[a/c + b*x^n/c]^m, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*ArcCsc[c_./(a_. + b_.*x_^n_.)]^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, m}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.2 Miscellaneous inverse secant.m",
       "filename": "5.5.2 Miscellaneous inverse secant.m",
            "rhs": "1/b*Subst[ Int[ReplaceAll[u, x -> -a/b + Sec[x]/b]*f^(c*x^n)*Sec[x]*Tan[x], x], x, ArcSec[a + b*x]]",
     "rulenumber": 0,
            "lhs": "Int[u_.*f_^(c_.*ArcSec[a_. + b_.*x_]^n_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, f}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.2 Miscellaneous inverse secant.m",
       "filename": "5.5.2 Miscellaneous inverse secant.m",
            "rhs": "-1/b* Subst[Int[ ReplaceAll[u, x -> -a/b + Csc[x]/b]*f^(c*x^n)*Csc[x]*Cot[x], x], x, ArcCsc[a + b*x]]",
     "rulenumber": 0,
            "lhs": "Int[u_.*f_^(c_.*ArcCsc[a_. + b_.*x_]^n_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, f}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.2 Miscellaneous inverse secant.m",
       "filename": "5.5.2 Miscellaneous inverse secant.m",
            "rhs": "x*ArcSec[u] - u/Sqrt[u^2]* Int[SimplifyIntegrand[x*D[u, x]/(u*Sqrt[u^2 - 1]), x], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcSec[u_], x_Symbol]",
        "comment": false,
         "givens": "InverseFunctionFreeQ[u, x] && Not[FunctionOfExponentialQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.2 Miscellaneous inverse secant.m",
       "filename": "5.5.2 Miscellaneous inverse secant.m",
            "rhs": "x*ArcCsc[u] + u/Sqrt[u^2]* Int[SimplifyIntegrand[x*D[u, x]/(u*Sqrt[u^2 - 1]), x], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCsc[u_], x_Symbol]",
        "comment": false,
         "givens": "InverseFunctionFreeQ[u, x] && Not[FunctionOfExponentialQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.2 Miscellaneous inverse secant.m",
       "filename": "5.5.2 Miscellaneous inverse secant.m",
            "rhs": "(c + d*x)^(m + 1)*(a + b*ArcSec[u])/(d*(m + 1)) - b*u/(d*(m + 1)*Sqrt[u^2])* Int[SimplifyIntegrand[(c + d*x)^(m + 1)*D[u, x]/(u*Sqrt[u^2 - 1]), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(a_. + b_.*ArcSec[u_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && Not[FunctionOfQ[(c + d*x)^(m + 1), u, x]] && Not[FunctionOfExponentialQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.2 Miscellaneous inverse secant.m",
       "filename": "5.5.2 Miscellaneous inverse secant.m",
            "rhs": "(c + d*x)^(m + 1)*(a + b*ArcCsc[u])/(d*(m + 1)) + b*u/(d*(m + 1)*Sqrt[u^2])* Int[SimplifyIntegrand[(c + d*x)^(m + 1)*D[u, x]/(u*Sqrt[u^2 - 1]), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(a_. + b_.*ArcCsc[u_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && Not[FunctionOfQ[(c + d*x)^(m + 1), u, x]] && Not[FunctionOfExponentialQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.2 Miscellaneous inverse secant.m",
       "filename": "5.5.2 Miscellaneous inverse secant.m",
            "rhs": "With[{w = IntHide[v, x]}, Dist[(a + b*ArcSec[u]), w, x] - b*u/Sqrt[u^2]* Int[SimplifyIntegrand[w*D[u, x]/(u*Sqrt[u^2 - 1]), x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && Not[MatchQ[v, (c_. + d_.*x)^m_.",
     "rulenumber": 0,
            "lhs": "Int[v_*(a_. + b_.*ArcSec[u_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, m}, x]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/5 Inverse trig functions/5.5 Inverse secant/5.5.2 Miscellaneous inverse secant.m",
       "filename": "5.5.2 Miscellaneous inverse secant.m",
            "rhs": "With[{w = IntHide[v, x]}, Dist[(a + b*ArcCsc[u]), w, x] + b*u/Sqrt[u^2]* Int[SimplifyIntegrand[w*D[u, x]/(u*Sqrt[u^2 - 1]), x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && Not[MatchQ[v, (c_. + d_.*x)^m_.",
     "rulenumber": 0,
            "lhs": "Int[v_*(a_. + b_.*ArcCsc[u_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, m}, x]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.10 (c+d x)^m (a+b sinh)^n.m",
       "filename": "6.1.10 (c+d x)^m (a+b sinh)^n.m",
            "rhs": "Int[ExpandToSum[u, x]^m*(a + b*Sinh[ExpandToSum[v, x]])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*(a_. + b_.*Sinh[v_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && LinearQ[{u, v}, x] && Not[LinearMatchQ[{u, v}, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.10 (c+d x)^m (a+b sinh)^n.m",
       "filename": "6.1.10 (c+d x)^m (a+b sinh)^n.m",
            "rhs": "Int[ExpandToSum[u, x]^m*(a + b*Cosh[ExpandToSum[v, x]])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*(a_. + b_.*Cosh[v_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && LinearQ[{u, v}, x] && Not[LinearMatchQ[{u, v}, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.11 (e x)^m (a+b x^n)^p sinh.m",
       "filename": "6.1.11 (e x)^m (a+b x^n)^p sinh.m",
            "rhs": "Int[ExpandIntegrand[Sinh[c + d*x], (a + b*x^n)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_.*Sinh[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.11 (e x)^m (a+b x^n)^p sinh.m",
       "filename": "6.1.11 (e x)^m (a+b x^n)^p sinh.m",
            "rhs": "Int[ExpandIntegrand[Cosh[c + d*x], (a + b*x^n)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_.*Cosh[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.11 (e x)^m (a+b x^n)^p sinh.m",
       "filename": "6.1.11 (e x)^m (a+b x^n)^p sinh.m",
            "rhs": "x^(-n + 1)*(a + b*x^n)^(p + 1)*Sinh[c + d*x]/(b*n*(p + 1)) - (-n + 1)/(b*n*(p + 1))* Int[x^(-n)*(a + b*x^n)^(p + 1)*Sinh[c + d*x], x] - d/(b*n*(p + 1))* Int[x^(-n + 1)*(a + b*x^n)^(p + 1)*Cosh[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*Sinh[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IntegerQ[p] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[n, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.11 (e x)^m (a+b x^n)^p sinh.m",
       "filename": "6.1.11 (e x)^m (a+b x^n)^p sinh.m",
            "rhs": "x^(-n + 1)*(a + b*x^n)^(p + 1)*Cosh[c + d*x]/(b*n*(p + 1)) - (-n + 1)/(b*n*(p + 1))* Int[x^(-n)*(a + b*x^n)^(p + 1)*Cosh[c + d*x], x] - d/(b*n*(p + 1))* Int[x^(-n + 1)*(a + b*x^n)^(p + 1)*Sinh[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*Cosh[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IntegerQ[p] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[n, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.11 (e x)^m (a+b x^n)^p sinh.m",
       "filename": "6.1.11 (e x)^m (a+b x^n)^p sinh.m",
            "rhs": "Int[ExpandIntegrand[Sinh[c + d*x], (a + b*x^n)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*Sinh[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.11 (e x)^m (a+b x^n)^p sinh.m",
       "filename": "6.1.11 (e x)^m (a+b x^n)^p sinh.m",
            "rhs": "Int[ExpandIntegrand[Cosh[c + d*x], (a + b*x^n)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*Cosh[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.11 (e x)^m (a+b x^n)^p sinh.m",
       "filename": "6.1.11 (e x)^m (a+b x^n)^p sinh.m",
            "rhs": "Int[x^(n*p)*(b + a*x^(-n))^p*Sinh[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*Sinh[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.11 (e x)^m (a+b x^n)^p sinh.m",
       "filename": "6.1.11 (e x)^m (a+b x^n)^p sinh.m",
            "rhs": "Int[x^(n*p)*(b + a*x^(-n))^p*Cosh[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*Cosh[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.11 (e x)^m (a+b x^n)^p sinh.m",
       "filename": "6.1.11 (e x)^m (a+b x^n)^p sinh.m",
            "rhs": "Unintegrable[(a + b*x^n)^p*Sinh[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*Sinh[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.11 (e x)^m (a+b x^n)^p sinh.m",
       "filename": "6.1.11 (e x)^m (a+b x^n)^p sinh.m",
            "rhs": "Unintegrable[(a + b*x^n)^p*Cosh[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*x_^n_)^p_*Cosh[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.11 (e x)^m (a+b x^n)^p sinh.m",
       "filename": "6.1.11 (e x)^m (a+b x^n)^p sinh.m",
            "rhs": "Int[ExpandIntegrand[Sinh[c + d*x], (e*x)^m*(a + b*x^n)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_.*Sinh[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.11 (e x)^m (a+b x^n)^p sinh.m",
       "filename": "6.1.11 (e x)^m (a+b x^n)^p sinh.m",
            "rhs": "Int[ExpandIntegrand[Cosh[c + d*x], (e*x)^m*(a + b*x^n)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_.*Cosh[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.11 (e x)^m (a+b x^n)^p sinh.m",
       "filename": "6.1.11 (e x)^m (a+b x^n)^p sinh.m",
            "rhs": "e^m*(a + b*x^n)^(p + 1)*Sinh[c + d*x]/(b*n*(p + 1)) - d*e^m/(b*n*(p + 1))*Int[(a + b*x^n)^(p + 1)*Cosh[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_ + b_.*x_^n_)^p_*Sinh[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && IntegerQ[p] && EqQ[m - n + 1, 0] && LtQ[p, -1] && (IntegerQ[n] || GtQ[e, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.11 (e x)^m (a+b x^n)^p sinh.m",
       "filename": "6.1.11 (e x)^m (a+b x^n)^p sinh.m",
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
       "filename": "6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
            "rhs": "1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Cosh[c + d*x])^p, x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*Cosh[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[ Simplify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
       "filename": "6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
            "rhs": "e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Sinh[c + d*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_*x_)^m_*(a_. + b_.*Sinh[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
       "filename": "6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
            "rhs": "e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Cosh[c + d*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_*x_)^m_*(a_. + b_.*Cosh[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
       "filename": "6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
            "rhs": "e^(n - 1)*(e*x)^(m - n + 1)*Cosh[c + d*x^n]/(d*n) - e^n*(m - n + 1)/(d*n)*Int[(e*x)^(m - n)*Cosh[c + d*x^n], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Sinh[c_. + d_.*x_^n_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[0, n, m + 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
       "filename": "6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
            "rhs": "e^(n - 1)*(e*x)^(m - n + 1)*Sinh[c + d*x^n]/(d*n) - e^n*(m - n + 1)/(d*n)*Int[(e*x)^(m - n)*Sinh[c + d*x^n], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Cosh[c_. + d_.*x_^n_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[0, n, m + 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
       "filename": "6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
            "rhs": "(e*x)^(m + 1)*Sinh[c + d*x^n]/(e*(m + 1)) - d*n/(e^n*(m + 1))*Int[(e*x)^(m + n)*Cosh[c + d*x^n], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_*Sinh[c_. + d_.*x_^n_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
       "filename": "6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
            "rhs": "(e*x)^(m + 1)*Cosh[c + d*x^n]/(e*(m + 1)) - d*n/(e^n*(m + 1))*Int[(e*x)^(m + n)*Sinh[c + d*x^n], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_*Cosh[c_. + d_.*x_^n_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
       "filename": "6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
            "rhs": "1/2*Int[(e*x)^m*E^(c + d*x^n), x] - 1/2*Int[(e*x)^m*E^(-c - d*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Sinh[c_. + d_.*x_^n_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e, m}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
       "filename": "6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
            "rhs": "1/2*Int[(e*x)^m*E^(c + d*x^n), x] + 1/2*Int[(e*x)^m*E^(-c - d*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Cosh[c_. + d_.*x_^n_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e, m}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
       "filename": "6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
            "rhs": "-Sinh[a + b*x^n]^p/((n - 1)*x^(n - 1)) + b*n*p/(n - 1)*Int[Sinh[a + b*x^n]^(p - 1)*Cosh[a + b*x^n], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Sinh[a_. + b_.*x_^n_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IntegersQ[n, p] && EqQ[m + n, 0] && GtQ[p, 1] && NeQ[n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
       "filename": "6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
            "rhs": "-Cosh[a + b*x^n]^p/((n - 1)*x^(n - 1)) + b*n*p/(n - 1)*Int[Cosh[a + b*x^n]^(p - 1)*Sinh[a + b*x^n], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Cosh[a_. + b_.*x_^n_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IntegersQ[n, p] && EqQ[m + n, 0] && GtQ[p, 1] && NeQ[n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
       "filename": "6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
            "rhs": "-n*Sinh[a + b*x^n]^p/(b^2*n^2*p^2) + x^n*Cosh[a + b*x^n]*Sinh[a + b*x^n]^(p - 1)/(b*n*p) - (p - 1)/p*Int[x^m*Sinh[a + b*x^n]^(p - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Sinh[a_. + b_.*x_^n_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && EqQ[m - 2*n + 1] && GtQ[p, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
       "filename": "6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
            "rhs": "-n*Cosh[a + b*x^n]^p/(b^2*n^2*p^2) + x^n*Sinh[a + b*x^n]*Cosh[a + b*x^n]^(p - 1)/(b*n*p) + (p - 1)/p*Int[x^m*Cosh[a + b*x^n]^(p - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Cosh[a_. + b_.*x_^n_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && EqQ[m - 2*n + 1] && GtQ[p, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
       "filename": "6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
            "rhs": "-(m - n + 1)*x^(m - 2*n + 1)* Sinh[a + b*x^n]^p/(b^2*n^2*p^2) + x^(m - n + 1)*Cosh[a + b*x^n]*Sinh[a + b*x^n]^(p - 1)/(b*n*p) - (p - 1)/p*Int[x^m*Sinh[a + b*x^n]^(p - 2), x] + (m - n + 1)*(m - 2*n + 1)/(b^2*n^2*p^2)* Int[x^(m - 2*n)*Sinh[a + b*x^n]^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Sinh[a_. + b_.*x_^n_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IntegersQ[m, n] && GtQ[p, 1] && LtQ[0, 2*n, m + 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
       "filename": "6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
            "rhs": "-(m - n + 1)*x^(m - 2*n + 1)* Cosh[a + b*x^n]^p/(b^2*n^2*p^2) + x^(m - n + 1)*Sinh[a + b*x^n]*Cosh[a + b*x^n]^(p - 1)/(b*n*p) + (p - 1)/p*Int[x^m*Cosh[a + b*x^n]^(p - 2), x] + (m - n + 1)*(m - 2*n + 1)/(b^2*n^2*p^2)* Int[x^(m - 2*n)*Cosh[a + b*x^n]^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Cosh[a_. + b_.*x_^n_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IntegersQ[m, n] && GtQ[p, 1] && LtQ[0, 2*n, m + 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
       "filename": "6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
            "rhs": "x^(m + 1)*Sinh[a + b*x^n]^p/(m + 1) - b*n*p*x^(m + n + 1)*Cosh[a + b*x^n]* Sinh[a + b*x^n]^(p - 1)/((m + 1)*(m + n + 1)) + b^2*n^2*p^2/((m + 1)*(m + n + 1))* Int[x^(m + 2*n)*Sinh[a + b*x^n]^p, x] + b^2*n^2*p*(p - 1)/((m + 1)*(m + n + 1))* Int[x^(m + 2*n)*Sinh[a + b*x^n]^(p - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Sinh[a_. + b_.*x_^n_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IntegersQ[m, n] && GtQ[p, 1] && LtQ[0, 2*n, 1 - m] && NeQ[m + n + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
       "filename": "6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
            "rhs": "x^(m + 1)*Cosh[a + b*x^n]^p/(m + 1) - b*n*p*x^(m + n + 1)*Sinh[a + b*x^n]* Cosh[a + b*x^n]^(p - 1)/((m + 1)*(m + n + 1)) + b^2*n^2*p^2/((m + 1)*(m + n + 1))* Int[x^(m + 2*n)*Cosh[a + b*x^n]^p, x] - b^2*n^2*p*(p - 1)/((m + 1)*(m + n + 1))* Int[x^(m + 2*n)*Cosh[a + b*x^n]^(p - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Cosh[a_. + b_.*x_^n_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IntegersQ[m, n] && GtQ[p, 1] && LtQ[0, 2*n, 1 - m] && NeQ[m + n + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
       "filename": "6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
            "rhs": "With[{k = Denominator[m]}, k/e*Subst[ Int[x^(k*(m + 1) - 1)*(a + b*Sinh[c + d*x^(k*n)/e^n])^p, x], x, (e*x)^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_*(a_. + b_.*Sinh[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IntegerQ[p] && IGtQ[n, 0] && FractionQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
       "filename": "6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
            "rhs": "With[{k = Denominator[m]}, k/e*Subst[ Int[x^(k*(m + 1) - 1)*(a + b*Cosh[c + d*x^(k*n)/e^n])^p, x], x, (e*x)^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_*(a_. + b_.*Cosh[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IntegerQ[p] && IGtQ[n, 0] && FractionQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
       "filename": "6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
            "rhs": "Int[ExpandTrigReduce[(e*x)^m, (a + b*Sinh[c + d*x^n])^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_. + b_.*Sinh[c_. + d_.*x_^n_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 1] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
       "filename": "6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
            "rhs": "Int[ExpandTrigReduce[(e*x)^m, (a + b*Cosh[c + d*x^n])^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*(a_. + b_.*Cosh[c_. + d_.*x_^n_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 1] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
       "filename": "6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
            "rhs": "x^n*Cosh[a + b*x^n]*Sinh[a + b*x^n]^(p + 1)/(b*n*(p + 1)) - n*Sinh[a + b*x^n]^(p + 2)/(b^2*n^2*(p + 1)*(p + 2)) - (p + 2)/(p + 1)*Int[x^m*Sinh[a + b*x^n]^(p + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Sinh[a_. + b_.*x_^n_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && EqQ[m - 2*n + 1, 0] && LtQ[p, -1] && NeQ[p, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
       "filename": "6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
            "rhs": "-x^n*Sinh[a + b*x^n]* Cosh[a + b*x^n]^(p + 1)/(b*n*(p + 1)) + n*Cosh[a + b*x^n]^(p + 2)/(b^2*n^2*(p + 1)*(p + 2)) + (p + 2)/(p + 1)*Int[x^m*Cosh[a + b*x^n]^(p + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Cosh[a_. + b_.*x_^n_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && EqQ[m - 2*n + 1, 0] && LtQ[p, -1] && NeQ[p, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
       "filename": "6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
            "rhs": "x^(m - n + 1)*Cosh[a + b*x^n]* Sinh[a + b*x^n]^(p + 1)/(b*n*(p + 1)) - (m - n + 1)*x^(m - 2*n + 1)* Sinh[a + b*x^n]^(p + 2)/(b^2*n^2*(p + 1)*(p + 2)) - (p + 2)/(p + 1)*Int[x^m*Sinh[a + b*x^n]^(p + 2), x] + (m - n + 1)*(m - 2*n + 1)/(b^2*n^2*(p + 1)*(p + 2))* Int[x^(m - 2*n)*Sinh[a + b*x^n]^(p + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Sinh[a_. + b_.*x_^n_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IntegersQ[m, n] && LtQ[p, -1] && NeQ[p, -2] && LtQ[0, 2*n, m + 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
       "filename": "6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
            "rhs": "-x^(m - n + 1)*Sinh[a + b*x^n]* Cosh[a + b*x^n]^(p + 1)/(b*n*(p + 1)) + (m - n + 1)*x^(m - 2*n + 1)* Cosh[a + b*x^n]^(p + 2)/(b^2*n^2*(p + 1)*(p + 2)) + (p + 2)/(p + 1)*Int[x^m*Cosh[a + b*x^n]^(p + 2), x] - (m - n + 1)*(m - 2*n + 1)/(b^2*n^2*(p + 1)*(p + 2))* Int[x^(m - 2*n)*Cosh[a + b*x^n]^(p + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Cosh[a_. + b_.*x_^n_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IntegersQ[m, n] && LtQ[p, -1] && NeQ[p, -2] && LtQ[0, 2*n, m + 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
       "filename": "6.1.12 (e x)^m (a+b sinh(c+d x^n))^p.m",
            "rhs": "-Subst[ Int[(a + b*Sinh[c + d*x^(-n)])^p/x^(m + 2), x], x, 1/x]",
     "rulenumber": 0,
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.5 Hyperbolic secant/6.5.11 (e x)^m (a+b sech(c+d x^n))^p.m",
       "filename": "6.5.11 (e x)^m (a+b sech(c+d x^n))^p.m",
            "rhs": "1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sech[c + d*x])^p, x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*Sech[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 1)/n], 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.5 Hyperbolic secant/6.5.11 (e x)^m (a+b sech(c+d x^n))^p.m",
       "filename": "6.5.11 (e x)^m (a+b sech(c+d x^n))^p.m",
            "rhs": "1/n*Subst[ Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Csch[c + d*x])^p, x], x, x^n]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*Csch[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 1)/n], 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.5 Hyperbolic secant/6.5.11 (e x)^m (a+b sech(c+d x^n))^p.m",
       "filename": "6.5.11 (e x)^m (a+b sech(c+d x^n))^p.m",
            "rhs": "Unintegrable[x^m*(a + b*Sech[c + d*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*Sech[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.5 Hyperbolic secant/6.5.11 (e x)^m (a+b sech(c+d x^n))^p.m",
       "filename": "6.5.11 (e x)^m (a+b sech(c+d x^n))^p.m",
            "rhs": "Unintegrable[x^m*(a + b*Csch[c + d*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*Csch[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.5 Hyperbolic secant/6.5.11 (e x)^m (a+b sech(c+d x^n))^p.m",
       "filename": "6.5.11 (e x)^m (a+b sech(c+d x^n))^p.m",
            "rhs": "e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Sech[c + d*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_*x_)^m_.*(a_. + b_.*Sech[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.5 Hyperbolic secant/6.5.11 (e x)^m (a+b sech(c+d x^n))^p.m",
       "filename": "6.5.11 (e x)^m (a+b sech(c+d x^n))^p.m",
            "rhs": "e^IntPart[m]*(e*x)^FracPart[m]/x^FracPart[m]* Int[x^m*(a + b*Csch[c + d*x^n])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_*x_)^m_.*(a_. + b_.*Csch[c_. + d_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.5 Hyperbolic secant/6.5.11 (e x)^m (a+b sech(c+d x^n))^p.m",
       "filename": "6.5.11 (e x)^m (a+b sech(c+d x^n))^p.m",
            "rhs": "Int[(e*x)^m*(a + b*Sech[ExpandToSum[u, x]])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_*x_)^m_.*(a_. + b_.*Sech[u_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.5 Hyperbolic secant/6.5.11 (e x)^m (a+b sech(c+d x^n))^p.m",
       "filename": "6.5.11 (e x)^m (a+b sech(c+d x^n))^p.m",
            "rhs": "Int[(e*x)^m*(a + b*Csch[ExpandToSum[u, x]])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_*x_)^m_.*(a_. + b_.*Csch[u_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, m, p}, x] && BinomialQ[u, x] && Not[BinomialMatchQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.5 Hyperbolic secant/6.5.11 (e x)^m (a+b sech(c+d x^n))^p.m",
       "filename": "6.5.11 (e x)^m (a+b sech(c+d x^n))^p.m",
            "rhs": "-x^(m - n + 1)* Sech[a + b*x^n]^(p - 1)/(b*n*(p - 1)) + (m - n + 1)/(b*n*(p - 1))* Int[x^(m - n)*Sech[a + b*x^n]^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Sech[a_. + b_.*x_^n_.]^p_*Sinh[a_. + b_.*x_^n_.], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m - n, 0] && NeQ[p, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.5 Hyperbolic secant/6.5.11 (e x)^m (a+b sech(c+d x^n))^p.m",
       "filename": "6.5.11 (e x)^m (a+b sech(c+d x^n))^p.m",
            "rhs": "-x^(m - n + 1)* Csch[a + b*x^n]^(p - 1)/(b*n*(p - 1)) + (m - n + 1)/(b*n*(p - 1))* Int[x^(m - n)*Csch[a + b*x^n]^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Csch[a_. + b_.*x_^n_.]^p_*Cosh[a_. + b_.*x_^n_.], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m - n, 0] && NeQ[p, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
       "filename": "6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
            "rhs": "(c + d*x)^m*Sinh[a + b*x]^(n + 1)/(b*(n + 1)) - d*m/(b*(n + 1))*Int[(c + d*x)^(m - 1)*Sinh[a + b*x]^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Sinh[a_. + b_.*x_]^n_.*Cosh[a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
       "filename": "6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
            "rhs": "(c + d*x)^m*Cosh[a + b*x]^(n + 1)/(b*(n + 1)) - d*m/(b*(n + 1))*Int[(c + d*x)^(m - 1)*Cosh[a + b*x]^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Sinh[a_. + b_.*x_]*Cosh[a_. + b_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
       "filename": "6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
            "rhs": "Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Sinh[a_. + b_.*x_]^n_.*Cosh[a_. + b_.*x_]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
       "filename": "6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
            "rhs": "Int[(c + d*x)^m*Sinh[a + b*x]^n*Tanh[a + b*x]^(p - 2), x] - Int[(c + d*x)^m*Sinh[a + b*x]^(n - 2)*Tanh[a + b*x]^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Sinh[a_. + b_.*x_]^n_.*Tanh[a_. + b_.*x_]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
       "filename": "6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
            "rhs": "Int[(c + d*x)^m*Cosh[a + b*x]^n*Coth[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Cosh[a + b*x]^(n - 2)*Coth[a + b*x]^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Cosh[a_. + b_.*x_]^n_.*Coth[a_. + b_.*x_]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
       "filename": "6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
            "rhs": "-(c + d*x)^m*Sech[a + b*x]^n/(b*n) + d*m/(b*n)*Int[(c + d*x)^(m - 1)*Sech[a + b*x]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Sech[a_. + b_.*x_]^n_.*Tanh[a_. + b_.*x_]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
       "filename": "6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
            "rhs": "-(c + d*x)^m*Csch[a + b*x]^n/(b*n) + d*m/(b*n)*Int[(c + d*x)^(m - 1)*Csch[a + b*x]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Csch[a_. + b_.*x_]^n_.*Coth[a_. + b_.*x_]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
       "filename": "6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
            "rhs": "(c + d*x)^m*Tanh[a + b*x]^(n + 1)/(b*(n + 1)) - d*m/(b*(n + 1))*Int[(c + d*x)^(m - 1)*Tanh[a + b*x]^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Sech[a_. + b_.*x_]^2*Tanh[a_. + b_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
       "filename": "6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
            "rhs": "-(c + d*x)^m*Coth[a + b*x]^(n + 1)/(b*(n + 1)) + d*m/(b*(n + 1))*Int[(c + d*x)^(m - 1)*Coth[a + b*x]^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Csch[a_. + b_.*x_]^2*Coth[a_. + b_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
       "filename": "6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
            "rhs": "Int[(c + d*x)^m*Sech[a + b*x]*Tanh[a + b*x]^(p - 2), x] - Int[(c + d*x)^m*Sech[a + b*x]^3*Tanh[a + b*x]^(p - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Sech[a_. + b_.*x_]*Tanh[a_. + b_.*x_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && IGtQ[p/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
       "filename": "6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
            "rhs": "Int[(c + d*x)^m*Sech[a + b*x]^n*Tanh[a + b*x]^(p - 2), x] - Int[(c + d*x)^m*Sech[a + b*x]^(n + 2)*Tanh[a + b*x]^(p - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Sech[a_. + b_.*x_]^n_.*Tanh[a_. + b_.*x_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
       "filename": "6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
            "rhs": "Int[(c + d*x)^m*Csch[a + b*x]*Coth[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Csch[a + b*x]^3*Coth[a + b*x]^(p - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Csch[a_. + b_.*x_]*Coth[a_. + b_.*x_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && IGtQ[p/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
       "filename": "6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
            "rhs": "Int[(c + d*x)^m*Csch[a + b*x]^n*Coth[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Csch[a + b*x]^(n + 2)*Coth[a + b*x]^(p - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Csch[a_. + b_.*x_]^n_.*Coth[a_. + b_.*x_]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
       "filename": "6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
            "rhs": "With[{u = IntHide[Sech[a + b*x]^n*Tanh[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - d*m*Int[(c + d*x)^(m - 1)*u, x]]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Sech[a_. + b_.*x_]^n_.*Tanh[a_. + b_.*x_]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x] && IGtQ[m, 0] && (IntegerQ[n/2] || IntegerQ[(p - 1)/2])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
       "filename": "6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
            "rhs": "With[{u = IntHide[Csch[a + b*x]^n*Coth[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - d*m*Int[(c + d*x)^(m - 1)*u, x]]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Csch[a_. + b_.*x_]^n_.*Coth[a_. + b_.*x_]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x] && IGtQ[m, 0] && (IntegerQ[n/2] || IntegerQ[(p - 1)/2])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
       "filename": "6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
            "rhs": "2^n*Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Csch[a_. + b_.*x_]^n_.*Sech[a_. + b_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
       "filename": "6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
            "rhs": "With[{u = IntHide[Csch[a + b*x]^n*Sech[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - d*m*Int[(c + d*x)^(m - 1)*u, x]]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Csch[a_. + b_.*x_]^n_.*Sech[a_. + b_.*x_]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n, p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
       "filename": "6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
            "rhs": "Int[ExpandToSum[u, x]^m*F[ExpandToSum[v, x]]^n* G[ExpandToSum[v, x]]^p, x]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*F_[v_]^n_.*G_[w_]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{m, n, p}, x] && HyperbolicQ[F] && HyperbolicQ[G] && EqQ[v, w] && LinearQ[{u, v, w}, x] && Not[LinearMatchQ[{u, v, w}, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
       "filename": "6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
            "rhs": "(e + f*x)^ m*(a + b*Sinh[c + d*x])^(n + 1)/(b*d*(n + 1)) - f*m/(b*d*(n + 1))* Int[(e + f*x)^(m - 1)*(a + b*Sinh[c + d*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Cosh[c_. + d_.*x_]*(a_ + b_.*Sinh[c_. + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
       "filename": "6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
            "rhs": "(e + f*x)^ m*(a + b*Cosh[c + d*x])^(n + 1)/(b*d*(n + 1)) - f*m/(b*d*(n + 1))* Int[(e + f*x)^(m - 1)*(a + b*Cosh[c + d*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Sinh[c_. + d_.*x_]*(a_ + b_.*Cosh[c_. + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
       "filename": "6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
            "rhs": "(e + f*x)^ m*(a + b*Tanh[c + d*x])^(n + 1)/(b*d*(n + 1)) - f*m/(b*d*(n + 1))* Int[(e + f*x)^(m - 1)*(a + b*Tanh[c + d*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Sech[c_. + d_.*x_]^2*(a_ + b_.*Tanh[c_. + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
       "filename": "6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
            "rhs": "-(e + f*x)^ m*(a + b*Coth[c + d*x])^(n + 1)/(b*d*(n + 1)) + f*m/(b*d*(n + 1))* Int[(e + f*x)^(m - 1)*(a + b*Coth[c + d*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Csch[c_. + d_.*x_]^2*(a_ + b_.*Coth[c_. + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
       "filename": "6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
            "rhs": "-(e + f*x)^ m*(a + b*Sech[c + d*x])^(n + 1)/(b*d*(n + 1)) + f*m/(b*d*(n + 1))* Int[(e + f*x)^(m - 1)*(a + b*Sech[c + d*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Sech[c_. + d_.*x_]* Tanh[c_. + d_.*x_]*(a_ + b_.*Sech[c_. + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
       "filename": "6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
            "rhs": "-(e + f*x)^ m*(a + b*Csch[c + d*x])^(n + 1)/(b*d*(n + 1)) + f*m/(b*d*(n + 1))* Int[(e + f*x)^(m - 1)*(a + b*Csch[c + d*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Csch[c_. + d_.*x_]* Coth[c_. + d_.*x_]*(a_ + b_.*Csch[c_. + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
       "filename": "6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
            "rhs": "Int[ExpandTrigReduce[(e + f*x)^m, Sinh[a + b*x]^p*Sinh[c + d*x]^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Sinh[a_. + b_.*x_]^p_.*Sinh[c_. + d_.*x_]^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IGtQ[q, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
       "filename": "6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
            "rhs": "Int[ExpandTrigReduce[(e + f*x)^m, Cosh[a + b*x]^p*Cosh[c + d*x]^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Cosh[a_. + b_.*x_]^p_.*Cosh[c_. + d_.*x_]^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IGtQ[q, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
       "filename": "6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
            "rhs": "Int[ExpandTrigReduce[(e + f*x)^m, Sinh[a + b*x]^p*Cosh[c + d*x]^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Sinh[a_. + b_.*x_]^p_.*Cosh[c_. + d_.*x_]^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] && IGtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
       "filename": "6.7.6 (c+d x)^m hyper(a+b x)^n hyper(a+b x)^p.m",
            "rhs": "Int[ExpandTrigExpand[(e + f*x)^m*G[c + d*x]^q, F, c + d*x, p, b/d, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*F_[a_. + b_.*x_]^p_.*G_[c_. + d_.*x_]^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && MemberQ[{Sinh, Cosh}, F] && MemberQ[{Sech, Csch}, G] && IGtQ[p, 0] && IGtQ[q, 0] && EqQ[b*c - a*d, 0] && IGtQ[b/d, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "-b*c*Log[F]*F^(c*(a + b*x))* Sinh[d + e*x]/(e^2 - b^2*c^2*Log[F]^2) + e*F^(c*(a + b*x))*Cosh[d + e*x]/(e^2 - b^2*c^2*Log[F]^2)",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Sinh[d_. + e_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 - b^2*c^2*Log[F]^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "-b*c*Log[F]*F^(c*(a + b*x))* Cosh[d + e*x]/(e^2 - b^2*c^2*Log[F]^2) + e*F^(c*(a + b*x))*Sinh[d + e*x]/(e^2 - b^2*c^2*Log[F]^2)",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Cosh[d_. + e_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 - b^2*c^2*Log[F]^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "-b*c*Log[F]*F^(c*(a + b*x))* Sinh[d + e*x]^n/(e^2*n^2 - b^2*c^2*Log[F]^2) + e*n*F^(c*(a + b*x))*Cosh[d + e*x]* Sinh[d + e*x]^(n - 1)/(e^2*n^2 - b^2*c^2*Log[F]^2) - n*(n - 1)*e^2/(e^2*n^2 - b^2*c^2*Log[F]^2)* Int[F^(c*(a + b*x))*Sinh[d + e*x]^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Sinh[d_. + e_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*n^2 - b^2*c^2*Log[F]^2, 0] && GtQ[n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "-b*c*Log[F]*F^(c*(a + b*x))* Cosh[d + e*x]^n/(e^2*n^2 - b^2*c^2*Log[F]^2) + e*n*F^(c*(a + b*x))*Sinh[d + e*x]* Cosh[d + e*x]^(n - 1)/(e^2*n^2 - b^2*c^2*Log[F]^2) + n*(n - 1)*e^2/(e^2*n^2 - b^2*c^2*Log[F]^2)* Int[F^(c*(a + b*x))*Cosh[d + e*x]^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Cosh[d_. + e_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*n^2 - b^2*c^2*Log[F]^2, 0] && GtQ[n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "-b*c*Log[F]*F^(c*(a + b*x))* Sinh[d + e*x]^(n + 2)/(e^2*(n + 1)*(n + 2)) + F^(c*(a + b*x))*Cosh[d + e*x]*Sinh[d + e*x]^(n + 1)/(e*(n + 1))",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Sinh[d_. + e_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, n}, x] && EqQ[e^2*(n + 2)^2 - b^2*c^2*Log[F]^2, 0] && NeQ[n, -1] && NeQ[n, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "b*c*Log[F]*F^(c*(a + b*x))* Cosh[d + e*x]^(n + 2)/(e^2*(n + 1)*(n + 2)) - F^(c*(a + b*x))*Sinh[d + e*x]*Cosh[d + e*x]^(n + 1)/(e*(n + 1))",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Cosh[d_. + e_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, n}, x] && EqQ[e^2*(n + 2)^2 - b^2*c^2*Log[F]^2, 0] && NeQ[n, -1] && NeQ[n, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "-b*c*Log[F]*F^(c*(a + b*x))* Sinh[d + e*x]^(n + 2)/(e^2*(n + 1)*(n + 2)) + F^(c*(a + b*x))*Cosh[d + e*x]*Sinh[d + e*x]^(n + 1)/(e*(n + 1)) - (e^2*(n + 2)^2 - b^2*c^2*Log[F]^2)/(e^2*(n + 1)*(n + 2))* Int[F^(c*(a + b*x))*Sinh[d + e*x]^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Sinh[d_. + e_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*(n + 2)^2 - b^2*c^2*Log[F]^2, 0] && LtQ[n, -1] && NeQ[n, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "b*c*Log[F]*F^(c*(a + b*x))* Cosh[d + e*x]^(n + 2)/(e^2*(n + 1)*(n + 2)) - F^(c*(a + b*x))*Sinh[d + e*x]*Cosh[d + e*x]^(n + 1)/(e*(n + 1)) + (e^2*(n + 2)^2 - b^2*c^2*Log[F]^2)/(e^2*(n + 1)*(n + 2))* Int[F^(c*(a + b*x))*Cosh[d + e*x]^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Cosh[d_. + e_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*(n + 2)^2 - b^2*c^2*Log[F]^2, 0] && LtQ[n, -1] && NeQ[n, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "E^(n*(d + e*x))*Sinh[d + e*x]^n/(-1 + E^(2*(d + e*x)))^n* Int[F^(c*(a + b*x))*(-1 + E^(2*(d + e*x)))^n/E^(n*(d + e*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Sinh[d_. + e_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, n}, x] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "E^(n*(d + e*x))*Cosh[d + e*x]^n/(1 + E^(2*(d + e*x)))^n* Int[F^(c*(a + b*x))*(1 + E^(2*(d + e*x)))^n/E^(n*(d + e*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Cosh[d_. + e_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, n}, x] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "Int[ExpandIntegrand[ F^(c*(a + b*x))*(-1 + E^(2*(d + e*x)))^n/(1 + E^(2*(d + e*x)))^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Tanh[d_. + e_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "Int[ExpandIntegrand[ F^(c*(a + b*x))*(1 + E^(2*(d + e*x)))^n/(-1 + E^(2*(d + e*x)))^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Coth[d_. + e_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "-b*c*Log[F]* F^(c*(a + b*x))*(Sech[d + e*x]^n/(e^2*n^2 - b^2*c^2*Log[F]^2)) - e*n*F^(c*(a + b*x))* Sech[d + e*x]^(n + 1)*(Sinh[ d + e*x]/(e^2*n^2 - b^2*c^2*Log[F]^2)) + e^2*n*((n + 1)/(e^2*n^2 - b^2*c^2*Log[F]^2))* Int[F^(c*(a + b*x))*Sech[d + e*x]^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Sech[d_. + e_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*n^2 + b^2*c^2*Log[F]^2, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "-b*c*Log[F]* F^(c*(a + b*x))*(Csch[d + e*x]^n/(e^2*n^2 - b^2*c^2*Log[F]^2)) - e*n*F^(c*(a + b*x))* Csch[d + e*x]^(n + 1)*(Cosh[ d + e*x]/(e^2*n^2 - b^2*c^2*Log[F]^2)) - e^2*n*((n + 1)/(e^2*n^2 - b^2*c^2*Log[F]^2))* Int[F^(c*(a + b*x))*Csch[d + e*x]^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Csch[d_. + e_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*n^2 + b^2*c^2*Log[F]^2, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "b*c*Log[F]*F^(c*(a + b*x))* Sech[d + e*x]^(n - 2)/(e^2*(n - 1)*(n - 2)) + F^(c*(a + b*x))*Sech[d + e*x]^(n - 1)*Sinh[d + e*x]/(e*(n - 1))",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Sech[d_. + e_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, n}, x] && EqQ[e^2*(n - 2)^2 - b^2*c^2*Log[F]^2, 0] && NeQ[n, 1] && NeQ[n, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "-b*c*Log[F]*F^(c*(a + b*x))* Csch[d + e*x]^(n - 2)/(e^2*(n - 1)*(n - 2)) - F^(c*(a + b*x))*Csch[d + e*x]^(n - 1)*Cosh[d + e*x]/(e*(n - 1))",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Csch[d_. + e_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, n}, x] && EqQ[e^2*(n - 2)^2 - b^2*c^2*Log[F]^2, 0] && NeQ[n, 1] && NeQ[n, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "b*c*Log[F]*F^(c*(a + b*x))* Sech[d + e*x]^(n - 2)/(e^2*(n - 1)*(n - 2)) + F^(c*(a + b*x))*Sech[d + e*x]^(n - 1)* Sinh[d + e*x]/(e*(n - 1)) + (e^2*(n - 2)^2 - b^2*c^2*Log[F]^2)/(e^2*(n - 1)*(n - 2))* Int[F^(c*(a + b*x))*Sech[d + e*x]^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Sech[d_. + e_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*(n - 2)^2 - b^2*c^2*Log[F]^2, 0] && GtQ[n, 1] && NeQ[n, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "-b*c*Log[F]*F^(c*(a + b*x))* Csch[d + e*x]^(n - 2)/(e^2*(n - 1)*(n - 2)) - F^(c*(a + b*x))*Csch[d + e*x]^(n - 1)* Cosh[d + e*x]/(e*(n - 1)) - (e^2*(n - 2)^2 - b^2*c^2*Log[F]^2)/(e^2*(n - 1)*(n - 2))* Int[F^(c*(a + b*x))*Csch[d + e*x]^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Csch[d_. + e_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*(n - 2)^2 - b^2*c^2*Log[F]^2, 0] && GtQ[n, 1] && NeQ[n, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": " 2^n*Int[SimplifyIntegrand[F^(c*(a+b*x))*E^(n*(d+e*x))/(1+E^(2*(d+e*x)) )^n,x],x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_.+b_.*x_))*Sech[d_.+e_.*x_]^n_.,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F,a,b,c,d,e},x] && IntegerQ[n] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": " 2^n*Int[SimplifyIntegrand[F^(c*(a+b*x))*E^(-n*(d+e*x))/(1-E^(-2*(d+e* x)))^n,x],x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_.+b_.*x_))*Csch[d_.+e_.*x_]^n_.,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F,a,b,c,d,e},x] && IntegerQ[n] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "2^n*E^(n*(d + e*x))*F^(c*(a + b*x))/(e*n + b*c*Log[F])* Hypergeometric2F1[n, n/2 + b*c*Log[F]/(2*e), 1 + n/2 + b*c*Log[F]/(2*e), -E^(2*(d + e*x))]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Sech[d_. + e_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "(-2)^n*E^(n*(d + e*x))* F^(c*(a + b*x))/(e*n + b*c*Log[F])* Hypergeometric2F1[n, n/2 + b*c*Log[F]/(2*e), 1 + n/2 + b*c*Log[F]/(2*e), E^(2*(d + e*x))]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Csch[d_. + e_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "(1 + E^(2*(d + e*x)))^n* Sech[d + e*x]^n/E^(n*(d + e*x))* Int[SimplifyIntegrand[ F^(c*(a + b*x))*E^(n*(d + e*x))/(1 + E^(2*(d + e*x)))^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Sech[d_. + e_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "(1 - E^(-2*(d + e*x)))^n* Csch[d + e*x]^n/E^(-n*(d + e*x))* Int[SimplifyIntegrand[ F^(c*(a + b*x))*E^(-n*(d + e*x))/(1 - E^(-2*(d + e*x)))^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Csch[d_. + e_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "2^n*f^n*Int[F^(c*(a + b*x))*Cosh[d/2 + e*x/2 - f*Pi/(4*g)]^(2*n), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*(f_ + g_.*Sinh[d_. + e_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f^2 + g^2, 0] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "2^n*g^n*Int[F^(c*(a + b*x))*Cosh[d/2 + e*x/2]^(2*n), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*(f_ + g_.*Cosh[d_. + e_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f - g, 0] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "2^n*g^n*Int[F^(c*(a + b*x))*Sinh[d/2 + e*x/2]^(2*n), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*(f_ + g_.*Cosh[d_. + e_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f + g, 0] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "g^n*Int[F^(c*(a + b*x))*Tanh[d/2 + e*x/2 - f*Pi/(4*g)]^m, x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))* Cosh[d_. + e_.*x_]^m_.*(f_ + g_.*Sinh[d_. + e_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f^2 + g^2, 0] && IntegersQ[m, n] && EqQ[m + n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "g^n*Int[F^(c*(a + b*x))*Tanh[d/2 + e*x/2]^m, x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))* Sinh[d_. + e_.*x_]^m_.*(f_ + g_.*Cosh[d_. + e_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f - g, 0] && IntegersQ[m, n] && EqQ[m + n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "g^n*Int[F^(c*(a + b*x))*Coth[d/2 + e*x/2]^m, x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))* Sinh[d_. + e_.*x_]^m_.*(f_ + g_.*Cosh[d_. + e_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g}, x] && EqQ[f + g, 0] && IntegersQ[m, n] && EqQ[m + n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "2*i*Int[F^(c*(a + b*x))*(Cosh[d + e*x]/(f + g*Sinh[d + e*x])), x] + Int[F^(c*(a + b*x))*((h - i*Cosh[d + e*x])/(f + g*Sinh[d + e*x])), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*(h_ + i_.*Cosh[d_. + e_.*x_])/(f_ + g_.*Sinh[d_. + e_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g, h, i}, x] && EqQ[f^2 + g^2, 0] && EqQ[h^2 - i^2, 0] && EqQ[g*h - f*i, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "2*i*Int[F^(c*(a + b*x))*(Sinh[d + e*x]/(f + g*Cosh[d + e*x])), x] + Int[F^(c*(a + b*x))*((h - i*Sinh[d + e*x])/(f + g*Cosh[d + e*x])), x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*(h_ + i_.*Sinh[d_. + e_.*x_])/(f_ + g_.*Cosh[d_. + e_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g, h, i}, x] && EqQ[f^2 - g^2, 0] && EqQ[h^2 + i^2, 0] && EqQ[g*h + f*i, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "Int[F^(c*ExpandToSum[u, x])*G[ExpandToSum[v, x]]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*u_)*G_[v_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, c, n}, x] && HyperbolicQ[G] && LinearQ[{u, v}, x] && Not[LinearMatchQ[{u, v}, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "Module[{u = IntHide[F^(c*(a + b*x))*Sinh[d + e*x]^n, x]}, Dist[(f*x)^m, u, x] - f*m*Int[(f*x)^(m - 1)*u, x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*F_^(c_.*(a_. + b_.*x_))*Sinh[d_. + e_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f}, x] && IGtQ[n, 0] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "Module[{u = IntHide[F^(c*(a + b*x))*Cosh[d + e*x]^n, x]}, Dist[(f*x)^m, u, x] - f*m*Int[(f*x)^(m - 1)*u, x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*F_^(c_.*(a_. + b_.*x_))*Cosh[d_. + e_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f}, x] && IGtQ[n, 0] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "(f*x)^(m + 1)/(f*(m + 1))*F^(c*(a + b*x))* Sinh[d + e*x] - e/(f*(m + 1))* Int[(f*x)^(m + 1)*F^(c*(a + b*x))*Cosh[d + e*x], x] - b*c*Log[F]/(f*(m + 1))* Int[(f*x)^(m + 1)*F^(c*(a + b*x))*Sinh[d + e*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*F_^(c_.*(a_. + b_.*x_))*Sinh[d_. + e_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, m}, x] && (LtQ[m, -1] || SumSimplerQ[m, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "(f*x)^(m + 1)/(f*(m + 1))*F^(c*(a + b*x))* Cosh[d + e*x] - e/(f*(m + 1))* Int[(f*x)^(m + 1)*F^(c*(a + b*x))*Sinh[d + e*x], x] - b*c*Log[F]/(f*(m + 1))* Int[(f*x)^(m + 1)*F^(c*(a + b*x))*Cosh[d + e*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*F_^(c_.*(a_. + b_.*x_))*Cosh[d_. + e_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, m}, x] && (LtQ[m, -1] || SumSimplerQ[m, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": " (-1)^n/2^n*Int[ExpandIntegrand[(f*x)^m*F^(c*(a+b*x)),(E^(-(d+e*x))-E^( d+e*x))^n,x],x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*F_^(c_.*(a_.+b_.*x_))*Sinh[d_.+e_.*x_]^n_.,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{F,a,b,c,d,e,f},x] && IGtQ[n,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": " 1/2^n*Int[ExpandIntegrand[(f*x)^m*F^(c*(a+b*x)),(E^(-(d+e*x))+E^(d+e* x))^n,x],x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*F_^(c_.*(a_.+b_.*x_))*Cosh[d_.+e_.*x_]^n_.,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{F,a,b,c,d,e,f},x] && IGtQ[n,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "Int[ExpandTrigReduce[F^(c*(a + b*x)), Sinh[d + e*x]^m*Cosh[f + g*x]^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*Sinh[d_. + e_.*x_]^m_.* Cosh[f_. + g_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "Int[ExpandTrigReduce[x^p*F^(c*(a + b*x)), Sinh[d + e*x]^m*Cosh[f + g*x]^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^p_.*F_^(c_.*(a_. + b_.*x_))*Sinh[d_. + e_.*x_]^m_.* Cosh[f_. + g_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "Int[ExpandTrigToExp[F^(c*(a + b*x)), G[d + e*x]^m*H[d + e*x]^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[F_^(c_.*(a_. + b_.*x_))*G_[d_. + e_.*x_]^m_.*H_[d_. + e_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e}, x] && IGtQ[m, 0] && IGtQ[n, 0] && HyperbolicQ[G] && HyperbolicQ[H]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "Int[ExpandTrigToExp[F^u, Sinh[v]^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[F_^u_*Sinh[v_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[F, x] && (LinearQ[u, x] || PolyQ[u, x, 2]) && (LinearQ[v, x] || PolyQ[v, x, 2]) && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "Int[ExpandTrigToExp[F^u, Cosh[v]^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[F_^u_*Cosh[v_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[F, x] && (LinearQ[u, x] || PolyQ[u, x, 2]) && (LinearQ[v, x] || PolyQ[v, x, 2]) && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
       "filename": "6.7.7 F^(c (a+b x)) hyper(d+e x)^n.m",
            "rhs": "Int[ExpandTrigToExp[F^u, Sinh[v]^m*Cosh[v]^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[F_^u_*Sinh[v_]^m_.*Cosh[v_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[F, x] && (LinearQ[u, x] || PolyQ[u, x, 2]) && (LinearQ[v, x] || PolyQ[v, x, 2]) && IGtQ[m, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
       "filename": "6.7.8 u hyper(a+b log(c x^n))^p.m",
            "rhs": "Int[((c*x^n)^b/2 - 1/(2*(c*x^n)^b))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[Sinh[b_.*Log[c_.*x_^n_.]]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[c, x] && RationalQ[b, n, p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
       "filename": "6.7.8 u hyper(a+b log(c x^n))^p.m",
            "rhs": "Int[((c*x^n)^b/2 + 1/(2*(c*x^n)^b))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[Cosh[b_.*Log[c_.*x_^n_.]]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[c, x] && RationalQ[b, n, p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
       "filename": "6.7.8 u hyper(a+b log(c x^n))^p.m",
            "rhs": "-x*Sinh[d*(a + b*Log[c*x^n])]/(b^2*d^2*n^2 - 1) + b*d*n*x*Cosh[d*(a + b*Log[c*x^n])]/(b^2*d^2*n^2 - 1)",
     "rulenumber": 0,
            "lhs": "Int[Sinh[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && NeQ[b^2*d^2*n^2 - 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
       "filename": "6.7.8 u hyper(a+b log(c x^n))^p.m",
            "rhs": "-x*Cosh[d*(a + b*Log[c*x^n])]/(b^2*d^2*n^2 - 1) + b*d*n*x*Sinh[d*(a + b*Log[c*x^n])]/(b^2*d^2*n^2 - 1)",
     "rulenumber": 0,
            "lhs": "Int[Cosh[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && NeQ[b^2*d^2*n^2 - 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
       "filename": "6.7.8 u hyper(a+b log(c x^n))^p.m",
            "rhs": "-x* Sinh[d*(a + b*Log[c*x^n])]^p/(b^2*d^2*n^2*p^2 - 1) + b*d*n*p*x*Cosh[d*(a + b*Log[c*x^n])]* Sinh[d*(a + b*Log[c*x^n])]^(p - 1)/(b^2*d^2*n^2*p^2 - 1) - b^2*d^2*n^2*p*(p - 1)/(b^2*d^2*n^2*p^2 - 1)* Int[Sinh[d*(a + b*Log[c*x^n])]^(p - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[Sinh[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 - 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
       "filename": "6.7.8 u hyper(a+b log(c x^n))^p.m",
            "rhs": "-x* Cosh[d*(a + b*Log[c*x^n])]^p/(b^2*d^2*n^2*p^2 - 1) + b*d*n*p*x*Cosh[d*(a + b*Log[c*x^n])]^(p - 1)* Sinh[d*(a + b*Log[c*x^n])]/(b^2*d^2*n^2*p^2 - 1) + b^2*d^2*n^2*p*(p - 1)/(b^2*d^2*n^2*p^2 - 1)* Int[Cosh[d*(a + b*Log[c*x^n])]^(p - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[Cosh[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 - 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
       "filename": "6.7.8 u hyper(a+b log(c x^n))^p.m",
            "rhs": "1/(2^p*b^p*d^p*p^p)* Int[ExpandIntegrand[(-E^(-a*b*d^2*p)*x^(-1/p) + E^(a*b*d^2*p)*x^(1/p))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Sinh[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x] && IGtQ[p, 0] && EqQ[b^2*d^2*p^2 - 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
       "filename": "6.7.8 u hyper(a+b log(c x^n))^p.m",
            "rhs": "1/2^p*Int[ ExpandIntegrand[(E^(-a*b*d^2*p)*x^(-1/p) + E^(a*b*d^2*p)*x^(1/p))^ p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Cosh[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x] && IGtQ[p, 0] && EqQ[b^2*d^2*p^2 - 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
       "filename": "6.7.8 u hyper(a+b log(c x^n))^p.m",
            "rhs": " E^(a*d*p)/2^p*Int[x^(b*d*p)*(1-1/(E^(2*a*d)*x^(2*b*d)))^p,x]",
     "rulenumber": 0,
            "lhs": "Int[Sinh[d_.*(a_.+b_.*Log[x_])]^p_.,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,d},x] && IntegerQ[p] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
       "filename": "6.7.8 u hyper(a+b log(c x^n))^p.m",
            "rhs": " E^(a*d*p)/2^p*Int[x^(b*d*p)*(1+1/(E^(2*a*d)*x^(2*b*d)))^p,x]",
     "rulenumber": 0,
            "lhs": "Int[Cosh[d_.*(a_.+b_.*Log[x_])]^p_.,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,d},x] && IntegerQ[p] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
       "filename": "6.7.8 u hyper(a+b log(c x^n))^p.m",
            "rhs": "Sinh[d*(a + b*Log[x])]^ p/(x^(b*d*p)*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p)* Int[x^(b*d*p)*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[Sinh[d_.*(a_. + b_.*Log[x_])]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, p}, x] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
       "filename": "6.7.8 u hyper(a+b log(c x^n))^p.m",
            "rhs": "Cosh[d*(a + b*Log[x])]^ p/(x^(b*d*p)*(1 + 1/(E^(2*a*d)*x^(2*b*d)))^p)* Int[x^(b*d*p)*(1 + 1/(E^(2*a*d)*x^(2*b*d)))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[Cosh[d_.*(a_. + b_.*Log[x_])]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, p}, x] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
       "filename": "6.7.8 u hyper(a+b log(c x^n))^p.m",
            "rhs": "x/(n*(c*x^n)^(1/n))* Subst[Int[x^(1/n - 1)*Sinh[d*(a + b*Log[x])]^p, x], x, c*x^n]",
     "rulenumber": 0,
            "lhs": "Int[Sinh[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
       "filename": "6.7.8 u hyper(a+b log(c x^n))^p.m",
            "rhs": "x/(n*(c*x^n)^(1/n))* Subst[Int[x^(1/n - 1)*Cosh[d*(a + b*Log[x])]^p, x], x, c*x^n]",
     "rulenumber": 0,
            "lhs": "Int[Cosh[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
       "filename": "6.7.8 u hyper(a+b log(c x^n))^p.m",
            "rhs": "-(m + 1)*(e*x)^(m + 1)* Sinh[d*(a + b*Log[c*x^n])]/(b^2*d^2*e*n^2 - e*(m + 1)^2) + b*d*n*(e*x)^(m + 1)* Cosh[d*(a + b*Log[c*x^n])]/(b^2*d^2*e*n^2 - e*(m + 1)^2)",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Sinh[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b^2*d^2*n^2 - (m + 1)^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
       "filename": "6.7.8 u hyper(a+b log(c x^n))^p.m",
            "rhs": "-(m + 1)*(e*x)^(m + 1)* Cosh[d*(a + b*Log[c*x^n])]/(b^2*d^2*e*n^2 - e*(m + 1)^2) + b*d*n*(e*x)^(m + 1)* Sinh[d*(a + b*Log[c*x^n])]/(b^2*d^2*e*n^2 - e*(m + 1)^2)",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Cosh[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b^2*d^2*n^2 - (m + 1)^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
       "filename": "6.7.8 u hyper(a+b log(c x^n))^p.m",
            "rhs": "-(m + 1)*(e*x)^(m + 1)* Sinh[d*(a + b*Log[c*x^n])]^p/(b^2*d^2*e*n^2*p^2 - e*(m + 1)^2) + b*d*n*p*(e*x)^(m + 1)*Cosh[d*(a + b*Log[c*x^n])]* Sinh[d*(a + b*Log[c*x^n])]^(p - 1)/(b^2*d^2*e*n^2*p^2 - e*(m + 1)^2) - b^2*d^2*n^2*p*(p - 1)/(b^2*d^2*n^2*p^2 - (m + 1)^2)* Int[(e*x)^m*Sinh[d*(a + b*Log[c*x^n])]^(p - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Sinh[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 - (m + 1)^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
       "filename": "6.7.8 u hyper(a+b log(c x^n))^p.m",
            "rhs": "-(m + 1)*(e*x)^(m + 1)* Cosh[d*(a + b*Log[c*x^n])]^p/(b^2*d^2*e*n^2*p^2 - e*(m + 1)^2) + b*d*n*p*(e*x)^(m + 1)*Sinh[d*(a + b*Log[c*x^n])]* Cosh[d*(a + b*Log[c*x^n])]^(p - 1)/(b^2*d^2*e*n^2*p^2 - e*(m + 1)^2) + b^2*d^2*n^2*p*(p - 1)/(b^2*d^2*n^2*p^2 - (m + 1)^2)* Int[(e*x)^m*Cosh[d*(a + b*Log[c*x^n])]^(p - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Cosh[d_.*(a_. + b_.*Log[c_.*x_^n_.])]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 - (m + 1)^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
       "filename": "6.7.8 u hyper(a+b log(c x^n))^p.m",
            "rhs": "(m + 1)^p/(2^p*b^p*d^p*p^p)* Int[ ExpandIntegrand[(e*x)^ m*(-E^(-a*b*d^2*p/(m + 1))*x^(-(m + 1)/p) + E^(a*b*d^2*p/(m + 1))*x^((m + 1)/p))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Sinh[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, m}, x] && IGtQ[p, 0] && EqQ[b^2*d^2*p^2 - (m + 1)^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
       "filename": "6.7.8 u hyper(a+b log(c x^n))^p.m",
            "rhs": "1/2^p*Int[ ExpandIntegrand[(e*x)^ m*(E^(-a*b*d^2*p/(m + 1))*x^(-(m + 1)/p) + E^(a*b*d^2*p/(m + 1))*x^((m + 1)/p))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Cosh[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, m}, x] && IGtQ[p, 0] && EqQ[b^2*d^2*p^2 - (m + 1)^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
       "filename": "6.7.8 u hyper(a+b log(c x^n))^p.m",
            "rhs": " E^(a*d*p)/2^p*Int[(e*x)^m*x^(b*d*p)*(1-1/(E^(2*a*d)*x^(2*b*d)))^p,x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Sinh[d_.*(a_.+b_.*Log[x_])]^p_.,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,d,e,m},x] && IntegerQ[p] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
       "filename": "6.7.8 u hyper(a+b log(c x^n))^p.m",
            "rhs": " E^(a*d*p)/2^p*Int[(e*x)^m*x^(b*d*p)*(1+1/(E^(2*a*d)*x^(2*b*d)))^p,x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Cosh[d_.*(a_.+b_.*Log[x_])]^p_.,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,d,e,m},x] && IntegerQ[p] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
       "filename": "6.7.8 u hyper(a+b log(c x^n))^p.m",
            "rhs": "Sinh[d*(a + b*Log[x])]^ p/(x^(b*d*p)*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p)* Int[(e*x)^m*x^(b*d*p)*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Sinh[d_.*(a_. + b_.*Log[x_])]^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, m, p}, x] && Not[IntegerQ[p]]"
  },
  {
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        "comment": false,
         "givens": "FreeQ[{a, b, d, e, m, p}, x] && Not[IntegerQ[p]]"
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            "rhs": "(e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n))* Subst[Int[x^((m + 1)/n - 1)*Sinh[d*(a + b*Log[x])]^p, x], x, c*x^n]",
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         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])"
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            "rhs": "(e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n))* Subst[Int[x^((m + 1)/n - 1)*Cosh[d*(a + b*Log[x])]^p, x], x, c*x^n]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])"
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n, q}, x]"
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n, q}, x]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
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            "lhs": "Int[(i_.*x_)^r_.*(h_.*(e_. + f_.*Log[g_.*x_^m_.]))^q_.* Sinh[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, i, m, n, q, r}, x]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
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            "rhs": "E^(-a*d)*(i*x)^r*(c*x^n)^(-b*d)/(2*x^(r - b*d*n))* Int[x^(r - b*d*n)*(h*(e + f*Log[g*x^m]))^q, x] + E^(a*d)*(i*x)^r*(c*x^n)^(b*d)/(2*x^(r + b*d*n))* Int[x^(r + b*d*n)*(h*(e + f*Log[g*x^m]))^q, x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, i, m, n, q, r}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
       "filename": "6.7.8 u hyper(a+b log(c x^n))^p.m",
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            "lhs": "Int[Sech[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x] && IntegerQ[p]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
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            "lhs": "Int[Csch[d_.*(a_. + b_.*Log[x_])]^p_., x_Symbol]",
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         "givens": "FreeQ[{a, b, d}, x] && IntegerQ[p]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
       "filename": "6.7.8 u hyper(a+b log(c x^n))^p.m",
            "rhs": "Sech[d*(a + b*Log[x])]^p*(1 + E^(-2*a*d)*x^(-2*b*d))^p/x^(-b*d*p)* Int[x^(-b*d*p)/(1 + E^(-2*a*d)*x^(-2*b*d))^p, x]",
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         "givens": "FreeQ[{a, b, d, p}, x] && Not[IntegerQ[p]]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
       "filename": "6.7.8 u hyper(a+b log(c x^n))^p.m",
            "rhs": "Csch[d*(a + b*Log[x])]^p*(1 - E^(-2*a*d)*x^(-2*b*d))^p/x^(-b*d*p)* Int[x^(-b*d*p)/(1 - E^(-2*a*d)*x^(-2*b*d))^p, x]",
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         "givens": "FreeQ[{a, b, d, p}, x] && Not[IntegerQ[p]]"
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       "filename": "6.7.8 u hyper(a+b log(c x^n))^p.m",
            "rhs": "x/(n*(c*x^n)^(1/n))* Subst[Int[x^(1/n - 1)*Sech[d*(a + b*Log[x])]^p, x], x, c*x^n]",
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         "givens": "FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
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         "givens": "FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])"
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         "givens": "FreeQ[{a, b, d, e, m}, x] && IntegerQ[p]"
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         "givens": "FreeQ[{a, b, d, e, m}, x] && IntegerQ[p]"
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            "rhs": "Csch[d*(a + b*Log[x])]^p*(1 - E^(-2*a*d)*x^(-2*b*d))^p/x^(-b*d*p)* Int[(e*x)^m*x^(-b*d*p)/(1 - E^(-2*a*d)*x^(-2*b*d))^p, x]",
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       "filename": "6.7.8 u hyper(a+b log(c x^n))^p.m",
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         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])"
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            "rhs": "(e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n))* Subst[Int[x^((m + 1)/n - 1)*Csch[d*(a + b*Log[x])]^p, x], x, c*x^n]",
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         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])"
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         "givens": "FreeQ[{a, b}, x]"
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       "filename": "6.7.8 u hyper(a+b log(c x^n))^p.m",
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         "givens": "FreeQ[{a, b}, x]"
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  {
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            "lhs": "Int[x_^m_.*Sinh[a_.*x_^n_.*Log[b_.*x_]]*Log[b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && EqQ[m, n - 1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.8 u hyper(a+b log(c x^n))^p.m",
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            "rhs": "Sinh[a*x^n*Log[b*x]]/(a*n) - 1/n*Int[x^m*Cosh[a*x^n*Log[b*x]], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Cosh[a_.*x_^n_.*Log[b_.*x_]]*Log[b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && EqQ[m, n - 1]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
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            "lhs": "Int[(e_. + f_.*x_)^m_.* Sinh[c_. + d_.*x_]^n_./(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/b*Int[(e + f*x)^m*Cosh[c + d*x]^(n - 1), x] - a/b*Int[(e + f*x)^m*Cosh[c + d*x]^(n - 1)/(a + b*Cosh[c + d*x]), x]",
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            "lhs": "Int[(e_. + f_.*x_)^m_.* Cosh[c_. + d_.*x_]^n_./(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "-(e + f*x)^(m + 1)/(b*f*(m + 1)) + 2*Int[(e + f*x)^m*E^(c + d*x)/(a + b*E^(c + d*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Cosh[c_. + d_.*x_]/(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[a^2 + b^2, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "-(e + f*x)^(m + 1)/(b*f*(m + 1)) + 2*Int[(e + f*x)^m*E^(c + d*x)/(a + b*E^(c + d*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Sinh[c_. + d_.*x_]/(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[a^2 - b^2, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "-(e + f*x)^(m + 1)/(b*f*(m + 1)) + Int[(e + f*x)^m* E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d*x)), x] + Int[(e + f*x)^m* E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Cosh[c_. + d_.*x_]/(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "-(e + f*x)^(m + 1)/(b*f*(m + 1)) + Int[(e + f*x)^m* E^(c + d*x)/(a - Rt[a^2 - b^2, 2] + b*E^(c + d*x)), x] + Int[(e + f*x)^m* E^(c + d*x)/(a + Rt[a^2 - b^2, 2] + b*E^(c + d*x)), x]",
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            "lhs": "Int[(e_. + f_.*x_)^m_.* Sinh[c_. + d_.*x_]/(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/a*Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2), x] + 1/b*Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2)*Sinh[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Cosh[c_. + d_.*x_]^n_/(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 1] && EqQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "-1/a*Int[(e + f*x)^m*Sinh[c + d*x]^(n - 2), x] + 1/b*Int[(e + f*x)^m*Sinh[c + d*x]^(n - 2)*Cosh[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Sinh[c_. + d_.*x_]^n_/(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 1] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "-a/b^2*Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2), x] + 1/b*Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2)*Sinh[c + d*x], x] + (a^2 + b^2)/b^2* Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2)/(a + b*Sinh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Cosh[c_. + d_.*x_]^n_/(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "-a/b^2*Int[(e + f*x)^m*Sinh[c + d*x]^(n - 2), x] + 1/b*Int[(e + f*x)^m*Sinh[c + d*x]^(n - 2)*Cosh[c + d*x], x] + (a^2 - b^2)/b^2* Int[(e + f*x)^m*Sinh[c + d*x]^(n - 2)/(a + b*Cosh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Sinh[c_. + d_.*x_]^n_/(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/b*Int[(e + f*x)^m*Sech[c + d*x]*Tanh[c + d*x]^(n - 1), x] - a/b*Int[(e + f*x)^m*Sech[c + d*x]* Tanh[c + d*x]^(n - 1)/(a + b*Sinh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Tanh[c_. + d_.*x_]^n_./(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/b*Int[(e + f*x)^m*Csch[c + d*x]*Coth[c + d*x]^(n - 1), x] - a/b*Int[(e + f*x)^m*Csch[c + d*x]* Coth[c + d*x]^(n - 1)/(a + b*Cosh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Coth[c_. + d_.*x_]^n_./(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/a*Int[(e + f*x)^m*Coth[c + d*x]^n, x] - b/a*Int[(e + f*x)^m*Cosh[c + d*x]* Coth[c + d*x]^(n - 1)/(a + b*Sinh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Coth[c_. + d_.*x_]^n_./(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/a*Int[(e + f*x)^m*Tanh[c + d*x]^n, x] - b/a*Int[(e + f*x)^m*Sinh[c + d*x]* Tanh[c + d*x]^(n - 1)/(a + b*Cosh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Tanh[c_. + d_.*x_]^n_./(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/a*Int[(e + f*x)^m*Sech[c + d*x]^(n + 2), x] + 1/b*Int[(e + f*x)^m*Sech[c + d*x]^(n + 1)*Tanh[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Sech[c_. + d_.*x_]^n_./(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && EqQ[a^2 + b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "-1/a*Int[(e + f*x)^m*Csch[c + d*x]^(n + 2), x] + 1/b*Int[(e + f*x)^m*Csch[c + d*x]^(n + 1)*Coth[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Csch[c_. + d_.*x_]^n_./(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "b^2/(a^2 + b^2)* Int[(e + f*x)^m*Sech[c + d*x]^(n - 2)/(a + b*Sinh[c + d*x]), x] + 1/(a^2 + b^2)* Int[(e + f*x)^m*Sech[c + d*x]^n*(a - b*Sinh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Sech[c_. + d_.*x_]^n_./(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "b^2/(a^2 - b^2)* Int[(e + f*x)^m*Csch[c + d*x]^(n - 2)/(a + b*Cosh[c + d*x]), x] + 1/(a^2 - b^2)* Int[(e + f*x)^m*Csch[c + d*x]^n*(a - b*Cosh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Csch[c_. + d_.*x_]^n_./(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 - b^2, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/a*Int[(e + f*x)^m*Csch[c + d*x]^n, x] - b/a*Int[(e + f*x)^m*Csch[c + d*x]^(n - 1)/(a + b*Sinh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Csch[c_. + d_.*x_]^n_./(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/a*Int[(e + f*x)^m*Sech[c + d*x]^n, x] - b/a*Int[(e + f*x)^m*Sech[c + d*x]^(n - 1)/(a + b*Cosh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* Sech[c_. + d_.*x_]^n_./(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Unintegrable[(e + f*x)^m*F[c + d*x]^n/(a + b*Sinh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* F_[c_. + d_.*x_]^n_./(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && HyperbolicQ[F]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Unintegrable[(e + f*x)^m*F[c + d*x]^n/(a + b*Cosh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* F_[c_. + d_.*x_]^n_./(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && HyperbolicQ[F]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/b*Int[(e + f*x)^m*Cosh[c + d*x]^p*Sinh[c + d*x]^(n - 1), x] - a/b*Int[(e + f*x)^m*Cosh[c + d*x]^p* Sinh[c + d*x]^(n - 1)/(a + b*Sinh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Cosh[c_. + d_.*x_]^p_.* Sinh[c_. + d_.*x_]^n_./(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/b*Int[(e + f*x)^m*Sinh[c + d*x]^p*Cosh[c + d*x]^(n - 1), x] - a/b*Int[(e + f*x)^m*Sinh[c + d*x]^p* Cosh[c + d*x]^(n - 1)/(a + b*Cosh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Sinh[c_. + d_.*x_]^p_.* Cosh[c_. + d_.*x_]^n_./(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/b*Int[(e + f*x)^m*Sinh[c + d*x]^(p - 1)*Tanh[c + d*x]^n, x] - a/b*Int[(e + f*x)^m*Sinh[c + d*x]^(p - 1)* Tanh[c + d*x]^n/(a + b*Sinh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Sinh[c_. + d_.*x_]^p_.* Tanh[c_. + d_.*x_]^n_./(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/b*Int[(e + f*x)^m*Cosh[c + d*x]^(p - 1)*Coth[c + d*x]^n, x] - a/b*Int[(e + f*x)^m*Cosh[c + d*x]^(p - 1)* Coth[c + d*x]^n/(a + b*Cosh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Cosh[c_. + d_.*x_]^p_.* Coth[c_. + d_.*x_]^n_./(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/b*Int[(e + f*x)^m*Sech[c + d*x]^(p + 1)*Tanh[c + d*x]^(n - 1), x] - a/b*Int[(e + f*x)^m*Sech[c + d*x]^(p + 1)* Tanh[c + d*x]^(n - 1)/(a + b*Sinh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Sech[c_. + d_.*x_]^p_.* Tanh[c_. + d_.*x_]^n_./(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/b*Int[(e + f*x)^m*Csch[c + d*x]^(p + 1)*Coth[c + d*x]^(n - 1), x] - a/b*Int[(e + f*x)^m*Csch[c + d*x]^(p + 1)* Coth[c + d*x]^(n - 1)/(a + b*Cosh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Csch[c_. + d_.*x_]^p_.* Coth[c_. + d_.*x_]^n_./(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/a*Int[(e + f*x)^m*Cosh[c + d*x]^p*Coth[c + d*x]^n, x] - b/a*Int[(e + f*x)^m*Cosh[c + d*x]^(p + 1)* Coth[c + d*x]^(n - 1)/(a + b*Sinh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Cosh[c_. + d_.*x_]^p_.* Coth[c_. + d_.*x_]^n_./(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/a*Int[(e + f*x)^m*Sinh[c + d*x]^p*Tanh[c + d*x]^n, x] - b/a*Int[(e + f*x)^m*Sinh[c + d*x]^(p + 1)* Tanh[c + d*x]^(n - 1)/(a + b*Cosh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Sinh[c_. + d_.*x_]^p_.* Tanh[c_. + d_.*x_]^n_./(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/a*Int[(e + f*x)^m*Csch[c + d*x]^p*Coth[c + d*x]^n, x] - b/a*Int[(e + f*x)^m*Csch[c + d*x]^(p - 1)* Coth[c + d*x]^n/(a + b*Sinh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Csch[c_. + d_.*x_]^p_.* Coth[c_. + d_.*x_]^n_./(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/a*Int[(e + f*x)^m*Sech[c + d*x]^p*Tanh[c + d*x]^n, x] - b/a*Int[(e + f*x)^m*Sech[c + d*x]^(p - 1)* Tanh[c + d*x]^n/(a + b*Cosh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Sech[c_. + d_.*x_]^p_.* Tanh[c_. + d_.*x_]^n_./(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/a*Int[(e + f*x)^m*Sech[c + d*x]^p*Csch[c + d*x]^n, x] - b/a*Int[(e + f*x)^m*Sech[c + d*x]^p* Csch[c + d*x]^(n - 1)/(a + b*Sinh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Sech[c_. + d_.*x_]^p_.* Csch[c_. + d_.*x_]^n_./(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/a*Int[(e + f*x)^m*Csch[c + d*x]^p*Sech[c + d*x]^n, x] - b/a*Int[(e + f*x)^m*Csch[c + d*x]^p* Sech[c + d*x]^(n - 1)/(a + b*Cosh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Csch[c_. + d_.*x_]^p_.* Sech[c_. + d_.*x_]^n_./(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Unintegrable[(e + f*x)^m*F[c + d*x]^n* G[c + d*x]^p/(a + b*Sinh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*F_[c_. + d_.*x_]^n_.* G_[c_. + d_.*x_]^p_./(a_ + b_.*Sinh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && HyperbolicQ[F] && HyperbolicQ[G]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Unintegrable[(e + f*x)^m*F[c + d*x]^n* G[c + d*x]^p/(a + b*Cosh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*F_[c_. + d_.*x_]^n_.* G_[c_. + d_.*x_]^p_./(a_ + b_.*Cosh[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && HyperbolicQ[F] && HyperbolicQ[G]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Int[(e + f*x)^m*Cosh[c + d*x]*F[c + d*x]^n/(b + a*Cosh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* F_[c_. + d_.*x_]^n_./(a_ + b_.*Sech[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && HyperbolicQ[F] && IntegersQ[m, n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Int[(e + f*x)^m*Sinh[c + d*x]*F[c + d*x]^n/(b + a*Sinh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.* F_[c_. + d_.*x_]^n_./(a_ + b_.*Csch[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && HyperbolicQ[F] && IntegersQ[m, n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Int[(e + f*x)^m*Cosh[c + d*x]*F[c + d*x]^n* G[c + d*x]^p/(b + a*Cosh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*F_[c_. + d_.*x_]^n_.* G_[c_. + d_.*x_]^p_./(a_ + b_.*Sech[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && HyperbolicQ[F] && HyperbolicQ[G] && IntegersQ[m, n, p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Int[(e + f*x)^m*Sinh[c + d*x]*F[c + d*x]^n* G[c + d*x]^p/(b + a*Sinh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*F_[c_. + d_.*x_]^n_.* G_[c_. + d_.*x_]^p_./(a_ + b_.*Csch[c_. + d_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && HyperbolicQ[F] && HyperbolicQ[G] && IntegersQ[m, n, p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/2^(p + q)* Int[ExpandIntegrand[(-E^(-c - d*x) + E^(c + d*x))^ q, (-E^(-a - b*x) + E^(a + b*x))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Sinh[a_. + b_.*x_]^p_.*Sinh[c_. + d_.*x_]^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, q}, x] && IGtQ[p, 0] && Not[IntegerQ[q]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/2^(p + q)* Int[ExpandIntegrand[(E^(-c - d*x) + E^(c + d*x))^ q, (E^(-a - b*x) + E^(a + b*x))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Cosh[a_. + b_.*x_]^p_.*Cosh[c_. + d_.*x_]^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, q}, x] && IGtQ[p, 0] && Not[IntegerQ[q]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/2^(p + q)* Int[ExpandIntegrand[(E^(-c - d*x) + E^(c + d*x))^ q, (-E^(-a - b*x) + E^(a + b*x))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Sinh[a_. + b_.*x_]^p_.*Cosh[c_. + d_.*x_]^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, q}, x] && IGtQ[p, 0] && Not[IntegerQ[q]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/2^(p + q)* Int[ExpandIntegrand[(-E^(-c - d*x) + E^(c + d*x))^ q, (E^(-a - b*x) + E^(a + b*x))^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Cosh[a_. + b_.*x_]^p_.*Sinh[c_. + d_.*x_]^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, q}, x] && IGtQ[p, 0] && Not[IntegerQ[q]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Int[-E^(-(a + b*x))/2 + E^(a + b*x)/2 + E^(-(a + b*x))/(1 + E^(2*(c + d*x))) - E^(a + b*x)/(1 + E^(2*(c + d*x))), x]",
     "rulenumber": 0,
            "lhs": "Int[Sinh[a_. + b_.*x_]*Tanh[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Int[E^(-(a + b*x))/2 + E^(a + b*x)/2 - E^(-(a + b*x))/(1 - E^(2*(c + d*x))) - E^(a + b*x)/(1 - E^(2*(c + d*x))), x]",
     "rulenumber": 0,
            "lhs": "Int[Cosh[a_. + b_.*x_]*Coth[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Int[-E^(-(a + b*x))/2 + E^(a + b*x)/2 + E^(-(a + b*x))/(1 - E^(2*(c + d*x))) - E^(a + b*x)/(1 - E^(2*(c + d*x))), x]",
     "rulenumber": 0,
            "lhs": "Int[Sinh[a_. + b_.*x_]*Coth[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Int[E^(-(a + b*x))/2 + E^(a + b*x)/2 - E^(-(a + b*x))/(1 + E^(2*(c + d*x))) - E^(a + b*x)/(1 + E^(2*(c + d*x))), x]",
     "rulenumber": 0,
            "lhs": "Int[Cosh[a_. + b_.*x_]*Tanh[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "-1/d* Subst[Int[Sinh[a*x]^n/x^2, x], x, 1/(c + d*x)]",
     "rulenumber": 0,
            "lhs": "Int[Sinh[a_./(c_. + d_.*x_)]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "-1/d* Subst[Int[Cosh[a*x]^n/x^2, x], x, 1/(c + d*x)]",
     "rulenumber": 0,
            "lhs": "Int[Cosh[a_./(c_. + d_.*x_)]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "-1/d* Subst[Int[Sinh[b*e/d - e*(b*c - a*d)*x/d]^n/x^2, x], x, 1/(c + d*x)]",
     "rulenumber": 0,
            "lhs": "Int[Sinh[e_.*(a_. + b_.*x_)/(c_. + d_.*x_)]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "-1/d* Subst[Int[Cosh[b*e/d - e*(b*c - a*d)*x/d]^n/x^2, x], x, 1/(c + d*x)]",
     "rulenumber": 0,
            "lhs": "Int[Cosh[e_.*(a_. + b_.*x_)/(c_. + d_.*x_)]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "With[{lst = QuotientOfLinearsParts[u, x]}, Int[Sinh[(lst[[1]] + lst[[2]]*x)/(lst[[3]] + lst[[4]]*x)]^n, x]]",
     "rulenumber": 0,
            "lhs": "Int[Sinh[u_]^n_., x_Symbol]",
        "comment": false,
         "givens": "IGtQ[n, 0] && QuotientOfLinearsQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "With[{lst = QuotientOfLinearsParts[u, x]}, Int[Cosh[(lst[[1]] + lst[[2]]*x)/(lst[[3]] + lst[[4]]*x)]^n, x]]",
     "rulenumber": 0,
            "lhs": "Int[Cosh[u_]^n_., x_Symbol]",
        "comment": false,
         "givens": "IGtQ[n, 0] && QuotientOfLinearsQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Int[u*Sinh[v]^(p + q), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*Sinh[v_]^p_.*Sinh[w_]^q_., x_Symbol]",
        "comment": false,
         "givens": "EqQ[w, v]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Int[u*Cosh[v]^(p + q), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*Cosh[v_]^p_.*Cosh[w_]^q_., x_Symbol]",
        "comment": false,
         "givens": "EqQ[w, v]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Int[ExpandTrigReduce[Sinh[v]^p*Sinh[w]^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Sinh[v_]^p_.*Sinh[w_]^q_., x_Symbol]",
        "comment": false,
         "givens": "IGtQ[p, 0] && IGtQ[q, 0] && (PolynomialQ[v, x] && PolynomialQ[w, x] || BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Int[ExpandTrigReduce[Cosh[v]^p*Cosh[w]^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Cosh[v_]^p_.*Cosh[w_]^q_., x_Symbol]",
        "comment": false,
         "givens": "IGtQ[p, 0] && IGtQ[q, 0] && (PolynomialQ[v, x] && PolynomialQ[w, x] || BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Int[ExpandTrigReduce[x^m, Sinh[v]^p*Sinh[w]^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Sinh[v_]^p_.*Sinh[w_]^q_., x_Symbol]",
        "comment": false,
         "givens": "IGtQ[m, 0] && IGtQ[p, 0] && IGtQ[q, 0] && (PolynomialQ[v, x] && PolynomialQ[w, x] || BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Int[ExpandTrigReduce[x^m, Cosh[v]^p*Cosh[w]^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Cosh[v_]^p_.*Cosh[w_]^q_., x_Symbol]",
        "comment": false,
         "givens": "IGtQ[m, 0] && IGtQ[p, 0] && IGtQ[q, 0] && (PolynomialQ[v, x] && PolynomialQ[w, x] || BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/2^p*Int[u*Sinh[2*v]^p, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*Sinh[v_]^p_.*Cosh[w_]^p_., x_Symbol]",
        "comment": false,
         "givens": "EqQ[w, v] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Int[ExpandTrigReduce[Sinh[v]^p*Cosh[w]^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Sinh[v_]^p_.*Cosh[w_]^q_., x_Symbol]",
        "comment": false,
         "givens": "IGtQ[p, 0] && IGtQ[q, 0] && (PolynomialQ[v, x] && PolynomialQ[w, x] || BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Int[ExpandTrigReduce[x^m, Sinh[v]^p*Cosh[w]^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Sinh[v_]^p_.*Cosh[w_]^q_., x_Symbol]",
        "comment": false,
         "givens": "IGtQ[m, 0] && IGtQ[p, 0] && IGtQ[q, 0] && (PolynomialQ[v, x] && PolynomialQ[w, x] || BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Int[Cosh[v]*Tanh[w]^(n - 1), x] - Cosh[v - w]*Int[Sech[w]*Tanh[w]^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[Sinh[v_]*Tanh[w_]^n_., x_Symbol]",
        "comment": false,
         "givens": "GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Int[Sinh[v]*Coth[w]^(n - 1), x] + Cosh[v - w]*Int[Csch[w]*Coth[w]^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[Cosh[v_]*Coth[w_]^n_., x_Symbol]",
        "comment": false,
         "givens": "GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Int[Cosh[v]*Coth[w]^(n - 1), x] + Sinh[v - w]*Int[Csch[w]*Coth[w]^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[Sinh[v_]*Coth[w_]^n_., x_Symbol]",
        "comment": false,
         "givens": "GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Int[Sinh[v]*Tanh[w]^(n - 1), x] - Sinh[v - w]*Int[Sech[w]*Tanh[w]^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[Cosh[v_]*Tanh[w_]^n_., x_Symbol]",
        "comment": false,
         "givens": "GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Cosh[v - w]*Int[Tanh[w]*Sech[w]^(n - 1), x] + Sinh[v - w]*Int[Sech[w]^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[Sinh[v_]*Sech[w_]^n_., x_Symbol]",
        "comment": false,
         "givens": "GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Cosh[v - w]*Int[Coth[w]*Csch[w]^(n - 1), x] + Sinh[v - w]*Int[Csch[w]^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[Cosh[v_]*Csch[w_]^n_., x_Symbol]",
        "comment": false,
         "givens": "GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Sinh[v - w]*Int[Coth[w]*Csch[w]^(n - 1), x] + Cosh[v - w]*Int[Csch[w]^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[Sinh[v_]*Csch[w_]^n_., x_Symbol]",
        "comment": false,
         "givens": "GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Sinh[v - w]*Int[Tanh[w]*Sech[w]^(n - 1), x] + Cosh[v - w]*Int[Sech[w]^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[Cosh[v_]*Sech[w_]^n_., x_Symbol]",
        "comment": false,
         "givens": "GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Int[(e + f*x)^m*(a + b*Sinh[2*c + 2*d*x]/2)^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^ m_.*(a_ + b_.*Sinh[c_. + d_.*x_]*Cosh[c_. + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/2^n*Int[x^m*(2*a - b + b*Cosh[2*c + 2*d*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*Sinh[c_. + d_.*x_]^2)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a - b, 0] && IGtQ[m, 0] && ILtQ[n, 0] && (EqQ[n, -1] || EqQ[m, 1] && EqQ[n, -2])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/2^n*Int[x^m*(2*a + b + b*Cosh[2*c + 2*d*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_ + b_.*Cosh[c_. + d_.*x_]^2)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[a - b, 0] && IGtQ[m, 0] && ILtQ[n, 0] && (EqQ[n, -1] || EqQ[m, 1] && EqQ[n, -2])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "2*Int[(f + g*x)^m/(2*a + b - c + (b + c)*Cosh[2*d + 2*e*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^ m_./(a_. + b_.*Cosh[d_. + e_.*x_]^2 + c_.*Sinh[d_. + e_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && NeQ[a + b, 0] && NeQ[a + c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "2*Int[(f + g*x)^m/(b - c + (b + c)*Cosh[2*d + 2*e*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^m_.* Sech[d_. + e_.*x_]^2/(b_ + c_.*Tanh[d_. + e_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f, g}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "2*Int[(f + g*x)^m/(2*a + b - c + (b + c)*Cosh[2*d + 2*e*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^m_.* Sech[d_. + e_.*x_]^2/(b_. + a_.*Sech[d_. + e_.*x_]^2 + c_.*Tanh[d_. + e_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && NeQ[a + b, 0] && NeQ[a + c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "2*Int[(f + g*x)^m/(b - c + (b + c)*Cosh[2*d + 2*e*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^m_.* Csch[d_. + e_.*x_]^2/(c_ + b_.*Coth[d_. + e_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d, e, f, g}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "2*Int[(f + g*x)^m/(2*a + b - c + (b + c)*Cosh[2*d + 2*e*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^m_.* Csch[d_. + e_.*x_]^2/(c_. + b_.*Coth[d_. + e_.*x_]^2 + a_.*Csch[d_. + e_.*x_]^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && NeQ[a + b, 0] && NeQ[a + c, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "B*(e + f*x)*Cosh[c + d*x]/(a*d*(a + b*Sinh[c + d*x])) - B*f/(a*d)*Int[Cosh[c + d*x]/(a + b*Sinh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)*(A_ + B_.*Sinh[c_. + d_.*x_])/(a_ + b_.*Sinh[c_. + d_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && EqQ[a*A + b*B, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "B*(e + f*x)*Sinh[c + d*x]/(a*d*(a + b*Cosh[c + d*x])) - B*f/(a*d)*Int[Sinh[c + d*x]/(a + b*Cosh[c + d*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)*(A_ + B_.*Cosh[c_. + d_.*x_])/(a_ + b_.*Cosh[c_. + d_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B}, x] && EqQ[a*A - b*B, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/d^(m + 1)* Subst[Int[(d*e - c*f + f*x)^m*Sinh[a + b*x^n]^p, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Sinh[a_. + b_.*(c_ + d_.*x_)^n_]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && RationalQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/d^(m + 1)* Subst[Int[(d*e - c*f + f*x)^m*Cosh[a + b*x^n]^p, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Cosh[a_. + b_.*(c_ + d_.*x_)^n_]^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && RationalQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Int[(a*Cosh[v] + b*Sinh[v])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[Sech[v_]^m_.*(a_ + b_.*Tanh[v_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IntegerQ[(m - 1)/2] && EqQ[m + n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Int[(b*Cosh[v] + a*Sinh[v])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[Csch[v_]^m_.*(a_ + b_.*Coth[v_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IntegerQ[(m - 1)/2] && EqQ[m + n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Int[ExpandTrigReduce[u, Sinh[a + b*x]^m*Sinh[c + d*x]^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*Sinh[a_. + b_.*x_]^m_.*Sinh[c_. + d_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Int[ExpandTrigReduce[u, Cosh[a + b*x]^m*Cosh[c + d*x]^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*Cosh[a_. + b_.*x_]^m_.*Cosh[c_. + d_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "-Csch[(b*c - a*d)/d]*Int[Tanh[a + b*x], x] + Csch[(b*c - a*d)/b]*Int[Tanh[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[Sech[a_. + b_.*x_]*Sech[c_ + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Csch[(b*c - a*d)/b]*Int[Coth[a + b*x], x] - Csch[(b*c - a*d)/d]*Int[Coth[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[Csch[a_. + b_.*x_]*Csch[c_ + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "b*x/d - b/d*Cosh[(b*c - a*d)/d]*Int[Sech[a + b*x]*Sech[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[Tanh[a_. + b_.*x_]*Tanh[c_ + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "b*x/d + Cosh[(b*c - a*d)/d]*Int[Csch[a + b*x]*Csch[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[Coth[a_. + b_.*x_]*Coth[c_ + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "Int[u*(a*E^(a/b*v))^n, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_.*Cosh[v_] + b_.*Sinh[v_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n}, x] && EqQ[a^2 - b^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "-1/2*Int[E^(-d*(a + b*Log[c*x^n])^2), x] + 1/2*Int[E^(d*(a + b*Log[c*x^n])^2), x]",
     "rulenumber": 0,
            "lhs": "Int[Sinh[d_.*(a_. + b_.*Log[c_.*x_^n_.])^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/2*Int[E^(-d*(a + b*Log[c*x^n])^2), x] + 1/2*Int[E^(d*(a + b*Log[c*x^n])^2), x]",
     "rulenumber": 0,
            "lhs": "Int[Cosh[d_.*(a_. + b_.*Log[c_.*x_^n_.])^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "-1/2*Int[(e*x)^m*E^(-d*(a + b*Log[c*x^n])^2), x] + 1/2*Int[(e*x)^m*E^(d*(a + b*Log[c*x^n])^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Sinh[d_.*(a_. + b_.*Log[c_.*x_^n_.])^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/6 Hyperbolic functions/6.7 Miscellaneous/6.7.9 Active hyperbolic functions.m",
       "filename": "6.7.9 Active hyperbolic functions.m",
            "rhs": "1/2*Int[(e*x)^m*E^(-d*(a + b*Log[c*x^n])^2), x] + 1/2*Int[(e*x)^m*E^(d*(a + b*Log[c*x^n])^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Cosh[d_.*(a_. + b_.*Log[c_.*x_^n_.])^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.1 (a+b arcsinh(c x))^n.m",
       "filename": "7.1.1 (a+b arcsinh(c x))^n.m",
            "rhs": "x*(a + b*ArcSinh[c*x])^n - b*c*n*Int[x*(a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.1 (a+b arcsinh(c x))^n.m",
       "filename": "7.1.1 (a+b arcsinh(c x))^n.m",
            "rhs": "x*(a + b*ArcCosh[c*x])^n - b*c*n* Int[x*(a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.1 (a+b arcsinh(c x))^n.m",
       "filename": "7.1.1 (a+b arcsinh(c x))^n.m",
            "rhs": "Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1)) - c/(b*(n + 1))* Int[x*(a + b*ArcSinh[c*x])^(n + 1)/Sqrt[1 + c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSinh[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.1 (a+b arcsinh(c x))^n.m",
       "filename": "7.1.1 (a+b arcsinh(c x))^n.m",
            "rhs": "Sqrt[-1 + c*x]* Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)) - c/(b*(n + 1))* Int[x*(a + b*ArcCosh[c*x])^(n + 1)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.1 (a+b arcsinh(c x))^n.m",
       "filename": "7.1.1 (a+b arcsinh(c x))^n.m",
            "rhs": "1/(b*c)*Subst[Int[x^n*Cosh[a/b - x/b], x], x, a + b*ArcSinh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSinh[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.1 (a+b arcsinh(c x))^n.m",
       "filename": "7.1.1 (a+b arcsinh(c x))^n.m",
            "rhs": "-1/(b*c)* Subst[Int[x^n*Sinh[a/b - x/b], x], x, a + b*ArcCosh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.2 (d x)^m (a+b arcsinh(c x))^n.m",
       "filename": "7.1.2 (d x)^m (a+b arcsinh(c x))^n.m",
            "rhs": "Subst[Int[(a + b*x)^n/Tanh[x], x], x, ArcSinh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSinh[c_.*x_])^n_./x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.2 (d x)^m (a+b arcsinh(c x))^n.m",
       "filename": "7.1.2 (d x)^m (a+b arcsinh(c x))^n.m",
            "rhs": "Subst[Int[(a + b*x)^n/Coth[x], x], x, ArcCosh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCosh[c_.*x_])^n_./x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.2 (d x)^m (a+b arcsinh(c x))^n.m",
       "filename": "7.1.2 (d x)^m (a+b arcsinh(c x))^n.m",
            "rhs": "(d*x)^(m + 1)*(a + b*ArcSinh[c*x])^n/(d*(m + 1)) - b*c*n/(d*(m + 1))* Int[(d*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.2 (d x)^m (a+b arcsinh(c x))^n.m",
       "filename": "7.1.2 (d x)^m (a+b arcsinh(c x))^n.m",
            "rhs": "(d*x)^(m + 1)*(a + b*ArcCosh[c*x])^n/(d*(m + 1)) - b*c*n/(d*(m + 1))* Int[(d*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[-1 + c*x]* Sqrt[1 + c*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.2 (d x)^m (a+b arcsinh(c x))^n.m",
       "filename": "7.1.2 (d x)^m (a+b arcsinh(c x))^n.m",
            "rhs": "x^(m + 1)*(a + b*ArcSinh[c*x])^n/(m + 1) - b*c*n/(m + 1)* Int[x^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*ArcSinh[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.2 (d x)^m (a+b arcsinh(c x))^n.m",
       "filename": "7.1.2 (d x)^m (a+b arcsinh(c x))^n.m",
            "rhs": "x^(m + 1)*(a + b*ArcCosh[c*x])^n/(m + 1) - b*c*n/(m + 1)* Int[x^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[-1 + c*x]* Sqrt[1 + c*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.2 (d x)^m (a+b arcsinh(c x))^n.m",
       "filename": "7.1.2 (d x)^m (a+b arcsinh(c x))^n.m",
            "rhs": "x^m*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1)) - 1/(b*c^(m + 1)*(n + 1))* Subst[Int[ ExpandTrigReduce[(a + b*x)^(n + 1), Sinh[x]^(m - 1)*(m + (m + 1)*Sinh[x]^2), x], x], x, ArcSinh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*ArcSinh[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.2 (d x)^m (a+b arcsinh(c x))^n.m",
       "filename": "7.1.2 (d x)^m (a+b arcsinh(c x))^n.m",
            "rhs": "x^m*Sqrt[-1 + c*x]* Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)) + 1/(b*c^(m + 1)*(n + 1))* Subst[Int[ ExpandTrigReduce[(a + b*x)^(n + 1)* Cosh[x]^(m - 1)*(m - (m + 1)*Cosh[x]^2), x], x], x, ArcCosh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.2 (d x)^m (a+b arcsinh(c x))^n.m",
       "filename": "7.1.2 (d x)^m (a+b arcsinh(c x))^n.m",
            "rhs": "x^m*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1)) - m/(b*c*(n + 1))* Int[x^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1)/Sqrt[1 + c^2*x^2], x] - c*(m + 1)/(b*(n + 1))* Int[x^(m + 1)*(a + b*ArcSinh[c*x])^(n + 1)/Sqrt[1 + c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*ArcSinh[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.2 (d x)^m (a+b arcsinh(c x))^n.m",
       "filename": "7.1.2 (d x)^m (a+b arcsinh(c x))^n.m",
            "rhs": "x^m*Sqrt[-1 + c*x]* Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)) + m/(b*c*(n + 1))* Int[x^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1)/(Sqrt[-1 + c*x]* Sqrt[1 + c*x]), x] - c*(m + 1)/(b*(n + 1))* Int[x^(m + 1)*(a + b*ArcCosh[c*x])^(n + 1)/(Sqrt[-1 + c*x]* Sqrt[1 + c*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.2 (d x)^m (a+b arcsinh(c x))^n.m",
       "filename": "7.1.2 (d x)^m (a+b arcsinh(c x))^n.m",
            "rhs": "1/c^(m + 1)* Subst[Int[(a + b*x)^n*Sinh[x]^m*Cosh[x], x], x, ArcSinh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*ArcSinh[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.2 (d x)^m (a+b arcsinh(c x))^n.m",
       "filename": "7.1.2 (d x)^m (a+b arcsinh(c x))^n.m",
            "rhs": "1/c^(m + 1)* Subst[Int[(a + b*x)^n*Cosh[x]^m*Sinh[x], x], x, ArcCosh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.2 (d x)^m (a+b arcsinh(c x))^n.m",
       "filename": "7.1.2 (d x)^m (a+b arcsinh(c x))^n.m",
            "rhs": "Unintegrable[(d*x)^m*(a + b*ArcSinh[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.2 (d x)^m (a+b arcsinh(c x))^n.m",
       "filename": "7.1.2 (d x)^m (a+b arcsinh(c x))^n.m",
            "rhs": "Unintegrable[(d*x)^m*(a + b*ArcCosh[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": " 1/(c*Sqrt[d])*Subst[Int[(a+b*x)^n,x],x,ArcSinh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_.+b_.*ArcSinh[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e,n},x] && EqQ[e,c^2*d] && GtQ[d,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": " 1/(c*Sqrt[-d1*d2])*Subst[Int[(a+b*x)^n,x],x,ArcCosh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_.+b_.*ArcCosh[c_.*x_])^n_./(Sqrt[d1_+e1_.*x_]*Sqrt[d2_+e2_.* x_]),x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d1,e1,d2,e2,n},x] && EqQ[e1,c*d1] && EqQ[e2,-c*d2] &&  GtQ[d1,0] && LtQ[d2,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "Log[a + b*ArcSinh[c*x]]/(b*c*Sqrt[d])",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcSinh[c_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "Log[a + b*ArcCosh[c*x]]/(b*c*Sqrt[-d1*d2])",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[d1_ + e1_.*x_]* Sqrt[d2_ + e2_.*x_]*(a_. + b_.*ArcCosh[c_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && GtQ[d1, 0] && LtQ[d2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(a + b*ArcSinh[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1))",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSinh[c_.*x_])^n_./Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && GtQ[d, 0] && NeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(a + b*ArcCosh[c*x])^(n + 1)/(b*c* Sqrt[-d1*d2]*(n + 1))",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCosh[c_.*x_])^ n_./(Sqrt[d1_ + e1_.*x_]*Sqrt[d2_ + e2_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && GtQ[d1, 0] && LtQ[d2, 0] && NeQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]* Int[(a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSinh[c_.*x_])^n_./Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && Not[GtQ[d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "Sqrt[1 + c*x]*Sqrt[-1 + c*x]/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])* Int[(a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCosh[c_.*x_])^ n_./(Sqrt[d1_ + e1_.*x_]*Sqrt[d2_ + e2_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && Not[GtQ[d1, 0] && LtQ[d2, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": " x*(d+e*x^2)^p*(a+b*ArcSinh[c*x])^n/(2*p+1) + 2*d*p/(2*p+1)*Int[(d+e*x^2)^(p-1)*(a+b*ArcSinh[c*x])^n,x] - b*c*n*d^p/(2*p+1)*Int[x*(1+c^2*x^2)^(p-1/2)*(a+b*ArcSinh[c*x])^(n-1) ,x]",
     "rulenumber": 0,
            "lhs": "Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e},x] && EqQ[e,c^2*d] && GtQ[n,0] && GtQ[p,0] &&  (IntegerQ[p] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "x*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n/(2*p + 1) + 2*d*p/(2*p + 1)* Int[(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x] - b*c*n*(-d)^p/((2*p + 1))* Int[x*(-1 + c*x)^(p - 1/2)*(1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": " x*(d1+e1*x)^p*(d2+e2*x)^p*(a+b*ArcCosh[c*x])^n/(2*p+1) + 2*d1*d2*p/(2*p+1)*Int[(d1+e1*x)^(p-1)*(d2+e2*x)^(p-1)*(a+b*ArcCosh[ c*x])^n,x] - b*c*n*(-d1*d2)^p/((2*p+1))*Int[x*(-1+c^2*x^2)^(p-1/2)*(a+b*ArcCosh[ c*x])^(n-1),x]",
     "rulenumber": 0,
            "lhs": "Int[(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_. ,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d1,e1,d2,e2},x] && EqQ[e1,c*d1] &&  EqQ[e2,-c*d2] && GtQ[n,0] && GtQ[p,0] && IntegerQ[p-1/2] &&  (GtQ[d1,0] && LtQ[d2,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "x*Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^n/2 - b*c*n*Sqrt[d + e*x^2]/(2*Sqrt[1 + c^2*x^2])* Int[x*(a + b*ArcSinh[c*x])^(n - 1), x] + Sqrt[d + e*x^2]/(2*Sqrt[1 + c^2*x^2])* Int[(a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "x*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n/2 - b*c*n*Sqrt[d1 + e1*x]* Sqrt[d2 + e2*x]/(2*Sqrt[1 + c*x]*Sqrt[-1 + c*x])* Int[x*(a + b*ArcCosh[c*x])^(n - 1), x] - Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]/(2*Sqrt[1 + c*x]*Sqrt[-1 + c*x])* Int[(a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[d1_ + e1_.*x_]* Sqrt[d2_ + e2_.*x_]*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "x*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n/(2*p + 1) + 2*d*p/(2*p + 1)* Int[(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x] - b*c*n* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/((2*p + 1)*(1 + c^2*x^2)^FracPart[p])* Int[x*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "x*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n/(2*p + 1) + 2*d1*d2*p/(2*p + 1)* Int[(d1 + e1*x)^(p - 1)*(d2 + e2*x)^(p - 1)*(a + b*ArcCosh[c*x])^ n, x] - b*c*n*(-d1*d2)^(p - 1/2)*Sqrt[d1 + e1*x]* Sqrt[d2 + e2*x]/((2*p + 1)*Sqrt[1 + c*x]*Sqrt[-1 + c*x])* Int[x*(-1 + c^2*x^2)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d1_ + e1_.*x_)^p_.*(d2_ + e2_.*x_)^ p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && GtQ[n, 0] && GtQ[p, 0] && IntegerQ[p - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "x*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n/(2*p + 1) + 2*d1*d2*p/(2*p + 1)* Int[(d1 + e1*x)^(p - 1)*(d2 + e2*x)^(p - 1)*(a + b*ArcCosh[c*x])^ n, x] - b*c*n*(-d1*d2)^IntPart[p]*(d1 + e1*x)^ FracPart[p]*(d2 + e2*x)^ FracPart[ p]/((2*p + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p])* Int[ x*(-1 + c*x)^(p - 1/2)*(1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d1_ + e1_.*x_)^p_.*(d2_ + e2_.*x_)^ p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && GtQ[n, 0] && GtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "  x*(a+b*ArcSinh[c*x])^n/(d*Sqrt[d+e*x^2]) - b*c*n/Sqrt[d]*Int[x*(a+b*ArcSinh[c*x])^(n-1)/(d+e*x^2),x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.+b_.*ArcSinh[c_.*x_])^n_./(d_+e_.*x_^2)^(3/2),x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d,e},x] && EqQ[e,c^2*d] && GtQ[n,0] && GtQ[d,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": " x*(a+b*ArcCosh[c*x])^n/(d1*d2*Sqrt[d1+e1*x]*Sqrt[d2+e2*x]) + b*c*n/Sqrt[-d1*d2]*Int[x*(a+b*ArcCosh[c*x])^(n-1)/(d1*d2+e1*e2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_.+b_.*ArcCosh[c_.*x_])^n_./((d1_+e1_.*x_)^(3/2)*(d2_+e2_.*x_ )^(3/2)),x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d1,e1,d2,e2},x] && EqQ[e1,c*d1] && EqQ[e2,-c*d2] &&  GtQ[n,0] && (GtQ[d1,0] && LtQ[d2,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "x*(a + b*ArcSinh[c*x])^n/(d*Sqrt[d + e*x^2]) - b*c*n*Sqrt[1 + c^2*x^2]/(d*Sqrt[d + e*x^2])* Int[x*(a + b*ArcSinh[c*x])^(n - 1)/(1 + c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSinh[c_.*x_])^n_./(d_ + e_.*x_^2)^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "x*(a + b*ArcCosh[c*x])^n/(d1*d2*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]) + b*c*n*Sqrt[1 + c*x]* Sqrt[-1 + c*x]/(d1*d2*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])* Int[x*(a + b*ArcCosh[c*x])^(n - 1)/(1 - c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCosh[c_.*x_])^ n_./((d1_ + e1_.*x_)^(3/2)*(d2_ + e2_.*x_)^(3/2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": " -x*(d+e*x^2)^(p+1)*(a+b*ArcSinh[c*x])^n/(2*d*(p+1)) + (2*p+3)/(2*d*(p+1))*Int[(d+e*x^2)^(p+1)*(a+b*ArcSinh[c*x])^n,x] + b*c*n*d^p/(2*(p+1))*Int[x*(1+c^2*x^2)^(p+1/2)*(a+b*ArcSinh[c*x])^(n- 1),x]",
     "rulenumber": 0,
            "lhs": "Int[(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e},x] && EqQ[e,c^2*d] && GtQ[n,0] &&  LtQ[p,-1] && NeQ[p,-3/2] && (IntegerQ[p] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "-x*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^ n/(2*d*(p + 1)) + (2*p + 3)/(2*d*(p + 1))* Int[(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n, x] - b*c*n*(-d)^p/(2*(p + 1))* Int[x*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": " -x*(d1+e1*x)^(p+1)*(d2+e2*x)^(p+1)*(a+b*ArcCosh[c*x])^n/(2*d1*d2*(p+1) ) + (2*p+3)/(2*d1*d2*(p+1))*Int[(d1+e1*x)^(p+1)*(d2+e2*x)^(p+1)*(a+b* ArcCosh[c*x])^n,x] - b*c*n*(-d1*d2)^p/(2*(p+1))*Int[x*(-1+c^2*x^2)^(p+1/2)*(a+b*ArcCosh[ c*x])^(n-1),x]",
     "rulenumber": 0,
            "lhs": "Int[(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*(a_.+b_.*ArcCosh[c_.*x_])^n_. ,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d1,e1,d2,e2},x] && EqQ[e1,c*d1] &&  EqQ[e2,-c*d2] && GtQ[n,0] && LtQ[p,-1] && NeQ[p,-3/2] && IntegerQ[p+1/2] && (GtQ[d1,0] && LtQ[d2,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "-x*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^ n/(2*d*(p + 1)) + (2*p + 3)/(2*d*(p + 1))* Int[(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x] + b*c*n* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/(2*(p + 1)*(1 + c^2*x^2)^FracPart[p])* Int[x*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "-x*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n/(2*d1*d2*(p + 1)) + (2*p + 3)/(2*d1*d2*(p + 1))* Int[(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^ n, x] - b*c*n*(-d1*d2)^(p + 1/2)*Sqrt[1 + c*x]* Sqrt[-1 + c*x]/(2*(p + 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])* Int[x*(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^p_*(a_. + b_.*ArcCosh[c_.*x_])^ n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[p + 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "-x*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n/(2*d1*d2*(p + 1)) + (2*p + 3)/(2*d1*d2*(p + 1))* Int[(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^ n, x] - b*c*n*(-d1*d2)^IntPart[p]*(d1 + e1*x)^ FracPart[p]*(d2 + e2*x)^ FracPart[ p]/(2*(p + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p])* Int[ x*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^p_*(a_. + b_.*ArcCosh[c_.*x_])^ n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "1/(c*d)*Subst[Int[(a + b*x)^n*Sech[x], x], x, ArcSinh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSinh[c_.*x_])^n_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "-1/(c*d)* Subst[Int[(a + b*x)^n*Csch[x], x], x, ArcCosh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCosh[c_.*x_])^n_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": " d^p*(1+c^2*x^2)^(p+1/2)*(a+b*ArcSinh[c*x])^(n+1)/(b*c*(n+1)) - c*d^p*(2*p+1)/(b*(n+1))*Int[x*(1+c^2*x^2)^(p-1/2)*(a+b*ArcSinh[c*x]) ^(n+1),x]",
     "rulenumber": 0,
            "lhs": "Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,p},x] && EqQ[e,c^2*d] && LtQ[n,-1] &&  (IntegerQ[p] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(-d)^ p*(-1 + c*x)^(p + 1/2)*(1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)) - c*(-d)^p*(2*p + 1)/(b*(n + 1))* Int[x*(-1 + c*x)^(p - 1/2)*(1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": " (-d1*d2)^p*(-1+c*x)^(p+1/2)*(1+c*x)^(p+1/2)*(a+b*ArcCosh[c*x])^(n+1)/( b*c*(n+1)) - c*(-d1*d2)^p*(2*p+1)/(b*(n+1))*Int[x*(-1+c^2*x^2)^(p-1/2)*(a+b* ArcCosh[c*x])^(n+1),x]",
     "rulenumber": 0,
            "lhs": "Int[(d1_+e1_.*x_)^p_.*(d2_+e2_.*x_)^p_.*(a_.+b_.*ArcCosh[c_.*x_])^ n_,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d1,e1,d2,e2,p},x] &&  EqQ[e1,c*d1] && EqQ[e2,-c*d2] && LtQ[n,-1] && IntegerQ[p-1/2] &&  (GtQ[d1,0] && LtQ[d2,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "Sqrt[1 + c^2*x^2]*(d + e*x^2)^ p*(a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1)) - c*(2*p + 1)* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/(b*(n + 1)*(1 + c^2*x^2)^FracPart[p])* Int[x*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "Sqrt[1 + c*x]* Sqrt[-1 + c*x]*(d1 + e1*x)^p*(d2 + e2*x)^ p*(a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)) - c*(2*p + 1)*(-d1*d2)^(p - 1/2)*Sqrt[d1 + e1*x]* Sqrt[d2 + e2*x]/(b*(n + 1)*Sqrt[1 + c*x]*Sqrt[-1 + c*x])* Int[x*(-1 + c^2*x^2)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d1_ + e1_.*x_)^p_.*(d2_ + e2_.*x_)^ p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, p}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && LtQ[n, -1] && IntegerQ[p - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "Sqrt[1 + c*x]* Sqrt[-1 + c*x]*(d1 + e1*x)^p*(d2 + e2*x)^ p*(a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)) - c*(2*p + 1)*(-d1*d2)^IntPart[p]*(d1 + e1*x)^ FracPart[p]*(d2 + e2*x)^ FracPart[ p]/(b*(n + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p])* Int[ x*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(d1_ + e1_.*x_)^p_.*(d2_ + e2_.*x_)^ p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, p}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "d^p/c*Subst[Int[(a + b*x)^n*Cosh[x]^(2*p + 1), x], x, ArcSinh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p, 0] && (IntegerQ[p] || GtQ[d, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(-d)^p/c* Subst[Int[(a + b*x)^n*Sinh[x]^(2*p + 1), x], x, ArcCosh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(-d1*d2)^p/c* Subst[Int[(a + b*x)^n*Sinh[x]^(2*p + 1), x], x, ArcCosh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(d1_ + e1_.*x_)^p_.*(d2_ + e2_.*x_)^ p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && IGtQ[p + 1/2, 0] && (GtQ[d1, 0] && LtQ[d2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "d^(p - 1/2)*Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]* Int[(1 + c^2*x^2)^p*(a + b*ArcSinh[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p, 0] && Not[IntegerQ[p] || GtQ[d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(-d1*d2)^(p - 1/2)*Sqrt[d1 + e1*x]* Sqrt[d2 + e2*x]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])* Int[(1 + c*x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(d1_ + e1_.*x_)^p_.*(d2_ + e2_.*x_)^ p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, -c*d2] && IGtQ[2*p, 0] && Not[GtQ[d1, 0] && LtQ[d2, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[e, c^2*d] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": " With[{u=IntHide[(d+e*x^2)^p,x]}, Dist[a+b*ArcCosh[c*x],u,x] -  b*c*Sqrt[1-c^2*x^2]/(Sqrt[1+c*x]*Sqrt[-1+c*x])*Int[SimplifyIntegrand[ u/Sqrt[1-c^2*x^2],x],x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcCosh[c_.*x_]),x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e},x] && (IGtQ[p,0] ||  ILtQ[p+1/2,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcSinh[c*x])^n, (d + e*x^2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && NeQ[e, c^2*d] && IntegerQ[p] && (p > 0 || IGtQ[n, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^n, (d + e*x^2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (p > 0 || IGtQ[n, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "Unintegrable[(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "Unintegrable[(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p}, x] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "Unintegrable[(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(d1_ + e1_.*x_)^p_.*(d2_ + e2_.*x_)^ p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(-d^2*g/e)^q* Int[(d + e*x)^(p - q)*(1 + c^2*x^2)^q*(a + b*ArcSinh[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^p_*(f_ + g_.*x_)^q_*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 + e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0] && GtQ[d, 0] && LtQ[g/e, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(d + e*x)^q*(f + g*x)^q/(1 + c^2*x^2)^q* Int[(d + e*x)^(p - q)*(1 + c^2*x^2)^q*(a + b*ArcSinh[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^p_*(f_ + g_.*x_)^q_*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 + e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.3 (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(-d)^ IntPart[p]*(d + e*x^2)^ FracPart[p]/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p])* Int[(1 + c*x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[c^2*d + e, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "1/e*Subst[Int[(a + b*x)^n*Tanh[x], x], x, ArcSinh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_*(a_. + b_.*ArcSinh[c_.*x_])^n_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "1/e*Subst[Int[(a + b*x)^n*Coth[x], x], x, ArcCosh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_*(a_. + b_.*ArcCosh[c_.*x_])^n_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "  (d+e*x^2)^(p+1)*(a+b*ArcSinh[c*x])^n/(2*e*(p+1)) - b*n*d^p/(2*c*(p+1))*Int[(1+c^2*x^2)^(p+1/2)*(a+b*ArcSinh[c*x])^(n-1) ,x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,p},x] && EqQ[e,c^2*d] && GtQ[n,0] &&  NeQ[p,-1] && (IntegerQ[p] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^ n/(2*e*(p + 1)) - b*n*(-d)^p/(2*c*(p + 1))* Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": " (d1+e1*x)^(p+1)*(d2+e2*x)^(p+1)*(a+b*ArcCosh[c*x])^n/(2*e1*e2*(p+1)) - b*n*(-d1*d2)^(p-1/2)/(2*c*(p+1))*Int[(-1+c^2*x^2)^(p+1/2)*(a+b* ArcCosh[c*x])^(n-1),x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d1_+e1_.*x_)^p_.*(d2_+e2_.*x_)^p_.*(a_.+b_.*ArcCosh[c_.*x_] )^n_.,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d1,e1,d2,e2,p},x] &&  EqQ[e1-c*d1,0] && EqQ[e2+c*d2,0] && GtQ[n,0] && IntegerQ[p+1/2] &&  (GtQ[d1,0] && LtQ[d2,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^ n/(2*e*(p + 1)) - b*n*d^ IntPart[p]*(d + e*x^2)^ FracPart[p]/(2*c*(p + 1)*(1 + c^2*x^2)^FracPart[p])* Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1)) - b*n*(-d1*d2)^IntPart[p]*(d1 + e1*x)^ FracPart[p]*(d2 + e2*x)^ FracPart[p]/(2* c*(p + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p])* Int[(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d1_ + e1_.*x_)^p_.*(d2_ + e2_.*x_)^ p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p + 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1)) - b*n*(-d1*d2)^IntPart[p]*(d1 + e1*x)^ FracPart[p]*(d2 + e2*x)^ FracPart[p]/(2* c*(p + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p])* Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d1_ + e1_.*x_)^p_.*(d2_ + e2_.*x_)^ p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "1/d*Subst[Int[(a + b*x)^n/(Cosh[x]*Sinh[x]), x], x, ArcSinh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSinh[c_.*x_])^n_./(x_*(d_ + e_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "-1/d* Subst[Int[(a + b*x)^n/(Cosh[x]*Sinh[x]), x], x, ArcCosh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCosh[c_.*x_])^n_./(x_*(d_ + e_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": " (f*x)^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcSinh[c*x])^n/(d*f*(m+1)) - b*c*n*d^p/(f*(m+1))*Int[(f*x)^(m+1)*(1+c^2*x^2)^(p+1/2)*(a+b* ArcSinh[c*x])^(n-1),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,m,p},x] && EqQ[e,c^2*d] &&  GtQ[n,0] && EqQ[m+2*p+3,0] && NeQ[m,-1] && (IntegerQ[p] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n/(d*f*(m + 1)) + b*c*n*(-d)^p/(f*(m + 1))* Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": " (f*x)^(m+1)*(d1+e1*x)^(p+1)*(d2+e2*x)^(p+1)*(a+b*ArcCosh[c*x])^n/(d1* d2*f*(m+1)) + b*c*n*(-d1*d2)^p/(f*(m+1))*Int[(f*x)^(m+1)*(-1+c^2*x^2)^(p+1/2)*(a+ b*ArcCosh[c*x])^(n-1),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d1_+e1_.*x_)^p_.*(d2_+e2_.*x_)^p_.*(a_.+b_.* ArcCosh[c_.*x_])^n_.,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d1,e1,d2,e2,f,m,p},x] &&  EqQ[e1-c*d1,0] && EqQ[e2+c*d2,0] && GtQ[n,0] && EqQ[m+2*p+3,0] && NeQ[m,-1] && IntegerQ[p+1/2] && (GtQ[d1,0] && LtQ[d2,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n/(d*f*(m + 1)) - b*c*n* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/(f*(m + 1)*(1 + c^2*x^2)^FracPart[p])* Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n/(d1*d2*f*(m + 1)) + b*c*n*(-d1*d2)^IntPart[p]*(d1 + e1*x)^ FracPart[p]*(d2 + e2*x)^ FracPart[ p]/(f*(m + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p])* Int[(f*x)^(m + 1)*(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d1_ + e1_.*x_)^p_.*(d2_ + e2_.*x_)^ p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, m, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1] && IntegerQ[p + 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n/(d1*d2*f*(m + 1)) + b*c*n*(-d1*d2)^IntPart[p]*(d1 + e1*x)^ FracPart[p]*(d2 + e2*x)^ FracPart[ p]/(f*(m + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p])* Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d1_ + e1_.*x_)^p_.*(d2_ + e2_.*x_)^ p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, m, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(d + e*x^2)^p*(a + b*ArcSinh[c*x])/(2*p) - b*c*d^p/(2*p)*Int[(1 + c^2*x^2)^(p - 1/2), x] + d*Int[(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])/x, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(d + e*x^2)^p*(a + b*ArcCosh[c*x])/(2*p) - b*c*(-d)^p/(2*p)* Int[(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2), x] + d*Int[(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])/x, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*(d + e*x^2)^ p*(a + b*ArcSinh[c*x])/(f*(m + 1)) - b*c*d^p/(f*(m + 1))* Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p - 1/2), x] - 2*e*p/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && ILtQ[(m + 1)/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*(d + e*x^2)^ p*(a + b*ArcCosh[c*x])/(f*(m + 1)) - b*c*(-d)^p/(f*(m + 1))* Int[(f*x)^(m + 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2), x] - 2*e*p/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && ILtQ[(m + 1)/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = IntHide[x^m*(1 + c^2*x^2)^p, x]}, Dist[d^p*(a + b*ArcSinh[c*x]), u, x] - b*c*d^p*Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegerQ[ p - 1/2] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0]) && NeQ[p, -1/2] && GtQ[d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = IntHide[x^m*(1 + c*x)^p*(-1 + c*x)^p, x]}, Dist[(-d1*d2)^p*(a + b*ArcCosh[c*x]), u, x] - b*c*(-d1*d2)^p* Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p - 1/2] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0]) && NeQ[p, -1/2] && GtQ[d1, 0] && LtQ[d2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = IntHide[x^m*(1 + c^2*x^2)^p, x]}, (a + b*ArcSinh[c*x])*Int[x^m*(d + e*x^2)^p, x] - b*c*d^(p - 1/2)*Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]* Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p + 1/2, 0] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = IntHide[x^m*(1 + c*x)^p*(-1 + c*x)^p, x]}, (a + b*ArcCosh[c*x])*Int[x^m*(d1 + e1*x)^p*(d2 + e2*x)^p, x] - b*c*(-d1*d2)^(p - 1/2)*Sqrt[d1 + e1*x]* Sqrt[d2 + e2*x]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])* Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[p + 1/2, 0] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^(m+1)*(d+e*x^2)^p*(a+b*ArcSinh[c*x])^n/(f*(m+1)) -   2*e*p/(f^2*(m+1))*Int[(f*x)^(m+2)*(d+e*x^2)^(p-1)*(a+b*ArcSinh[c*x]) ^n,x] - b*c*n*d^p/(f*(m+1))*Int[(f*x)^(m+1)*(1+c^2*x^2)^(p-1/2)*(a+b* ArcSinh[c*x])^(n-1),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f},x] && EqQ[e,c^2*d] &&  GtQ[n,0] && GtQ[p,0] && LtQ[m,-1] && (IntegerQ[p] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*(d + e*x^2)^ p*(a + b*ArcCosh[c*x])^n/(f*(m + 1)) - 2*e*p/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x] - b*c*n*(-d)^p/(f*(m + 1))* Int[(f*x)^(m + 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": " (f*x)^(m+1)*(d1+e1*x)^p*(d2+e2*x)^p*(a+b*ArcCosh[c*x])^n/(f*(m+1)) - 2*e1*e2*p/(f^2*(m+1))*Int[(f*x)^(m+2)*(d1+e1*x)^(p-1)*(d2+e2*x)^(p- 1)*(a+b*ArcCosh[c*x])^n,x] - b*c*n*(-d1*d2)^p/(f*(m+1))*Int[(f*x)^(m+1)*(-1+c^2*x^2)^(p-1/2)*(a+ b*ArcCosh[c*x])^(n-1),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d1_+e1_.*x_)^p_*(d2_+e2_.*x_)^p_*(a_.+b_.*ArcCosh[ c_.*x_])^n_.,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d1,e1,d2,e2,f},x] &&  EqQ[e1-c*d1,0] && EqQ[e2+c*d2,0] && GtQ[n,0] && GtQ[p,0] && LtQ[m,-1]  && IntegerQ[p-1/2] && (GtQ[d1,0] && LtQ[d2,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^(m + 1)* Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^n/(f*(m + 1)) - b*c*n*Sqrt[d + e*x^2]/(f*(m + 1)*Sqrt[1 + c^2*x^2])* Int[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x] - c^2*Sqrt[d + e*x^2]/(f^2*(m + 1)*Sqrt[1 + c^2*x^2])* Int[(f*x)^(m + 2)*(a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*Sqrt[d1 + e1*x]* Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n/(f*(m + 1)) - b*c*n*Sqrt[d1 + e1*x]* Sqrt[d2 + e2*x]/(f*(m + 1)*Sqrt[1 + c*x]*Sqrt[-1 + c*x])* Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1), x] - c^2*Sqrt[d1 + e1*x]* Sqrt[d2 + e2*x]/(f^2*(m + 1)*Sqrt[1 + c*x]*Sqrt[-1 + c*x])* Int[((f*x)^(m + 2)*(a + b*ArcCosh[c*x])^n)/(Sqrt[1 + c*x]* Sqrt[-1 + c*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*Sqrt[d1_ + e1_.*x_]* Sqrt[d2_ + e2_.*x_]*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*(d + e*x^2)^ p*(a + b*ArcSinh[c*x])^n/(f*(m + 1)) - 2*e*p/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x] - b*c*n* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/(f*(m + 1)*(1 + c^2*x^2)^FracPart[p])* Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*(d1 + e1*x)^p*(d2 + e2*x)^ p*(a + b*ArcCosh[c*x])^n/(f*(m + 1)) - 2*e1*e2*p/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(d1 + e1*x)^(p - 1)*(d2 + e2*x)^(p - 1)*(a + b*ArcCosh[c*x])^n, x] - b*c*n*(-d1*d2)^(p - 1/2)*Sqrt[d1 + e1*x]* Sqrt[d2 + e2*x]/(f*(m + 1)*Sqrt[1 + c*x]*Sqrt[-1 + c*x])* Int[(f*x)^(m + 1)*(-1 + c^2*x^2)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] && IntegerQ[p - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": " (f*x)^(m+1)*(d+e*x^2)^p*(a+b*ArcSinh[c*x])^n/(f*(m+2*p+1)) + 2*d*p/(m+2*p+1)*Int[(f*x)^m*(d+e*x^2)^(p-1)*(a+b*ArcSinh[c*x])^n,x]  - b*c*n*d^p/(f*(m+2*p+1))*Int[(f*x)^(m+1)*(1+c^2*x^2)^(p-1/2)*(a+b* ArcSinh[c*x])^(n-1),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,m},x] && EqQ[e,c^2*d] &&  GtQ[n,0] && GtQ[p,0] && Not[LtQ[m,-1]] && (IntegerQ[p] || GtQ[d,0]) &&  (RationalQ[m] || EqQ[n,1]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*(d + e*x^2)^ p*(a + b*ArcCosh[c*x])^n/(f*(m + 2*p + 1)) + 2*d*p/(m + 2*p + 1)* Int[(f*x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x] - b*c*n*(-d)^p/(f*(m + 2*p + 1))* Int[(f*x)^(m + 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && Not[LtQ[m, -1]] && IntegerQ[p] && (RationalQ[m] || EqQ[n, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^(m + 1)* Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^n/(f*(m + 2)) - b*c*n*Sqrt[d + e*x^2]/(f*(m + 2)*Sqrt[1 + c^2*x^2])* Int[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x] + Sqrt[d + e*x^2]/((m + 2)*Sqrt[1 + c^2*x^2])* Int[(f*x)^m*(a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && Not[LtQ[m, -1]] && (RationalQ[m] || EqQ[n, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*Sqrt[d1 + e1*x]* Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n/(f*(m + 2)) - b*c*n*Sqrt[d1 + e1*x]* Sqrt[d2 + e2*x]/(f*(m + 2)*Sqrt[1 + c*x]*Sqrt[-1 + c*x])* Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1), x] - Sqrt[d1 + e1*x]* Sqrt[d2 + e2*x]/((m + 2)*Sqrt[1 + c*x]*Sqrt[-1 + c*x])* Int[(f*x)^m*(a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*Sqrt[d1_ + e1_.*x_]* Sqrt[d2_ + e2_.*x_]*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && Not[LtQ[m, -1]] && (RationalQ[m] || EqQ[n, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*(d + e*x^2)^ p*(a + b*ArcSinh[c*x])^n/(f*(m + 2*p + 1)) + 2*d*p/(m + 2*p + 1)* Int[(f*x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x] - b*c*n* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/(f*(m + 2*p + 1)*(1 + c^2*x^2)^FracPart[p])* Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && Not[LtQ[m, -1]] && (RationalQ[m] || EqQ[n, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*(d1 + e1*x)^p*(d2 + e2*x)^ p*(a + b*ArcCosh[c*x])^n/(f*(m + 2*p + 1)) + 2*d1*d2*p/(m + 2*p + 1)* Int[(f*x)^ m*(d1 + e1*x)^(p - 1)*(d2 + e2*x)^(p - 1)*(a + b*ArcCosh[c*x])^ n, x] - b*c*n*(-d1*d2)^(p - 1/2)*Sqrt[d1 + e1*x]* Sqrt[d2 + e2*x]/(f*(m + 2*p + 1)*Sqrt[1 + c*x]*Sqrt[-1 + c*x])* Int[(f*x)^(m + 1)*(-1 + c^2*x^2)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[p, 0] && Not[LtQ[m, -1]] && IntegerQ[p - 1/2] && (RationalQ[m] || EqQ[n, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": " (f*x)^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcSinh[c*x])^n/(d*f*(m+1)) - c^2*(m+2*p+3)/(f^2*(m+1))*Int[(f*x)^(m+2)*(d+e*x^2)^p*(a+b*ArcSinh[ c*x])^n,x] - b*c*n*d^p/(f*(m+1))*Int[(f*x)^(m+1)*(1+c^2*x^2)^(p+1/2)*(a+b* ArcSinh[c*x])^(n-1),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,p},x] && EqQ[e,c^2*d] &&  GtQ[n,0] && LtQ[m,-1] && IntegerQ[m] && (IntegerQ[p] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n/(d*f*(m + 1)) + c^2*(m + 2*p + 3)/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x] + b*c*n*(-d)^p/(f*(m + 1))* Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[m, -1] && IntegerQ[m] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n/(d*f*(m + 1)) - c^2*(m + 2*p + 3)/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x] - b*c*n* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/(f*(m + 1)*(1 + c^2*x^2)^FracPart[p])* Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[m, -1] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n/(d1*d2*f*(m + 1)) + c^2*(m + 2*p + 3)/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(d1 + e1*x)^p*(d2 + e2*x)^ p*(a + b*ArcCosh[c*x])^n, x] + b*c*n*(-d1*d2)^IntPart[p]*(d1 + e1*x)^ FracPart[p]*(d2 + e2*x)^ FracPart[ p]/(f*(m + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p])* Int[(f*x)^(m + 1)*(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && LtQ[m, -1] && IntegerQ[m] && IntegerQ[p + 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n/(d1*d2*f*(m + 1)) + c^2*(m + 2*p + 3)/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(d1 + e1*x)^p*(d2 + e2*x)^ p*(a + b*ArcCosh[c*x])^n, x] + b*c*n*(-d1*d2)^IntPart[p]*(d1 + e1*x)^ FracPart[p]*(d2 + e2*x)^ FracPart[ p]/(f*(m + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p])* Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && LtQ[m, -1] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": " f*(f*x)^(m-1)*(d+e*x^2)^(p+1)*(a+b*ArcSinh[c*x])^n/(2*e*(p+1)) - f^2*(m-1)/(2*e*(p+1))*Int[(f*x)^(m-2)*(d+e*x^2)^(p+1)*(a+b*ArcSinh[ c*x])^n,x] - b*f*n*d^p/(2*c*(p+1))*Int[(f*x)^(m-1)*(1+c^2*x^2)^(p+1/2)*(a+b* ArcSinh[c*x])^(n-1),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f},x] && EqQ[e,c^2*d] &&  GtQ[n,0] && LtQ[p,-1] && GtQ[m,1] && (IntegerQ[p] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^ n/(2*e*(p + 1)) - f^2*(m - 1)/(2*e*(p + 1))* Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n, x] - b*f*n*(-d)^p/(2*c*(p + 1))* Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^ n/(2*e*(p + 1)) - f^2*(m - 1)/(2*e*(p + 1))* Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x] - b*f*n* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/(2*c*(p + 1)*(1 + c^2*x^2)^FracPart[p])* Int[(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && GtQ[m, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1)) - f^2*(m - 1)/(2*e1*e2*(p + 1))* Int[(f*x)^(m - 2)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n, x] - b*f*n*(-d1*d2)^IntPart[p]*(d1 + e1*x)^ FracPart[p]*(d2 + e2*x)^ FracPart[p]/(2* c*(p + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p])* Int[(f*x)^(m - 1)*(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[p + 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1)) - f^2*(m - 1)/(2*e1*e2*(p + 1))* Int[(f*x)^(m - 2)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n, x] - b*f*n*(-d1*d2)^IntPart[p]*(d1 + e1*x)^ FracPart[p]*(d2 + e2*x)^ FracPart[p]/(2* c*(p + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p])* Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && LtQ[p, -1] && Not[IntegerQ[p]] && GtQ[m, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": " -(f*x)^(m+1)*(d+e*x^2)^(p+1)*(a+b*ArcSinh[c*x])^n/(2*d*f*(p+1)) + (m+2*p+3)/(2*d*(p+1))*Int[(f*x)^m*(d+e*x^2)^(p+1)*(a+b*ArcSinh[c*x]) ^n,x] + b*c*n*d^p/(2*f*(p+1))*Int[(f*x)^(m+1)*(1+c^2*x^2)^(p+1/2)*(a+b* ArcSinh[c*x])^(n-1),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,m},x] && EqQ[e,c^2*d] &&  GtQ[n,0] && LtQ[p,-1] && Not[GtQ[m,1]] && (IntegerQ[p] || GtQ[d,0]) &&   (IntegerQ[m] || IntegerQ[p] || EqQ[n,1]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "-(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n/(2*d*f*(p + 1)) + (m + 2*p + 3)/(2*d*(p + 1))* Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n, x] - b*c*n*(-d)^p/(2*f*(p + 1))* Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && Not[GtQ[m, 1]] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "-(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n/(2*d*f*(p + 1)) + (m + 2*p + 3)/(2*d*(p + 1))* Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x] + b*c*n* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/(2*f*(p + 1)*(1 + c^2*x^2)^FracPart[p])* Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && Not[GtQ[m, 1]] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "-(f*x)^(m + 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n/(2*d1*d2*f*(p + 1)) + (m + 2*p + 3)/(2*d1*d2*(p + 1))* Int[(f*x)^ m*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^ n, x] - b*c*n*(-d1*d2)^IntPart[p]*(d1 + e1*x)^ FracPart[p]*(d2 + e2*x)^ FracPart[p]/(2* f*(p + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p])* Int[(f*x)^(m + 1)*(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && LtQ[p, -1] && Not[GtQ[m, 1]] && (IntegerQ[m] || EqQ[n, 1]) && IntegerQ[p + 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "-(f*x)^(m + 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n/(2*d1*d2*f*(p + 1)) + (m + 2*p + 3)/(2*d1*d2*(p + 1))* Int[(f*x)^ m*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^ n, x] - b*c*n*(-d1*d2)^IntPart[p]*(d1 + e1*x)^ FracPart[p]*(d2 + e2*x)^ FracPart[p]/(2* f*(p + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p])* Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && LtQ[p, -1] && Not[GtQ[m, 1]] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "f*(f*x)^(m-1)*Sqrt[d+e*x^2]*(a+b*ArcSinh[c*x])^n/(e*m) -   b*f*n*Sqrt[1+c^2*x^2]/(c*m*Sqrt[d+e*x^2])*Int[(f*x)^(m-1)*(a+b* ArcSinh[c*x])^(n-1),x] - f^2*(m-1)/(c^2*m)*Int[((f*x)^(m-2)*(a+b*ArcSinh[c*x])^n)/Sqrt[d+e*x^ 2],x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_.+b_.*ArcSinh[c_.*x_])^n_./Sqrt[d_+e_.*x_^2],x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f},x] && EqQ[e,c^2*d] && GtQ[n,0] &&  GtQ[m,1] && GtQ[d,0] && IntegerQ[m] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": " f*(f*x)^(m-1)*Sqrt[d1+e1*x]*Sqrt[d2+e2*x]*(a+b*ArcCosh[c*x])^n/(e1*e2* m) + b*f*n*Sqrt[d1+e1*x]*Sqrt[d2+e2*x]/(c*d1*d2*m*Sqrt[1+c*x]*Sqrt[-1+c* x])*Int[(f*x)^(m-1)*(a+b*ArcCosh[c*x])^(n-1),x] + f^2*(m-1)/(c^2*m)*Int[(f*x)^(m-2)*(a+b*ArcCosh[c*x])^n/(Sqrt[d1+e1* x]*Sqrt[d2+e2*x]),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_.+b_.*ArcCosh[c_.*x_])^n_./(Sqrt[d1_+e1_.*x_]* Sqrt[d2_+e2_.*x_]),x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d1,e1,d2,e2,f},x] &&  EqQ[e1-c*d1,0] && EqQ[e2+c*d2,0] && GtQ[n,0] && GtQ[m,1] &&  IntegerQ[m] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^n/(e*m) - b*f*n*Sqrt[1 + c^2*x^2]/(c*m*Sqrt[d + e*x^2])* Int[(f*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n - 1), x] - f^2*(m - 1)/(c^2*m)* Int[((f*x)^(m - 2)*(a + b*ArcSinh[c*x])^n)/Sqrt[d + e*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcSinh[c_.*x_])^n_./Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "f*(f*x)^(m - 1)*Sqrt[d1 + e1*x]* Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n/(e1*e2*m) + b*f*n*Sqrt[d1 + e1*x]* Sqrt[d2 + e2*x]/(c*d1*d2*m*Sqrt[1 + c*x]*Sqrt[-1 + c*x])* Int[(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n - 1), x] + f^2*(m - 1)/(c^2*m)* Int[(f*x)^(m - 2)*(a + b*ArcCosh[c*x])^ n/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^ m_*(a_. + b_.*ArcCosh[c_.*x_])^ n_./(Sqrt[d1_ + e1_.*x_]*Sqrt[d2_ + e2_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "1/(c^(m + 1)*Sqrt[d])* Subst[Int[(a + b*x)^n*Sinh[x]^m, x], x, ArcSinh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*(a_. + b_.*ArcSinh[c_.*x_])^n_./Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0] && IGtQ[n, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "1/(c^(m + 1)*Sqrt[-d1*d2])* Subst[Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*(a_. + b_.*ArcCosh[c_.*x_])^ n_./(Sqrt[d1_ + e1_.*x_]*Sqrt[d2_ + e2_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[n, 0] && GtQ[d1, 0] && LtQ[d2, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^(m + 1)*(a + b*ArcSinh[c*x])* Hypergeometric2F1[ 1/2, (1 + m)/2, (3 + m)/2, -c^2*x^2]/(Sqrt[d]*f*(m + 1)) - b*c*(f*x)^(m + 2)* HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, -c^2*x^2]/(Sqrt[d]*f^2*(m + 1)*(m + 2))",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcSinh[c_.*x_])/Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[d, 0] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^(m + 1)* Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])* Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2]/ (f*(m + 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]) + b*c*(f*x)^(m + 2)* HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2]/(Sqrt[-d1*d2]*f^2*(m + 1)*(m + 2))",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^ m_*(a_. + b_.*ArcCosh[c_.*x_])/(Sqrt[d1_ + e1_.*x_]* Sqrt[d2_ + e2_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[d1, 0] && LtQ[d2, 0] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]* Int[(f*x)^m*(a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcSinh[c_.*x_])^n_./Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && Not[GtQ[d, 0]] && (IntegerQ[m] || EqQ[n, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "Sqrt[1 + c*x]*Sqrt[-1 + c*x]/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])* Int[(f*x)^m*(a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^ m_*(a_. + b_.*ArcCosh[c_.*x_])^ n_./(Sqrt[d1_ + e1_.*x_]*Sqrt[d2_ + e2_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && Not[GtQ[d1, 0] && LtQ[d2, 0]] && (IntegerQ[m] || EqQ[n, 1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": " f*(f*x)^(m-1)*(d+e*x^2)^(p+1)*(a+b*ArcSinh[c*x])^n/(e*(m+2*p+1)) - f^2*(m-1)/(c^2*(m+2*p+1))*Int[(f*x)^(m-2)*(d+e*x^2)^p*(a+b*ArcSinh[ c*x])^n,x] - b*f*n*d^p/(c*(m+2*p+1))*Int[(f*x)^(m-1)*(1+c^2*x^2)^(p+1/2)*(a+b* ArcSinh[c*x])^(n-1),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_+e_.*x_^2)^p_*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,p},x] && EqQ[e,c^2*d] &&  GtQ[n,0] && GtQ[m,1] && NeQ[m+2*p+1,0] && (IntegerQ[p] || GtQ[d,0]) &&  IntegerQ[m] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^ n/(e*(m + 2*p + 1)) + f^2*(m - 1)/(c^2*(m + 2*p + 1))* Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x] - b*f*n*(-d)^p/(c*(m + 2*p + 1))* Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[p] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^ n/(e*(m + 2*p + 1)) - f^2*(m - 1)/(c^2*(m + 2*p + 1))* Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x] - b*f*n* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/(c*(m + 2*p + 1)*(1 + c^2*x^2)^FracPart[p])* Int[(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n/(e1*e2*(m + 2*p + 1)) + f^2*(m - 1)/(c^2*(m + 2*p + 1))* Int[(f*x)^(m - 2)*(d1 + e1*x)^p*(d2 + e2*x)^ p*(a + b*ArcCosh[c*x])^n, x] - b*f*n*(-d1*d2)^IntPart[p]*(d1 + e1*x)^ FracPart[p]*(d2 + e2*x)^ FracPart[ p]/(c*(m + 2*p + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^ FracPart[p])* Int[(f*x)^(m - 1)*(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[m] && IntegerQ[p + 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n/(e1*e2*(m + 2*p + 1)) + f^2*(m - 1)/(c^2*(m + 2*p + 1))* Int[(f*x)^(m - 2)*(d1 + e1*x)^p*(d2 + e2*x)^ p*(a + b*ArcCosh[c*x])^n, x] - b*f*n*(-d1*d2)^IntPart[p]*(d1 + e1*x)^ FracPart[p]*(d2 + e2*x)^ FracPart[ p]/(c*(m + 2*p + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^ FracPart[p])* Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": " d^p*(f*x)^m*(1+c^2*x^2)^(p+1/2)*(a+b*ArcSinh[c*x])^(n+1)/(b*c*(n+1)) - f*m*d^p/(b*c*(n+1))*Int[(f*x)^(m-1)*(1+c^2*x^2)^(p-1/2)*(a+b* ArcSinh[c*x])^(n+1),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,m,p},x] && EqQ[e,c^2*d] &&  LtQ[n,-1] && EqQ[m+2*p+1,0] && (IntegerQ[p] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^m*Sqrt[1 + c*x]* Sqrt[-1 + c*x]*(d + e*x^2)^ p*(a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)) + f*m*(-d)^p/(b*c*(n + 1))* Int[(f*x)^(m - 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && EqQ[m + 2*p + 1, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^m* Sqrt[1 + c^2*x^2]*(d + e*x^2)^ p*(a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1)) - f*m*d^ IntPart[p]*(d + e*x^2)^ FracPart[p]/(b*c*(n + 1)*(1 + c^2*x^2)^FracPart[p])* Int[(f*x)^(m - 1)*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[e, c^2*d] && LtQ[n, -1] && EqQ[m + 2*p + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^m*Sqrt[1 + c*x]* Sqrt[-1 + c*x]*(d1 + e1*x)^p*(d2 + e2*x)^ p*(a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)) + f*m*(-d1*d2)^IntPart[p]*(d1 + e1*x)^ FracPart[p]*(d2 + e2*x)^ FracPart[p]/(b* c*(n + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p])* Int[(f*x)^(m - 1)*(-1 + c^2*x^2)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d1_ + e1_.*x_)^p_.*(d2_ + e2_.*x_)^ p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, m, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && LtQ[n, -1] && EqQ[m + 2*p + 1, 0] && IntegerQ[p - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^m*Sqrt[1 + c*x]* Sqrt[-1 + c*x]*(d1 + e1*x)^p*(d2 + e2*x)^ p*(a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)) + f*m*(-d1*d2)^IntPart[p]*(d1 + e1*x)^ FracPart[p]*(d2 + e2*x)^ FracPart[p]/(b* c*(n + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p])* Int[(f*x)^(m - 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d1_ + e1_.*x_)^p_.*(d2_ + e2_.*x_)^ p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, m, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && LtQ[n, -1] && EqQ[m + 2*p + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^ m*(a + b*ArcSinh[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1)) - f*m/(b*c*Sqrt[d]*(n + 1))* Int[(f*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(a_. + b_.*ArcSinh[c_.*x_])^n_/Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && LtQ[n, -1] && GtQ[d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^ m*(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-d1*d2]*(n + 1)) - (f*m)/(b*c*Sqrt[-d1*d2]*(n + 1))* Int[(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^ m_.*(a_. + b_.*ArcCosh[c_.*x_])^ n_/(Sqrt[d1_ + e1_.*x_]*Sqrt[d2_ + e2_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && LtQ[n, -1] && GtQ[d1, 0] && LtQ[d2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": " Sqrt[1+c^2*x^2]/Sqrt[d+e*x^2]*Int[(f*x)^m*(a+b*ArcSinh[c*x])^n/Sqrt[1+ c^2*x^2],x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(a_.+b_.*ArcSinh[c_.*x_])^n_/Sqrt[d_+e_.*x_^2],x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f,m},x] && EqQ[e,c^2*d] && LtQ[n,-1] &&  Not[GtQ[d,0]] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": " Sqrt[1+c*x]*Sqrt[-1+c*x]/(Sqrt[d1+e1*x]*Sqrt[d2+e2*x])*Int[(f*x)^m*(a+ b*ArcCosh[c*x])^n/(Sqrt[1+c*x]*Sqrt[-1+c*x]),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(a_.+b_.*ArcCosh[c_.*x_])^n_/(Sqrt[d1_+e1_.*x_]* Sqrt[d2_+e2_.*x_]),x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c,d1,e1,d2,e2,f,m},x] && EqQ[e1-c*d1,0] && EqQ[e2+c*d2,0] &&  LtQ[n,-1] && Not[GtQ[d1,0] && LtQ[d2,0]] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": " d^p*(f*x)^m*(1+c^2*x^2)^(p+1/2)*(a+b*ArcSinh[c*x])^(n+1)/(b*c*(n+1)) - f*m*d^p/(b*c*(n+1))*Int[(f*x)^(m-1)*(1+c^2*x^2)^(p-1/2)*(a+b* ArcSinh[c*x])^(n+1),x] - c*d^p*(m+2*p+1)/(b*f*(n+1))*Int[(f*x)^(m+1)*(1+c^2*x^2)^(p-1/2)*(a+ b*ArcSinh[c*x])^(n+1),x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_+e_.*x_^2)^p_.*(a_.+b_.*ArcSinh[c_.*x_])^n_,x_ Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,f},x] && EqQ[e,c^2*d] &&  LtQ[n,-1] && IGtQ[m,-3] && IGtQ[2*p,0] && (IntegerQ[p] || GtQ[d,0]) *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^m*Sqrt[1 + c*x]* Sqrt[-1 + c*x]*(d + e*x^2)^ p*(a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)) + f*m*(-d)^p/(b*c*(n + 1))* Int[(f*x)^(m - 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x] - c*(-d)^p*(m + 2*p + 1)/(b*f*(n + 1))* Int[(f*x)^(m + 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IGtQ[m, -3] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^m* Sqrt[1 + c^2*x^2]*(d + e*x^2)^ p*(a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1)) - f*m*d^ IntPart[p]*(d + e*x^2)^ FracPart[p]/(b*c*(n + 1)*(1 + c^2*x^2)^FracPart[p])* Int[(f*x)^(m - 1)*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n + 1), x] - c*(m + 2*p + 1)* d^IntPart[p]*(d + e*x^2)^ FracPart[p]/(b*f*(n + 1)*(1 + c^2*x^2)^FracPart[p])* Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && LtQ[n, -1] && IGtQ[m, -3] && IGtQ[2*p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(f*x)^m*Sqrt[1 + c*x]* Sqrt[-1 + c*x]*(d1 + e1*x)^p*(d2 + e2*x)^ p*(a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)) + f*m*(-d1*d2)^IntPart[p]*(d1 + e1*x)^ FracPart[p]*(d2 + e2*x)^ FracPart[p]/(b* c*(n + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p])* Int[(f*x)^(m - 1)*(-1 + c^2*x^2)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x] - c*(m + 2*p + 1)*(-d1*d2)^IntPart[p]*(d1 + e1*x)^ FracPart[p]*(d2 + e2*x)^ FracPart[p]/(b* f*(n + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p])* Int[(f*x)^(m + 1)*(-1 + c^2*x^2)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d1_ + e1_.*x_)^p_.*(d2_ + e2_.*x_)^ p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && LtQ[n, -1] && IGtQ[m, -3] && IGtQ[p + 1/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "d^p/c^(m + 1)* Subst[Int[(a + b*x)^n*Sinh[x]^m*Cosh[x]^(2*p + 1), x], x, ArcSinh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(-d)^p/c^(m + 1)* Subst[Int[(a + b*x)^n*Cosh[x]^m*Sinh[x]^(2*p + 1), x], x, ArcCosh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(-d1*d2)^p/c^(m + 1)* Subst[Int[(a + b*x)^n*Cosh[x]^m*Sinh[x]^(2*p + 1), x], x, ArcCosh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d1_ + e1_.*x_)^p_.*(d2_ + e2_.*x_)^ p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p + 1/2] && GtQ[p, -1] && IGtQ[m, 0] && (GtQ[d1, 0] && LtQ[d2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "d^IntPart[p]*(d + e*x^2)^FracPart[p]/(1 + c^2*x^2)^FracPart[p]* Int[x^m*(1 + c^2*x^2)^p*(a + b*ArcSinh[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && Not[(IntegerQ[p] || GtQ[d, 0])]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(-d1*d2)^IntPart[p]*(d1 + e1*x)^ FracPart[p]*(d2 + e2*x)^ FracPart[p]/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p])* Int[x^m*(1 + c*x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d1_ + e1_.*x_)^p_.*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && Not[IntegerQ[p] || GtQ[d1, 0] && LtQ[d2, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcSinh[c*x])^n/ Sqrt[d + e*x^2], (f*x)^m*(d + e*x^2)^(p + 1/2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[e, c^2*d] && GtQ[d, 0] && IGtQ[p + 1/2, 0] && Not[IGtQ[(m + 1)/2, 0]] && (EqQ[m, -1] || EqQ[m, -2])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^ n/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), (f*x)^ m*(d1 + e1*x)^(p + 1/2)*(d2 + e2*x)^(p + 1/2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, m, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[p + 1/2, 0] && Not[IGtQ[(m + 1)/2, 0]] && (EqQ[m, -1] || EqQ[m, -2])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "d*(f*x)^(m + 1)*(a + b*ArcCosh[c*x])/(f*(m + 1)) + e*(f*x)^(m + 3)*(a + b*ArcCosh[c*x])/(f^3*(m + 3)) - b*c/(f*(m + 1)*(m + 3))* Int[(f*x)^(m + 1)*(d*(m + 3) + e*(m + 1)*x^2)/(Sqrt[1 + c*x]* Sqrt[-1 + c*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)*(a_. + b_.*ArcCosh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && NeQ[m, -1] && NeQ[m, -3]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])/(2* e*(p + 1)) - b*c/(2*e*(p + 1))*Int[(d + e*x^2)^(p + 1)/Sqrt[1 + c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && NeQ[e, c^2*d] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])/(2* e*(p + 1)) - b*c/(2*e*(p + 1))* Int[(d + e*x^2)^(p + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && NeQ[c^2*d + e, 0] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[e, c^2*d] && IntegerQ[p] && (GtQ[p, 0] || IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcSinh[c*x])^n, (f*x)^m*(d + e*x^2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[e, c^2*d] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^n, (f*x)^m*(d + e*x^2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && NeQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "Unintegrable[(f*x)^m*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "Unintegrable[(f*x)^m*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "Unintegrable[(f*x)^m*(d1 + e1*x)^p*(d2 + e2*x)^ p*(a + b*ArcCosh[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d1_ + e1_.*x_)^p_.*(d2_ + e2_.*x_)^ p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, m, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(-d^2*g/e)^q* Int[(h*x)^m*(d + e*x)^(p - q)*(1 + c^2*x^2)^q*(a + b*ArcSinh[c*x])^ n, x]",
     "rulenumber": 0,
            "lhs": "Int[(h_.*x_)^m_.*(d_ + e_.*x_)^p_*(f_ + g_.*x_)^ q_*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 + e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0] && GtQ[d, 0] && LtQ[g/e, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(-d^2*g/e)^IntPart[q]*(d + e*x)^ FracPart[q]*(f + g*x)^FracPart[q]/(1 + c^2*x^2)^FracPart[q]* Int[(h*x)^m*(d + e*x)^(p - q)*(1 + c^2*x^2)^ q*(a + b*ArcSinh[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(h_.*x_)^m_.*(d_ + e_.*x_)^p_*(f_ + g_.*x_)^ q_*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 + e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
       "filename": "7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m",
            "rhs": "(-d)^ IntPart[p]*(d + e*x^2)^ FracPart[p]/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p])* Int[(f*x)^m*(1 + c*x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[c^2*d + e, 0] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "Subst[Int[(a + b*x)^n*Cosh[x]/(c*d + e*Sinh[x]), x], x, ArcSinh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSinh[c_.*x_])^n_./(d_. + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "Subst[Int[(a + b*x)^n*Sinh[x]/(c*d + e*Cosh[x]), x], x, ArcCosh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCosh[c_.*x_])^n_./(d_. + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "(d + e*x)^(m + 1)*(a + b*ArcSinh[c*x])^ n/(e*(m + 1)) - b*c*n/(e*(m + 1))* Int[(d + e*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1)/ Sqrt[1 + c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "(d + e*x)^(m + 1)*(a + b*ArcCosh[c*x])^ n/(e*(m + 1)) - b*c*n/(e*(m + 1))* Int[(d + e*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[-1 + c*x]* Sqrt[1 + c*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x)^m*(a + b*ArcSinh[c*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_.*(a_. + b_.*ArcSinh[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[m, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x)^m*(a + b*ArcCosh[c*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_.*(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[m, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "1/c^(m + 1)* Subst[Int[(a + b*x)^n*Cosh[x]*(c*d + e*Sinh[x])^m, x], x, ArcSinh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "1/c^(m + 1)* Subst[Int[(a + b*x)^n*(c*d + e*Cosh[x])^m*Sinh[x], x], x, ArcCosh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = IntHide[ExpandExpression[Px, x], x]}, Dist[a + b*ArcSinh[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_. + b_.*ArcSinh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && PolynomialQ[Px, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = IntHide[ExpandExpression[Px, x], x]}, Dist[a + b*ArcCosh[c*x], u, x] - b*c*Sqrt[1 - c^2*x^2]/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])* Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_. + b_.*ArcCosh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && PolynomialQ[Px, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": " With[{u=IntHide[Px,x]}, Dist[(a+b*ArcSinh[c*x])^n,u,x] -  b*c*n*Int[SimplifyIntegrand[u*(a+b*ArcSinh[c*x])^(n-1)/Sqrt[1+c^2*x^2] ,x],x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_.+b_.*ArcSinh[c_.*x_])^n_.,x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c},x] && PolynomialQ[Px,x] && IGtQ[n,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": " With[{u=IntHide[Px,x]}, Dist[(a+b*ArcCosh[c*x])^n,u,x] - b*c*n*Sqrt[1-c^2*x^2]/(Sqrt[-1+c*x]*Sqrt[1+c*x])*Int[ SimplifyIntegrand[u*(a+b*ArcCosh[c*x])^(n-1)/Sqrt[1-c^2*x^2],x],x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_.+b_.*ArcCosh[c_.*x_])^n_.,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,c},x] && PolynomialQ[Px,x] && IGtQ[n,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[Px*(a + b*ArcSinh[c*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_. + b_.*ArcSinh[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && PolynomialQ[Px, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[Px*(a + b*ArcCosh[c*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && PolynomialQ[Px, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = IntHide[Px*(d + e*x)^m, x]}, Dist[a + b*ArcSinh[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_. + e_.*x_)^m_.*(a_. + b_.*ArcSinh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && PolynomialQ[Px, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = IntHide[Px*(d + e*x)^m, x]}, Dist[a + b*ArcCosh[c*x], u, x] - b*c*Sqrt[1 - c^2*x^2]/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])* Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_. + e_.*x_)^m_.*(a_. + b_.*ArcCosh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && PolynomialQ[Px, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = IntHide[(f + g*x)^p*(d + e*x)^m, x]}, Dist[(a + b*ArcSinh[c*x])^n, u, x] - b*c*n*Int[ SimplifyIntegrand[ u*(a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^p_.*(d_ + e_.*x_)^m_*(a_. + b_.*ArcSinh[c_.*x_])^ n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[m, 0] && LtQ[m + p + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = IntHide[(f + g*x)^p*(d + e*x)^m, x]}, Dist[(a + b*ArcCosh[c*x])^n, u, x] - b*c*n*Int[ SimplifyIntegrand[ u*(a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^p_.*(d_ + e_.*x_)^m_*(a_. + b_.*ArcCosh[c_.*x_])^ n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[m, 0] && LtQ[m + p + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = IntHide[(f + g*x + h*x^2)^p/(d + e*x)^2, x]}, Dist[(a + b*ArcSinh[c*x])^n, u, x] - b*c*n*Int[ SimplifyIntegrand[ u*(a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2], x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_ + h_.*x_^2)^ p_.*(a_. + b_.*ArcSinh[c_.*x_])^n_/(d_ + e_.*x_)^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h}, x] && IGtQ[n, 0] && IGtQ[p, 0] && EqQ[e*g - 2*d*h, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = IntHide[(f + g*x + h*x^2)^p/(d + e*x)^2, x]}, Dist[(a + b*ArcCosh[c*x])^n, u, x] - b*c*n*Int[ SimplifyIntegrand[ u*(a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_ + h_.*x_^2)^ p_.*(a_. + b_.*ArcCosh[c_.*x_])^n_/(d_ + e_.*x_)^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h}, x] && IGtQ[n, 0] && IGtQ[p, 0] && EqQ[e*g - 2*d*h, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[Px*(d + e*x)^m*(a + b*ArcSinh[c*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_ + e_.*x_)^m_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && PolynomialQ[Px, x] && IGtQ[n, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[Px*(d + e*x)^m*(a + b*ArcCosh[c*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_ + e_.*x_)^m_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && PolynomialQ[Px, x] && IGtQ[n, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = IntHide[(f + g*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - b*c*Int[Dist[1/Sqrt[1 + c^2*x^2], u, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && (LtQ[m, -2*p - 1] || GtQ[m, 3])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = IntHide[(f + g*x)^m*(d1 + e1*x)^p*(d2 + e2*x)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - b*c*Int[Dist[1/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), u, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[m, 0] && ILtQ[p + 1/2, 0] && GtQ[d1, 0] && LtQ[d2, 0] && (LtQ[m, -2*p - 1] || GtQ[m, 3])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, (f + g*x)^ m, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_])^ n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n, 0] && (EqQ[n, 1] && GtQ[p, -1] || GtQ[p, 0] || EqQ[m, 1] || EqQ[m, 2] && LtQ[p, -2])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^ n, (f + g*x)^m, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n, 0] && (EqQ[n, 1] && GtQ[p, -1] || GtQ[p, 0] || EqQ[m, 1] || EqQ[m, 2] && LtQ[p, -2])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "(f + g*x)^ m*(d + e*x^2)*(a + b*ArcSinh[c*x])^(n + 1)/(b*c* Sqrt[d]*(n + 1)) - 1/(b*c*Sqrt[d]*(n + 1))* Int[(d*g*m + 2*e*f*x + e*g*(m + 2)*x^2)*(f + g*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_. + g_.*x_)^m_* Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && ILtQ[m, 0] && GtQ[d, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "(f + g*x)^ m*(d1*d2 + e1*e2*x^2)*(a + b*ArcCosh[c*x])^(n + 1)/(b*c* Sqrt[-d1*d2]*(n + 1)) - 1/(b*c*Sqrt[-d1*d2]*(n + 1))* Int[(d1*d2*g*m + 2*e1*e2*f*x + e1*e2*g*(m + 2)*x^2)*(f + g*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_*Sqrt[d1_ + e1_.*x_]* Sqrt[d2_ + e2_.*x_]*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && ILtQ[m, 0] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[ Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^n, (f + g*x)^ m*(d + e*x^2)^(p - 1/2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_])^ n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IntegerQ[m] && IGtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[ Sqrt[d1 + e1*x]* Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n, (f + g*x)^ m*(d1 + e1*x)^(p - 1/2)*(d2 + e2*x)^(p - 1/2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[m] && IGtQ[p + 1/2, 0] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "(f + g*x)^ m*(d + e*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n + 1)/(b*c* Sqrt[d]*(n + 1)) - 1/(b*c*Sqrt[d]*(n + 1))* Int[ ExpandIntegrand[(f + g*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1), (d*g*m + e*f*(2*p + 1)*x + e*g*(m + 2*p + 1)*x^2)*(d + e*x^2)^(p - 1/2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_])^ n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && ILtQ[m, 0] && IGtQ[p - 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "(f + g*x)^ m*(d1 + e1*x)^(p + 1/2)*(d2 + e2*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n + 1)/(b*c* Sqrt[-d1*d2]*(n + 1)) - 1/(b*c*Sqrt[-d1*d2]*(n + 1))* Int[ ExpandIntegrand[(f + g*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), (d1*d2*g*m + e1*e2*f*(2*p + 1)*x + e1*e2*g*(m + 2*p + 1)*x^2)*(d1 + e1*x)^(p - 1/2)*(d2 + e2*x)^(p - 1/2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && ILtQ[m, 0] && IGtQ[p - 1/2, 0] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "(f + g*x)^ m*(a + b*ArcSinh[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1)) - g*m/(b*c*Sqrt[d]*(n + 1))* Int[(f + g*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(a_. + b_.*ArcSinh[c_.*x_])^n_/ Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && GtQ[d, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "(f + g*x)^ m*(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-d1*d2]*(n + 1)) - g*m/(b*c*Sqrt[-d1*d2]*(n + 1))* Int[(f + g*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^ m_.*(a_. + b_.*ArcCosh[c_.*x_])^ n_/(Sqrt[d1_ + e1_.*x_]*Sqrt[d2_ + e2_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[m, 0] && GtQ[d1, 0] && LtQ[d2, 0] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "1/(c^(m + 1)*Sqrt[d])* Subst[Int[(a + b*x)^n*(c*f + g*Sinh[x])^m, x], x, ArcSinh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(a_. + b_.*ArcSinh[c_.*x_])^n_./ Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e, c^2*d] && IntegerQ[m] && GtQ[d, 0] && (GtQ[m, 0] || IGtQ[n, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "1/(c^(m + 1)*Sqrt[-d1*d2])* Subst[Int[(a + b*x)^n*(c*f + g*Cosh[x])^m, x], x, ArcCosh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^ m_.*(a_. + b_.*ArcCosh[c_.*x_])^ n_./(Sqrt[d1_ + e1_.*x_]*Sqrt[d2_ + e2_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, g, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[m] && GtQ[d1, 0] && LtQ[d2, 0] && (GtQ[m, 0] || IGtQ[n, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcSinh[c*x])^n/ Sqrt[d + e*x^2], (f + g*x)^m*(d + e*x^2)^(p + 1/2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_])^ n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IntegerQ[m] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^ n/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), (f + g*x)^ m*(d1 + e1*x)^(p + 1/2)*(d2 + e2*x)^(p + 1/2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[m] && ILtQ[p + 1/2, 0] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "d^IntPart[p]*(d + e*x^2)^FracPart[p]/(1 + c^2*x^2)^FracPart[p]* Int[(f + g*x)^m*(1 + c^2*x^2)^p*(a + b*ArcSinh[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_])^ n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e, c^2*d] && IntegerQ[m] && IntegerQ[p - 1/2] && Not[GtQ[d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "(-d)^ IntPart[p]*(d + e*x^2)^ FracPart[p]/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p])* Int[(f + g*x)^m*(1 + c*x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^ n, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcCosh[c_.*x_])^ n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "(-d1*d2)^IntPart[p]*(d1 + e1*x)^ FracPart[p]*(d2 + e2*x)^FracPart[p]/(1 - c^2*x^2)^FracPart[p]* Int[(f + g*x)^m*(1 + c*x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^ n, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, g, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[m] && IntegerQ[p - 1/2] && Not[GtQ[d1, 0] && LtQ[d2, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "Log[h*(f + g*x)^m]*(a + b*ArcSinh[c*x])^(n + 1)/(b*c* Sqrt[d]*(n + 1)) - g*m/(b*c*Sqrt[d]*(n + 1))* Int[(a + b*ArcSinh[c*x])^(n + 1)/(f + g*x), x]",
     "rulenumber": 0,
            "lhs": "Int[Log[h_.*(f_. + g_.*x_)^m_.]*(a_. + b_.*ArcSinh[c_.*x_])^n_./ Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m}, x] && EqQ[e, c^2*d] && GtQ[d, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "Log[h*(f + g*x)^m]*(a + b*ArcCosh[c*x])^(n + 1)/(b*c* Sqrt[-d1*d2]*(n + 1)) - g*m/(b*c*Sqrt[-d1*d2]*(n + 1))* Int[(a + b*ArcCosh[c*x])^(n + 1)/(f + g*x), x]",
     "rulenumber": 0,
            "lhs": "Int[Log[h_.*(f_. + g_.*x_)^m_.]*(a_. + b_.*ArcCosh[c_.*x_])^ n_./(Sqrt[d1_ + e1_.*x_]*Sqrt[d2_ + e2_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, g, h, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "d^IntPart[p]*(d + e*x^2)^FracPart[p]/(1 + c^2*x^2)^FracPart[p]* Int[Log[h*(f + g*x)^m]*(1 + c^2*x^2)^p*(a + b*ArcSinh[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[Log[h_.*(f_. + g_.*x_)^m_.]*(d_ + e_.*x_^2)^ p_*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e, c^2*d] && IntegerQ[p - 1/2] && Not[GtQ[d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "(-d)^ IntPart[p]*(d + e*x^2)^ FracPart[p]/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p])* Int[ Log[h*(f + g*x)^m]*(1 + c*x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^ n, x]",
     "rulenumber": 0,
            "lhs": "Int[Log[h_.*(f_. + g_.*x_)^m_.]*(d_ + e_.*x_^2)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "(-d1*d2)^IntPart[p]*(d1 + e1*x)^ FracPart[p]*(d2 + e2*x)^ FracPart[p]/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p])* Int[ Log[h*(f + g*x)^m]*(1 + c*x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^ n, x]",
     "rulenumber": 0,
            "lhs": "Int[Log[h_.*(f_. + g_.*x_)^m_.]*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, g, h, m, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p - 1/2] && Not[GtQ[d1, 0] && LtQ[d2, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = IntHide[(d + e*x)^m*(f + g*x)^m, x]}, Dist[a + b*ArcSinh[c*x], u, x] - b*c*Int[Dist[1/Sqrt[1 + c^2*x^2], u, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(f_ + g_.*x_)^m_*(a_. + b_.*ArcSinh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && ILtQ[m + 1/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = IntHide[(d + e*x)^m*(f + g*x)^m, x]}, Dist[a + b*ArcCosh[c*x], u, x] - b*c*Int[Dist[1/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), u, x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_*(f_ + g_.*x_)^m_*(a_. + b_.*ArcCosh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && ILtQ[m + 1/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcSinh[c*x])^n, (d + e*x)^m*(f + g*x)^m, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_.*(f_ + g_.*x_)^m_.*(a_. + b_.*ArcSinh[c_.*x_])^ n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n}, x] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^n, (d + e*x)^m*(f + g*x)^m, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^m_.*(f_ + g_.*x_)^m_.*(a_. + b_.*ArcCosh[c_.*x_])^ n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, n}, x] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "With[{v = IntHide[u, x]}, Dist[a + b*ArcSinh[c*x], v, x] - b*c*Int[SimplifyIntegrand[v/Sqrt[1 + c^2*x^2], x], x] /; InverseFunctionFreeQ[v, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*(a_. + b_.*ArcSinh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "With[{v = IntHide[u, x]}, Dist[a + b*ArcCosh[c*x], v, x] - b*c*Sqrt[1 - c^2*x^2]/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])* Int[SimplifyIntegrand[v/Sqrt[1 - c^2*x^2], x], x] /; InverseFunctionFreeQ[v, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*(a_. + b_.*ArcCosh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = ExpandIntegrand[Px*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d_ + e_.*x_^2)^p_*(a_. + b_.*ArcSinh[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && PolynomialQ[Px, x] && EqQ[e, c^2*d] && IntegerQ[p - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = ExpandIntegrand[ Px*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[Px_*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_. + b_.*ArcCosh[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && PolynomialQ[Px, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = ExpandIntegrand[ Px*(f + g*(d + e*x^2)^p)^m*(a + b*ArcSinh[c*x])^n, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[Px_.*(f_ + g_.*(d_ + e_.*x_^2)^p_)^ m_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && PolynomialQ[Px, x] && EqQ[e, c^2*d] && IGtQ[p + 1/2, 0] && IntegersQ[m, n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = ExpandIntegrand[ Px*(f + g*(d1 + e1*x)^p*(d2 + e2*x)^p)^m*(a + b*ArcCosh[c*x])^n, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[Px_.*(f_ + g_.*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^p_)^ m_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && PolynomialQ[Px, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[p + 1/2, 0] && IntegersQ[m, n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = ExpandIntegrand[ArcSinh[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[RFx_*ArcSinh[c_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[c, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = ExpandIntegrand[ArcCosh[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[RFx_*ArcCosh[c_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[c, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[RFx*(a + b*ArcSinh[c*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[RFx_*(a_ + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[RFx*(a + b*ArcCosh[c*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[RFx_*(a_ + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = ExpandIntegrand[(d + e*x^2)^p*ArcSinh[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[RFx_*(d_ + e_.*x_^2)^p_*ArcSinh[c_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[e, c^2*d] && IntegerQ[p - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "With[{u = ExpandIntegrand[(d1 + e1*x)^p*(d2 + e2*x)^p*ArcCosh[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[RFx_*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^p_*ArcCosh[c_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d1, e1, d2, e2}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(d + e*x^2)^p, RFx*(a + b*ArcSinh[c*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[RFx_*(d_ + e_.*x_^2)^p_*(a_ + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[e, c^2*d] && IntegerQ[p - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "Int[ExpandIntegrand[(d1 + e1*x)^p*(d2 + e2*x)^p, RFx*(a + b*ArcCosh[c*x])^n, x], x]",
     "rulenumber": 0,
            "lhs": "Int[RFx_*(d1_ + e1_.*x_)^p_*(d2_ + e2_.*x_)^ p_*(a_ + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d1, e1, d2, e2}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "Unintegrable[u*(a + b*ArcSinh[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_. + b_.*ArcSinh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 u (a+b arcsinh(c x))^n.m",
       "filename": "7.1.5 u (a+b arcsinh(c x))^n.m",
            "rhs": "Unintegrable[u*(a + b*ArcCosh[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_. + b_.*ArcCosh[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "1/d*Subst[Int[(a + b*ArcSinh[x])^n, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSinh[c_ + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "1/d*Subst[Int[(a + b*ArcCosh[x])^n, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCosh[c_ + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "1/d*Subst[Int[((d*e - c*f)/d + f*x/d)^m*(a + b*ArcSinh[x])^n, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcSinh[c_ + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "1/d*Subst[Int[((d*e - c*f)/d + f*x/d)^m*(a + b*ArcCosh[x])^n, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCosh[c_ + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "1/d*Subst[Int[(C/d^2 + C/d^2*x^2)^p*(a + b*ArcSinh[x])^n, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*x_ + C_.*x_^2)^p_.*(a_. + b_.*ArcSinh[c_ + d_.*x_])^ n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, A, B, C, n, p}, x] && EqQ[B*(1 + c^2) - 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "1/d*Subst[Int[(-C/d^2 + C/d^2*x^2)^p*(a + b*ArcCosh[x])^n, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(A_. + B_.*x_ + C_.*x_^2)^p_.*(a_. + b_.*ArcCosh[c_ + d_.*x_])^ n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, A, B, C, n, p}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "1/d*Subst[ Int[((d*e - c*f)/d + f*x/d)^m*(C/d^2 + C/d^2*x^2)^ p*(a + b*ArcSinh[x])^n, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(A_. + B_.*x_ + C_.*x_^2)^ p_.*(a_. + b_.*ArcSinh[c_ + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, m, n, p}, x] && EqQ[B*(1 + c^2) - 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "1/d*Subst[ Int[((d*e - c*f)/d + f*x/d)^m*(-C/d^2 + C/d^2*x^2)^ p*(a + b*ArcCosh[x])^n, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(A_. + B_.*x_ + C_.*x_^2)^ p_.*(a_. + b_.*ArcCosh[c_ + d_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, A, B, C, m, n, p}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "x*Sqrt[a + b*ArcSinh[c + d*x^2]] - Sqrt[Pi]*x*(Cosh[a/(2*b)] - c*Sinh[a/(2*b)])* FresnelC[Sqrt[-c/(Pi*b)]*Sqrt[a + b*ArcSinh[c + d*x^2]]]/ (Sqrt[-(c/b)]*(Cosh[ArcSinh[c + d*x^2]/2] + c*Sinh[ArcSinh[c + d*x^2]/2])) + Sqrt[Pi]*x*(Cosh[a/(2*b)] + c*Sinh[a/(2*b)])* FresnelS[Sqrt[-c/(Pi*b)]*Sqrt[a + b*ArcSinh[c + d*x^2]]]/ (Sqrt[-(c/b)]*(Cosh[ArcSinh[c + d*x^2]/2] + c*Sinh[ArcSinh[c + d*x^2]/2]))",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_. + b_.*ArcSinh[c_ + d_.*x_^2]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[c^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "x*(a + b*ArcSinh[c + d*x^2])^n - 2*b*n* Sqrt[2*c*d*x^2 + d^2*x^4]*(a + b*ArcSinh[c + d*x^2])^(n - 1)/(d*x) + 4*b^2*n*(n - 1)*Int[(a + b*ArcSinh[c + d*x^2])^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSinh[c_ + d_.*x_^2])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[c^2, -1] && GtQ[n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "x*(c*Cosh[a/(2*b)] - Sinh[a/(2*b)])* CoshIntegral[(a + b*ArcSinh[c + d*x^2])/(2*b)]/ (2* b*(Cosh[ArcSinh[c + d*x^2]/2] + c*Sinh[(1/2)*ArcSinh[c + d*x^2]])) + x*(Cosh[a/(2*b)] - c*Sinh[a/(2*b)])* SinhIntegral[(a + b*ArcSinh[c + d*x^2])/(2*b)]/ (2* b*(Cosh[ArcSinh[c + d*x^2]/2] + c*Sinh[(1/2)*ArcSinh[c + d*x^2]]))",
     "rulenumber": 0,
            "lhs": "Int[1/(a_. + b_.*ArcSinh[c_ + d_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[c^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "(c + 1)*Sqrt[Pi/2]* x*(Cosh[a/(2*b)] - Sinh[a/(2*b)])* Erfi[Sqrt[a + b*ArcSinh[c + d*x^2]]/Sqrt[2*b]]/ (2* Sqrt[b]*(Cosh[ArcSinh[c + d*x^2]/2] + c*Sinh[ArcSinh[c + d*x^2]/2])) + (c - 1)*Sqrt[Pi/2]*x*(Cosh[a/(2*b)] + Sinh[a/(2*b)])* Erf[Sqrt[a + b*ArcSinh[c + d*x^2]]/Sqrt[2*b]]/ (2* Sqrt[b]*(Cosh[ArcSinh[c + d*x^2]/2] + c*Sinh[ArcSinh[c + d*x^2]/2]))",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_. + b_.*ArcSinh[c_ + d_.*x_^2]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[c^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "-Sqrt[2*c*d*x^2 + d^2*x^4]/(b*d*x* Sqrt[a + b*ArcSinh[c + d*x^2]]) - (-c/b)^(3/2)*Sqrt[Pi]*x*(Cosh[a/(2*b)] - c*Sinh[a/(2*b)])* FresnelC[Sqrt[-c/(Pi*b)]*Sqrt[a + b*ArcSinh[c + d*x^2]]]/ (Cosh[ArcSinh[c + d*x^2]/2] + c*Sinh[ArcSinh[c + d*x^2]/2]) + (-c/b)^(3/2)*Sqrt[Pi]*x*(Cosh[a/(2*b)] + c*Sinh[a/(2*b)])* FresnelS[Sqrt[-c/(Pi*b)]*Sqrt[a + b*ArcSinh[c + d*x^2]]]/ (Cosh[ArcSinh[c + d*x^2]/2] + c*Sinh[ArcSinh[c + d*x^2]/2])",
     "rulenumber": 0,
            "lhs": "Int[1/(a_. + b_.*ArcSinh[c_ + d_.*x_^2])^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[c^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "-Sqrt[2*c*d*x^2 + d^2*x^4]/(2*b*d* x*(a + b*ArcSinh[c + d*x^2])) + x*(Cosh[a/(2*b)] - c*Sinh[a/(2*b)])* CoshIntegral[(a + b*ArcSinh[c + d*x^2])/(2*b)]/ (4* b^2*(Cosh[ArcSinh[c + d*x^2]/2] + c*Sinh[ArcSinh[c + d*x^2]/2])) + x*(c*Cosh[a/(2*b)] - Sinh[a/(2*b)])* SinhIntegral[(a + b*ArcSinh[c + d*x^2])/(2*b)]/ (4* b^2*(Cosh[ArcSinh[c + d*x^2]/2] + c*Sinh[ArcSinh[c + d*x^2]/2]))",
     "rulenumber": 0,
            "lhs": "Int[1/(a_. + b_.*ArcSinh[c_ + d_.*x_^2])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[c^2, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "-x*(a + b*ArcSinh[c + d*x^2])^(n + 2)/(4* b^2*(n + 1)*(n + 2)) + Sqrt[ 2*c*d*x^2 + d^2*x^4]*(a + b*ArcSinh[c + d*x^2])^(n + 1)/(2*b*d*(n + 1)* x) + 1/(4*b^2*(n + 1)*(n + 2))* Int[(a + b*ArcSinh[c + d*x^2])^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSinh[c_ + d_.*x_^2])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[c^2, -1] && LtQ[n, -1] && NeQ[n, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "2*Sqrt[a + b*ArcCosh[1 + d*x^2]]* Sinh[(1/2)*ArcCosh[1 + d*x^2]]^2/(d*x) - Sqrt[b]*Sqrt[Pi/2]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])* Sinh[(1/2)*ArcCosh[1 + d*x^2]]* Erfi[(1/Sqrt[2*b])*Sqrt[a + b*ArcCosh[1 + d*x^2]]]/(d*x) + Sqrt[b]*Sqrt[Pi/2]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])* Sinh[(1/2)*ArcCosh[1 + d*x^2]]* Erf[(1/Sqrt[2*b])*Sqrt[a + b*ArcCosh[1 + d*x^2]]]/(d*x)",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_. + b_.*ArcCosh[1 + d_.*x_^2]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "2*Sqrt[a + b*ArcCosh[-1 + d*x^2]]* Cosh[(1/2)*ArcCosh[-1 + d*x^2]]^2/(d*x) - Sqrt[b]*Sqrt[Pi/2]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])* Cosh[(1/2)*ArcCosh[-1 + d*x^2]]* Erfi[(1/Sqrt[2*b])*Sqrt[a + b*ArcCosh[-1 + d*x^2]]]/(d*x) - Sqrt[b]*Sqrt[Pi/2]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])* Cosh[(1/2)*ArcCosh[-1 + d*x^2]]* Erf[(1/Sqrt[2*b])*Sqrt[a + b*ArcCosh[-1 + d*x^2]]]/(d*x)",
     "rulenumber": 0,
            "lhs": "Int[Sqrt[a_. + b_.*ArcCosh[-1 + d_.*x_^2]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "x*(a + b*ArcCosh[c + d*x^2])^n - 2*b*n*(2*c*d*x^2 + d^2*x^4)*(a + b*ArcCosh[c + d*x^2])^(n - 1)/(d*x* Sqrt[-1 + c + d*x^2]*Sqrt[1 + c + d*x^2]) + 4*b^2*n*(n - 1)*Int[(a + b*ArcCosh[c + d*x^2])^(n - 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCosh[c_ + d_.*x_^2])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1] && GtQ[n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "x*Cosh[a/(2*b)]* CoshIntegral[(a + b*ArcCosh[1 + d*x^2])/(2*b)]/(Sqrt[2]*b* Sqrt[d*x^2]) - x*Sinh[a/(2*b)]* SinhIntegral[(a + b*ArcCosh[1 + d*x^2])/(2*b)]/(Sqrt[2]*b* Sqrt[d*x^2])",
     "rulenumber": 0,
            "lhs": "Int[1/(a_. + b_.*ArcCosh[1 + d_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "-x*Sinh[a/(2*b)]* CoshIntegral[(a + b*ArcCosh[-1 + d*x^2])/(2*b)]/(Sqrt[2]*b* Sqrt[d*x^2]) + x*Cosh[a/(2*b)]* SinhIntegral[(a + b*ArcCosh[-1 + d*x^2])/(2*b)]/(Sqrt[2]*b* Sqrt[d*x^2])",
     "rulenumber": 0,
            "lhs": "Int[1/(a_. + b_.*ArcCosh[-1 + d_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "Sqrt[Pi/2]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])* Sinh[ArcCosh[1 + d*x^2]/2]* Erfi[Sqrt[a + b*ArcCosh[1 + d*x^2]]/Sqrt[2*b]]/(Sqrt[b]*d*x) + Sqrt[Pi/2]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])* Sinh[ArcCosh[1 + d*x^2]/2]* Erf[Sqrt[a + b*ArcCosh[1 + d*x^2]]/Sqrt[2*b]]/(Sqrt[b]*d*x)",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_. + b_.*ArcCosh[1 + d_.*x_^2]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "Sqrt[Pi/2]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])* Cosh[ArcCosh[-1 + d*x^2]/2]* Erfi[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/Sqrt[2*b]]/(Sqrt[b]*d*x) - Sqrt[Pi/2]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])* Cosh[ArcCosh[-1 + d*x^2]/2]* Erf[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/Sqrt[2*b]]/(Sqrt[b]*d*x)",
     "rulenumber": 0,
            "lhs": "Int[1/Sqrt[a_. + b_.*ArcCosh[-1 + d_.*x_^2]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "-Sqrt[d*x^2]* Sqrt[2 + d*x^2]/(b*d*x*Sqrt[a + b*ArcCosh[1 + d*x^2]]) + Sqrt[Pi/2]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])* Sinh[ArcCosh[1 + d*x^2]/2]* Erfi[Sqrt[a + b*ArcCosh[1 + d*x^2]]/Sqrt[2*b]]/(b^(3/2)*d*x) - Sqrt[Pi/2]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])* Sinh[ArcCosh[1 + d*x^2]/2]* Erf[Sqrt[a + b*ArcCosh[1 + d*x^2]]/Sqrt[2*b]]/(b^(3/2)*d*x)",
     "rulenumber": 0,
            "lhs": "Int[1/(a_. + b_.*ArcCosh[1 + d_.*x_^2])^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "-Sqrt[d*x^2]* Sqrt[-2 + d*x^2]/(b*d*x*Sqrt[a + b*ArcCosh[-1 + d*x^2]]) + Sqrt[Pi/2]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])* Cosh[ArcCosh[-1 + d*x^2]/2]* Erfi[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/Sqrt[2*b]]/(b^(3/2)*d*x) + Sqrt[Pi/2]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])* Cosh[ArcCosh[-1 + d*x^2]/2]* Erf[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/Sqrt[2*b]]/(b^(3/2)*d*x)",
     "rulenumber": 0,
            "lhs": "Int[1/(a_. + b_.*ArcCosh[-1 + d_.*x_^2])^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "-Sqrt[d*x^2]* Sqrt[2 + d*x^2]/(2*b*d*x*(a + b*ArcCosh[1 + d*x^2])) - x*Sinh[a/(2*b)]* CoshIntegral[(a + b*ArcCosh[1 + d*x^2])/(2*b)]/(2*Sqrt[2]*b^2* Sqrt[d*x^2]) + x*Cosh[a/(2*b)]* SinhIntegral[(a + b*ArcCosh[1 + d*x^2])/(2*b)]/(2*Sqrt[2]*b^2* Sqrt[d*x^2])",
     "rulenumber": 0,
            "lhs": "Int[1/(a_. + b_.*ArcCosh[1 + d_.*x_^2])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "-Sqrt[d*x^2]* Sqrt[-2 + d*x^2]/(2*b*d*x*(a + b*ArcCosh[-1 + d*x^2])) + x*Cosh[a/(2*b)]* CoshIntegral[(a + b*ArcCosh[-1 + d*x^2])/(2*b)]/(2*Sqrt[2]*b^2* Sqrt[d*x^2]) - x*Sinh[a/(2*b)]* SinhIntegral[(a + b*ArcCosh[-1 + d*x^2])/(2*b)]/(2*Sqrt[2]*b^2* Sqrt[d*x^2])",
     "rulenumber": 0,
            "lhs": "Int[1/(a_. + b_.*ArcCosh[-1 + d_.*x_^2])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "-x*(a + b*ArcCosh[c + d*x^2])^(n + 2)/(4* b^2*(n + 1)*(n + 2)) + (2*c*x^2 + d*x^4)*(a + b*ArcCosh[c + d*x^2])^(n + 1)/(2*b*(n + 1)*x* Sqrt[-1 + c + d*x^2]*Sqrt[1 + c + d*x^2]) + 1/(4*b^2*(n + 1)*(n + 2))* Int[(a + b*ArcCosh[c + d*x^2])^(n + 2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCosh[c_ + d_.*x_^2])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1] && LtQ[n, -1] && NeQ[n, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "1/p*Subst[Int[x^n*Coth[x], x], x, ArcSinh[a*x^p]]",
     "rulenumber": 0,
            "lhs": "Int[ArcSinh[a_.*x_^p_]^n_./x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, p}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "1/p*Subst[Int[x^n*Tanh[x], x], x, ArcCosh[a*x^p]]",
     "rulenumber": 0,
            "lhs": "Int[ArcCosh[a_.*x_^p_]^n_./x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, p}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "Int[u*ArcCsch[a/c + b*x^n/c]^m, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*ArcSinh[c_./(a_. + b_.*x_^n_.)]^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, m}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "Int[u*ArcSech[a/c + b*x^n/c]^m, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*ArcCosh[c_./(a_. + b_.*x_^n_.)]^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, m}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "Sqrt[b*x^2]/(b*x)* Subst[Int[ArcSinh[x]^n/Sqrt[1 + x^2], x], x, Sqrt[-1 + b*x^2]]",
     "rulenumber": 0,
            "lhs": "Int[ArcSinh[Sqrt[-1 + b_.*x_^2]]^n_./Sqrt[-1 + b_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "Sqrt[-1 + Sqrt[1 + b*x^2]]*Sqrt[1 + Sqrt[1 + b*x^2]]/(b*x)* Subst[Int[ArcCosh[x]^n/(Sqrt[-1 + x]*Sqrt[1 + x]), x], x, Sqrt[1 + b*x^2]]",
     "rulenumber": 0,
            "lhs": "Int[ArcCosh[Sqrt[1 + b_.*x_^2]]^n_./Sqrt[1 + b_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "1/b*Subst[Int[f^(c*x^n)*Cosh[x], x], x, ArcSinh[a + b*x]]",
     "rulenumber": 0,
            "lhs": "Int[f_^(c_.*ArcSinh[a_. + b_.*x_]^n_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, f}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "1/b*Subst[Int[f^(c*x^n)*Sinh[x], x], x, ArcCosh[a + b*x]]",
     "rulenumber": 0,
            "lhs": "Int[f_^(c_.*ArcCosh[a_. + b_.*x_]^n_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, f}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "1/b*Subst[Int[(-a/b + Sinh[x]/b)^m*f^(c*x^n)*Cosh[x], x], x, ArcSinh[a + b*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*f_^(c_.*ArcSinh[a_. + b_.*x_]^n_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "1/b*Subst[Int[(-a/b + Cosh[x]/b)^m*f^(c*x^n)*Sinh[x], x], x, ArcCosh[a + b*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*f_^(c_.*ArcCosh[a_. + b_.*x_]^n_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "x*ArcSinh[u] - Int[SimplifyIntegrand[x*D[u, x]/Sqrt[1 + u^2], x], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcSinh[u_], x_Symbol]",
        "comment": false,
         "givens": "InverseFunctionFreeQ[u, x] && Not[FunctionOfExponentialQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "x*ArcCosh[u] - Int[SimplifyIntegrand[x*D[u, x]/(Sqrt[-1 + u]*Sqrt[1 + u]), x], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCosh[u_], x_Symbol]",
        "comment": false,
         "givens": "InverseFunctionFreeQ[u, x] && Not[FunctionOfExponentialQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "(c + d*x)^(m + 1)*(a + b*ArcSinh[u])/(d*(m + 1)) - b/(d*(m + 1))* Int[SimplifyIntegrand[(c + d*x)^(m + 1)*D[u, x]/Sqrt[1 + u^2], x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(a_. + b_.*ArcSinh[u_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && Not[FunctionOfQ[(c + d*x)^(m + 1), u, x]] && Not[FunctionOfExponentialQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "(c + d*x)^(m + 1)*(a + b*ArcCosh[u])/(d*(m + 1)) - b/(d*(m + 1))* Int[SimplifyIntegrand[(c + d*x)^(m + 1)* D[u, x]/(Sqrt[-1 + u]*Sqrt[1 + u]), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(a_. + b_.*ArcCosh[u_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && Not[FunctionOfQ[(c + d*x)^(m + 1), u, x]] && Not[FunctionOfExponentialQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "With[{w = IntHide[v, x]}, Dist[(a + b*ArcSinh[u]), w, x] - b*Int[SimplifyIntegrand[w*D[u, x]/Sqrt[1 + u^2], x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && Not[MatchQ[v, (c_. + d_.*x)^m_.",
     "rulenumber": 0,
            "lhs": "Int[v_*(a_. + b_.*ArcSinh[u_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, m}, x]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "With[{w = IntHide[v, x]}, Dist[(a + b*ArcCosh[u]), w, x] - b*Int[SimplifyIntegrand[w*D[u, x]/(Sqrt[-1 + u]*Sqrt[1 + u]), x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && Not[MatchQ[v, (c_. + d_.*x)^m_.",
     "rulenumber": 0,
            "lhs": "Int[v_*(a_. + b_.*ArcCosh[u_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, m}, x]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "Int[(u + Sqrt[1 + u^2])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[E^(n_.*ArcSinh[u_]), x_Symbol]",
        "comment": false,
         "givens": "IntegerQ[n] && PolynomialQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "Int[x^m*(u + Sqrt[1 + u^2])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*E^(n_.*ArcSinh[u_]), x_Symbol]",
        "comment": false,
         "givens": "RationalQ[m] && IntegerQ[n] && PolynomialQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "Int[(u + Sqrt[-1 + u]*Sqrt[1 + u])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[E^(n_.*ArcCosh[u_]), x_Symbol]",
        "comment": false,
         "givens": "IntegerQ[n] && PolynomialQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.6 Miscellaneous inverse hyperbolic sine.m",
       "filename": "7.1.6 Miscellaneous inverse hyperbolic sine.m",
            "rhs": "Int[x^m*(u + Sqrt[-1 + u]*Sqrt[1 + u])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*E^(n_.*ArcCosh[u_]), x_Symbol]",
        "comment": false,
         "givens": "RationalQ[m] && IntegerQ[n] && PolynomialQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "x*(a + b*ArcTanh[c*x])^p - b*c*p*Int[x*(a + b*ArcTanh[c*x])^(p - 1)/(1 - c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "x*(a + b*ArcCoth[c*x])^p - b*c*p*Int[x*(a + b*ArcCoth[c*x])^(p - 1)/(1 - c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "a*Log[x] - b/2*PolyLog[2, -c*x] + b/2*PolyLog[2, c*x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTanh[c_.*x_])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "a*Log[x] + b/2*PolyLog[2, -1/(c*x)] - b/2*PolyLog[2, 1/(c*x)]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCoth[c_.*x_])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "2*(a + b*ArcTanh[c*x])^p*ArcTanh[1 - 2/(1 - c*x)] - 2*b*c*p* Int[(a + b*ArcTanh[c*x])^(p - 1)* ArcTanh[1 - 2/(1 - c*x)]/(1 - c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTanh[c_.*x_])^p_/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IGtQ[p, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "2*(a + b*ArcCoth[c*x])^p*ArcCoth[1 - 2/(1 - c*x)] - 2*b*c*p* Int[(a + b*ArcCoth[c*x])^(p - 1)* ArcCoth[1 - 2/(1 - c*x)]/(1 - c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCoth[c_.*x_])^p_/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IGtQ[p, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(d*x)^(m + 1)*(a + b*ArcTanh[c*x])^p/(d*(m + 1)) - b*c*p/(d*(m + 1))* Int[(d*x)^(m + 1)*(a + b*ArcTanh[c*x])^(p - 1)/(1 - c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(d*x)^(m + 1)*(a + b*ArcCoth[c*x])^p/(d*(m + 1)) - b*c*p/(d*(m + 1))* Int[(d*x)^(m + 1)*(a + b*ArcCoth[c*x])^(p - 1)/(1 - c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "-(a + b*ArcTanh[c*x])^p*Log[2/(1 + e*x/d)]/e + b*c*p/e* Int[(a + b*ArcTanh[c*x])^(p - 1)*Log[2/(1 + e*x/d)]/(1 - c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "-(a + b*ArcCoth[c*x])^p*Log[2/(1 + e*x/d)]/e + b*c*p/e* Int[(a + b*ArcCoth[c*x])^(p - 1)*Log[2/(1 + e*x/d)]/(1 - c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "-(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)]/e + b*c/e*Int[Log[2/(1 + c*x)]/(1 - c^2*x^2), x] + (a + b*ArcTanh[c*x])*Log[2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/e - b*c/e* Int[Log[2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/(1 - c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTanh[c_.*x_])/(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "-(a + b*ArcCoth[c*x])*Log[2/(1 + c*x)]/e + b*c/e*Int[Log[2/(1 + c*x)]/(1 - c^2*x^2), x] + (a + b*ArcCoth[c*x])*Log[2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/e - b*c/e* Int[Log[2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/(1 - c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCoth[c_.*x_])/(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "-(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)]/e + b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)]/e + b^2*PolyLog[3, 1 - 2/(1 + c*x)]/(2*e) + (a + b*ArcTanh[c*x])^2* Log[2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/e - b*(a + b*ArcTanh[c*x])* PolyLog[2, 1 - 2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/e - b^2*PolyLog[3, 1 - 2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/(2*e)",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTanh[c_.*x_])^2/(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "-(a + b*ArcCoth[c*x])^2*Log[2/(1 + c*x)]/e + b*(a + b*ArcCoth[c*x])*PolyLog[2, 1 - 2/(1 + c*x)]/e + b^2*PolyLog[3, 1 - 2/(1 + c*x)]/(2*e) + (a + b*ArcCoth[c*x])^2* Log[2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/e - b*(a + b*ArcCoth[c*x])* PolyLog[2, 1 - 2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/e - b^2*PolyLog[3, 1 - 2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/(2*e)",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCoth[c_.*x_])^2/(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "-(a + b*ArcTanh[c*x])^3*Log[2/(1 + c*x)]/e + 3*b*(a + b*ArcTanh[c*x])^2*PolyLog[2, 1 - 2/(1 + c*x)]/(2*e) + 3*b^2*(a + b*ArcTanh[c*x])*PolyLog[3, 1 - 2/(1 + c*x)]/(2*e) + 3*b^3*PolyLog[4, 1 - 2/(1 + c*x)]/(4*e) + (a + b*ArcTanh[c*x])^3* Log[2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/e - 3*b*(a + b*ArcTanh[c*x])^2* PolyLog[2, 1 - 2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/(2*e) - 3*b^2*(a + b*ArcTanh[c*x])* PolyLog[3, 1 - 2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/(2*e) - 3*b^3*PolyLog[4, 1 - 2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/(4*e)",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTanh[c_.*x_])^3/(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "-(a + b*ArcCoth[c*x])^3*Log[2/(1 + c*x)]/e + 3*b*(a + b*ArcCoth[c*x])^2*PolyLog[2, 1 - 2/(1 + c*x)]/(2*e) + 3*b^2*(a + b*ArcCoth[c*x])*PolyLog[3, 1 - 2/(1 + c*x)]/(2*e) + 3*b^3*PolyLog[4, 1 - 2/(1 + c*x)]/(4*e) + (a + b*ArcCoth[c*x])^3* Log[2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/e - 3*b*(a + b*ArcCoth[c*x])^2* PolyLog[2, 1 - 2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/(2*e) - 3*b^2*(a + b*ArcCoth[c*x])* PolyLog[3, 1 - 2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/(2*e) - 3*b^3*PolyLog[4, 1 - 2*c*(d + e*x)/((c*d + e)*(1 + c*x))]/(4*e)",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCoth[c_.*x_])^3/(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(d + e*x)^(q + 1)*(a + b*ArcTanh[c*x])/(e*(q + 1)) - b*c/(e*(q + 1))*Int[(d + e*x)^(q + 1)/(1 - c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^q_.*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(d + e*x)^(q + 1)*(a + b*ArcCoth[c*x])/(e*(q + 1)) - b*c/(e*(q + 1))*Int[(d + e*x)^(q + 1)/(1 - c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^q_.*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(d + e*x)^(q + 1)*(a + b*ArcTanh[c*x])^ p/(e*(q + 1)) - b*c*p/(e*(q + 1))* Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^(p - 1), (d + e*x)^(q + 1)/(1 - c^2*x^2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^q_.*(a_. + b_.*ArcTanh[c_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(d + e*x)^(q + 1)*(a + b*ArcCoth[c*x])^ p/(e*(q + 1)) - b*c*p/(e*(q + 1))* Int[ExpandIntegrand[(a + b*ArcCoth[c*x])^(p - 1), (d + e*x)^(q + 1)/(1 - c^2*x^2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_)^q_.*(a_. + b_.*ArcCoth[c_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "f/e*Int[(f*x)^(m - 1)*(a + b*ArcTanh[c*x])^p, x] - d*f/e*Int[(f*x)^(m - 1)*(a + b*ArcTanh[c*x])^p/(d + e*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "f/e*Int[(f*x)^(m - 1)*(a + b*ArcCoth[c*x])^p, x] - d*f/e*Int[(f*x)^(m - 1)*(a + b*ArcCoth[c*x])^p/(d + e*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(a + b*ArcTanh[c*x])^p*Log[2 - 2/(1 + e*x/d)]/d - b*c*p/d* Int[(a + b*ArcTanh[c*x])^(p - 1)* Log[2 - 2/(1 + e*x/d)]/(1 - c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTanh[c_.*x_])^p_./(x_*(d_ + e_.*x_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(a + b*ArcCoth[c*x])^p*Log[2 - 2/(1 + e*x/d)]/d - b*c*p/d* Int[(a + b*ArcCoth[c*x])^(p - 1)* Log[2 - 2/(1 + e*x/d)]/(1 - c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCoth[c_.*x_])^p_./(x_*(d_ + e_.*x_)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "1/d*Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x] - e/(d*f)*Int[(f*x)^(m + 1)*(a + b*ArcTanh[c*x])^p/(d + e*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "1/d*Int[(f*x)^m*(a + b*ArcCoth[c*x])^p, x] - e/(d*f)*Int[(f*x)^(m + 1)*(a + b*ArcCoth[c*x])^p/(d + e*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[(a + b*ArcTanh[c*x]), u] - b*c*Int[SimplifyIntegrand[u/(1 - c^2*x^2), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_)^q_.*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[q, -1] && IntegerQ[ 2*m] && (IGtQ[m, 0] && IGtQ[q, 0] || ILtQ[m + q + 1, 0] && LtQ[m*q, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[(a + b*ArcCoth[c*x]), u] - b*c*Int[SimplifyIntegrand[u/(1 - c^2*x^2), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_)^q_.*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[q, -1] && IntegerQ[ 2*m] && (IGtQ[m, 0] && IGtQ[q, 0] || ILtQ[m + q + 1, 0] && LtQ[m*q, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[(a + b*ArcTanh[c*x])^p, u] - b*c*p*Int[ ExpandIntegrand[(a + b*ArcTanh[c*x])^(p - 1), u/(1 - c^2*x^2), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, q}, x] && IGtQ[p, 1] && EqQ[c^2*d^2 - e^2, 0] && IntegersQ[m, q] && NeQ[m, -1] && NeQ[q, -1] && ILtQ[m + q + 1, 0] && LtQ[m*q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[(a + b*ArcCoth[c*x])^p, u] - b*c*p*Int[ ExpandIntegrand[(a + b*ArcCoth[c*x])^(p - 1), u/(1 - c^2*x^2), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, q}, x] && IGtQ[p, 1] && EqQ[c^2*d^2 - e^2, 0] && IntegersQ[m, q] && NeQ[m, -1] && NeQ[q, -1] && ILtQ[m + q + 1, 0] && LtQ[m*q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^p, (f*x)^m*(d + e*x)^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_)^q_.*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcCoth[c*x])^p, (f*x)^m*(d + e*x)^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_)^q_.*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "b*(d + e*x^2)^q/(2*c*q*(2*q + 1)) + x*(d + e*x^2)^q*(a + b*ArcTanh[c*x])/(2*q + 1) + 2*d*q/(2*q + 1)* Int[(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "b*(d + e*x^2)^q/(2*c*q*(2*q + 1)) + x*(d + e*x^2)^q*(a + b*ArcCoth[c*x])/(2*q + 1) + 2*d*q/(2*q + 1)* Int[(d + e*x^2)^(q - 1)*(a + b*ArcCoth[c*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0]"
  },
  {
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0] && GtQ[p, 1]"
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0] && GtQ[p, 1]"
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         "givens": " FreeQ[{a,b,c,d,e,p},x] && EqQ[c^2*d+e,0] *)"
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]"
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]"
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            "rhs": "(a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1))",
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         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]"
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            "rhs": "(a + b*ArcCoth[c*x])^(p + 1)/(b*c*d*(p + 1))",
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            "lhs": "Int[(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]"
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            "rhs": "-2*(a + b*ArcTanh[c*x])* ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]]/(c*Sqrt[d]) - I*b*PolyLog[2, -I*Sqrt[1 - c*x]/Sqrt[1 + c*x]]/(c*Sqrt[d]) + I*b*PolyLog[2, I*Sqrt[1 - c*x]/Sqrt[1 + c*x]]/(c*Sqrt[d])",
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            "lhs": "Int[(a_. + b_.*ArcTanh[c_.*x_])/Sqrt[d_ + e_.*x_^2], x_Symbol]",
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         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0]"
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       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "1/(c*Sqrt[d])*Subst[Int[(a + b*x)^p*Sech[x], x], x, ArcTanh[c*x]]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && GtQ[d, 0]"
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       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && GtQ[d, 0]"
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       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && Not[GtQ[d, 0]]"
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       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]* Int[(a + b*ArcCoth[c*x])^p/Sqrt[1 - c^2*x^2], x]",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && Not[GtQ[d, 0]]"
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]"
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       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "x*(a + b*ArcCoth[c*x])^p/(2*d*(d + e*x^2)) + (a + b*ArcCoth[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)) - b*c*p/2*Int[x*(a + b*ArcCoth[c*x])^(p - 1)/(d + e*x^2)^2, x]",
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            "lhs": "Int[(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2)^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]"
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            "lhs": "Int[(a_. + b_.*ArcTanh[c_.*x_])/(d_ + e_.*x_^2)^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]"
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]"
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         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]"
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            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]"
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && LtQ[p, -1]"
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "d^q/c*Subst[Int[(a + b*x)^p/Cosh[x]^(2*(q + 1)), x], x, ArcTanh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && ILtQ[2*(q + 1), 0] && (IntegerQ[q] || GtQ[d, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "d^(q + 1/2)*Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]* Int[(1 - c^2*x^2)^q*(a + b*ArcTanh[c*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && ILtQ[2*(q + 1), 0] && Not[IntegerQ[q] || GtQ[d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "-(-d)^q/c* Subst[Int[(a + b*x)^p/Sinh[x]^(2*(q + 1)), x], x, ArcCoth[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && ILtQ[2*(q + 1), 0] && IntegerQ[q]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "-(-d)^(q + 1/2)*x* Sqrt[(c^2*x^2 - 1)/(c^2*x^2)]/Sqrt[d + e*x^2]* Subst[Int[(a + b*x)^p/Sinh[x]^(2*(q + 1)), x], x, ArcCoth[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && ILtQ[2*(q + 1), 0] && Not[IntegerQ[q]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "1/2*Int[Log[1 + c*x]/(d + e*x^2), x] - 1/2*Int[Log[1 - c*x]/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTanh[c_.*x_]/(d_. + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "1/2*Int[Log[1 + 1/(c*x)]/(d + e*x^2), x] - 1/2*Int[Log[1 - 1/(c*x)]/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCoth[c_.*x_]/(d_. + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, e}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "a*Int[1/(d + e*x^2), x] + b*Int[ArcTanh[c*x]/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*ArcTanh[c_.*x_])/(d_. + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "a*Int[1/(d + e*x^2), x] + b*Int[ArcCoth[c*x]/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_ + b_.*ArcCoth[c_.*x_])/(d_. + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "With[{u = IntHide[(d + e*x^2)^q, x]}, Dist[a + b*ArcTanh[c*x], u, x] - b*c*Int[u/(1 - c^2*x^2), x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && (IntegerQ[q] || ILtQ[q + 1/2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "With[{u = IntHide[(d + e*x^2)^q, x]}, Dist[a + b*ArcCoth[c*x], u, x] - b*c*Int[u/(1 - c^2*x^2), x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && (IntegerQ[q] || ILtQ[q + 1/2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^p, (d + e*x^2)^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IntegerQ[q] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcCoth[c*x])^p, (d + e*x^2)^q, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IntegerQ[q] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "f^2/e*Int[(f*x)^(m - 2)*(a + b*ArcTanh[c*x])^p, x] - d*f^2/e* Int[(f*x)^(m - 2)*(a + b*ArcTanh[c*x])^p/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "f^2/e*Int[(f*x)^(m - 2)*(a + b*ArcCoth[c*x])^p, x] - d*f^2/e* Int[(f*x)^(m - 2)*(a + b*ArcCoth[c*x])^p/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "1/d*Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x] - e/(d*f^2)* Int[(f*x)^(m + 2)*(a + b*ArcTanh[c*x])^p/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "1/d*Int[(f*x)^m*(a + b*ArcCoth[c*x])^p, x] - e/(d*f^2)* Int[(f*x)^(m + 2)*(a + b*ArcCoth[c*x])^p/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(a + b*ArcTanh[c*x])^(p + 1)/(b*e*(p + 1)) + 1/(c*d)*Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(a + b*ArcCoth[c*x])^(p + 1)/(b*e*(p + 1)) + 1/(c*d)*Int[(a + b*ArcCoth[c*x])^p/(1 - c*x), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "x*(a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1)) - 1/(b*c*d*(p + 1))*Int[(a + b*ArcTanh[c*x])^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(a_. + b_.*ArcTanh[c_.*x_])^p_/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && Not[IGtQ[p, 0]] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "-x*(a + b*ArcCoth[c*x])^(p + 1)/(b*c*d*(p + 1)) - 1/(b*c*d*(p + 1))*Int[(a + b*ArcCoth[c*x])^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(a_. + b_.*ArcCoth[c_.*x_])^p_/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && Not[IGtQ[p, 0]] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(a + b*ArcTanh[c*x])^(p + 1)/(b*d*(p + 1)) + 1/d*Int[(a + b*ArcTanh[c*x])^p/(x*(1 + c*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTanh[c_.*x_])^p_./(x_*(d_ + e_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(a + b*ArcCoth[c*x])^(p + 1)/(b*d*(p + 1)) + 1/d*Int[(a + b*ArcCoth[c*x])^p/(x*(1 + c*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCoth[c_.*x_])^p_./(x_*(d_ + e_.*x_^2)), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(f*x)^ m*(a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1)) - f*m/(b*c*d*(p + 1))* Int[(f*x)^(m - 1)*(a + b*ArcTanh[c*x])^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcTanh[c_.*x_])^p_/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(f*x)^ m*(a + b*ArcCoth[c*x])^(p + 1)/(b*c*d*(p + 1)) - f*m/(b*c*d*(p + 1))* Int[(f*x)^(m - 1)*(a + b*ArcCoth[c*x])^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcCoth[c_.*x_])^p_/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcTanh[c*x]), x^m/(d + e*x^2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*ArcTanh[c_.*x_])/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] && Not[EqQ[m, 1] && NeQ[a, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcCoth[c*x]), x^m/(d + e*x^2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*ArcCoth[c_.*x_])/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] && Not[EqQ[m, 1] && NeQ[a, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^ p/(2*e*(q + 1)) + b*p/(2*c*(q + 1))* Int[(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, q}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && NeQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^ p/(2*e*(q + 1)) + b*p/(2*c*(q + 1))* Int[(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, q}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && NeQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "x*(a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1)*(d + e*x^2)) + (1 + c^2*x^2)*(a + b*ArcTanh[c*x])^(p + 2)/(b^2* e*(p + 1)*(p + 2)*(d + e*x^2)) + 4/(b^2*(p + 1)*(p + 2))* Int[x*(a + b*ArcTanh[c*x])^(p + 2)/(d + e*x^2)^2, x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(a_. + b_.*ArcTanh[c_.*x_])^p_/(d_ + e_.*x_^2)^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[p, -1] && NeQ[p, -2]"
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            "rhs": "x*(a + b*ArcCoth[c*x])^(p + 1)/(b*c*d*(p + 1)*(d + e*x^2)) + (1 + c^2*x^2)*(a + b*ArcCoth[c*x])^(p + 2)/(b^2* e*(p + 1)*(p + 2)*(d + e*x^2)) + 4/(b^2*(p + 1)*(p + 2))* Int[x*(a + b*ArcCoth[c*x])^(p + 2)/(d + e*x^2)^2, x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(a_. + b_.*ArcCoth[c_.*x_])^p_/(d_ + e_.*x_^2)^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[p, -1] && NeQ[p, -2]"
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            "lhs": "Int[x_^2*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && NeQ[q, -5/2]"
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     "rulenumber": 0,
            "lhs": "Int[x_^2*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && NeQ[q, -5/2]"
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            "lhs": "Int[x_^2*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2)^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "-(a + b*ArcCoth[c*x])^(p + 1)/(2*b*c^3* d^2*(p + 1)) + x*(a + b*ArcCoth[c*x])^p/(2*c^2*d*(d + e*x^2)) - b*p/(2*c)*Int[x*(a + b*ArcCoth[c*x])^(p - 1)/(d + e*x^2)^2, x]",
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            "lhs": "Int[x_^2*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2)^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
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            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1]"
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            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1]"
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            "rhs": "-b* p*(f*x)^m*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^(p - 1)/(c*d* m^2) + f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^ p/(c^2*d*m) + b^2*p*(p - 1)/m^2* Int[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 2), x] - f^2*(m - 1)/(c^2*d*m)* Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1] && GtQ[p, 1]"
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  {
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            "rhs": "-b* p*(f*x)^m*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^(p - 1)/(c*d* m^2) + f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^ p/(c^2*d*m) + b^2*p*(p - 1)/m^2* Int[(f*x)^m*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p - 2), x] - f^2*(m - 1)/(c^2*d*m)* Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1] && GtQ[p, 1]"
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            "rhs": "(f*x)^ m*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^(p + 1)/(b*c* d*(p + 1)) - f*m/(b*c*(p + 1))* Int[(f*x)^(m - 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p + 1), x]",
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            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 2, 0] && LtQ[p, -1]"
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            "rhs": "(f*x)^ m*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^(p + 1)/(b*c* d*(p + 1)) - f*m/(b*c*(p + 1))* Int[(f*x)^(m - 1)*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 2, 0] && LtQ[p, -1]"
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            "rhs": "(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p/(d*(m + 1)) - b*c*p/(m + 1)* Int[(f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]"
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  {
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       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^p/(d*f*(m + 1)) - b*c*p/(f*(m + 1))* Int[(f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p - 1), x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]"
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       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(f*x)^(m + 1)* Sqrt[d + e*x^2]*(a + b*ArcTanh[c*x])/(f*(m + 2)) - b*c*d/(f*(m + 2))*Int[(f*x)^(m + 1)/Sqrt[d + e*x^2], x] + d/(m + 2)*Int[(f*x)^m*(a + b*ArcTanh[c*x])/Sqrt[d + e*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && NeQ[m, -2]"
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       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(f*x)^(m + 1)* Sqrt[d + e*x^2]*(a + b*ArcCoth[c*x])/(f*(m + 2)) - b*c*d/(f*(m + 2))*Int[(f*x)^(m + 1)/Sqrt[d + e*x^2], x] + d/(m + 2)*Int[(f*x)^m*(a + b*ArcCoth[c*x])/Sqrt[d + e*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*Sqrt[d_ + e_.*x_^2]*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && NeQ[m, -2]"
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            "rhs": "Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^p, x], x]",
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            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && IGtQ[q, 1]"
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            "rhs": "Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^p, x], x]",
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            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && IGtQ[q, 1]"
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       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "d*Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x] - c^2*d/f^2* Int[(f*x)^(m + 2)*(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x]",
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            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || EqQ[p, 1] && IntegerQ[q])"
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            "rhs": "d*Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcCoth[c*x])^p, x] - c^2*d/f^2* Int[(f*x)^(m + 2)*(d + e*x^2)^(q - 1)*(a + b*ArcCoth[c*x])^p, x]",
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            "lhs": "Int[(f_.*x_)^m_*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || EqQ[p, 1] && IntegerQ[q])"
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            "rhs": "-f*(f*x)^(m - 1)* Sqrt[d + e*x^2]*(a + b*ArcTanh[c*x])^p/(c^2*d*m) + b*f*p/(c*m)* Int[(f*x)^(m - 1)*(a + b*ArcTanh[c*x])^(p - 1)/Sqrt[d + e*x^2], x] + f^2*(m - 1)/(c^2*m)* Int[(f*x)^(m - 2)*(a + b*ArcTanh[c*x])^p/Sqrt[d + e*x^2], x]",
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            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcTanh[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && GtQ[m, 1]"
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            "rhs": "-f*(f*x)^(m - 1)* Sqrt[d + e*x^2]*(a + b*ArcCoth[c*x])^p/(c^2*d*m) + b*f*p/(c*m)* Int[(f*x)^(m - 1)*(a + b*ArcCoth[c*x])^(p - 1)/Sqrt[d + e*x^2], x] + f^2*(m - 1)/(c^2*m)* Int[(f*x)^(m - 2)*(a + b*ArcCoth[c*x])^p/Sqrt[d + e*x^2], x]",
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            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcCoth[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && GtQ[m, 1]"
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       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "-2/Sqrt[d]*(a + b*ArcTanh[c*x])* ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]] + b/Sqrt[d]*PolyLog[2, -Sqrt[1 - c*x]/Sqrt[1 + c*x]] - b/Sqrt[d]*PolyLog[2, Sqrt[1 - c*x]/Sqrt[1 + c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTanh[c_.*x_])/(x_*Sqrt[d_ + e_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "-2/Sqrt[d]*(a + b*ArcCoth[c*x])* ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]] + b/Sqrt[d]*PolyLog[2, -Sqrt[1 - c*x]/Sqrt[1 + c*x]] - b/Sqrt[d]*PolyLog[2, Sqrt[1 - c*x]/Sqrt[1 + c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCoth[c_.*x_])/(x_*Sqrt[d_ + e_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "1/Sqrt[d]*Subst[Int[(a + b*x)^p*Csch[x], x], x, ArcTanh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTanh[c_.*x_])^p_/(x_*Sqrt[d_ + e_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && GtQ[d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "-c*x*Sqrt[1 - 1/(c^2*x^2)]/Sqrt[d + e*x^2]* Subst[Int[(a + b*x)^p*Sech[x], x], x, ArcCoth[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCoth[c_.*x_])^p_/(x_*Sqrt[d_ + e_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && GtQ[d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]* Int[(a + b*ArcTanh[c*x])^p/(x*Sqrt[1 - c^2*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTanh[c_.*x_])^p_./(x_*Sqrt[d_ + e_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && Not[GtQ[d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]* Int[(a + b*ArcCoth[c*x])^p/(x*Sqrt[1 - c^2*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCoth[c_.*x_])^p_./(x_*Sqrt[d_ + e_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && Not[GtQ[d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "-Sqrt[d + e*x^2]*(a + b*ArcTanh[c*x])^p/(d*x) + b*c*p*Int[(a + b*ArcTanh[c*x])^(p - 1)/(x*Sqrt[d + e*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTanh[c_.*x_])^p_./(x_^2*Sqrt[d_ + e_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "-Sqrt[d + e*x^2]*(a + b*ArcCoth[c*x])^p/(d*x) + b*c*p*Int[(a + b*ArcCoth[c*x])^(p - 1)/(x*Sqrt[d + e*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCoth[c_.*x_])^p_./(x_^2*Sqrt[d_ + e_.*x_^2]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(f*x)^(m + 1)* Sqrt[d + e*x^2]*(a + b*ArcTanh[c*x])^p/(d*f*(m + 1)) - b*c*p/(f*(m + 1))* Int[(f*x)^(m + 1)*(a + b*ArcTanh[c*x])^(p - 1)/Sqrt[d + e*x^2], x] + c^2*(m + 2)/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(a + b*ArcTanh[c*x])^p/Sqrt[d + e*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcTanh[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && LtQ[m, -1] && NeQ[m, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(f*x)^(m + 1)* Sqrt[d + e*x^2]*(a + b*ArcCoth[c*x])^p/(d*f*(m + 1)) - b*c*p/(f*(m + 1))* Int[(f*x)^(m + 1)*(a + b*ArcCoth[c*x])^(p - 1)/Sqrt[d + e*x^2], x] + c^2*(m + 2)/(f^2*(m + 1))* Int[(f*x)^(m + 2)*(a + b*ArcCoth[c*x])^p/Sqrt[d + e*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_*(a_. + b_.*ArcCoth[c_.*x_])^p_./Sqrt[d_ + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && LtQ[m, -1] && NeQ[m, -2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "1/e*Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p, x] - d/e*Int[x^(m - 2)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegersQ[p, 2*q] && LtQ[q, -1] && IGtQ[m, 1] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "1/e*Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^p, x] - d/e*Int[x^(m - 2)*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegersQ[p, 2*q] && LtQ[q, -1] && IGtQ[m, 1] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "1/d*Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p, x] - e/d*Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegersQ[p, 2*q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "1/d*Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^p, x] - e/d*Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegersQ[p, 2*q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^(p + 1)/(b*c* d*(p + 1)) - m/(b*c*(p + 1))* Int[x^(m - 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p + 1), x] + c*(m + 2*q + 2)/(b*(p + 1))* Int[x^(m + 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && LtQ[q, -1] && LtQ[p, -1] && NeQ[m + 2*q + 2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "x^m*(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])^(p + 1)/(b*c* d*(p + 1)) - m/(b*c*(p + 1))* Int[x^(m - 1)*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p + 1), x] + c*(m + 2*q + 2)/(b*(p + 1))* Int[x^(m + 1)*(d + e*x^2)^q*(a + b*ArcCoth[c*x])^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && LtQ[q, -1] && LtQ[p, -1] && NeQ[m + 2*q + 2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "d^q/c^(m + 1)* Subst[Int[(a + b*x)^p*Sinh[x]^m/Cosh[x]^(m + 2*(q + 1)), x], x, ArcTanh[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && (IntegerQ[q] || GtQ[d, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "d^(q + 1/2)*Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]* Int[x^m*(1 - c^2*x^2)^q*(a + b*ArcTanh[c*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && Not[IntegerQ[q] || GtQ[d, 0]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "-(-d)^q/c^(m + 1)* Subst[Int[(a + b*x)^p*Cosh[x]^m/Sinh[x]^(m + 2*(q + 1)), x], x, ArcCoth[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && IntegerQ[q]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "-(-d)^(q + 1/2)*x* Sqrt[(c^2*x^2 - 1)/(c^2*x^2)]/(c^m*Sqrt[d + e*x^2])* Subst[Int[(a + b*x)^p*Cosh[x]^m/Sinh[x]^(m + 2*(q + 1)), x], x, ArcCoth[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_ + e_.*x_^2)^q_*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && Not[IntegerQ[q]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])/(2* e*(q + 1)) - b*c/(2*e*(q + 1))*Int[(d + e*x^2)^(q + 1)/(1 - c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d_. + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(d + e*x^2)^(q + 1)*(a + b*ArcCoth[c*x])/(2* e*(q + 1)) - b*c/(2*e*(q + 1))*Int[(d + e*x^2)^(q + 1)/(1 - c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d_. + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "With[{u = IntHide[(f*x)^m*(d + e*x^2)^q, x]}, Dist[a + b*ArcTanh[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/(1 - c^2*x^2), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, q}, x] && ( IGtQ[q, 0] && Not[ILtQ[(m - 1)/2, 0] && GtQ[m + 2*q + 3, 0]] || IGtQ[(m + 1)/2, 0] && Not[ILtQ[q, 0] && GtQ[m + 2*q + 3, 0]] || ILtQ[(m + 2*q + 1)/2, 0] && Not[ILtQ[(m - 1)/2, 0]] )"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "With[{u = IntHide[(f*x)^m*(d + e*x^2)^q, x]}, Dist[a + b*ArcCoth[c*x], u, x] - b*c*Int[SimplifyIntegrand[u/(1 - c^2*x^2), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, q}, x] && ( IGtQ[q, 0] && Not[ILtQ[(m - 1)/2, 0] && GtQ[m + 2*q + 3, 0]] || IGtQ[(m + 1)/2, 0] && Not[ILtQ[q, 0] && GtQ[m + 2*q + 3, 0]] || ILtQ[(m + 2*q + 1)/2, 0] && Not[ILtQ[(m - 1)/2, 0]] )"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "1/(4*d^2*Rt[-e/d, 2])* Int[(a + b*ArcTanh[c*x])^p/(1 - Rt[-e/d, 2]*x)^2, x] - 1/(4*d^2*Rt[-e/d, 2])* Int[(a + b*ArcTanh[c*x])^p/(1 + Rt[-e/d, 2]*x)^2, x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2)^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "1/(4*d^2*Rt[-e/d, 2])* Int[(a + b*ArcCoth[c*x])^p/(1 - Rt[-e/d, 2]*x)^2, x] - 1/(4*d^2*Rt[-e/d, 2])* Int[(a + b*ArcCoth[c*x])^p/(1 + Rt[-e/d, 2]*x)^2, x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2)^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "With[{u = ExpandIntegrand[(a + b*ArcTanh[c*x])^p, (f*x)^m*(d + e*x^2)^q, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && IntegerQ[q] && IGtQ[p, 0] && (GtQ[q, 0] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "With[{u = ExpandIntegrand[(a + b*ArcCoth[c*x])^p, (f*x)^m*(d + e*x^2)^q, x]}, Int[u, x] /; SumQ[u]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && IntegerQ[q] && IGtQ[p, 0] && (GtQ[q, 0] || IntegerQ[m])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "a*Int[(f*x)^m*(d + e*x^2)^q, x] + b*Int[(f*x)^m*(d + e*x^2)^q*ArcTanh[c*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_ + b_.*ArcTanh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, q}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "a*Int[(f*x)^m*(d + e*x^2)^q, x] + b*Int[(f*x)^m*(d + e*x^2)^q*ArcCoth[c*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_ + e_.*x_^2)^q_.*(a_ + b_.*ArcCoth[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, q}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^p/(d + e*x^2), (f + g*x)^m, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*ArcCoth[c*x])^p/(d + e*x^2), (f + g*x)^m, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(f_ + g_.*x_)^m_.*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "1/2*Int[Log[1 + u]*(a + b*ArcTanh[c*x])^p/(d + e*x^2), x] - 1/2*Int[Log[1 - u]*(a + b*ArcTanh[c*x])^p/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTanh[u_]*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 + c*x))^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "1/2*Int[Log[ SimplifyIntegrand[1 + 1/u, x]]*(a + b*ArcCoth[c*x])^ p/(d + e*x^2), x] - 1/2*Int[ Log[SimplifyIntegrand[1 - 1/u, x]]*(a + b*ArcCoth[c*x])^ p/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCoth[u_]*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 + c*x))^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "1/2*Int[Log[1 + u]*(a + b*ArcTanh[c*x])^p/(d + e*x^2), x] - 1/2*Int[Log[1 - u]*(a + b*ArcTanh[c*x])^p/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTanh[u_]*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 - c*x))^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "1/2*Int[Log[ SimplifyIntegrand[1 + 1/u, x]]*(a + b*ArcCoth[c*x])^ p/(d + e*x^2), x] - 1/2*Int[ Log[SimplifyIntegrand[1 - 1/u, x]]*(a + b*ArcCoth[c*x])^ p/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCoth[u_]*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 - c*x))^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(a + b*ArcTanh[c*x])^(p + 1)* Log[f + g*x]/(b*c*d*(p + 1)) - g/(b*c*d*(p + 1))* Int[(a + b*ArcTanh[c*x])^(p + 1)/(f + g*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTanh[c_.*x_])^p_.*Log[f_ + g_.*x_]/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[c^2*f^2 - g^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(a + b*ArcCoth[c*x])^(p + 1)* Log[f + g*x]/(b*c*d*(p + 1)) - g/(b*c*d*(p + 1))* Int[(a + b*ArcCoth[c*x])^(p + 1)/(f + g*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCoth[c_.*x_])^p_.*Log[f_ + g_.*x_]/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[c^2*f^2 - g^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(a + b*ArcTanh[c*x])^p*PolyLog[2, 1 - u]/(2*c*d) - b*p/2* Int[(a + b*ArcTanh[c*x])^(p - 1)*PolyLog[2, 1 - u]/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTanh[c_.*x_])^p_.*Log[u_]/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 + c*x))^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(a + b*ArcCoth[c*x])^p*PolyLog[2, 1 - u]/(2*c*d) - b*p/2* Int[(a + b*ArcCoth[c*x])^(p - 1)*PolyLog[2, 1 - u]/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCoth[c_.*x_])^p_.*Log[u_]/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 + c*x))^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "-(a + b*ArcTanh[c*x])^p* PolyLog[2, 1 - u]/(2*c*d) + b*p/2* Int[(a + b*ArcTanh[c*x])^(p - 1)*PolyLog[2, 1 - u]/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTanh[c_.*x_])^p_.*Log[u_]/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 - c*x))^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "-(a + b*ArcCoth[c*x])^p* PolyLog[2, 1 - u]/(2*c*d) + b*p/2* Int[(a + b*ArcCoth[c*x])^(p - 1)*PolyLog[2, 1 - u]/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCoth[c_.*x_])^p_.*Log[u_]/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 - c*x))^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "-(a + b*ArcTanh[c*x])^p*PolyLog[k + 1, u]/(2*c*d) + b*p/2* Int[(a + b*ArcTanh[c*x])^(p - 1)*PolyLog[k + 1, u]/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTanh[c_.*x_])^p_.*PolyLog[k_, u_]/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 + c*x))^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "-(a + b*ArcCoth[c*x])^p*PolyLog[k + 1, u]/(2*c*d) + b*p/2* Int[(a + b*ArcCoth[c*x])^(p - 1)*PolyLog[k + 1, u]/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCoth[c_.*x_])^p_.*PolyLog[k_, u_]/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 + c*x))^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(a + b*ArcTanh[c*x])^p*PolyLog[k + 1, u]/(2*c*d) - b*p/2* Int[(a + b*ArcTanh[c*x])^(p - 1)*PolyLog[k + 1, u]/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTanh[c_.*x_])^p_.*PolyLog[k_, u_]/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 - c*x))^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(a + b*ArcCoth[c*x])^p*PolyLog[k + 1, u]/(2*c*d) - b*p/2* Int[(a + b*ArcCoth[c*x])^(p - 1)*PolyLog[k + 1, u]/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCoth[c_.*x_])^p_.*PolyLog[k_, u_]/(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 - c*x))^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(-Log[a + b*ArcCoth[c*x]] + Log[a + b*ArcTanh[c*x]])/(b^2*c* d*(ArcCoth[c*x] - ArcTanh[c*x]))",
     "rulenumber": 0,
            "lhs": "Int[1/((d_ + e_.*x_^2)*(a_. + b_.*ArcCoth[c_.*x_])*(a_. + b_.*ArcTanh[c_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(a + b*ArcCoth[c*x])^(m + 1)*(a + b*ArcTanh[c*x])^ p/(b*c*d*(m + 1)) - p/(m + 1)* Int[(a + b*ArcCoth[c*x])^(m + 1)*(a + b*ArcTanh[c*x])^(p - 1)/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCoth[c_.*x_])^ m_.*(a_. + b_.*ArcTanh[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && IGeQ[m, p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(a + b*ArcTanh[c*x])^(m + 1)*(a + b*ArcCoth[c*x])^ p/(b*c*d*(m + 1)) - p/(m + 1)* Int[(a + b*ArcTanh[c*x])^(m + 1)*(a + b*ArcCoth[c*x])^(p - 1)/(d + e*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTanh[c_.*x_])^ m_.*(a_. + b_.*ArcCoth[c_.*x_])^p_./(d_ + e_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && IGtQ[m, p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "1/2*Int[Log[1 + a*x]/(c + d*x^n), x] - 1/2*Int[Log[1 - a*x]/(c + d*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTanh[a_.*x_]/(c_ + d_.*x_^n_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d}, x] && IntegerQ[n] && Not[EqQ[n, 2] && EqQ[a^2*c + d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "1/2*Int[Log[1 + 1/(a*x)]/(c + d*x^n), x] - 1/2*Int[Log[1 - 1/(a*x)]/(c + d*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCoth[a_.*x_]/(c_ + d_.*x_^n_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d}, x] && IntegerQ[n] && Not[EqQ[n, 2] && EqQ[a^2*c + d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "1/2*Int[Log[d*x^m]*Log[1 + c*x^n]/x, x] - 1/2*Int[Log[d*x^m]*Log[1 - c*x^n]/x, x]",
     "rulenumber": 0,
            "lhs": "Int[Log[d_.*x_^m_.]*ArcTanh[c_.*x_^n_.]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "1/2*Int[Log[d*x^m]*Log[1 + 1/(c*x^n)]/x, x] - 1/2*Int[Log[d*x^m]*Log[1 - 1/(c*x^n)]/x, x]",
     "rulenumber": 0,
            "lhs": "Int[Log[d_.*x_^m_.]*ArcCoth[c_.*x_^n_.]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "a*Int[Log[d*x^m]/x, x] + b*Int[(Log[d*x^m]*ArcTanh[c*x^n])/x, x]",
     "rulenumber": 0,
            "lhs": "Int[Log[d_.*x_^m_.]*(a_ + b_.*ArcTanh[c_.*x_^n_.])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "a*Int[Log[d*x^m]/x, x] + b*Int[(Log[d*x^m]*ArcCoth[c*x^n])/x, x]",
     "rulenumber": 0,
            "lhs": "Int[Log[d_.*x_^m_.]*(a_ + b_.*ArcCoth[c_.*x_^n_.])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "x*(d + e*Log[f + g*x^2])*(a + b*ArcTanh[c*x]) - 2*e*g*Int[x^2*(a + b*ArcTanh[c*x])/(f + g*x^2), x] - b*c*Int[x*(d + e*Log[f + g*x^2])/(1 - c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "x*(d + e*Log[f + g*x^2])*(a + b*ArcCoth[c*x]) - 2*e*g*Int[x^2*(a + b*ArcCoth[c*x])/(f + g*x^2), x] - b*c*Int[x*(d + e*Log[f + g*x^2])/(1 - c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(Log[f + g*x^2] - Log[1 - c*x] - Log[1 + c*x])* Int[ArcTanh[c*x]/x, x] - 1/2*Int[Log[1 - c*x]^2/x, x] + 1/2*Int[Log[1 + c*x]^2/x, x]",
     "rulenumber": 0,
            "lhs": "Int[Log[f_. + g_.*x_^2]*ArcTanh[c_.*x_]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, f, g}, x] && EqQ[c^2*f + g, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(Log[f + g*x^2] - Log[-c^2*x^2] - Log[1 - 1/(c*x)] - Log[1 + 1/(c*x)])*Int[ArcCoth[c*x]/x, x] + Int[Log[-c^2*x^2]*ArcCoth[c*x]/x, x] - 1/2*Int[Log[1 - 1/(c*x)]^2/x, x] + 1/2*Int[Log[1 + 1/(c*x)]^2/x, x]",
     "rulenumber": 0,
            "lhs": "Int[Log[f_. + g_.*x_^2]*ArcCoth[c_.*x_]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, f, g}, x] && EqQ[c^2*f + g, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "a*Int[Log[f + g*x^2]/x, x] + b*Int[Log[f + g*x^2]*ArcTanh[c*x]/x, x]",
     "rulenumber": 0,
            "lhs": "Int[Log[f_. + g_.*x_^2]*(a_ + b_.*ArcTanh[c_.*x_])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, f, g}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "a*Int[Log[f + g*x^2]/x, x] + b*Int[Log[f + g*x^2]*ArcCoth[c*x]/x, x]",
     "rulenumber": 0,
            "lhs": "Int[Log[f_. + g_.*x_^2]*(a_ + b_.*ArcCoth[c_.*x_])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, f, g}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "d*Int[(a + b*ArcTanh[c*x])/x, x] + e*Int[Log[f + g*x^2]*(a + b*ArcTanh[c*x])/x, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcTanh[c_.*x_])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "d*Int[(a + b*ArcCoth[c*x])/x, x] + e*Int[Log[f + g*x^2]*(a + b*ArcCoth[c*x])/x, x]",
     "rulenumber": 0,
            "lhs": "Int[(d_ + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcCoth[c_.*x_])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "x^(m + 1)*(d + e*Log[f + g*x^2])*(a + b*ArcTanh[c*x])/(m + 1) - 2*e*g/(m + 1)* Int[x^(m + 2)*(a + b*ArcTanh[c*x])/(f + g*x^2), x] - b*c/(m + 1)* Int[x^(m + 1)*(d + e*Log[f + g*x^2])/(1 - c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && ILtQ[m/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "x^(m + 1)*(d + e*Log[f + g*x^2])*(a + b*ArcCoth[c*x])/(m + 1) - 2*e*g/(m + 1)* Int[x^(m + 2)*(a + b*ArcCoth[c*x])/(f + g*x^2), x] - b*c/(m + 1)* Int[x^(m + 1)*(d + e*Log[f + g*x^2])/(1 - c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && ILtQ[m/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "With[{u = IntHide[x^m*(d + e*Log[f + g*x^2]), x]}, Dist[a + b*ArcTanh[c*x], u, x] - b*c*Int[ExpandIntegrand[u/(1 - c^2*x^2), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[(m + 1)/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "With[{u = IntHide[x^m*(d + e*Log[f + g*x^2]), x]}, Dist[a + b*ArcCoth[c*x], u, x] - b*c*Int[ExpandIntegrand[u/(1 - c^2*x^2), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[(m + 1)/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "With[{u = IntHide[x^m*(a + b*ArcTanh[c*x]), x]}, Dist[d + e*Log[f + g*x^2], u, x] - 2*e*g*Int[ExpandIntegrand[x*u/(f + g*x^2), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcTanh[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && IntegerQ[m] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "With[{u = IntHide[x^m*(a + b*ArcCoth[c*x]), x]}, Dist[d + e*Log[f + g*x^2], u, x] - 2*e*g*Int[ExpandIntegrand[x*u/(f + g*x^2), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_. + e_.*Log[f_. + g_.*x_^2])*(a_. + b_.*ArcCoth[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && IntegerQ[m] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(f + g*x^2)*(d + e*Log[f + g*x^2])*(a + b*ArcTanh[c*x])^2/(2*g) - e*x^2*(a + b*ArcTanh[c*x])^2/2 + b/c*Int[(d + e*Log[f + g*x^2])*(a + b*ArcTanh[c*x]), x] + b*c*e*Int[x^2*(a + b*ArcTanh[c*x])/(1 - c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d_. + e_.*Log[f_ + g_.*x_^2])*(a_. + b_.*ArcTanh[c_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*f + g, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "(f + g*x^2)*(d + e*Log[f + g*x^2])*(a + b*ArcCoth[c*x])^2/(2*g) - e*x^2*(a + b*ArcCoth[c*x])^2/2 + b/c*Int[(d + e*Log[f + g*x^2])*(a + b*ArcCoth[c*x]), x] + b*c*e*Int[x^2*(a + b*ArcCoth[c*x])/(1 - c^2*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d_. + e_.*Log[f_ + g_.*x_^2])*(a_. + b_.*ArcCoth[c_.*x_])^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*f + g, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "Unintegrable[u*(a + b*ArcTanh[c*x])^p, x] /; FreeQ[{a, b, c, p}, x] && (EqQ[u, 1] || MatchQ[u, (d_. + e_.*x)^q_. /; FreeQ[{d, e, q}, x]] || MatchQ[ u, (f_.*x)^m_.*(d_. + e_.*x)^q_. /; FreeQ[{d, e, f, m, q}, x]] || MatchQ[u, (d_. + e_.*x^2)^q_. /; FreeQ[{d, e, q}, x]] || MatchQ[ u, (f_.*x)^m_.*(d_. + e_.*x^2)^q_.",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_. + b_.*ArcTanh[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{d, e, f, m, q}, x]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "Unintegrable[u*(a + b*ArcCoth[c*x])^p, x] /; FreeQ[{a, b, c, p}, x] && (EqQ[u, 1] || MatchQ[u, (d_. + e_.*x)^q_. /; FreeQ[{d, e, q}, x]] || MatchQ[ u, (f_.*x)^m_.*(d_. + e_.*x)^q_. /; FreeQ[{d, e, f, m, q}, x]] || MatchQ[u, (d_. + e_.*x^2)^q_. /; FreeQ[{d, e, q}, x]] || MatchQ[ u, (f_.*x)^m_.*(d_. + e_.*x^2)^q_.",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_. + b_.*ArcCoth[c_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{d, e, f, m, q}, x]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "x*ArcTanh[c*x^n] - c*n*Int[x^n/(1 - c^2*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTanh[c_.*x_^n_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "x*ArcCoth[c*x^n] - c*n*Int[x^n/(1 - c^2*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCoth[c_.*x_^n_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*Log[1 + c*x^n]/2 - b*Log[1 - c*x^n]/2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcTanh[c_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
            "rhs": "Int[ExpandIntegrand[(a + b*Log[1 + x^(-n)/c]/2 - b*Log[1 - x^(-n)/c]/2)^p, x], x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCoth[c_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.1 u (a+b arctanh(c x^n))^p.m",
       "filename": "7.3.1 u (a+b arctanh(c x^n))^p.m",
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         "givens": "FreeQ[{a, c, d, e, f, m, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p - n/2 - 1, 0]) && IntegerQ[2*p]"
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            "lhs": "Int[u_.*(c_ + d_.*x_)^p_.*E^(n_.*ArcTanh[a_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ[p] || GtQ[c, 0])"
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         "givens": "FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && Not[IntegerQ[p] || GtQ[c, 0]]"
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            "rhs": "d^p*Int[u*(1 + c*x/d)^p*E^(n*ArcTanh[a*x])/x^p, x]",
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         "givens": "FreeQ[{a, c, d, n}, x] && EqQ[c^2 - a^2*d^2, 0] && IntegerQ[p]"
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         "givens": "FreeQ[{a, c, d, p}, x] && EqQ[c^2 - a^2*d^2, 0] && Not[IntegerQ[p]] && IntegerQ[n/2] && GtQ[c, 0]"
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         "givens": "FreeQ[{a, c, d, p}, x] && EqQ[c^2 - a^2*d^2, 0] && Not[IntegerQ[p]] && IntegerQ[n/2] && Not[GtQ[c, 0]]"
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            "rhs": "x^p*(c + d/x)^p/(1 + c*x/d)^p* Int[u*(1 + c*x/d)^p*E^(n*ArcTanh[a*x])/x^p, x]",
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         "givens": "FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] && Not[IntegerQ[p]]"
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            "rhs": "(n - a*x)* E^(n*ArcTanh[a*x])/(a*c*(n^2 - 1)*Sqrt[c + d*x^2])",
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            "lhs": "Int[E^(n_*ArcTanh[a_.*x_])/(c_ + d_.*x_^2)^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n]]"
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            "rhs": "(n + 2*a*(p + 1)*x)*(c + d*x^2)^(p + 1)* E^(n*ArcTanh[a*x])/(a*c*(n^2 - 4*(p + 1)^2)) - 2*(p + 1)*(2*p + 3)/(c*(n^2 - 4*(p + 1)^2))* Int[(c + d*x^2)^(p + 1)*E^(n*ArcTanh[a*x]), x]",
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         "givens": "FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && LtQ[p, -1] && Not[IntegerQ[n]] && NeQ[n^2 - 4*(p + 1)^2, 0] && IntegerQ[2*p]"
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            "lhs": "Int[E^(n_.*ArcTanh[a_.*x_])/(c_ + d_.*x_^2), x_Symbol]",
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         "givens": "FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n/2]]"
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         "givens": "FreeQ[{a, c, d, p}, x] && EqQ[a^2*c + d, 0] && IntegerQ[p] && IGtQ[(n + 1)/2, 0] && Not[IntegerQ[p - n/2]]"
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        "comment": false,
         "givens": "FreeQ[{a, c, d, p}, x] && EqQ[a^2*c + d, 0] && IntegerQ[p] && ILtQ[(n - 1)/2, 0] && Not[IntegerQ[p - n/2]]"
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            "lhs": "Int[(c_ + d_.*x_^2)^p_.*E^(n_.*ArcTanh[a_.*x_]), x_Symbol]",
        "comment": false,
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         "givens": "FreeQ[{a, c, d, p}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[p] || GtQ[c, 0]] && IGtQ[n/2, 0]"
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            "rhs": "c^IntPart[p]*(c + d*x^2)^FracPart[p]/(1 - a^2*x^2)^FracPart[p]* Int[(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x]",
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         "givens": "FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && EqQ[n^2 + 2*(p + 1), 0] && Not[IntegerQ[n]]"
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         "givens": "FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0]) && ILtQ[(n - 1)/2, 0] && Not[IntegerQ[p - n/2]]"
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         "givens": "FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])"
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         "givens": "FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[p] || GtQ[c, 0]] && IGtQ[n/2, 0]"
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            "rhs": "E^(n*ArcCoth[a*x])/(a*c*n)",
     "rulenumber": 0,
            "lhs": "Int[E^(n_.*ArcCoth[a_.*x_])/(c_ + d_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n/2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 Exponentials of inverse hyperbolic tangent.m",
       "filename": "7.3.3 Exponentials of inverse hyperbolic tangent.m",
            "rhs": "(n - a*x)* E^(n*ArcCoth[a*x])/(a*c*(n^2 - 1)*Sqrt[c + d*x^2])",
     "rulenumber": 0,
            "lhs": "Int[E^(n_*ArcCoth[a_.*x_])/(c_ + d_.*x_^2)^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 Exponentials of inverse hyperbolic tangent.m",
       "filename": "7.3.3 Exponentials of inverse hyperbolic tangent.m",
            "rhs": "(n + 2*a*(p + 1)*x)*(c + d*x^2)^(p + 1)* E^(n*ArcCoth[a*x])/(a*c*(n^2 - 4*(p + 1)^2)) - 2*(p + 1)*(2*p + 3)/(c*(n^2 - 4*(p + 1)^2))* Int[(c + d*x^2)^(p + 1)*E^(n*ArcCoth[a*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_ + d_.*x_^2)^p_*E^(n_.*ArcCoth[a_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n/2]] && LtQ[p, -1] && NeQ[p, -3/2] && NeQ[n^2 - 4*(p + 1)^2, 0] && (IntegerQ[p] || Not[IntegerQ[n]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 Exponentials of inverse hyperbolic tangent.m",
       "filename": "7.3.3 Exponentials of inverse hyperbolic tangent.m",
            "rhs": "-(1 - a*n*x)* E^(n*ArcCoth[a*x])/(a^2*c*(n^2 - 1)*Sqrt[c + d*x^2])",
     "rulenumber": 0,
            "lhs": "Int[x_*E^(n_*ArcCoth[a_.*x_])/(c_ + d_.*x_^2)^(3/2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 Exponentials of inverse hyperbolic tangent.m",
       "filename": "7.3.3 Exponentials of inverse hyperbolic tangent.m",
            "rhs": "(2*(p + 1) + a*n*x)*(c + d*x^2)^(p + 1)* E^(n*ArcCoth[a*x])/(a^2*c*(n^2 - 4*(p + 1)^2)) - n*(2*p + 3)/(a*c*(n^2 - 4*(p + 1)^2))* Int[(c + d*x^2)^(p + 1)*E^(n*ArcCoth[a*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(c_ + d_.*x_^2)^p_*E^(n_.*ArcCoth[a_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n/2]] && LeQ[p, -1] && NeQ[p, -3/2] && NeQ[n^2 - 4*(p + 1)^2, 0] && (IntegerQ[p] || Not[IntegerQ[n]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 Exponentials of inverse hyperbolic tangent.m",
       "filename": "7.3.3 Exponentials of inverse hyperbolic tangent.m",
            "rhs": "-(n + 2*(p + 1)*a*x)*(c + d*x^2)^(p + 1)* E^(n*ArcCoth[a*x])/(a^3*c*n^2*(n^2 - 1))",
     "rulenumber": 0,
            "lhs": "Int[x_^2*(c_ + d_.*x_^2)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n/2]] && EqQ[n^2 + 2*(p + 1), 0] && NeQ[n^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 Exponentials of inverse hyperbolic tangent.m",
       "filename": "7.3.3 Exponentials of inverse hyperbolic tangent.m",
            "rhs": "(n + 2*(p + 1)*a*x)*(c + d*x^2)^(p + 1)* E^(n*ArcCoth[a*x])/(a^3*c*(n^2 - 4*(p + 1)^2)) - (n^2 + 2*(p + 1))/(a^2*c*(n^2 - 4*(p + 1)^2))* Int[(c + d*x^2)^(p + 1)*E^(n*ArcCoth[a*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^2*(c_ + d_.*x_^2)^p_*E^(n_.*ArcCoth[a_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n/2]] && LeQ[p, -1] && NeQ[n^2 + 2*(p + 1), 0] && NeQ[n^2 - 4*(p + 1)^2, 0] && (IntegerQ[p] || Not[IntegerQ[n]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 Exponentials of inverse hyperbolic tangent.m",
       "filename": "7.3.3 Exponentials of inverse hyperbolic tangent.m",
            "rhs": "-(-c)^p/a^(m + 1)* Subst[Int[E^(n*x)*Coth[x]^(m + 2*(p + 1))/Cosh[x]^(2*(p + 1)), x], x, ArcCoth[a*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(c_ + d_.*x_^2)^p_*E^(n_.*ArcCoth[a_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n/2]] && IntegerQ[m] && LeQ[3, m, -2 (p + 1)] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 Exponentials of inverse hyperbolic tangent.m",
       "filename": "7.3.3 Exponentials of inverse hyperbolic tangent.m",
            "rhs": "d^p*Int[u*x^(2*p)*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(c_ + d_.*x_^2)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n/2]] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 Exponentials of inverse hyperbolic tangent.m",
       "filename": "7.3.3 Exponentials of inverse hyperbolic tangent.m",
            "rhs": "(c + d*x^2)^p/(x^(2*p)*(1 - 1/(a^2*x^2))^p)* Int[u*x^(2*p)*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(c_ + d_.*x_^2)^p_*E^(n_.*ArcCoth[a_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && Not[IntegerQ[n/2]] && Not[IntegerQ[p]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 Exponentials of inverse hyperbolic tangent.m",
       "filename": "7.3.3 Exponentials of inverse hyperbolic tangent.m",
            "rhs": "c^p/a^(2*p)* Int[u/x^(2*p)*(-1 + a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(c_ + d_./x_^2)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && Not[IntegerQ[n/2]] && (IntegerQ[p] || GtQ[c, 0]) && IntegersQ[2*p, p + n/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 Exponentials of inverse hyperbolic tangent.m",
       "filename": "7.3.3 Exponentials of inverse hyperbolic tangent.m",
            "rhs": "-c^p* Subst[Int[(1 - x/a)^(p - n/2)*(1 + x/a)^(p + n/2)/x^2, x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(c_ + d_./x_^2)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && Not[IntegerQ[n/2]] && (IntegerQ[p] || GtQ[c, 0]) && Not[IntegersQ[2*p, p + n/2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 Exponentials of inverse hyperbolic tangent.m",
       "filename": "7.3.3 Exponentials of inverse hyperbolic tangent.m",
            "rhs": "-c^p* Subst[Int[(1 - x/a)^(p - n/2)*(1 + x/a)^(p + n/2)/x^(m + 2), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(c_ + d_./x_^2)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && Not[IntegerQ[n/2]] && (IntegerQ[p] || GtQ[c, 0]) && Not[IntegersQ[2*p, p + n/2]] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 Exponentials of inverse hyperbolic tangent.m",
       "filename": "7.3.3 Exponentials of inverse hyperbolic tangent.m",
            "rhs": "-c^p*x^m*(1/x)^m* Subst[Int[(1 - x/a)^(p - n/2)*(1 + x/a)^(p + n/2)/x^(m + 2), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*(c_ + d_./x_^2)^p_.*E^(n_.*ArcCoth[a_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, m, n, p}, x] && EqQ[c + a^2*d, 0] && Not[IntegerQ[n/2]] && (IntegerQ[p] || GtQ[c, 0]) && Not[IntegersQ[2*p, p + n/2]] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 Exponentials of inverse hyperbolic tangent.m",
       "filename": "7.3.3 Exponentials of inverse hyperbolic tangent.m",
            "rhs": "c^IntPart[p]*(c + d/x^2)^FracPart[p]/(1 - 1/(a^2*x^2))^FracPart[p]* Int[u*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(c_ + d_./x_^2)^p_*E^(n_.*ArcCoth[a_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && Not[IntegerQ[n/2]] && Not[IntegerQ[p] || GtQ[c, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 Exponentials of inverse hyperbolic tangent.m",
       "filename": "7.3.3 Exponentials of inverse hyperbolic tangent.m",
            "rhs": "(-1)^(n/2)*Int[u*E^(n*ArcTanh[c*(a + b*x)]), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*E^(n_*ArcCoth[c_.*(a_ + b_.*x_)]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IntegerQ[n/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 Exponentials of inverse hyperbolic tangent.m",
       "filename": "7.3.3 Exponentials of inverse hyperbolic tangent.m",
            "rhs": "(c*(a + b*x))^(n/ 2)*(1 + 1/(c*(a + b*x)))^(n/2)/(1 + a*c + b*c*x)^(n/2)* Int[(1 + a*c + b*c*x)^(n/2)/(-1 + a*c + b*c*x)^(n/2), x]",
     "rulenumber": 0,
            "lhs": "Int[E^(n_.*ArcCoth[c_.*(a_ + b_.*x_)]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && Not[IntegerQ[n/2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 Exponentials of inverse hyperbolic tangent.m",
       "filename": "7.3.3 Exponentials of inverse hyperbolic tangent.m",
            "rhs": "-4/(n*b^(m + 1)*c^(m + 1))* Subst[ Int[x^(2/n)*(1 + a*c + (1 - a*c)*x^(2/n))^ m/(-1 + x^(2/n))^(m + 2), x], x, (1 + 1/(c*(a + b*x)))^(n/2)/(1 - 1/(c*(a + b*x)))^(n/2)]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*E^(n_*ArcCoth[c_.*(a_ + b_.*x_)]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && ILtQ[m, 0] && LtQ[-1, n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 Exponentials of inverse hyperbolic tangent.m",
       "filename": "7.3.3 Exponentials of inverse hyperbolic tangent.m",
            "rhs": "(c*(a + b*x))^(n/ 2)*(1 + 1/(c*(a + b*x)))^(n/2)/(1 + a*c + b*c*x)^(n/2)* Int[(d + e*x)^m*(1 + a*c + b*c*x)^(n/2)/(-1 + a*c + b*c*x)^(n/2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*E^(n_.*ArcCoth[c_.*(a_ + b_.*x_)]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && Not[IntegerQ[n/2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 Exponentials of inverse hyperbolic tangent.m",
       "filename": "7.3.3 Exponentials of inverse hyperbolic tangent.m",
            "rhs": "(c/(1 - a^2))^ p*((a + b*x)/(1 + a + b*x))^(n/2)*((1 + a + b*x)/(a + b*x))^(n/ 2)*((1 - a - b*x)^(n/2)/(-1 + a + b*x)^(n/2))* Int[u*(1 - a - b*x)^(p - n/2)*(1 + a + b*x)^(p + n/2), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(c_ + d_.*x_ + e_.*x_^2)^p_.*E^(n_.*ArcCoth[a_ + b_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p}, x] && Not[IntegerQ[n/2]] && EqQ[b*d - 2*a*e, 0] && EqQ[b^2*c + e (1 - a^2), 0] && (IntegerQ[p] || GtQ[c/(1 - a^2), 0])"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 Exponentials of inverse hyperbolic tangent.m",
       "filename": "7.3.3 Exponentials of inverse hyperbolic tangent.m",
            "rhs": "(c + d*x + e*x^2)^p/(1 - a^2 - 2*a*b*x - b^2*x^2)^p* Int[u*(1 - a^2 - 2*a*b*x - b^2*x^2)^p*E^(n*ArcCoth[a*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(c_ + d_.*x_ + e_.*x_^2)^p_.*E^(n_.*ArcCoth[a_ + b_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p}, x] && Not[IntegerQ[n/2]] && EqQ[b*d - 2*a*e, 0] && EqQ[b^2*c + e (1 - a^2), 0] && Not[IntegerQ[p] || GtQ[c/(1 - a^2), 0]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 Exponentials of inverse hyperbolic tangent.m",
       "filename": "7.3.3 Exponentials of inverse hyperbolic tangent.m",
            "rhs": "Int[u*E^(n*ArcTanh[a/c + b*x/c]), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*E^(n_.*ArcCoth[c_./(a_. + b_.*x_)]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "x*ArcTanh[a + b*x^n] - b*n*Int[x^n/(1 - a^2 - 2*a*b*x^n - b^2*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTanh[a_ + b_.*x_^n_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "x*ArcCoth[a + b*x^n] - b*n*Int[x^n/(1 - a^2 - 2*a*b*x^n - b^2*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCoth[a_ + b_.*x_^n_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "1/2*Int[Log[1 + a + b*x^n]/x, x] - 1/2*Int[Log[1 - a - b*x^n]/x, x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTanh[a_. + b_.*x_^n_.]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "1/2*Int[Log[1 + 1/(a + b*x^n)]/x, x] - 1/2*Int[Log[1 - 1/(a + b*x^n)]/x, x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCoth[a_. + b_.*x_^n_.]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "x^(m + 1)*ArcTanh[a + b*x^n]/(m + 1) - b*n/(m + 1)* Int[x^(m + n)/(1 - a^2 - 2*a*b*x^n - b^2*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*ArcTanh[a_ + b_.*x_^n_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && RationalQ[m, n] && NeQ[m, -1] && NeQ[m + 1, n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "x^(m + 1)*ArcCoth[a + b*x^n]/(m + 1) - b*n/(m + 1)* Int[x^(m + n)/(1 - a^2 - 2*a*b*x^n - b^2*x^(2*n)), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*ArcCoth[a_ + b_.*x_^n_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && RationalQ[m, n] && NeQ[m, -1] && NeQ[m + 1, n]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "1/2*Int[Log[1 + a + b*f^(c + d*x)], x] - 1/2*Int[Log[1 - a - b*f^(c + d*x)], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTanh[a_. + b_.*f_^(c_. + d_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f}, x] "
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "1/2*Int[Log[1 + 1/(a + b*f^(c + d*x))], x] - 1/2*Int[Log[1 - 1/(a + b*f^(c + d*x))], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCoth[a_. + b_.*f_^(c_. + d_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f}, x] "
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "1/2*Int[x^m*Log[1 + a + b*f^(c + d*x)], x] - 1/2*Int[x^m*Log[1 - a - b*f^(c + d*x)], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*ArcTanh[a_. + b_.*f_^(c_. + d_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "1/2*Int[x^m*Log[1 + 1/(a + b*f^(c + d*x))], x] - 1/2*Int[x^m*Log[1 - 1/(a + b*f^(c + d*x))], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*ArcCoth[a_. + b_.*f_^(c_. + d_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, f}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "Int[u*ArcCoth[a/c + b*x^n/c]^m, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*ArcTanh[c_./(a_. + b_.*x_^n_.)]^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, m}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "Int[u*ArcTanh[a/c + b*x^n/c]^m, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*ArcCoth[c_./(a_. + b_.*x_^n_.)]^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, m}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "x*ArcTanh[(c*x)/Sqrt[a + b*x^2]] - c*Int[x/Sqrt[a + b*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTanh[c_.*x_/Sqrt[a_. + b_.*x_^2]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[b, c^2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "x*ArcCoth[(c*x)/Sqrt[a + b*x^2]] - c*Int[x/Sqrt[a + b*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCoth[c_.*x_/Sqrt[a_. + b_.*x_^2]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[b, c^2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "ArcTanh[c*x/Sqrt[a + b*x^2]]*Log[x] - c*Int[Log[x]/Sqrt[a + b*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTanh[c_.*x_/Sqrt[a_. + b_.*x_^2]]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[b, c^2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "ArcCoth[c*x/Sqrt[a + b*x^2]]*Log[x] - c*Int[Log[x]/Sqrt[a + b*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCoth[c_.*x_/Sqrt[a_. + b_.*x_^2]]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[b, c^2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "(d*x)^(m + 1)* ArcTanh[(c*x)/Sqrt[a + b*x^2]]/(d*(m + 1)) - c/(d*(m + 1))*Int[(d*x)^(m + 1)/Sqrt[a + b*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*ArcTanh[c_.*x_/Sqrt[a_. + b_.*x_^2]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && EqQ[b, c^2] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "(d*x)^(m + 1)* ArcCoth[(c*x)/Sqrt[a + b*x^2]]/(d*(m + 1)) - c/(d*(m + 1))*Int[(d*x)^(m + 1)/Sqrt[a + b*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*ArcCoth[c_.*x_/Sqrt[a_. + b_.*x_^2]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && EqQ[b, c^2] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "1/c*Log[ArcTanh[c*x/Sqrt[a + b*x^2]]]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_. + b_.*x_^2]*ArcTanh[c_.*x_/Sqrt[a_. + b_.*x_^2]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[b, c^2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "-1/c*Log[ArcCoth[c*x/Sqrt[a + b*x^2]]]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[a_. + b_.*x_^2]*ArcCoth[c_.*x_/Sqrt[a_. + b_.*x_^2]]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[b, c^2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "ArcTanh[c*x/Sqrt[a + b*x^2]]^(m + 1)/(c*(m + 1))",
     "rulenumber": 0,
            "lhs": "Int[ArcTanh[c_.*x_/Sqrt[a_. + b_.*x_^2]]^m_./Sqrt[a_. + b_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m}, x] && EqQ[b, c^2] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "-ArcCoth[c*x/Sqrt[a + b*x^2]]^(m + 1)/(c*(m + 1))",
     "rulenumber": 0,
            "lhs": "Int[ArcCoth[c_.*x_/Sqrt[a_. + b_.*x_^2]]^m_./Sqrt[a_. + b_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m}, x] && EqQ[b, c^2] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "Sqrt[a + b*x^2]/Sqrt[d + e*x^2]* Int[ArcTanh[c*x/Sqrt[a + b*x^2]]^m/Sqrt[a + b*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTanh[c_.*x_/Sqrt[a_. + b_.*x_^2]]^m_./Sqrt[d_. + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && EqQ[b, c^2] && EqQ[b*d - a*e, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "Sqrt[a + b*x^2]/Sqrt[d + e*x^2]* Int[ArcCoth[c*x/Sqrt[a + b*x^2]]^m/Sqrt[a + b*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCoth[c_.*x_/Sqrt[a_. + b_.*x_^2]]^m_./Sqrt[d_. + e_.*x_^2], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && EqQ[b, c^2] && EqQ[b*d - a*e, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "With[{tmp = InverseFunctionOfLinear[u, x]}, (-Discriminant[v, x]/(4*Coefficient[v, x, 2]))^n/ Coefficient[tmp[[1]], x, 1]* Subst[ Int[SimplifyIntegrand[ SubstForInverseFunction[u, tmp, x]*Sech[x]^(2*(n + 1)), x], x], x, tmp] /; Not[FalseQ[tmp]] && EqQ[Head[tmp], ArcTanh] && EqQ[Discriminant[v, x]*tmp[[1]]^2 - D[v, x]^2, 0]] /; QuadraticQ[v, x] && ILtQ[n, 0] && PosQ[Discriminant[v, x]] && MatchQ[u, r_.*f_^w_",
     "rulenumber": 0,
            "lhs": "Int[u_*v_^n_., x_Symbol] := With[{tmp = InverseFunctionOfLinear[u, x]}, ShowStep[\"\", \"Int[f[x,ArcTanh[a+b*x]]/(1-(a+b*x)^2),x]\", \"Subst[Int[f[-a/b+Tanh[x]/b,x],x],x,ArcTanh[a+b*x]]/b\", Hold[ (-Discriminant[v, x]/(4*Coefficient[v, x, 2]))^n/ Coefficient[tmp[[1]], x, 1]* Subst[ Int[SimplifyIntegrand[ SubstForInverseFunction[u, tmp, x]*Sech[x]^(2*(n + 1)), x], x], x, tmp]]] /; Not[FalseQ[tmp]] && EqQ[Head[tmp], ArcTanh] && EqQ[Discriminant[v, x]*tmp[[1]]^2 - D[v, x]^2, 0]] /; SimplifyFlag && QuadraticQ[v, x] && ILtQ[n, 0] && PosQ[Discriminant[v, x]] && MatchQ[u, r_.*f_^w_ /; FreeQ[f, x]], Int[u_*v_^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[f, x]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "With[{tmp = InverseFunctionOfLinear[u, x]}, (-Discriminant[v, x]/(4*Coefficient[v, x, 2]))^n/ Coefficient[tmp[[1]], x, 1]* Subst[ Int[SimplifyIntegrand[ SubstForInverseFunction[u, tmp, x]*(-Csch[x]^2)^(n + 1), x], x], x, tmp] /; Not[FalseQ[tmp]] && EqQ[Head[tmp], ArcCoth] && EqQ[Discriminant[v, x]*tmp[[1]]^2 - D[v, x]^2, 0]] /; QuadraticQ[v, x] && ILtQ[n, 0] && PosQ[Discriminant[v, x]] && MatchQ[u, r_.*f_^w_",
     "rulenumber": 0,
            "lhs": "Int[u_*v_^n_., x_Symbol] := With[{tmp = InverseFunctionOfLinear[u, x]}, ShowStep[\"\", \"Int[f[x,ArcCoth[a+b*x]]/(1-(a+b*x)^2),x]\", \"Subst[Int[f[-a/b+Coth[x]/b,x],x],x,ArcCoth[a+b*x]]/b\", Hold[ (-Discriminant[v, x]/(4*Coefficient[v, x, 2]))^n/ Coefficient[tmp[[1]], x, 1]* Subst[ Int[SimplifyIntegrand[ SubstForInverseFunction[u, tmp, x]*(-Csch[x]^2)^(n + 1), x], x], x, tmp]]] /; Not[FalseQ[tmp]] && EqQ[Head[tmp], ArcCoth] && EqQ[Discriminant[v, x]*tmp[[1]]^2 - D[v, x]^2, 0]] /; SimplifyFlag && QuadraticQ[v, x] && ILtQ[n, 0] && PosQ[Discriminant[v, x]] && MatchQ[u, r_.*f_^w_ /; FreeQ[f, x]], Int[u_*v_^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[f, x]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "x*ArcTanh[c + d*Tanh[a + b*x]] + b*Int[x/(c - d + c*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTanh[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[(c - d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "x*ArcCoth[c + d*Tanh[a + b*x]] + b*Int[x/(c - d + c*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCoth[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[(c - d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "x*ArcTanh[c + d*Coth[a + b*x]] + b*Int[x/(c - d - c*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTanh[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[(c - d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "x*ArcCoth[c + d*Coth[a + b*x]] + b*Int[x/(c - d - c*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCoth[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[(c - d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "x*ArcTanh[c + d*Tanh[a + b*x]] + b*(1 - c - d)* Int[x*E^(2*a + 2*b*x)/(1 - c + d + (1 - c - d)*E^(2*a + 2*b*x)), x] - b*(1 + c + d)* Int[x*E^(2*a + 2*b*x)/(1 + c - d + (1 + c + d)*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTanh[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[(c - d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "x*ArcCoth[c + d*Tanh[a + b*x]] + b*(1 - c - d)* Int[x*E^(2*a + 2*b*x)/(1 - c + d + (1 - c - d)*E^(2*a + 2*b*x)), x] - b*(1 + c + d)* Int[x*E^(2*a + 2*b*x)/(1 + c - d + (1 + c + d)*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCoth[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[(c - d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "x*ArcTanh[c + d*Coth[a + b*x]] + b*(1 + c + d)* Int[x*E^(2*a + 2*b*x)/(1 + c - d - (1 + c + d)*E^(2*a + 2*b*x)), x] - b*(1 - c - d)* Int[x*E^(2*a + 2*b*x)/(1 - c + d - (1 - c - d)*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTanh[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[(c - d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "x*ArcCoth[c + d*Coth[a + b*x]] + b*(1 + c + d)* Int[x*E^(2*a + 2*b*x)/(1 + c - d - (1 + c + d)*E^(2*a + 2*b*x)), x] - b*(1 - c - d)* Int[x*E^(2*a + 2*b*x)/(1 - c + d - (1 - c - d)*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCoth[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[(c - d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcTanh[c + d*Tanh[a + b*x]]/(f*(m + 1)) + b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c - d + c*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcTanh[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcCoth[c + d*Tanh[a + b*x]]/(f*(m + 1)) + b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c - d + c*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcCoth[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcTanh[c + d*Coth[a + b*x]]/(f*(m + 1)) + b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c - d - c*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcTanh[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcCoth[c + d*Coth[a + b*x]]/(f*(m + 1)) + b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c - d - c*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcCoth[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcTanh[c + d*Tanh[a + b*x]]/(f*(m + 1)) + b*(1 - c - d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(1 - c + d + (1 - c - d)*E^(2*a + 2*b*x)), x] - b*(1 + c + d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(1 + c - d + (1 + c + d)*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcTanh[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcCoth[c + d*Tanh[a + b*x]]/(f*(m + 1)) + b*(1 - c - d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(1 - c + d + (1 - c - d)*E^(2*a + 2*b*x)), x] - b*(1 + c + d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(1 + c - d + (1 + c + d)*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcCoth[c_. + d_.*Tanh[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcTanh[c + d*Coth[a + b*x]]/(f*(m + 1)) + b*(1 + c + d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(1 + c - d - (1 + c + d)*E^(2*a + 2*b*x)), x] - b*(1 - c - d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(1 - c + d - (1 - c - d)*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcTanh[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcCoth[c + d*Coth[a + b*x]]/(f*(m + 1)) + b*(1 + c + d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(1 + c - d - (1 + c + d)*E^(2*a + 2*b*x)), x] - b*(1 - c - d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*a + 2*b*x)/(1 - c + d - (1 - c - d)*E^(2*a + 2*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcCoth[c_. + d_.*Coth[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "x*ArcTanh[Tan[a + b*x]] - b*Int[x*Sec[2*a + 2*b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTanh[Tan[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "x*ArcCoth[Tan[a + b*x]] - b*Int[x*Sec[2*a + 2*b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCoth[Tan[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "x*ArcTanh[Cot[a + b*x]] - b*Int[x*Sec[2*a + 2*b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTanh[Cot[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "x*ArcCoth[Cot[a + b*x]] - b*Int[x*Sec[2*a + 2*b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCoth[Cot[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcTanh[Tan[a + b*x]]/(f*(m + 1)) - b/(f*(m + 1))*Int[(e + f*x)^(m + 1)*Sec[2*a + 2*b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcTanh[Tan[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcCoth[Tan[a + b*x]]/(f*(m + 1)) - b/(f*(m + 1))*Int[(e + f*x)^(m + 1)*Sec[2*a + 2*b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcCoth[Tan[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcTanh[Cot[a + b*x]]/(f*(m + 1)) - b/(f*(m + 1))*Int[(e + f*x)^(m + 1)*Sec[2*a + 2*b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcTanh[Cot[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcCoth[Cot[a + b*x]]/(f*(m + 1)) - b/(f*(m + 1))*Int[(e + f*x)^(m + 1)*Sec[2*a + 2*b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcCoth[Cot[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, e, f}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "x*ArcTanh[c + d*Tan[a + b*x]] + I*b*Int[x/(c + I*d + c*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTanh[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[(c + I*d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "x*ArcCoth[c + d*Tan[a + b*x]] + I*b*Int[x/(c + I*d + c*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCoth[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[(c + I*d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "x*ArcTanh[c + d*Cot[a + b*x]] + I*b*Int[x/(c - I*d - c*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTanh[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[(c - I*d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "x*ArcCoth[c + d*Cot[a + b*x]] + I*b*Int[x/(c - I*d - c*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCoth[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && EqQ[(c - I*d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "x*ArcTanh[c + d*Tan[a + b*x]] + I*b*(1 - c + I*d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 - c - I*d + (1 - c + I*d)*E^(2*I*a + 2*I*b*x)), x] - I*b*(1 + c - I*d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 + c + I*d + (1 + c - I*d)*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTanh[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[(c + I*d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "x*ArcCoth[c + d*Tan[a + b*x]] + I*b*(1 - c + I*d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 - c - I*d + (1 - c + I*d)*E^(2*I*a + 2*I*b*x)), x] - I*b*(1 + c - I*d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 + c + I*d + (1 + c - I*d)*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCoth[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[(c + I*d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "x*ArcTanh[c + d*Cot[a + b*x]] - I*b*(1 - c - I*d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 - c + I*d - (1 - c - I*d)*E^(2*I*a + 2*I*b*x)), x] + I*b*(1 + c + I*d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 + c - I*d - (1 + c + I*d)*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTanh[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[(c - I*d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "x*ArcCoth[c + d*Cot[a + b*x]] - I*b*(1 - c - I*d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 - c + I*d - (1 - c - I*d)*E^(2*I*a + 2*I*b*x)), x] + I*b*(1 + c + I*d)* Int[x*E^(2*I*a + 2*I*b*x)/(1 + c - I*d - (1 + c + I*d)*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCoth[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NeQ[(c - I*d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcTanh[c + d*Tan[a + b*x]]/(f*(m + 1)) + I*b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c + I*d + c*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcTanh[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c + I*d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcCoth[c + d*Tan[a + b*x]]/(f*(m + 1)) + I*b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c + I*d + c*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcCoth[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c + I*d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcTanh[c + d*Cot[a + b*x]]/(f*(m + 1)) + I*b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c - I*d - c*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcTanh[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - I*d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcCoth[c + d*Cot[a + b*x]]/(f*(m + 1)) + I*b/(f*(m + 1))* Int[(e + f*x)^(m + 1)/(c - I*d - c*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcCoth[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - I*d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcTanh[c + d*Tan[a + b*x]]/(f*(m + 1)) + I*b*(1 - c + I*d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 - c - I*d + (1 - c + I*d)*E^(2*I*a + 2*I*b*x)), x] - I*b*(1 + c - I*d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 + c + I*d + (1 + c - I*d)*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcTanh[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c + I*d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcCoth[c + d*Tan[a + b*x]]/(f*(m + 1)) + I*b*(1 - c + I*d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 - c - I*d + (1 - c + I*d)*E^(2*I*a + 2*I*b*x)), x] - I*b*(1 + c - I*d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 + c + I*d + (1 + c - I*d)*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcCoth[c_. + d_.*Tan[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c + I*d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcTanh[c + d*Cot[a + b*x]]/(f*(m + 1)) - I*b*(1 - c - I*d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 - c + I*d - (1 - c - I*d)*E^(2*I*a + 2*I*b*x)), x] + I*b*(1 + c + I*d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 + c - I*d - (1 + c + I*d)*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcTanh[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - I*d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "(e + f*x)^(m + 1)* ArcCoth[c + d*Cot[a + b*x]]/(f*(m + 1)) - I*b*(1 - c - I*d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 - c + I*d - (1 - c - I*d)*E^(2*I*a + 2*I*b*x)), x] + I*b*(1 + c + I*d)/(f*(m + 1))* Int[(e + f*x)^(m + 1)* E^(2*I*a + 2*I*b*x)/(1 + c - I*d - (1 + c + I*d)*E^(2*I*a + 2*I*b*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*ArcCoth[c_. + d_.*Cot[a_. + b_.*x_]], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[(c - I*d)^2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "x*ArcTanh[u] - Int[SimplifyIntegrand[x*D[u, x]/(1 - u^2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcTanh[u_], x_Symbol]",
        "comment": false,
         "givens": "InverseFunctionFreeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "x*ArcCoth[u] - Int[SimplifyIntegrand[x*D[u, x]/(1 - u^2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCoth[u_], x_Symbol]",
        "comment": false,
         "givens": "InverseFunctionFreeQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "(c + d*x)^(m + 1)*(a + b*ArcTanh[u])/(d*(m + 1)) - b/(d*(m + 1))* Int[SimplifyIntegrand[(c + d*x)^(m + 1)*D[u, x]/(1 - u^2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(a_. + b_.*ArcTanh[u_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && Not[FunctionOfQ[(c + d*x)^(m + 1), u, x]] && FalseQ[PowerVariableExpn[u, m + 1, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "(c + d*x)^(m + 1)*(a + b*ArcCoth[u])/(d*(m + 1)) - b/(d*(m + 1))* Int[SimplifyIntegrand[(c + d*x)^(m + 1)*D[u, x]/(1 - u^2), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(a_. + b_.*ArcCoth[u_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && Not[FunctionOfQ[(c + d*x)^(m + 1), u, x]] && FalseQ[PowerVariableExpn[u, m + 1, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "With[{w = IntHide[v, x]}, Dist[(a + b*ArcTanh[u]), w, x] - b*Int[SimplifyIntegrand[w*D[u, x]/(1 - u^2), x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && Not[MatchQ[v, (c_. + d_.*x)^m_.",
     "rulenumber": 0,
            "lhs": "Int[v_*(a_. + b_.*ArcTanh[u_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, m}, x]]] && FalseQ[FunctionOfLinear[v*(a + b*ArcTanh[u]), x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 Miscellaneous inverse hyperbolic tangent.m",
       "filename": "7.3.4 Miscellaneous inverse hyperbolic tangent.m",
            "rhs": "With[{w = IntHide[v, x]}, Dist[(a + b*ArcCoth[u]), w, x] - b*Int[SimplifyIntegrand[w*D[u, x]/(1 - u^2), x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && Not[MatchQ[v, (c_. + d_.*x)^m_.",
     "rulenumber": 0,
            "lhs": "Int[v_*(a_. + b_.*ArcCoth[u_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, m}, x]]] && FalseQ[FunctionOfLinear[v*(a + b*ArcCoth[u]), x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "x*ArcSech[c*x] + Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)]*Int[1/Sqrt[1 - c^2*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcSech[c_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[c, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "x*ArcCsch[c*x] + 1/c*Int[1/(x*Sqrt[1 + 1/(c^2*x^2)]), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCsch[c_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[c, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "-1/c* Subst[Int[(a + b*x)^n*Sech[x]*Tanh[x], x], x, ArcSech[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSech[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "-1/c* Subst[Int[(a + b*x)^n*Csch[x]*Coth[x], x], x, ArcCsch[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCsch[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "-Subst[Int[(a + b*ArcCosh[x/c])/x, x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSech[c_.*x_])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "-Subst[Int[(a + b*ArcSinh[x/c])/x, x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCsch[c_.*x_])/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "(d*x)^(m + 1)*(a + b*ArcSech[c*x])/(d*(m + 1)) + b*Sqrt[1 + c*x]/(m + 1)*Sqrt[1/(1 + c*x)]* Int[(d*x)^m/(Sqrt[1 - c*x]*Sqrt[1 + c*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_. + b_.*ArcSech[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "(d*x)^(m + 1)*(a + b*ArcCsch[c*x])/(d*(m + 1)) + b*d/(c*(m + 1))*Int[(d*x)^(m - 1)/Sqrt[1 + 1/(c^2*x^2)], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*(a_. + b_.*ArcCsch[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "-1/c^(m + 1)* Subst[Int[(a + b*x)^n*Sech[x]^(m + 1)*Tanh[x], x], x, ArcSech[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*ArcSech[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (GtQ[n, 0] || LtQ[m, -1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "-1/c^(m + 1)* Subst[Int[(a + b*x)^n*Csch[x]^(m + 1)*Coth[x], x], x, ArcCsch[c*x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(a_. + b_.*ArcCsch[c_.*x_])^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (GtQ[n, 0] || LtQ[m, -1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "(a + b*ArcSech[c*x])* Log[1 + (e - Sqrt[-c^2*d^2 + e^2])/(c*d*E^ArcSech[c*x])]/e + (a + b*ArcSech[c*x])* Log[1 + (e + Sqrt[-c^2*d^2 + e^2])/(c*d*E^ArcSech[c*x])]/e - (a + b*ArcSech[c*x])*Log[1 + 1/E^(2*ArcSech[c*x])]/e + b/e*Int[(Sqrt[(1 - c*x)/(1 + c*x)]* Log[1 + (e - Sqrt[-c^2*d^2 + e^2])/(c*d* E^ArcSech[c*x])])/(x*(1 - c*x)), x] + b/e*Int[(Sqrt[(1 - c*x)/(1 + c*x)]* Log[1 + (e + Sqrt[-c^2*d^2 + e^2])/(c*d* E^ArcSech[c*x])])/(x*(1 - c*x)), x] - b/e*Int[(Sqrt[(1 - c*x)/(1 + c*x)]* Log[1 + 1/E^(2*ArcSech[c*x])])/(x*(1 - c*x)), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSech[c_.*x_])/(d_. + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "(d + e*x)^(m + 1)*(a + b*ArcSech[c*x])/(e*(m + 1)) + b*Sqrt[1 + c*x]/(e*(m + 1))*Sqrt[1/(1 + c*x)]* Int[(d + e*x)^(m + 1)/(x*Sqrt[1 - c^2*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*(a_. + b_.*ArcSech[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "(a + b*ArcCsch[c*x])* Log[1 - (e - Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x]/(c*d)]/e + (a + b*ArcCsch[c*x])* Log[1 - (e + Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x]/(c*d)]/e - (a + b*ArcCsch[c*x])*Log[1 - E^(2*ArcCsch[c*x])]/e + b/(c*e)* Int[Log[1 - (e - Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x]/(c*d)]/(x^2* Sqrt[1 + 1/(c^2*x^2)]), x] + b/(c*e)* Int[Log[1 - (e + Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x]/(c*d)]/(x^2* Sqrt[1 + 1/(c^2*x^2)]), x] - b/(c*e)* Int[Log[1 - E^(2*ArcCsch[c*x])]/(x^2*Sqrt[1 + 1/(c^2*x^2)]), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCsch[c_.*x_])/(d_. + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "(d + e*x)^(m + 1)*(a + b*ArcCsch[c*x])/(e*(m + 1)) + b/(c*e*(m + 1))* Int[(d + e*x)^(m + 1)/(x^2*Sqrt[1 + 1/(c^2*x^2)]), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*(a_. + b_.*ArcCsch[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[(a + b*ArcSech[c*x]), u, x] + b*Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)]* Int[SimplifyIntegrand[u/(x*Sqrt[1 - c*x]*Sqrt[1 + c*x]), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcSech[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "With[{u = IntHide[(d + e*x^2)^p, x]}, Dist[(a + b*ArcCsch[c*x]), u, x] - b*c*x/Sqrt[-c^2*x^2]* Int[SimplifyIntegrand[u/(x*Sqrt[-1 - c^2*x^2]), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcCsch[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "-Subst[ Int[(e + d*x^2)^p*(a + b*ArcCosh[x/c])^n/x^(2*(p + 1)), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcSech[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "-Subst[ Int[(e + d*x^2)^p*(a + b*ArcSinh[x/c])^n/x^(2*(p + 1)), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcCsch[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "-Sqrt[x^2]/x* Subst[Int[(e + d*x^2)^p*(a + b*ArcCosh[x/c])^n/x^(2*(p + 1)), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcSech[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p + 1/2] && GtQ[e, 0] && LtQ[d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "-Sqrt[x^2]/x* Subst[Int[(e + d*x^2)^p*(a + b*ArcSinh[x/c])^n/x^(2*(p + 1)), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcCsch[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[e - c^2*d, 0] && IntegerQ[p + 1/2] && GtQ[e, 0] && LtQ[d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "-Sqrt[d + e*x^2]/(x*Sqrt[e + d/x^2])* Subst[Int[(e + d*x^2)^p*(a + b*ArcCosh[x/c])^n/x^(2*(p + 1)), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcSech[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p + 1/2] && Not[GtQ[e, 0] && LtQ[d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "-Sqrt[d + e*x^2]/(x*Sqrt[e + d/x^2])* Subst[Int[(e + d*x^2)^p*(a + b*ArcSinh[x/c])^n/x^(2*(p + 1)), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcCsch[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[e - c^2*d, 0] && IntegerQ[p + 1/2] && Not[GtQ[e, 0] && LtQ[d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "(d + e*x^2)^(p + 1)*(a + b*ArcSech[c*x])/(2* e*(p + 1)) + b*Sqrt[1 + c*x]/(2*e*(p + 1))*Sqrt[1/(1 + c*x)]* Int[(d + e*x^2)^(p + 1)/(x*Sqrt[1 - c*x]*Sqrt[1 + c*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcSech[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "(d + e*x^2)^(p + 1)*(a + b*ArcCsch[c*x])/(2* e*(p + 1)) - b*c*x/(2*e*(p + 1)*Sqrt[-c^2*x^2])* Int[(d + e*x^2)^(p + 1)/(x*Sqrt[-1 - c^2*x^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[x_*(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcCsch[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[(a + b*ArcSech[c*x]), u, x] + b*Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)]* Int[SimplifyIntegrand[u/(x*Sqrt[1 - c*x]*Sqrt[1 + c*x]), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcSech[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p}, x] && ( IGtQ[p, 0] && Not[ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0]] || IGtQ[(m + 1)/2, 0] && Not[ILtQ[p, 0] && GtQ[m + 2*p + 3, 0]] || ILtQ[(m + 2*p + 1)/2, 0] && Not[ILtQ[(m - 1)/2, 0]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[(a + b*ArcCsch[c*x]), u, x] - b*c*x/Sqrt[-c^2*x^2]* Int[SimplifyIntegrand[u/(x*Sqrt[-1 - c^2*x^2]), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_.*x_)^m_.*(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcCsch[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p}, x] && ( IGtQ[p, 0] && Not[ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0]] || IGtQ[(m + 1)/2, 0] && Not[ILtQ[p, 0] && GtQ[m + 2*p + 3, 0]] || ILtQ[(m + 2*p + 1)/2, 0] && Not[ILtQ[(m - 1)/2, 0]] )"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "-Subst[ Int[(e + d*x^2)^p*(a + b*ArcCosh[x/c])^n/x^(m + 2*(p + 1)), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcSech[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && IntegersQ[m, p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "-Subst[ Int[(e + d*x^2)^p*(a + b*ArcSinh[x/c])^n/x^(m + 2*(p + 1)), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_. + e_.*x_^2)^p_.*(a_. + b_.*ArcCsch[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && IntegersQ[m, p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "-Sqrt[x^2]/x* Subst[Int[(e + d*x^2)^p*(a + b*ArcCosh[x/c])^n/x^(m + 2*(p + 1)), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcSech[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p + 1/2] && GtQ[e, 0] && LtQ[d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "-Sqrt[x^2]/x* Subst[Int[(e + d*x^2)^p*(a + b*ArcSinh[x/c])^n/x^(m + 2*(p + 1)), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcCsch[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[e - c^2*d, 0] && IntegerQ[m] && IntegerQ[p + 1/2] && GtQ[e, 0] && LtQ[d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "-Sqrt[d + e*x^2]/(x*Sqrt[e + d/x^2])* Subst[Int[(e + d*x^2)^p*(a + b*ArcCosh[x/c])^n/x^(m + 2*(p + 1)), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcSech[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p + 1/2] && Not[GtQ[e, 0] && LtQ[d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "-Sqrt[d + e*x^2]/(x*Sqrt[e + d/x^2])* Subst[Int[(e + d*x^2)^p*(a + b*ArcSinh[x/c])^n/x^(m + 2*(p + 1)), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(d_. + e_.*x_^2)^p_*(a_. + b_.*ArcCsch[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && EqQ[e - c^2*d, 0] && IntegerQ[m] && IntegerQ[p + 1/2] && Not[GtQ[e, 0] && LtQ[d, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "With[{v = IntHide[u, x]}, Dist[(a + b*ArcSech[c*x]), v, x] + b*Sqrt[ 1 - c^2*x^2]/(c*x*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])* Int[SimplifyIntegrand[v/(x*Sqrt[1 - c^2*x^2]), x], x] /; InverseFunctionFreeQ[v, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*(a_. + b_.*ArcSech[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "With[{v = IntHide[u, x]}, Dist[(a + b*ArcCsch[c*x]), v, x] + b/c* Int[SimplifyIntegrand[v/(x^2*Sqrt[1 + 1/(c^2*x^2)]), x], x] /; InverseFunctionFreeQ[v, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*(a_. + b_.*ArcCsch[c_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "Unintegrable[u*(a + b*ArcSech[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_. + b_.*ArcSech[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m",
       "filename": "7.5.1 u (a+b arcsech(c x))^n.m",
            "rhs": "Unintegrable[u*(a + b*ArcCsch[c*x])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_. + b_.*ArcCsch[c_.*x_])^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "(c + d*x)*ArcSech[c + d*x]/d + Int[Sqrt[(1 - c - d*x)/(1 + c + d*x)]/(1 - c - d*x), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcSech[c_ + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "(c + d*x)*ArcCsch[c + d*x]/d + Int[1/((c + d*x)*Sqrt[1 + 1/(c + d*x)^2]), x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCsch[c_ + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "1/d*Subst[Int[(a + b*ArcSech[x])^p, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSech[c_ + d_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "1/d*Subst[Int[(a + b*ArcCsch[x])^p, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCsch[c_ + d_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "Unintegrable[(a + b*ArcSech[c + d*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcSech[c_ + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, p}, x] && Not[IGtQ[p, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "Unintegrable[(a + b*ArcCsch[c + d*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*ArcCsch[c_ + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, p}, x] && Not[IGtQ[p, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "1/d*Subst[Int[(f*x/d)^m*(a + b*ArcSech[x])^p, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcSech[c_ + d_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "1/d*Subst[Int[(f*x/d)^m*(a + b*ArcCsch[x])^p, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCsch[c_ + d_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": " -((-a)^(m+1)-b^(m+1)*x^(m+1))*ArcSech[a+b*x]/(b^(m+1)*(m+1)) + 1/(b^(m+1)*(m+1))*Subst[Int[((-a*x)^(m+1)-(1-a*x)^(m+1))/(x^(m+1)* Sqrt[-1+x]*Sqrt[1+x]),x],x,1/(a+b*x)]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*ArcSech[a_+b_.*x_],x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b},x] &&  IntegerQ[m] && NeQ[m,-1] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": " -((-a)^(m+1)-b^(m+1)*x^(m+1))*ArcCsch[a+b*x]/(b^(m+1)*(m+1)) + 1/(b^(m+1)*(m+1))*Subst[Int[((-a*x)^(m+1)-(1-a*x)^(m+1))/(x^(m+1)* Sqrt[1+x^2]),x],x,1/(a+b*x)]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*ArcCsch[a_+b_.*x_],x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b},x] && IntegerQ[m] &&  NeQ[m,-1] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "-1/d^(m + 1)* Subst[Int[(a + b*x)^p*Sech[x]*Tanh[x]*(d*e - c*f + f*Sech[x])^m, x], x, ArcSech[c + d*x]]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcSech[c_ + d_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "-1/d^(m + 1)* Subst[Int[(a + b*x)^p*Csch[x]*Coth[x]*(d*e - c*f + f*Csch[x])^m, x], x, ArcCsch[c + d*x]]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCsch[c_ + d_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "1/d*Subst[Int[((d*e - c*f)/d + f*x/d)^m*(a + b*ArcSech[x])^p, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcSech[c_ + d_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "1/d*Subst[Int[((d*e - c*f)/d + f*x/d)^m*(a + b*ArcCsch[x])^p, x], x, c + d*x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCsch[c_ + d_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "Unintegrable[(e + f*x)^m*(a + b*ArcSech[c + d*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcSech[c_ + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p}, x] && Not[IGtQ[p, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "Unintegrable[(e + f*x)^m*(a + b*ArcCsch[c + d*x])^p, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(a_. + b_.*ArcCsch[c_ + d_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, p}, x] && Not[IGtQ[p, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "Int[u*ArcCosh[a/c + b*x^n/c]^m, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*ArcSech[c_./(a_. + b_.*x_^n_.)]^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, m}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "Int[u*ArcSinh[a/c + b*x^n/c]^m, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*ArcCsch[c_./(a_. + b_.*x_^n_.)]^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, m}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "x*E^ArcSech[a*x] + Log[x]/a + 1/a*Int[1/(x*(1 - a*x))*Sqrt[(1 - a*x)/(1 + a*x)], x]",
     "rulenumber": 0,
            "lhs": "Int[E^ArcSech[a_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[a, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "x*E^ArcSech[a*x^p] + p/a*Int[1/x^p, x] + p*Sqrt[1 + a*x^p]/a*Sqrt[1/(1 + a*x^p)]* Int[1/(x^p*Sqrt[1 + a*x^p]*Sqrt[1 - a*x^p]), x]",
     "rulenumber": 0,
            "lhs": "Int[E^ArcSech[a_.*x_^p_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "1/a*Int[1/x^p, x] + Int[Sqrt[1 + 1/(a^2*x^(2*p))], x]",
     "rulenumber": 0,
            "lhs": "Int[E^ArcCsch[a_.*x_^p_.], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "Int[(1/u + Sqrt[(1 - u)/(1 + u)] + 1/u*Sqrt[(1 - u)/(1 + u)])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[E^(n_.*ArcSech[u_]), x_Symbol]",
        "comment": false,
         "givens": "IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "Int[(1/u + Sqrt[1 + 1/u^2])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[E^(n_.*ArcCsch[u_]), x_Symbol]",
        "comment": false,
         "givens": "IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "-1/(a*p*x^p) + Sqrt[1 + a*x^p]/a*Sqrt[1/(1 + a*x^p)]* Int[Sqrt[1 + a*x^p]*Sqrt[1 - a*x^p]/x^(p + 1), x]",
     "rulenumber": 0,
            "lhs": "Int[E^ArcSech[a_.*x_^p_.]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "x^(m + 1)*E^ArcSech[a*x^p]/(m + 1) + p/(a*(m + 1))*Int[x^(m - p), x] + p*Sqrt[1 + a*x^p]/(a*(m + 1))*Sqrt[1/(1 + a*x^p)]* Int[x^(m - p)/(Sqrt[1 + a*x^p]*Sqrt[1 - a*x^p]), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*E^ArcSech[a_.*x_^p_.], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, m, p}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "1/a*Int[x^(m - p), x] + Int[x^m*Sqrt[1 + 1/(a^2*x^(2*p))], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*E^ArcCsch[a_.*x_^p_.], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, m, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "Int[x^m*(1/u + Sqrt[(1 - u)/(1 + u)] + 1/u*Sqrt[(1 - u)/(1 + u)])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*E^(n_.*ArcSech[u_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[m, x] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "Int[x^m*(1/u + Sqrt[1 + 1/u^2])^n, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*E^(n_.*ArcCsch[u_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[m, x] && IntegerQ[n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "1/(a*c)*Int[Sqrt[1/(1 + c*x)]/(x*Sqrt[1 - c*x]), x] + 1/c*Int[1/(x*(a + b*x^2)), x]",
     "rulenumber": 0,
            "lhs": "Int[E^(ArcSech[c_.*x_])/(a_ + b_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[b + a*c^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "1/(a*c^2)*Int[1/(x^2*Sqrt[1 + 1/(c^2*x^2)]), x] + 1/c*Int[1/(x*(a + b*x^2)), x]",
     "rulenumber": 0,
            "lhs": "Int[E^(ArcCsch[c_.*x_])/(a_ + b_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[b - a*c^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "d/(a*c)*Int[(d*x)^(m - 1)*Sqrt[1/(1 + c*x)]/Sqrt[1 - c*x], x] + d/c*Int[(d*x)^(m - 1)/(a + b*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*E^(ArcSech[c_.*x_])/(a_ + b_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && EqQ[b + a*c^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "d^2/(a*c^2)*Int[(d*x)^(m - 2)/Sqrt[1 + 1/(c^2*x^2)], x] + d/c*Int[(d*x)^(m - 1)/(a + b*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*E^(ArcCsch[c_.*x_])/(a_ + b_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && EqQ[b - a*c^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "x*ArcSech[u] + Sqrt[1 - u^2]/(u*Sqrt[-1 + 1/u]*Sqrt[1 + 1/u])* Int[SimplifyIntegrand[x*D[u, x]/(u*Sqrt[1 - u^2]), x], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcSech[u_], x_Symbol]",
        "comment": false,
         "givens": "InverseFunctionFreeQ[u, x] && Not[FunctionOfExponentialQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "x*ArcCsch[u] - u/Sqrt[-u^2]* Int[SimplifyIntegrand[x*D[u, x]/(u*Sqrt[-1 - u^2]), x], x]",
     "rulenumber": 0,
            "lhs": "Int[ArcCsch[u_], x_Symbol]",
        "comment": false,
         "givens": "InverseFunctionFreeQ[u, x] && Not[FunctionOfExponentialQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "(c + d*x)^(m + 1)*(a + b*ArcSech[u])/(d*(m + 1)) + b*Sqrt[1 - u^2]/(d*(m + 1)*u*Sqrt[-1 + 1/u]*Sqrt[1 + 1/u])* Int[SimplifyIntegrand[(c + d*x)^(m + 1)*D[u, x]/(u*Sqrt[1 - u^2]), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(a_. + b_.*ArcSech[u_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && Not[FunctionOfQ[(c + d*x)^(m + 1), u, x]] && Not[FunctionOfExponentialQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "(c + d*x)^(m + 1)*(a + b*ArcCsch[u])/(d*(m + 1)) - b*u/(d*(m + 1)*Sqrt[-u^2])* Int[SimplifyIntegrand[(c + d*x)^(m + 1)* D[u, x]/(u*Sqrt[-1 - u^2]), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*(a_. + b_.*ArcCsch[u_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && Not[FunctionOfQ[(c + d*x)^(m + 1), u, x]] && Not[FunctionOfExponentialQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "With[{w = IntHide[v, x]}, Dist[(a + b*ArcSech[u]), w, x] + b*Sqrt[1 - u^2]/(u*Sqrt[-1 + 1/u]*Sqrt[1 + 1/u])* Int[SimplifyIntegrand[w*D[u, x]/(u*Sqrt[1 - u^2]), x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && Not[MatchQ[v, (c_. + d_.*x)^m_.",
     "rulenumber": 0,
            "lhs": "Int[v_*(a_. + b_.*ArcSech[u_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, m}, x]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Miscellaneous inverse hyperbolic secant.m",
       "filename": "7.5.2 Miscellaneous inverse hyperbolic secant.m",
            "rhs": "With[{w = IntHide[v, x]}, Dist[(a + b*ArcCsch[u]), w, x] - b*u/Sqrt[-u^2]* Int[SimplifyIntegrand[w*D[u, x]/(u*Sqrt[-1 - u^2]), x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x] && InverseFunctionFreeQ[u, x] && Not[MatchQ[v, (c_. + d_.*x)^m_.",
     "rulenumber": 0,
            "lhs": "Int[v_*(a_. + b_.*ArcCsch[u_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, d, m}, x]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.1 Error functions.m",
       "filename": "8.1 Error functions.m",
            "rhs": "(a + b*x)*Erf[a + b*x]/b + 1/(b*Sqrt[Pi]*E^(a + b*x)^2)",
     "rulenumber": 0,
            "lhs": "Int[Erf[a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.1 Error functions.m",
       "filename": "8.1 Error functions.m",
            "rhs": "(a + b*x)*Erfc[a + b*x]/b - 1/(b*Sqrt[Pi]*E^(a + b*x)^2)",
     "rulenumber": 0,
            "lhs": "Int[Erfc[a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.1 Error functions.m",
       "filename": "8.1 Error functions.m",
            "rhs": "(a + b*x)*Erfi[a + b*x]/b - E^(a + b*x)^2/(b*Sqrt[Pi])",
     "rulenumber": 0,
            "lhs": "Int[Erfi[a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.1 Error functions.m",
       "filename": "8.1 Error functions.m",
            "rhs": "(a + b*x)*Erf[a + b*x]^2/b - 4/Sqrt[Pi]*Int[(a + b*x)*Erf[a + b*x]/E^(a + b*x)^2, x]",
     "rulenumber": 0,
            "lhs": "Int[Erf[a_. + b_.*x_]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.1 Error functions.m",
       "filename": "8.1 Error functions.m",
            "rhs": "(a + b*x)*Erfc[a + b*x]^2/b + 4/Sqrt[Pi]*Int[(a + b*x)*Erfc[a + b*x]/E^(a + b*x)^2, x]",
     "rulenumber": 0,
            "lhs": "Int[Erfc[a_. + b_.*x_]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.1 Error functions.m",
       "filename": "8.1 Error functions.m",
            "rhs": "(a + b*x)*Erfi[a + b*x]^2/b - 4/Sqrt[Pi]*Int[(a + b*x)*E^(a + b*x)^2*Erfi[a + b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[Erfi[a_. + b_.*x_]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.1 Error functions.m",
       "filename": "8.1 Error functions.m",
            "rhs": "Unintegrable[Erf[a + b*x]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[Erf[a_. + b_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n}, x] && NeQ[n, 1] && NeQ[n, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.1 Error functions.m",
       "filename": "8.1 Error functions.m",
            "rhs": "Unintegrable[Erfc[a + b*x]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[Erfc[a_. + b_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n}, x] && NeQ[n, 1] && NeQ[n, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.1 Error functions.m",
       "filename": "8.1 Error functions.m",
            "rhs": "Unintegrable[Erfi[a + b*x]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[Erfi[a_. + b_.*x_]^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n}, x] && NeQ[n, 1] && NeQ[n, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.1 Error functions.m",
       "filename": "8.1 Error functions.m",
            "rhs": "2*b*x/Sqrt[Pi]*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, -b^2*x^2]",
     "rulenumber": 0,
            "lhs": "Int[Erf[b_.*x_]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[b, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.1 Error functions.m",
       "filename": "8.1 Error functions.m",
            "rhs": "Log[x] - Int[Erf[b*x]/x, x]",
     "rulenumber": 0,
            "lhs": "Int[Erfc[b_.*x_]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[b, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.1 Error functions.m",
       "filename": "8.1 Error functions.m",
            "rhs": "2*b*x/Sqrt[Pi]*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, b^2*x^2]",
     "rulenumber": 0,
            "lhs": "Int[Erfi[b_.*x_]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[b, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.1 Error functions.m",
       "filename": "8.1 Error functions.m",
            "rhs": "(c + d*x)^(m + 1)*Erf[a + b*x]/(d*(m + 1)) - 2*b/(Sqrt[Pi]*d*(m + 1))* Int[(c + d*x)^(m + 1)/E^(a + b*x)^2, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Erf[a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.1 Error functions.m",
       "filename": "8.1 Error functions.m",
            "rhs": "(c + d*x)^(m + 1)*Erfc[a + b*x]/(d*(m + 1)) + 2*b/(Sqrt[Pi]*d*(m + 1))* Int[(c + d*x)^(m + 1)/E^(a + b*x)^2, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Erfc[a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.1 Error functions.m",
       "filename": "8.1 Error functions.m",
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       "filename": "8.2 Fresnel integral functions.m",
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            "lhs": "Int[E^(c_. + d_.*x_^2)*FresnelS[b_.*x_], x_Symbol]",
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         "givens": "FreeQ[{b, c, d}, x] && EqQ[d^2, -Pi^2/4*b^4]"
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            "lhs": "Int[E^(c_. + d_.*x_^2)*FresnelC[b_.*x_], x_Symbol]",
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         "givens": "FreeQ[{b, c, d}, x] && EqQ[d^2, -Pi^2/4*b^4]"
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     "rulenumber": 0,
            "lhs": "Int[E^(c_. + d_.*x_^2)*FresnelS[a_. + b_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x]"
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       "filename": "8.2 Fresnel integral functions.m",
            "rhs": "Unintegrable[E^(c + d*x^2)*FresnelC[a + b*x]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[E^(c_. + d_.*x_^2)*FresnelC[a_. + b_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x]"
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       "filename": "8.2 Fresnel integral functions.m",
            "rhs": "Pi*b/(2*d)*Subst[Int[x^n, x], x, FresnelS[b*x]]",
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            "lhs": "Int[Sin[d_.*x_^2]*FresnelS[b_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, d, n}, x] && EqQ[d^2, Pi^2/4*b^4]"
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       "filename": "8.2 Fresnel integral functions.m",
            "rhs": "Pi*b/(2*d)*Subst[Int[x^n, x], x, FresnelC[b*x]]",
     "rulenumber": 0,
            "lhs": "Int[Cos[d_.*x_^2]*FresnelC[b_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, d, n}, x] && EqQ[d^2, Pi^2/4*b^4]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.2 Fresnel integral functions.m",
       "filename": "8.2 Fresnel integral functions.m",
            "rhs": "Sin[c]*Int[Cos[d*x^2]*FresnelS[b*x], x] + Cos[c]*Int[Sin[d*x^2]*FresnelS[b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[Sin[c_ + d_.*x_^2]*FresnelS[b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d}, x] && EqQ[d^2, Pi^2/4*b^4]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.2 Fresnel integral functions.m",
       "filename": "8.2 Fresnel integral functions.m",
            "rhs": "Cos[c]*Int[Cos[d*x^2]*FresnelC[b*x], x] - Sin[c]*Int[Sin[d*x^2]*FresnelC[b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[Cos[c_ + d_.*x_^2]*FresnelC[b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d}, x] && EqQ[d^2, Pi^2/4*b^4]"
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       "filename": "8.2 Fresnel integral functions.m",
            "rhs": "Unintegrable[Sin[c + d*x^2]*FresnelS[a + b*x]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[Sin[c_. + d_.*x_^2]*FresnelS[a_. + b_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x]"
  },
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       "filename": "8.2 Fresnel integral functions.m",
            "rhs": "Unintegrable[Cos[c + d*x^2]*FresnelC[a + b*x]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[Cos[c_. + d_.*x_^2]*FresnelC[a_. + b_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x]"
  },
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.2 Fresnel integral functions.m",
       "filename": "8.2 Fresnel integral functions.m",
            "rhs": "FresnelC[b*x]*FresnelS[b*x]/(2*b) - 1/8*I*b*x^2* HypergeometricPFQ[{1, 1}, {3/2, 2}, -1/2*I*b^2*Pi*x^2] + 1/8*I*b*x^2* HypergeometricPFQ[{1, 1}, {3/2, 2}, 1/2*I*b^2*Pi*x^2]",
     "rulenumber": 0,
            "lhs": "Int[Cos[d_.*x_^2]*FresnelS[b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, d}, x] && EqQ[d^2, Pi^2/4*b^4]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.2 Fresnel integral functions.m",
       "filename": "8.2 Fresnel integral functions.m",
            "rhs": "b*Pi*FresnelC[b*x]*FresnelS[b*x]/(4*d) + 1/8*I*b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, -I*d*x^2] - 1/8*I*b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, I*d*x^2]",
     "rulenumber": 0,
            "lhs": "Int[Sin[d_.*x_^2]*FresnelC[b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, d}, x] && EqQ[d^2, Pi^2/4*b^4]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.2 Fresnel integral functions.m",
       "filename": "8.2 Fresnel integral functions.m",
            "rhs": "Cos[c]*Int[Cos[d*x^2]*FresnelS[b*x], x] - Sin[c]*Int[Sin[d*x^2]*FresnelS[b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[Cos[c_ + d_.*x_^2]*FresnelS[b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d}, x] && EqQ[d^2, Pi^2/4*b^4]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.2 Fresnel integral functions.m",
       "filename": "8.2 Fresnel integral functions.m",
            "rhs": "Sin[c]*Int[Cos[d*x^2]*FresnelC[b*x], x] + Cos[c]*Int[Sin[d*x^2]*FresnelC[b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[Sin[c_ + d_.*x_^2]*FresnelC[b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, c, d}, x] && EqQ[d^2, Pi^2/4*b^4]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.2 Fresnel integral functions.m",
       "filename": "8.2 Fresnel integral functions.m",
            "rhs": "Unintegrable[Cos[c + d*x^2]*FresnelS[a + b*x]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[Cos[c_. + d_.*x_^2]*FresnelS[a_. + b_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x]"
  },
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.2 Fresnel integral functions.m",
       "filename": "8.2 Fresnel integral functions.m",
            "rhs": "Unintegrable[Sin[c + d*x^2]*FresnelC[a + b*x]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[Sin[c_. + d_.*x_^2]*FresnelC[a_. + b_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.2 Fresnel integral functions.m",
       "filename": "8.2 Fresnel integral functions.m",
            "rhs": "-Cos[d*x^2]*FresnelS[b*x]/(2*d) + 1/(2*b*Pi)*Int[Sin[2*d*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[x_*Sin[d_.*x_^2]*FresnelS[b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, d}, x] && EqQ[d^2, Pi^2/4*b^4]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.2 Fresnel integral functions.m",
       "filename": "8.2 Fresnel integral functions.m",
            "rhs": "Sin[d*x^2]*FresnelC[b*x]/(2*d) - b/(4*d)*Int[Sin[2*d*x^2], x]",
     "rulenumber": 0,
            "lhs": "Int[x_*Cos[d_.*x_^2]*FresnelC[b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, d}, x] && EqQ[d^2, Pi^2/4*b^4]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.2 Fresnel integral functions.m",
       "filename": "8.2 Fresnel integral functions.m",
            "rhs": "-x^(m - 1)*Cos[d*x^2]*FresnelS[b*x]/(2*d) + 1/(2*b*Pi)*Int[x^(m - 1)*Sin[2*d*x^2], x] + (m - 1)/(2*d)*Int[x^(m - 2)*Cos[d*x^2]*FresnelS[b*x], x]",
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            "lhs": "Int[x_^m_*Sin[d_.*x_^2]*FresnelS[b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, d}, x] && EqQ[d^2, Pi^2/4*b^4] && IGtQ[m, 1]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.2 Fresnel integral functions.m",
       "filename": "8.2 Fresnel integral functions.m",
            "rhs": "x^(m - 1)*Sin[d*x^2]*FresnelC[b*x]/(2*d) - b/(4*d)*Int[x^(m - 1)*Sin[2*d*x^2], x] - (m - 1)/(2*d)*Int[x^(m - 2)*Sin[d*x^2]*FresnelC[b*x], x]",
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            "lhs": "Int[x_^m_*Cos[d_.*x_^2]*FresnelC[b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, d}, x] && EqQ[d^2, Pi^2/4*b^4] && IGtQ[m, 1]"
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       "filename": "8.2 Fresnel integral functions.m",
            "rhs": "x^(m + 1)*Sin[d*x^2]*FresnelS[b*x]/(m + 1) - d*x^(m + 2)/(Pi*b*(m + 1)*(m + 2)) + d/(Pi*b*(m + 1))*Int[x^(m + 1)*Cos[2*d*x^2], x] - 2*d/(m + 1)*Int[x^(m + 2)*Cos[d*x^2]*FresnelS[b*x], x]",
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            "lhs": "Int[x_^m_*Sin[d_.*x_^2]*FresnelS[b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, d}, x] && EqQ[d^2, Pi^2/4*b^4] && ILtQ[m, -2]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.2 Fresnel integral functions.m",
       "filename": "8.2 Fresnel integral functions.m",
            "rhs": "x^(m + 1)*Cos[d*x^2]*FresnelC[b*x]/(m + 1) - b*x^(m + 2)/(2*(m + 1)*(m + 2)) - b/(2*(m + 1))*Int[x^(m + 1)*Cos[2*d*x^2], x] + 2*d/(m + 1)*Int[x^(m + 2)*Sin[d*x^2]*FresnelC[b*x], x]",
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            "lhs": "Int[x_^m_*Cos[d_.*x_^2]*FresnelC[b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, d}, x] && EqQ[d^2, Pi^2/4*b^4] && ILtQ[m, -2]"
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       "filename": "8.2 Fresnel integral functions.m",
            "rhs": "Unintegrable[(e*x)^m*Sin[c + d*x^2]*FresnelS[a + b*x]^n, x]",
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            "lhs": "Int[(e_.*x_)^m_.*Sin[c_. + d_.*x_^2]*FresnelS[a_. + b_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.2 Fresnel integral functions.m",
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     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Cos[c_. + d_.*x_^2]*FresnelC[a_. + b_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.2 Fresnel integral functions.m",
       "filename": "8.2 Fresnel integral functions.m",
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            "lhs": "Int[x_*Cos[d_.*x_^2]*FresnelS[b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, d}, x] && EqQ[d^2, Pi^2/4*b^4]"
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            "lhs": "Int[x_*Sin[d_.*x_^2]*FresnelC[b_.*x_], x_Symbol]",
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         "givens": "FreeQ[{b, d}, x] && EqQ[d^2, Pi^2/4*b^4]"
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         "givens": "FreeQ[{b, d}, x] && EqQ[d^2, Pi^2/4*b^4] && IGtQ[m, 1]"
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            "lhs": "Int[x_^m_*Sin[d_.*x_^2]*FresnelC[b_.*x_], x_Symbol]",
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         "givens": "FreeQ[{b, d}, x] && EqQ[d^2, Pi^2/4*b^4] && IGtQ[m, 1]"
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       "filename": "8.2 Fresnel integral functions.m",
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            "lhs": "Int[x_^m_*Cos[d_.*x_^2]*FresnelS[b_.*x_], x_Symbol]",
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         "givens": "FreeQ[{b, d}, x] && EqQ[d^2, Pi^2/4*b^4] && ILtQ[m, -1]"
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       "filename": "8.2 Fresnel integral functions.m",
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            "lhs": "Int[x_^m_*Sin[d_.*x_^2]*FresnelC[b_.*x_], x_Symbol]",
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         "givens": "FreeQ[{b, d}, x] && EqQ[d^2, Pi^2/4*b^4] && ILtQ[m, -1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.2 Fresnel integral functions.m",
       "filename": "8.2 Fresnel integral functions.m",
            "rhs": "Unintegrable[(e*x)^m*Cos[c + d*x^2]*FresnelS[a + b*x]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Cos[c_. + d_.*x_^2]*FresnelS[a_. + b_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.2 Fresnel integral functions.m",
       "filename": "8.2 Fresnel integral functions.m",
            "rhs": "Unintegrable[(e*x)^m*Sin[c + d*x^2]*FresnelC[a + b*x]^n, x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*Sin[c_. + d_.*x_^2]*FresnelC[a_. + b_.*x_]^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.2 Fresnel integral functions.m",
       "filename": "8.2 Fresnel integral functions.m",
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     "rulenumber": 0,
            "lhs": "Int[FresnelS[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.2 Fresnel integral functions.m",
       "filename": "8.2 Fresnel integral functions.m",
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     "rulenumber": 0,
            "lhs": "Int[FresnelC[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.2 Fresnel integral functions.m",
       "filename": "8.2 Fresnel integral functions.m",
            "rhs": "1/n*Subst[F[d*(a + b*x)], x, Log[c*x^n]]",
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            "lhs": "Int[F_[d_.*(a_. + b_.*Log[c_.*x_^n_.])]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && MemberQ[{FresnelS, FresnelC}, F]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.2 Fresnel integral functions.m",
       "filename": "8.2 Fresnel integral functions.m",
            "rhs": "(e*x)^(m + 1)* FresnelS[d*(a + b*Log[c*x^n])]/(e*(m + 1)) - b*d*n/(m + 1)* Int[(e*x)^m*Sin[Pi/2*(d*(a + b*Log[c*x^n]))^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*FresnelS[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.2 Fresnel integral functions.m",
       "filename": "8.2 Fresnel integral functions.m",
            "rhs": "(e*x)^(m + 1)* FresnelC[d*(a + b*Log[c*x^n])]/(e*(m + 1)) - b*d*n/(m + 1)* Int[(e*x)^m*Cos[Pi/2*(d*(a + b*Log[c*x^n]))^2], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*FresnelC[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.3 Exponential integral functions.m",
       "filename": "8.3 Exponential integral functions.m",
            "rhs": "-ExpIntegralE[n + 1, a + b*x]/b",
     "rulenumber": 0,
            "lhs": "Int[ExpIntegralE[n_, a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.3 Exponential integral functions.m",
       "filename": "8.3 Exponential integral functions.m",
            "rhs": "-x^m*ExpIntegralE[n + 1, b*x]/b + m/b*Int[x^(m - 1)*ExpIntegralE[n + 1, b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*ExpIntegralE[n_, b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[b, x] && EqQ[m + n, 0] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.3 Exponential integral functions.m",
       "filename": "8.3 Exponential integral functions.m",
            "rhs": "b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -b*x] - EulerGamma*Log[x] - 1/2*Log[b*x]^2",
     "rulenumber": 0,
            "lhs": "Int[ExpIntegralE[1, b_.*x_]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[b, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.3 Exponential integral functions.m",
       "filename": "8.3 Exponential integral functions.m",
            "rhs": "x^(m + 1)*ExpIntegralE[n, b*x]/(m + 1) + b/(m + 1)*Int[x^(m + 1)*ExpIntegralE[n - 1, b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*ExpIntegralE[n_, b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[b, x] && EqQ[m + n, 0] && ILtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.3 Exponential integral functions.m",
       "filename": "8.3 Exponential integral functions.m",
            "rhs": "(d*x)^m*Gamma[m + 1]* Log[x]/(b*(b*x)^m) - (d*x)^(m + 1)* HypergeometricPFQ[{m + 1, m + 1}, {m + 2, m + 2}, -b* x]/(d*(m + 1)^2)",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_*ExpIntegralE[n_, b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, d, m, n}, x] && EqQ[m + n, 0] && Not[IntegerQ[m]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.3 Exponential integral functions.m",
       "filename": "8.3 Exponential integral functions.m",
            "rhs": "(d*x)^(m + 1)* ExpIntegralE[n, b*x]/(d*(m + n)) - (d*x)^(m + 1)* ExpIntegralE[-m, b*x]/(d*(m + n))",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*ExpIntegralE[n_, b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, d, m, n}, x] && NeQ[m + n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.3 Exponential integral functions.m",
       "filename": "8.3 Exponential integral functions.m",
            "rhs": "-(c + d*x)^m*ExpIntegralE[n + 1, a + b*x]/b + d*m/b*Int[(c + d*x)^(m - 1)*ExpIntegralE[n + 1, a + b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*ExpIntegralE[n_, a_ + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && (IGtQ[m, 0] || ILtQ[n, 0] || GtQ[m, 0] && LtQ[n, -1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.3 Exponential integral functions.m",
       "filename": "8.3 Exponential integral functions.m",
            "rhs": "(c + d*x)^(m + 1)* ExpIntegralE[n, a + b*x]/(d*(m + 1)) + b/(d*(m + 1))* Int[(c + d*x)^(m + 1)*ExpIntegralE[n - 1, a + b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*ExpIntegralE[n_, a_ + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && (IGtQ[n, 0] || LtQ[m, -1] && GtQ[n, 0]) && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.3 Exponential integral functions.m",
       "filename": "8.3 Exponential integral functions.m",
            "rhs": "Unintegrable[(c + d*x)^m*ExpIntegralE[n, a + b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*ExpIntegralE[n_, a_ + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.3 Exponential integral functions.m",
       "filename": "8.3 Exponential integral functions.m",
            "rhs": "(a + b*x)*ExpIntegralEi[a + b*x]/b - E^(a + b*x)/b",
     "rulenumber": 0,
            "lhs": "Int[ExpIntegralEi[a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.3 Exponential integral functions.m",
       "filename": "8.3 Exponential integral functions.m",
            "rhs": "Log[x]*(ExpIntegralEi[b*x] + ExpIntegralE[1, -b*x]) - Int[ExpIntegralE[1, -b*x]/x, x]",
     "rulenumber": 0,
            "lhs": "Int[ExpIntegralEi[b_.*x_]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[b, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.3 Exponential integral functions.m",
       "filename": "8.3 Exponential integral functions.m",
            "rhs": "Unintegrable[ExpIntegralEi[a + b*x]/(c + d*x), x]",
     "rulenumber": 0,
            "lhs": "Int[ExpIntegralEi[a_. + b_.*x_]/(c_. + d_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.3 Exponential integral functions.m",
       "filename": "8.3 Exponential integral functions.m",
            "rhs": "(c + d*x)^(m + 1)* ExpIntegralEi[a + b*x]/(d*(m + 1)) - b/(d*(m + 1))*Int[(c + d*x)^(m + 1)*E^(a + b*x)/(a + b*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*ExpIntegralEi[a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.3 Exponential integral functions.m",
       "filename": "8.3 Exponential integral functions.m",
            "rhs": "(a + b*x)*ExpIntegralEi[a + b*x]^2/b - 2*Int[E^(a + b*x)*ExpIntegralEi[a + b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[ExpIntegralEi[a_. + b_.*x_]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.3 Exponential integral functions.m",
       "filename": "8.3 Exponential integral functions.m",
            "rhs": "x^(m + 1)*ExpIntegralEi[b*x]^2/(m + 1) - 2/(m + 1)*Int[x^m*E^(b*x)*ExpIntegralEi[b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*ExpIntegralEi[b_.*x_]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[b, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.3 Exponential integral functions.m",
       "filename": "8.3 Exponential integral functions.m",
            "rhs": "x^(m + 1)*ExpIntegralEi[a + b*x]^2/(m + 1) + a*x^m*ExpIntegralEi[a + b*x]^2/(b*(m + 1)) - 2/(m + 1)*Int[x^m*E^(a + b*x)*ExpIntegralEi[a + b*x], x] - a*m/(b*(m + 1))*Int[x^(m - 1)*ExpIntegralEi[a + b*x]^2, x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*ExpIntegralEi[a_ + b_.*x_]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.3 Exponential integral functions.m",
       "filename": "8.3 Exponential integral functions.m",
            "rhs": " b*x^(m+2)*ExpIntegralEi[a+b*x]^2/(a*(m+1)) + x^(m+1)*ExpIntegralEi[a+b*x]^2/(m+1) - 2*b/(a*(m+1))*Int[x^(m+1)*E^(a+b*x)*ExpIntegralEi[a+b*x],x] - b*(m+2)/(a*(m+1))*Int[x^(m+1)*ExpIntegralEi[a+b*x]^2,x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*ExpIntegralEi[a_+b_.*x_]^2,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b},x] && ILtQ[m,-2] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.3 Exponential integral functions.m",
       "filename": "8.3 Exponential integral functions.m",
            "rhs": "E^(a + b*x)*ExpIntegralEi[c + d*x]/b - d/b*Int[E^(a + c + (b + d)*x)/(c + d*x), x]",
     "rulenumber": 0,
            "lhs": "Int[E^(a_. + b_.*x_)*ExpIntegralEi[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.3 Exponential integral functions.m",
       "filename": "8.3 Exponential integral functions.m",
            "rhs": "x^m*E^(a + b*x)*ExpIntegralEi[c + d*x]/b - d/b*Int[x^m*E^(a + c + (b + d)*x)/(c + d*x), x] - m/b*Int[x^(m - 1)*E^(a + b*x)*ExpIntegralEi[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*E^(a_. + b_.*x_)*ExpIntegralEi[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.3 Exponential integral functions.m",
       "filename": "8.3 Exponential integral functions.m",
            "rhs": "x^(m + 1)*E^(a + b*x)*ExpIntegralEi[c + d*x]/(m + 1) - d/(m + 1)*Int[x^(m + 1)*E^(a + c + (b + d)*x)/(c + d*x), x] - b/(m + 1)*Int[x^(m + 1)*E^(a + b*x)*ExpIntegralEi[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*E^(a_. + b_.*x_)*ExpIntegralEi[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && ILtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.3 Exponential integral functions.m",
       "filename": "8.3 Exponential integral functions.m",
            "rhs": "x*ExpIntegralEi[d*(a + b*Log[c*x^n])] - b*n*E^(a*d)*Int[(c*x^n)^(b*d)/(a + b*Log[c*x^n]), x]",
     "rulenumber": 0,
            "lhs": "Int[ExpIntegralEi[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.3 Exponential integral functions.m",
       "filename": "8.3 Exponential integral functions.m",
            "rhs": "1/n*Subst[ExpIntegralEi[d*(a + b*x)], x, Log[c*x^n]]",
     "rulenumber": 0,
            "lhs": "Int[ExpIntegralEi[d_.*(a_. + b_.*Log[c_.*x_^n_.])]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.3 Exponential integral functions.m",
       "filename": "8.3 Exponential integral functions.m",
            "rhs": "(e*x)^(m + 1)* ExpIntegralEi[d*(a + b*Log[c*x^n])]/(e*(m + 1)) - b*n*E^(a*d)*(c*x^n)^(b*d)/((m + 1)*(e*x)^(b*d*n))* Int[(e*x)^(m + b*d*n)/(a + b*Log[c*x^n]), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*ExpIntegralEi[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.3 Exponential integral functions.m",
       "filename": "8.3 Exponential integral functions.m",
            "rhs": "(a + b*x)*LogIntegral[a + b*x]/b - ExpIntegralEi[2*Log[a + b*x]]/b",
     "rulenumber": 0,
            "lhs": "Int[LogIntegral[a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.3 Exponential integral functions.m",
       "filename": "8.3 Exponential integral functions.m",
            "rhs": "-b*x + Log[b*x]*LogIntegral[b*x]",
     "rulenumber": 0,
            "lhs": "Int[LogIntegral[b_.*x_]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[b, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.3 Exponential integral functions.m",
       "filename": "8.3 Exponential integral functions.m",
            "rhs": "Unintegrable[LogIntegral[a + b*x]/(c + d*x), x]",
     "rulenumber": 0,
            "lhs": "Int[LogIntegral[a_. + b_.*x_]/(c_. + d_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.3 Exponential integral functions.m",
       "filename": "8.3 Exponential integral functions.m",
            "rhs": "(c + d*x)^(m + 1)* LogIntegral[a + b*x]/(d*(m + 1)) - b/(d*(m + 1))*Int[(c + d*x)^(m + 1)/Log[a + b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*LogIntegral[a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": "(a + b*x)*SinIntegral[a + b*x]/b + Cos[a + b*x]/b",
     "rulenumber": 0,
            "lhs": "Int[SinIntegral[a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": "(a + b*x)*CosIntegral[a + b*x]/b - Sin[a + b*x]/b",
     "rulenumber": 0,
            "lhs": "Int[CosIntegral[a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": "1/2*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -I*b*x] + 1/2*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, I*b*x]",
     "rulenumber": 0,
            "lhs": "Int[SinIntegral[b_.*x_]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[b, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": "-1/2*I*b*x* HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -I*b*x] + 1/2*I*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, I*b*x] + EulerGamma*Log[x] + 1/2*Log[b*x]^2",
     "rulenumber": 0,
            "lhs": "Int[CosIntegral[b_.*x_]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[b, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": "(c + d*x)^(m + 1)*SinIntegral[a + b*x]/(d*(m + 1)) - b/(d*(m + 1))*Int[(c + d*x)^(m + 1)*Sin[a + b*x]/(a + b*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*SinIntegral[a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": "(c + d*x)^(m + 1)*CosIntegral[a + b*x]/(d*(m + 1)) - b/(d*(m + 1))*Int[(c + d*x)^(m + 1)*Cos[a + b*x]/(a + b*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*CosIntegral[a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": "(a + b*x)*SinIntegral[a + b*x]^2/b - 2*Int[Sin[a + b*x]*SinIntegral[a + b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[SinIntegral[a_. + b_.*x_]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": "(a + b*x)*CosIntegral[a + b*x]^2/b - 2*Int[Cos[a + b*x]*CosIntegral[a + b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[CosIntegral[a_. + b_.*x_]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": "x^(m + 1)*SinIntegral[b*x]^2/(m + 1) - 2/(m + 1)*Int[x^m*Sin[b*x]*SinIntegral[b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*SinIntegral[b_.*x_]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[b, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": "x^(m + 1)*CosIntegral[b*x]^2/(m + 1) - 2/(m + 1)*Int[x^m*Cos[b*x]*CosIntegral[b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*CosIntegral[b_.*x_]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[b, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": "(a + b*x)*(c + d*x)^m* SinIntegral[a + b*x]^2/(b*(m + 1)) - 2/(m + 1)*Int[(c + d*x)^m*Sin[a + b*x]*SinIntegral[a + b*x], x] + (b*c - a*d)*m/(b*(m + 1))* Int[(c + d*x)^(m - 1)*SinIntegral[a + b*x]^2, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*SinIntegral[a_ + b_.*x_]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": "(a + b*x)*(c + d*x)^m* CosIntegral[a + b*x]^2/(b*(m + 1)) - 2/(m + 1)*Int[(c + d*x)^m*Cos[a + b*x]*CosIntegral[a + b*x], x] + (b*c - a*d)*m/(b*(m + 1))* Int[(c + d*x)^(m - 1)*CosIntegral[a + b*x]^2, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*CosIntegral[a_ + b_.*x_]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": " b*x^(m+2)*SinIntegral[a+b*x]^2/(a*(m+1)) + x^(m+1)*SinIntegral[a+b*x]^2/(m+1) - 2*b/(a*(m+1))*Int[x^(m+1)*Sin[a+b*x]*SinIntegral[a+b*x],x] - b*(m+2)/(a*(m+1))*Int[x^(m+1)*SinIntegral[a+b*x]^2,x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*SinIntegral[a_+b_.*x_]^2,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b},x] && ILtQ[m,-2] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": " b*x^(m+2)*CosIntegral[a+b*x]^2/(a*(m+1)) + x^(m+1)*CosIntegral[a+b*x]^2/(m+1) - 2*b/(a*(m+1))*Int[x^(m+1)*Cos[a+b*x]*CosIntegral[a+b*x],x] - b*(m+2)/(a*(m+1))*Int[x^(m+1)*CosIntegral[a+b*x]^2,x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*CosIntegral[a_+b_.*x_]^2,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b},x] && ILtQ[m,-2] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": "-Cos[a + b*x]*SinIntegral[c + d*x]/b + d/b*Int[Cos[a + b*x]*Sin[c + d*x]/(c + d*x), x]",
     "rulenumber": 0,
            "lhs": "Int[Sin[a_. + b_.*x_]*SinIntegral[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": "Sin[a + b*x]*CosIntegral[c + d*x]/b - d/b*Int[Sin[a + b*x]*Cos[c + d*x]/(c + d*x), x]",
     "rulenumber": 0,
            "lhs": "Int[Cos[a_. + b_.*x_]*CosIntegral[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": "-(e + f*x)^m*Cos[a + b*x]*SinIntegral[c + d*x]/b + d/b*Int[(e + f*x)^m*Cos[a + b*x]*Sin[c + d*x]/(c + d*x), x] + f*m/b* Int[(e + f*x)^(m - 1)*Cos[a + b*x]*SinIntegral[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Sin[a_. + b_.*x_]*SinIntegral[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": "(e + f*x)^m*Sin[a + b*x]*CosIntegral[c + d*x]/b - d/b*Int[(e + f*x)^m*Sin[a + b*x]*Cos[c + d*x]/(c + d*x), x] - f*m/b* Int[(e + f*x)^(m - 1)*Sin[a + b*x]*CosIntegral[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Cos[a_. + b_.*x_]*CosIntegral[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": "(e + f*x)^(m + 1)*Sin[a + b*x]* SinIntegral[c + d*x]/(f*(m + 1)) - d/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Sin[a + b*x]*Sin[c + d*x]/(c + d*x), x] - b/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Cos[a + b*x]*SinIntegral[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_*Sin[a_. + b_.*x_]*SinIntegral[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && ILtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": "(e + f*x)^(m + 1)*Cos[a + b*x]* CosIntegral[c + d*x]/(f*(m + 1)) - d/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Cos[a + b*x]*Cos[c + d*x]/(c + d*x), x] + b/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Sin[a + b*x]*CosIntegral[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Cos[a_. + b_.*x_]*CosIntegral[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && ILtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": "Sin[a + b*x]*SinIntegral[c + d*x]/b - d/b*Int[Sin[a + b*x]*Sin[c + d*x]/(c + d*x), x]",
     "rulenumber": 0,
            "lhs": "Int[Cos[a_. + b_.*x_]*SinIntegral[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": "-Cos[a + b*x]*CosIntegral[c + d*x]/b + d/b*Int[Cos[a + b*x]*Cos[c + d*x]/(c + d*x), x]",
     "rulenumber": 0,
            "lhs": "Int[Sin[a_. + b_.*x_]*CosIntegral[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": "(e + f*x)^m*Sin[a + b*x]*SinIntegral[c + d*x]/b - d/b*Int[(e + f*x)^m*Sin[a + b*x]*Sin[c + d*x]/(c + d*x), x] - f*m/b* Int[(e + f*x)^(m - 1)*Sin[a + b*x]*SinIntegral[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Cos[a_. + b_.*x_]*SinIntegral[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": "-(e + f*x)^m*Cos[a + b*x]*CosIntegral[c + d*x]/b + d/b*Int[(e + f*x)^m*Cos[a + b*x]*Cos[c + d*x]/(c + d*x), x] + f*m/b* Int[(e + f*x)^(m - 1)*Cos[a + b*x]*CosIntegral[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Sin[a_. + b_.*x_]*CosIntegral[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": "(e + f*x)^(m + 1)*Cos[a + b*x]* SinIntegral[c + d*x]/(f*(m + 1)) - d/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Cos[a + b*x]*Sin[c + d*x]/(c + d*x), x] + b/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Sin[a + b*x]*SinIntegral[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Cos[a_. + b_.*x_]*SinIntegral[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && ILtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": "(e + f*x)^(m + 1)*Sin[a + b*x]* CosIntegral[c + d*x]/(f*(m + 1)) - d/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Sin[a + b*x]*Cos[c + d*x]/(c + d*x), x] - b/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Cos[a + b*x]*CosIntegral[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_*Sin[a_. + b_.*x_]*CosIntegral[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && ILtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": "x*SinIntegral[d*(a + b*Log[c*x^n])] - b*d*n*Int[Sin[d*(a + b*Log[c*x^n])]/(d*(a + b*Log[c*x^n])), x]",
     "rulenumber": 0,
            "lhs": "Int[SinIntegral[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": "x*CosIntegral[d*(a + b*Log[c*x^n])] - b*d*n*Int[Cos[d*(a + b*Log[c*x^n])]/(d*(a + b*Log[c*x^n])), x]",
     "rulenumber": 0,
            "lhs": "Int[CosIntegral[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": "1/n*Subst[F[d*(a + b*x)], x, Log[c*x^n]]",
     "rulenumber": 0,
            "lhs": "Int[F_[d_.*(a_. + b_.*Log[c_.*x_^n_.])]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && MemberQ[{SinIntegral, CosIntegral}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": "(e*x)^(m + 1)* SinIntegral[d*(a + b*Log[c*x^n])]/(e*(m + 1)) - b*d*n/(m + 1)* Int[(e*x)^m*Sin[d*(a + b*Log[c*x^n])]/(d*(a + b*Log[c*x^n])), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*SinIntegral[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.4 Trig integral functions.m",
       "filename": "8.4 Trig integral functions.m",
            "rhs": "(e*x)^(m + 1)* CosIntegral[d*(a + b*Log[c*x^n])]/(e*(m + 1)) - b*d*n/(m + 1)* Int[(e*x)^m*Cos[d*(a + b*Log[c*x^n])]/(d*(a + b*Log[c*x^n])), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*CosIntegral[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": "(a + b*x)*SinhIntegral[a + b*x]/b - Cosh[a + b*x]/b",
     "rulenumber": 0,
            "lhs": "Int[SinhIntegral[a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": "(a + b*x)*CoshIntegral[a + b*x]/b - Sinh[a + b*x]/b",
     "rulenumber": 0,
            "lhs": "Int[CoshIntegral[a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": "1/2*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -b*x] + 1/2*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, b*x]",
     "rulenumber": 0,
            "lhs": "Int[SinhIntegral[b_.*x_]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[b, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": "-1/2*b*x* HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -b*x] + 1/2*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, b*x] + EulerGamma*Log[x] + 1/2*Log[b*x]^2",
     "rulenumber": 0,
            "lhs": "Int[CoshIntegral[b_.*x_]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[b, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": "(c + d*x)^(m + 1)* SinhIntegral[a + b*x]/(d*(m + 1)) - b/(d*(m + 1))*Int[(c + d*x)^(m + 1)*Sinh[a + b*x]/(a + b*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*SinhIntegral[a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": "(c + d*x)^(m + 1)* CoshIntegral[a + b*x]/(d*(m + 1)) - b/(d*(m + 1))*Int[(c + d*x)^(m + 1)*Cosh[a + b*x]/(a + b*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*CoshIntegral[a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": "(a + b*x)*SinhIntegral[a + b*x]^2/b - 2*Int[Sinh[a + b*x]*SinhIntegral[a + b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[SinhIntegral[a_. + b_.*x_]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": "(a + b*x)*CoshIntegral[a + b*x]^2/b - 2*Int[Cosh[a + b*x]*CoshIntegral[a + b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[CoshIntegral[a_. + b_.*x_]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": "x^(m + 1)*SinhIntegral[b*x]^2/(m + 1) - 2/(m + 1)*Int[x^m*Sinh[b*x]*SinhIntegral[b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*SinhIntegral[b_.*x_]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[b, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": "x^(m + 1)*CoshIntegral[b*x]^2/(m + 1) - 2/(m + 1)*Int[x^m*Cosh[b*x]*CoshIntegral[b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*CoshIntegral[b_.*x_]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[b, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": "(a + b*x)*(c + d*x)^m* SinhIntegral[a + b*x]^2/(b*(m + 1)) - 2/(m + 1)* Int[(c + d*x)^m*Sinh[a + b*x]*SinhIntegral[a + b*x], x] + (b*c - a*d)*m/(b*(m + 1))* Int[(c + d*x)^(m - 1)*SinhIntegral[a + b*x]^2, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*SinhIntegral[a_ + b_.*x_]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": "(a + b*x)*(c + d*x)^m* CoshIntegral[a + b*x]^2/(b*(m + 1)) - 2/(m + 1)* Int[(c + d*x)^m*Cosh[a + b*x]*CoshIntegral[a + b*x], x] + (b*c - a*d)*m/(b*(m + 1))* Int[(c + d*x)^(m - 1)*CoshIntegral[a + b*x]^2, x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*CoshIntegral[a_ + b_.*x_]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": " b*x^(m+2)*SinhIntegral[a+b*x]^2/(a*(m+1)) + x^(m+1)*SinhIntegral[a+b*x]^2/(m+1) - 2*b/(a*(m+1))*Int[x^(m+1)*Sinh[a+b*x]*SinhIntegral[a+b*x],x] - b*(m+2)/(a*(m+1))*Int[x^(m+1)*SinhIntegral[a+b*x]^2,x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*SinhIntegral[a_+b_.*x_]^2,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b},x] && ILtQ[m,-2] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": " b*x^(m+2)*CoshIntegral[a+b*x]^2/(a*(m+1)) + x^(m+1)*CoshIntegral[a+b*x]^2/(m+1) - 2*b/(a*(m+1))*Int[x^(m+1)*Cosh[a+b*x]*CoshIntegral[a+b*x],x] - b*(m+2)/(a*(m+1))*Int[x^(m+1)*CoshIntegral[a+b*x]^2,x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*CoshIntegral[a_+b_.*x_]^2,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b},x] && ILtQ[m,-2] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": "Cosh[a + b*x]*SinhIntegral[c + d*x]/b - d/b*Int[Cosh[a + b*x]*Sinh[c + d*x]/(c + d*x), x]",
     "rulenumber": 0,
            "lhs": "Int[Sinh[a_. + b_.*x_]*SinhIntegral[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": "Sinh[a + b*x]*CoshIntegral[c + d*x]/b - d/b*Int[Sinh[a + b*x]*Cosh[c + d*x]/(c + d*x), x]",
     "rulenumber": 0,
            "lhs": "Int[Cosh[a_. + b_.*x_]*CoshIntegral[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": "(e + f*x)^m*Cosh[a + b*x]* SinhIntegral[c + d*x]/b - d/b*Int[(e + f*x)^m*Cosh[a + b*x]*Sinh[c + d*x]/(c + d*x), x] - f*m/b* Int[(e + f*x)^(m - 1)*Cosh[a + b*x]*SinhIntegral[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Sinh[a_. + b_.*x_]*SinhIntegral[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": "(e + f*x)^m*Sinh[a + b*x]* CoshIntegral[c + d*x]/b - d/b*Int[(e + f*x)^m*Sinh[a + b*x]*Cosh[c + d*x]/(c + d*x), x] - f*m/b* Int[(e + f*x)^(m - 1)*Sinh[a + b*x]*CoshIntegral[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Cosh[a_. + b_.*x_]*CoshIntegral[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": "(e + f*x)^(m + 1)*Sinh[a + b*x]* SinhIntegral[c + d*x]/(f*(m + 1)) - d/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Sinh[a + b*x]*Sinh[c + d*x]/(c + d*x), x] - b/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Cosh[a + b*x]*SinhIntegral[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_*Sinh[a_. + b_.*x_]*SinhIntegral[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && ILtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": "(e + f*x)^(m + 1)*Cosh[a + b*x]* CoshIntegral[c + d*x]/(f*(m + 1)) - d/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Cosh[a + b*x]*Cosh[c + d*x]/(c + d*x), x] - b/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Sinh[a + b*x]*CoshIntegral[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Cosh[a_. + b_.*x_]*CoshIntegral[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && ILtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": "Sinh[a + b*x]*SinhIntegral[c + d*x]/b - d/b*Int[Sinh[a + b*x]*Sinh[c + d*x]/(c + d*x), x]",
     "rulenumber": 0,
            "lhs": "Int[Cosh[a_. + b_.*x_]*SinhIntegral[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": "Cosh[a + b*x]*CoshIntegral[c + d*x]/b - d/b*Int[Cosh[a + b*x]*Cosh[c + d*x]/(c + d*x), x]",
     "rulenumber": 0,
            "lhs": "Int[Sinh[a_. + b_.*x_]*CoshIntegral[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": "(e + f*x)^m*Sinh[a + b*x]*SinhIntegral[c + d*x]/b - d/b*Int[(e + f*x)^m*Sinh[a + b*x]*Sinh[c + d*x]/(c + d*x), x] - f*m/b* Int[(e + f*x)^(m - 1)*Sinh[a + b*x]*SinhIntegral[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Cosh[a_. + b_.*x_]*SinhIntegral[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": "(e + f*x)^m*Cosh[a + b*x]*CoshIntegral[c + d*x]/b - d/b*Int[(e + f*x)^m*Cosh[a + b*x]*Cosh[c + d*x]/(c + d*x), x] - f*m/b* Int[(e + f*x)^(m - 1)*Cosh[a + b*x]*CoshIntegral[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Sinh[a_. + b_.*x_]*CoshIntegral[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": "(e + f*x)^(m + 1)*Cosh[a + b*x]* SinhIntegral[c + d*x]/(f*(m + 1)) - d/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Cosh[a + b*x]*Sinh[c + d*x]/(c + d*x), x] - b/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Sinh[a + b*x]*SinhIntegral[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*Cosh[a_. + b_.*x_]*SinhIntegral[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && ILtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": "(e + f*x)^(m + 1)*Sinh[a + b*x]* CoshIntegral[c + d*x]/(f*(m + 1)) - d/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Sinh[a + b*x]*Cosh[c + d*x]/(c + d*x), x] - b/(f*(m + 1))* Int[(e + f*x)^(m + 1)*Cosh[a + b*x]*CoshIntegral[c + d*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_*Sinh[a_. + b_.*x_]*CoshIntegral[c_. + d_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f}, x] && ILtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": "x*SinhIntegral[d*(a + b*Log[c*x^n])] - b*d*n*Int[Sinh[d*(a + b*Log[c*x^n])]/(d*(a + b*Log[c*x^n])), x]",
     "rulenumber": 0,
            "lhs": "Int[SinhIntegral[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": "x*CoshIntegral[d*(a + b*Log[c*x^n])] - b*d*n*Int[Cosh[d*(a + b*Log[c*x^n])]/(d*(a + b*Log[c*x^n])), x]",
     "rulenumber": 0,
            "lhs": "Int[CoshIntegral[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": "1/n*Subst[F[d*(a + b*x)], x, Log[c*x^n]]",
     "rulenumber": 0,
            "lhs": "Int[F_[d_.*(a_. + b_.*Log[c_.*x_^n_.])]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && MemberQ[{SinhIntegral, CoshIntegral}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": "(e*x)^(m + 1)* SinhIntegral[d*(a + b*Log[c*x^n])]/(e*(m + 1)) - b*d*n/(m + 1)* Int[(e*x)^m*Sinh[d*(a + b*Log[c*x^n])]/(d*(a + b*Log[c*x^n])), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*SinhIntegral[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.5 Hyperbolic integral functions.m",
       "filename": "8.5 Hyperbolic integral functions.m",
            "rhs": "(e*x)^(m + 1)* CoshIntegral[d*(a + b*Log[c*x^n])]/(e*(m + 1)) - b*d*n/(m + 1)* Int[(e*x)^m*Cosh[d*(a + b*Log[c*x^n])]/(d*(a + b*Log[c*x^n])), x]",
     "rulenumber": 0,
            "lhs": "Int[(e_.*x_)^m_.*CoshIntegral[d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.6 Gamma functions.m",
       "filename": "8.6 Gamma functions.m",
            "rhs": "(a + b*x)*Gamma[n, a + b*x]/b - Gamma[n + 1, a + b*x]/b",
     "rulenumber": 0,
            "lhs": "Int[Gamma[n_, a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.6 Gamma functions.m",
       "filename": "8.6 Gamma functions.m",
            "rhs": "b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -b*x] - EulerGamma*Log[x] - 1/2*Log[b*x]^2",
     "rulenumber": 0,
            "lhs": "Int[Gamma[0, b_.*x_]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[b, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.6 Gamma functions.m",
       "filename": "8.6 Gamma functions.m",
            "rhs": "Int[1/(x*E^(b*x)),x]",
     "rulenumber": 0,
            "lhs": "Int[Gamma[1,b_.*x_]/x_,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[b,x] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.6 Gamma functions.m",
       "filename": "8.6 Gamma functions.m",
            "rhs": "-Gamma[n - 1, b*x] + (n - 1)* Int[Gamma[n - 1, b*x]/x, x]",
     "rulenumber": 0,
            "lhs": "Int[Gamma[n_, b_.*x_]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[b, x] && IGtQ[n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.6 Gamma functions.m",
       "filename": "8.6 Gamma functions.m",
            "rhs": "Gamma[n, b*x]/n + 1/n*Int[Gamma[n + 1, b*x]/x, x]",
     "rulenumber": 0,
            "lhs": "Int[Gamma[n_, b_.*x_]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[b, x] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.6 Gamma functions.m",
       "filename": "8.6 Gamma functions.m",
            "rhs": "Gamma[n]*Log[x] - (b*x)^n/n^2* HypergeometricPFQ[{n, n}, {1 + n, 1 + n}, -b*x]",
     "rulenumber": 0,
            "lhs": "Int[Gamma[n_, b_.*x_]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, n}, x] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.6 Gamma functions.m",
       "filename": "8.6 Gamma functions.m",
            "rhs": "(d*x)^(m + 1)*Gamma[n, b*x]/(d*(m + 1)) - (d*x)^m*Gamma[m + n + 1, b*x]/(b*(m + 1)*(b*x)^m)",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*Gamma[n_, b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{b, d, m, n}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.6 Gamma functions.m",
       "filename": "8.6 Gamma functions.m",
            "rhs": "1/b*Subst[Int[(d*x/b)^m*Gamma[n, x], x], x, a + b*x]",
     "rulenumber": 0,
            "lhs": "Int[(c_ + d_.*x_)^m_.*Gamma[n_, a_ + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.6 Gamma functions.m",
       "filename": "8.6 Gamma functions.m",
            "rhs": "Int[(a + b*x)^(n - 1)/((c + d*x)*E^(a + b*x)), x] + (n - 1)* Int[Gamma[n - 1, a + b*x]/(c + d*x), x]",
     "rulenumber": 0,
            "lhs": "Int[Gamma[n_, a_. + b_.*x_]/(c_. + d_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IGtQ[n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.6 Gamma functions.m",
       "filename": "8.6 Gamma functions.m",
            "rhs": "Block[{$UseGamma = True}, (c + d*x)^(m + 1)*Gamma[n, a + b*x]/(d*(m + 1)) + b/(d*(m + 1))* Int[(c + d*x)^(m + 1)*(a + b*x)^(n - 1)/E^(a + b*x), x]]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Gamma[n_, a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && (IGtQ[m, 0] || IGtQ[n, 0] || IntegersQ[m, n]) && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.6 Gamma functions.m",
       "filename": "8.6 Gamma functions.m",
            "rhs": "Unintegrable[(c + d*x)^m*Gamma[n, a + b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Gamma[n_, a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.6 Gamma functions.m",
       "filename": "8.6 Gamma functions.m",
            "rhs": "PolyGamma[-2, a + b*x]/b",
     "rulenumber": 0,
            "lhs": "Int[LogGamma[a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.6 Gamma functions.m",
       "filename": "8.6 Gamma functions.m",
            "rhs": "(c + d*x)^m*PolyGamma[-2, a + b*x]/b - d*m/b*Int[(c + d*x)^(m - 1)*PolyGamma[-2, a + b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*LogGamma[a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.6 Gamma functions.m",
       "filename": "8.6 Gamma functions.m",
            "rhs": "Unintegrable[(c + d*x)^m*LogGamma[a + b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*LogGamma[a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.6 Gamma functions.m",
       "filename": "8.6 Gamma functions.m",
            "rhs": "PolyGamma[n - 1, a + b*x]/b",
     "rulenumber": 0,
            "lhs": "Int[PolyGamma[n_, a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.6 Gamma functions.m",
       "filename": "8.6 Gamma functions.m",
            "rhs": "(c + d*x)^m*PolyGamma[n - 1, a + b*x]/b - d*m/b*Int[(c + d*x)^(m - 1)*PolyGamma[n - 1, a + b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*PolyGamma[n_, a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.6 Gamma functions.m",
       "filename": "8.6 Gamma functions.m",
            "rhs": "(c + d*x)^(m + 1)*PolyGamma[n, a + b*x]/(d*(m + 1)) - b/(d*(m + 1))* Int[(c + d*x)^(m + 1)*PolyGamma[n + 1, a + b*x], x]",
     "rulenumber": 0,
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         "givens": "FreeQ[{a, b, n}, x]"
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       "filename": "8.6 Gamma functions.m",
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         "givens": "FreeQ[{a, b, c, n}, x] && EqQ[c, a + 1]"
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  {
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       "filename": "8.6 Gamma functions.m",
            "rhs": "x*Gamma[p, d*(a + b*Log[c*x^n])] + b*d*n*E^(-a*d)* Int[(d*(a + b*Log[c*x^n]))^(p - 1)/(c*x^n)^(b*d), x]",
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         "givens": "FreeQ[{a, b, c, d, n, p}, x]"
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       "filename": "8.6 Gamma functions.m",
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         "givens": "FreeQ[{a, b, c, d, n, p}, x]"
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            "rhs": "(e*x)^(m + 1)* Gamma[p, d*(a + b*Log[c*x^n])]/(e*(m + 1)) + b*d*n*E^(-a*d)*(e*x)^(b*d*n)/((m + 1)*(c*x^n)^(b*d))* Int[(e*x)^(m - b*d*n)*(d*(a + b*Log[c*x^n]))^(p - 1), x]",
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            "lhs": "Int[(e_.*x_)^m_.*Gamma[p_, d_.*(a_. + b_.*Log[c_.*x_^n_.])], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[m, -1]"
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  {
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       "filename": "8.6 Gamma functions.m",
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.6 Gamma functions.m",
       "filename": "8.6 Gamma functions.m",
            "rhs": "1/e*Subst[Int[(g*x/d)^m*Gamma[p, f*(a + b*Log[c*x^n])], x], x, d + e*x]",
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            "lhs": "Int[(g_ + h_. x_)^m_.* Gamma[p_, f_.*(a_. + b_.*Log[c_.*(d_ + e_.*x_)^n_.])], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n, p}, x] && EqQ[e*g - d*h, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.7 Zeta function.m",
       "filename": "8.7 Zeta function.m",
            "rhs": "Int[PolyGamma[1, a + b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[Zeta[2, a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.7 Zeta function.m",
       "filename": "8.7 Zeta function.m",
            "rhs": "-Zeta[s - 1, a + b*x]/(b*(s - 1))",
     "rulenumber": 0,
            "lhs": "Int[Zeta[s_, a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, s}, x] && NeQ[s, 1] && NeQ[s, 2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.7 Zeta function.m",
       "filename": "8.7 Zeta function.m",
            "rhs": "Int[(c + d*x)^m*PolyGamma[1, a + b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Zeta[2, a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && RationalQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.7 Zeta function.m",
       "filename": "8.7 Zeta function.m",
            "rhs": "-(c + d*x)^m*Zeta[s - 1, a + b*x]/(b*(s - 1)) + d*m/(b*(s - 1))*Int[(c + d*x)^(m - 1)*Zeta[s - 1, a + b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Zeta[s_, a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, s}, x] && NeQ[s, 1] && NeQ[s, 2] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.7 Zeta function.m",
       "filename": "8.7 Zeta function.m",
            "rhs": "(c + d*x)^(m + 1)*Zeta[s, a + b*x]/(d*(m + 1)) + b*s/(d*(m + 1))*Int[(c + d*x)^(m + 1)*Zeta[s + 1, a + b*x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_. + d_.*x_)^m_.*Zeta[s_, a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, s}, x] && NeQ[s, 1] && NeQ[s, 2] && LtQ[m, -1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m",
       "filename": "8.8 Polylogarithm function.m",
            "rhs": " x*PolyLog[2,a*(b*x^p)^q] + p*q*Int[Log[1-a*(b*x^p)^q],x]",
     "rulenumber": 0,
            "lhs": "Int[PolyLog[2,a_.*(b_.*x_^p_.)^q_.],x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,p,q},x] *)"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m",
       "filename": "8.8 Polylogarithm function.m",
            "rhs": "x*PolyLog[n, a*(b*x^p)^q] - p*q*Int[PolyLog[n - 1, a*(b*x^p)^q], x]",
     "rulenumber": 0,
            "lhs": "Int[PolyLog[n_, a_.*(b_.*x_^p_.)^q_.], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, p, q}, x] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m",
       "filename": "8.8 Polylogarithm function.m",
            "rhs": "x*PolyLog[n + 1, a*(b*x^p)^q]/(p*q) - 1/(p*q)*Int[PolyLog[n + 1, a*(b*x^p)^q], x]",
     "rulenumber": 0,
            "lhs": "Int[PolyLog[n_, a_.*(b_.*x_^p_.)^q_.], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, p, q}, x] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m",
       "filename": "8.8 Polylogarithm function.m",
            "rhs": "Unintegrable[PolyLog[n, a*(b*x^p)^q], x]",
     "rulenumber": 0,
            "lhs": "Int[PolyLog[n_, a_.*(b_.*x_^p_.)^q_.], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n, p, q}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m",
       "filename": "8.8 Polylogarithm function.m",
            "rhs": "PolyLog[n + 1, c*(a + b*x)^p]/(e*p)",
     "rulenumber": 0,
            "lhs": "Int[PolyLog[n_, c_.*(a_. + b_.*x_)^p_.]/(d_. + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m",
       "filename": "8.8 Polylogarithm function.m",
            "rhs": "PolyLog[n + 1, a*(b*x^p)^q]/(p*q)",
     "rulenumber": 0,
            "lhs": "Int[PolyLog[n_, a_.*(b_.*x_^p_.)^q_.]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n, p, q}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m",
       "filename": "8.8 Polylogarithm function.m",
            "rhs": "(d*x)^(m + 1)* PolyLog[n, a*(b*x^p)^q]/(d*(m + 1)) - p*q/(m + 1)*Int[(d*x)^m*PolyLog[n - 1, a*(b*x^p)^q], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*PolyLog[n_, a_.*(b_.*x_^p_.)^q_.], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, m, p, q}, x] && NeQ[m, -1] && GtQ[n, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m",
       "filename": "8.8 Polylogarithm function.m",
            "rhs": "(d*x)^(m + 1)* PolyLog[n + 1, a*(b*x^p)^q]/(d*p*q) - (m + 1)/(p*q)*Int[(d*x)^m*PolyLog[n + 1, a*(b*x^p)^q], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*PolyLog[n_, a_.*(b_.*x_^p_.)^q_.], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, m, p, q}, x] && NeQ[m, -1] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m",
       "filename": "8.8 Polylogarithm function.m",
            "rhs": "Unintegrable[(d*x)^m*PolyLog[n, a*(b*x^p)^q], x]",
     "rulenumber": 0,
            "lhs": "Int[(d_.*x_)^m_.*PolyLog[n_, a_.*(b_.*x_^p_.)^q_.], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, m, n, p, q}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m",
       "filename": "8.8 Polylogarithm function.m",
            "rhs": "Log[c*x^m]^r*PolyLog[n + 1, a*(b*x^p)^q]/(p*q) - m*r/(p*q)* Int[Log[c*x^m]^(r - 1)*PolyLog[n + 1, a*(b*x^p)^q]/x, x]",
     "rulenumber": 0,
            "lhs": "Int[Log[c_.*x_^m_.]^r_.*PolyLog[n_, a_.*(b_.*x_^p_.)^q_.]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n, q, r}, x] && GtQ[r, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m",
       "filename": "8.8 Polylogarithm function.m",
            "rhs": "x*PolyLog[n, c*(a + b*x)^p] - p*Int[PolyLog[n - 1, c*(a + b*x)^p], x] + a*p*Int[PolyLog[n - 1, c*(a + b*x)^p]/(a + b*x), x]",
     "rulenumber": 0,
            "lhs": "Int[PolyLog[n_, c_.*(a_. + b_.*x_)^p_.], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && GtQ[n, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m",
       "filename": "8.8 Polylogarithm function.m",
            "rhs": "Log[1 - a*c - b*c*x]*PolyLog[2, c*(a + b*x)]/e + b/e*Int[Log[1 - a*c - b*c*x]^2/(a + b*x), x]",
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            "lhs": "Int[PolyLog[2, c_.*(a_. + b_.*x_)]/(d_. + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && EqQ[c*(b*d - a*e) + e, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m",
       "filename": "8.8 Polylogarithm function.m",
            "rhs": "Log[d + e*x]*PolyLog[2, c*(a + b*x)]/e + b/e*Int[Log[d + e*x]*Log[1 - a*c - b*c*x]/(a + b*x), x]",
     "rulenumber": 0,
            "lhs": "Int[PolyLog[2, c_.*(a_. + b_.*x_)]/(d_. + e_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e}, x] && NeQ[c*(b*d - a*e) + e, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m",
       "filename": "8.8 Polylogarithm function.m",
            "rhs": "(d + e*x)^(m + 1)* PolyLog[2, c*(a + b*x)]/(e*(m + 1)) + b/(e*(m + 1))* Int[(d + e*x)^(m + 1)*Log[1 - a*c - b*c*x]/(a + b*x), x]",
     "rulenumber": 0,
            "lhs": "Int[(d_. + e_.*x_)^m_.*PolyLog[2, c_.*(a_. + b_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m",
       "filename": "8.8 Polylogarithm function.m",
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     "rulenumber": 0,
            "lhs": "Int[(d_.+e_.*x_)^m_.*PolyLog[n_,c_.*(a_.+b_.*x_)^p_.],x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c,d,e,m,p},x] && GtQ[n,0] && IGtQ[m,0] *)"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m",
       "filename": "8.8 Polylogarithm function.m",
            "rhs": "-(a^(m + 1) - b^(m + 1)*x^(m + 1))* PolyLog[n, c*(a + b*x)^p]/((m + 1)*b^(m + 1)) + p/((m + 1)*b^m)* Int[ExpandIntegrand[ PolyLog[n - 1, c*(a + b*x)^p], (a^(m + 1) - b^(m + 1)*x^(m + 1))/(a + b*x), x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*PolyLog[n_, c_.*(a_. + b_.*x_)^p_.], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, p}, x] && GtQ[n, 0] && IntegerQ[m] && NeQ[m, -1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m",
       "filename": "8.8 Polylogarithm function.m",
            "rhs": "x*(g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)] + b*Int[(g + h*Log[f*(d + e*x)^n])*Log[1 - a*c - b*c*x]* ExpandIntegrand[x/(a + b*x), x], x] - e*h*n* Int[PolyLog[2, c*(a + b*x)]*ExpandIntegrand[x/(d + e*x), x], x]",
     "rulenumber": 0,
            "lhs": "Int[(g_. + h_.*Log[f_.*(d_. + e_.*x_)^n_.])* PolyLog[2, c_.*(a_. + b_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, n}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m",
       "filename": "8.8 Polylogarithm function.m",
            "rhs": "-PolyLog[2, c*x]^2/2",
     "rulenumber": 0,
            "lhs": "Int[Log[1 + e_.*x_]*PolyLog[2, c_.*x_]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, e}, x] && EqQ[c + e, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m",
       "filename": "8.8 Polylogarithm function.m",
            "rhs": "g*Int[PolyLog[2, c*x]/x, x] + h*Int[(Log[1 + e*x]*PolyLog[2, c*x])/x, x]",
     "rulenumber": 0,
            "lhs": "Int[(g_ + h_.*Log[1 + e_.*x_])*PolyLog[2, c_.*x_]/x_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, e, g, h}, x] && EqQ[c + e, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m",
       "filename": "8.8 Polylogarithm function.m",
            "rhs": "x^(m + 1)*(g + h*Log[f*(d + e*x)^n])* PolyLog[2, c*(a + b*x)]/(m + 1) + b/(m + 1)* Int[ExpandIntegrand[(g + h*Log[f*(d + e*x)^n])* Log[1 - a*c - b*c*x], x^(m + 1)/(a + b*x), x], x] - e*h*n/(m + 1)* Int[ExpandIntegrand[PolyLog[2, c*(a + b*x)], x^(m + 1)/(d + e*x), x], x]",
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            "lhs": "Int[x_^m_.*(g_. + h_.*Log[f_.*(d_. + e_.*x_)^n_.])* PolyLog[2, c_.*(a_. + b_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, n}, x] && IntegerQ[m] && NeQ[m, -1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m",
       "filename": "8.8 Polylogarithm function.m",
            "rhs": "With[{u = IntHide[Px, x]}, u*(g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)] + b*Int[ ExpandIntegrand[(g + h*Log[f*(d + e*x)^n])*Log[1 - a*c - b*c*x], u/(a + b*x), x], x] - e*h*n* Int[ExpandIntegrand[PolyLog[2, c*(a + b*x)], u/(d + e*x), x], x]]",
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            "lhs": "Int[Px_*(g_. + h_.*Log[f_.*(d_. + e_.*x_)^n_.])* PolyLog[2, c_.*(a_. + b_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, n}, x] && PolyQ[Px, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m",
       "filename": "8.8 Polylogarithm function.m",
            "rhs": "Coeff[Px, x, -m - 1]*Int[(g + h*Log[1 + e*x])*PolyLog[2, c*x]/x, x] + Int[x^ m*(Px - Coeff[Px, x, -m - 1]*x^(-m - 1))*(g + h*Log[1 + e*x])* PolyLog[2, c*x], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*Px_*(g_. + h_.*Log[1 + e_.*x_])*PolyLog[2, c_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{c, e, g, h}, x] && PolyQ[Px, x] && ILtQ[m, 0] && EqQ[c + e, 0] && NeQ[Coeff[Px, x, -m - 1], 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m",
       "filename": "8.8 Polylogarithm function.m",
            "rhs": "With[{u = IntHide[x^m*Px, x]}, u*(g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)] + b*Int[ ExpandIntegrand[(g + h*Log[f*(d + e*x)^n])*Log[1 - a*c - b*c*x], u/(a + b*x), x], x] - e*h*n* Int[ExpandIntegrand[PolyLog[2, c*(a + b*x)], u/(d + e*x), x], x]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*Px_*(g_. + h_.*Log[f_.*(d_. + e_.*x_)^n_.])* PolyLog[2, c_.*(a_. + b_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, n}, x] && PolyQ[Px, x] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m",
       "filename": "8.8 Polylogarithm function.m",
            "rhs": "Unintegrable[ x^m*Px*(g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)], x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*Px_.*(g_. + h_.*Log[f_.*(d_. + e_.*x_)^n_.])* PolyLog[2, c_.*(a_. + b_.*x_)], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && PolyQ[Px, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m",
       "filename": "8.8 Polylogarithm function.m",
            "rhs": "PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])",
     "rulenumber": 0,
            "lhs": "Int[PolyLog[n_, d_.*(F_^(c_.*(a_. + b_.*x_)))^p_.], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m",
       "filename": "8.8 Polylogarithm function.m",
            "rhs": "(e + f*x)^m* PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F]) - f*m/(b*c*p*Log[F])* Int[(e + f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*PolyLog[n_, d_.*(F_^(c_.*(a_. + b_.*x_)))^p_.], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m",
       "filename": "8.8 Polylogarithm function.m",
            "rhs": "With[{w = DerivativeDivides[v, u*v, x]}, w*PolyLog[n + 1, v] /; Not[FalseQ[w]]]",
     "rulenumber": 0,
            "lhs": "Int[u_*PolyLog[n_, v_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[n, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.8 Polylogarithm function.m",
       "filename": "8.8 Polylogarithm function.m",
            "rhs": "With[{z = DerivativeDivides[v, u*v, x]}, z*Log[w]*PolyLog[n + 1, v] - Int[SimplifyIntegrand[z*D[w, x]*PolyLog[n + 1, v]/w, x], x] /; Not[FalseQ[z]]]",
     "rulenumber": 0,
            "lhs": "Int[u_*Log[w_]*PolyLog[n_, v_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[n, x] && InverseFunctionFreeQ[w, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "(a + b*x)*(c*ProductLog[a + b*x])^p/(b*(p + 1)) + p/(c*(p + 1))* Int[(c*ProductLog[a + b*x])^(p + 1)/(1 + ProductLog[a + b*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*ProductLog[a_. + b_.*x_])^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "(a + b*x)*(c*ProductLog[a + b*x])^p/b - p*Int[(c*ProductLog[a + b*x])^p/(1 + ProductLog[a + b*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*ProductLog[a_. + b_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && Not[LtQ[p, -1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "1/b^(m + 1)* Subst[Int[ ExpandIntegrand[(c*ProductLog[x])^p, (b*e - a*f + f*x)^m, x], x], x, a + b*x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_.*(c_.*ProductLog[a_ + b_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, e, f, p}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "x*(c*ProductLog[a*x^n])^p - n*p*Int[(c*ProductLog[a*x^n])^p/(1 + ProductLog[a*x^n]), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*ProductLog[a_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, n, p}, x] && (EqQ[n*(p - 1), -1] || IntegerQ[p - 1/2] && EqQ[n*(p - 1/2), -1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "x*(c*ProductLog[a*x^n])^p/(n*p + 1) + n*p/(c*(n*p + 1))* Int[(c*ProductLog[a*x^n])^(p + 1)/(1 + ProductLog[a*x^n]), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*ProductLog[a_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, n}, x] && (IntegerQ[p] && EqQ[n*(p + 1), -1] || IntegerQ[p - 1/2] && EqQ[n*(p + 1/2), -1])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "-Subst[Int[(c*ProductLog[a*x^(-n)])^p/x^2, x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*ProductLog[a_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, p}, x] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "x^(m + 1)*(c*ProductLog[a*x^n])^p/(m + 1) - n*p/(m + 1)* Int[x^m*(c*ProductLog[a*x^n])^p/(1 + ProductLog[a*x^n]), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(c_.*ProductLog[a_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, m, n, p}, x] && NeQ[m, -1] && (IntegerQ[p - 1/2] && IGtQ[2*Simplify[p + (m + 1)/n], 0] || Not[IntegerQ[p - 1/2]] && IGtQ[Simplify[p + (m + 1)/n] + 1, 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "x^(m + 1)*(c*ProductLog[a*x^n])^p/(m + n*p + 1) + n*p/(c*(m + n*p + 1))* Int[x^m*(c*ProductLog[a*x^n])^(p + 1)/(1 + ProductLog[a*x^n]), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(c_.*ProductLog[a_.*x_^n_.])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, m, n, p}, x] && (EqQ[m, -1] || IntegerQ[p - 1/2] && ILtQ[Simplify[p + (m + 1)/n] - 1/2, 0] || Not[IntegerQ[p - 1/2]] && ILtQ[Simplify[p + (m + 1)/n], 0])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "Int[x^m*(c*ProductLog[a*x])^p/(1 + ProductLog[a*x]), x] + 1/c*Int[x^m*(c*ProductLog[a*x])^(p + 1)/(1 + ProductLog[a*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(c_.*ProductLog[a_.*x_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, m}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "-Subst[Int[(c*ProductLog[a*x^(-n)])^p/x^(m + 2), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(c_.*ProductLog[a_.*x_^n_])^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, p}, x] && ILtQ[n, 0] && IntegerQ[m] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "(a + b*x)/(b*d*ProductLog[a + b*x])",
     "rulenumber": 0,
            "lhs": "Int[1/(d_ + d_.*ProductLog[a_. + b_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "d*x - Int[1/(d + d*ProductLog[a + b*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[ProductLog[a_. + b_.*x_]/(d_ + d_.*ProductLog[a_. + b_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "c*(a + b*x)*(c*ProductLog[a + b*x])^(p - 1)/(b*d) - c*p*Int[(c*ProductLog[a + b*x])^(p - 1)/(d + d*ProductLog[a + b*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*ProductLog[a_. + b_.*x_])^ p_/(d_ + d_.*ProductLog[a_. + b_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && GtQ[p, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "ExpIntegralEi[ProductLog[a + b*x]]/(b*d)",
     "rulenumber": 0,
            "lhs": "Int[1/(ProductLog[a_. + b_.*x_]*(d_ + d_.*ProductLog[a_. + b_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "Rt[Pi*c, 2]*Erfi[Sqrt[c*ProductLog[a + b*x]]/Rt[c, 2]]/(b*c*d)",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[ c_.*ProductLog[a_. + b_.*x_]]*(d_ + d_.*ProductLog[a_. + b_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && PosQ[c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "Rt[-Pi*c, 2]*Erf[Sqrt[c*ProductLog[a + b*x]]/Rt[-c, 2]]/(b*c*d)",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[ c_.*ProductLog[a_. + b_.*x_]]*(d_ + d_.*ProductLog[a_. + b_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && NegQ[c]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "(a + b*x)*(c*ProductLog[a + b*x])^p/(b*d*(p + 1)) - 1/(c*(p + 1))* Int[(c*ProductLog[a + b*x])^(p + 1)/(d + d*ProductLog[a + b*x]), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*ProductLog[a_. + b_.*x_])^ p_/(d_ + d_.*ProductLog[a_. + b_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d}, x] && LtQ[p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "Gamma[p + 1, -ProductLog[a + b*x]]*(c*ProductLog[a + b*x])^ p/(b*d*(-ProductLog[a + b*x])^p)",
     "rulenumber": 0,
            "lhs": "Int[(c_.*ProductLog[a_. + b_.*x_])^ p_./(d_ + d_.*ProductLog[a_. + b_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "1/b^(m + 1)* Subst[Int[ ExpandIntegrand[1/(d + d*ProductLog[x]), (b*e - a*f + f*x)^m, x], x], x, a + b*x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^m_./(d_ + d_.*ProductLog[a_ + b_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, d, e, f}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "1/b^(m + 1)* Subst[Int[ ExpandIntegrand[(c*ProductLog[x])^ p/(d + d*ProductLog[x]), (b*e - a*f + f*x)^m, x], x], x, a + b*x]",
     "rulenumber": 0,
            "lhs": "Int[(e_. + f_.*x_)^ m_.*(c_.*ProductLog[a_ + b_.*x_])^ p_./(d_ + d_.*ProductLog[a_ + b_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, p}, x] && IGtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "-Subst[Int[1/(x^2*(d + d*ProductLog[a*x^(-n)])), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[1/(d_ + d_.*ProductLog[a_.*x_^n_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, d}, x] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "c*x*(c*ProductLog[a*x^n])^(p - 1)/d",
     "rulenumber": 0,
            "lhs": "Int[(c_.*ProductLog[a_.*x_^n_.])^ p_./(d_ + d_.*ProductLog[a_.*x_^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, n, p}, x] && EqQ[n*(p - 1), -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "a^p*ExpIntegralEi[-p*ProductLog[a*x^n]]/(d*n)",
     "rulenumber": 0,
            "lhs": "Int[ProductLog[a_.*x_^n_.]^p_./(d_ + d_.*ProductLog[a_.*x_^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, d}, x] && IntegerQ[p] && EqQ[n*p, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "Rt[Pi*c*n, 2]/(d*n*a^(1/n)*c^(1/n))* Erfi[Sqrt[c*ProductLog[a*x^n]]/Rt[c*n, 2]]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*ProductLog[a_.*x_^n_.])^p_/(d_ + d_.*ProductLog[a_.*x_^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d}, x] && IntegerQ[1/n] && EqQ[p, 1/2 - 1/n] && PosQ[c*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "Rt[-Pi*c*n, 2]/(d*n*a^(1/n)*c^(1/n))* Erf[Sqrt[c*ProductLog[a*x^n]]/Rt[-c*n, 2]]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*ProductLog[a_.*x_^n_.])^p_/(d_ + d_.*ProductLog[a_.*x_^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d}, x] && IntegerQ[1/n] && EqQ[p, 1/2 - 1/n] && NegQ[c*n]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "c*x*(c*ProductLog[a*x^n])^(p - 1)/d - c*(n*(p - 1) + 1)* Int[(c*ProductLog[a*x^n])^(p - 1)/(d + d*ProductLog[a*x^n]), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*ProductLog[a_.*x_^n_.])^ p_./(d_ + d_.*ProductLog[a_.*x_^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d}, x] && GtQ[n, 0] && GtQ[n*(p - 1) + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "x*(c*ProductLog[a*x^n])^p/(d*(n*p + 1)) - 1/(c*(n*p + 1))* Int[(c*ProductLog[a*x^n])^(p + 1)/(d + d*ProductLog[a*x^n]), x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*ProductLog[a_.*x_^n_.])^ p_./(d_ + d_.*ProductLog[a_.*x_^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d}, x] && GtQ[n, 0] && LtQ[n*p + 1, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "-Subst[ Int[(c*ProductLog[a*x^(-n)])^p/(x^2*(d + d*ProductLog[a*x^(-n)])), x], x, 1/x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*ProductLog[a_.*x_^n_])^p_./(d_ + d_.*ProductLog[a_.*x_^n_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, p}, x] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "x^(m + 1)/(d*(m + 1)*ProductLog[a*x]) - m/(m + 1)*Int[x^m/(ProductLog[a*x]*(d + d*ProductLog[a*x])), x]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_./(d_ + d_.*ProductLog[a_.*x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, d}, x] && GtQ[m, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "Log[ProductLog[a*x]]/d",
     "rulenumber": 0,
            "lhs": "Int[1/(x_*(d_ + d_.*ProductLog[a_.*x_])), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, d}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
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     "rulenumber": 0,
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         "givens": "FreeQ[{a, d}, x] && LtQ[m, -1]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "x^m*Gamma[m + 1, -(m + 1)*ProductLog[a*x]]/ (a*d*(m + 1)* E^(m*ProductLog[a*x])*(-(m + 1)*ProductLog[a*x])^m)",
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         "givens": "FreeQ[{a, d, m}, x] && Not[IntegerQ[m]]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
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            "lhs": "Int[1/(x_*(d_ + d_.*ProductLog[a_.*x_^n_.])), x_Symbol]",
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         "givens": "FreeQ[{a, d, n}, x]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
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         "givens": "FreeQ[{a, d}, x] && IntegerQ[m] && ILtQ[n, 0] && NeQ[m, -1]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
            "rhs": "(c*ProductLog[a*x^n])^p/(d*n*p)",
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            "lhs": "Int[(c_.*ProductLog[a_.*x_^n_.])^ p_./(x_*(d_ + d_.*ProductLog[a_.*x_^n_.])), x_Symbol]",
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         "givens": "FreeQ[{a, c, d, n, p}, x]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
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            "lhs": "Int[x_^m_.*(c_.*ProductLog[a_.*x_^n_.])^ p_./(d_ + d_.*ProductLog[a_.*x_^n_.]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, d, m, n, p}, x] && NeQ[m, -1] && EqQ[m + n*(p - 1), -1]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
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            "lhs": "Int[x_^m_.* ProductLog[a_.*x_^n_.]^p_./(d_ + d_.*ProductLog[a_.*x_^n_.]), x_Symbol]",
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         "givens": "FreeQ[{a, d, m, n}, x] && IntegerQ[p] && EqQ[m + n*p, -1]"
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         "givens": "FreeQ[{a, c, d, m, n}, x] && NeQ[m, -1] && IntegerQ[p - 1/2] && EqQ[m + n*(p - 1/2), -1] && PosQ[c/(p - 1/2)]"
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         "givens": "FreeQ[{a, c, d, m, n}, x] && NeQ[m, -1] && IntegerQ[p - 1/2] && EqQ[m + n*(p - 1/2), -1] && NegQ[c/(p - 1/2)]"
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            "lhs": "Int[x_^m_.*(c_.*ProductLog[a_.*x_^n_.])^ p_./(d_ + d_.*ProductLog[a_.*x_^n_.]), x_Symbol]",
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         "givens": "FreeQ[{a, c, d, m, n, p}, x] && NeQ[m, -1] && GtQ[Simplify[p + (m + 1)/n], 1]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
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            "lhs": "Int[x_^m_.*(c_.*ProductLog[a_.*x_^n_.])^ p_./(d_ + d_.*ProductLog[a_.*x_^n_.]), x_Symbol]",
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         "givens": "FreeQ[{a, c, d, m, n, p}, x] && NeQ[m, -1] && LtQ[Simplify[p + (m + 1)/n], 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
       "filename": "8.9 Product logarithm function.m",
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     "rulenumber": 0,
            "lhs": "Int[x_^m_.*(c_.*ProductLog[a_.*x_])^p_./(d_ + d_.*ProductLog[a_.*x_]), x_Symbol]",
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         "givens": "FreeQ[{a, c, d, m, p}, x] && NeQ[m, -1]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
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            "lhs": "Int[x_^m_.*(c_.*ProductLog[a_.*x_^n_.])^ p_./(d_ + d_.*ProductLog[a_.*x_^n_.]), x_Symbol]",
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         "givens": "FreeQ[{a, c, d, p}, x] && NeQ[m, -1] && IntegerQ[m] && LtQ[n, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/8 Special functions/8.9 Product logarithm function.m",
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     "rulenumber": 0,
            "lhs": "Int[u_, x_Symbol]",
        "comment": false,
         "givens": "FunctionOfQ[ProductLog[x], u, x]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.1 Integrand simplification rules.m",
       "filename": "9.1 Integrand simplification rules.m",
            "rhs": "Int[u*w^p,x]",
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            "lhs": "Int[u_.*(v_+w_)^p_.,x_Symbol]",
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         "givens": "FreeQ[p,x] &&  EqQ[v,0] *)"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.1 Integrand simplification rules.m",
       "filename": "9.1 Integrand simplification rules.m",
            "rhs": "Int[u*(b*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_ + b_.*x_^n_.)^p_., x_Symbol]",
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         "givens": "FreeQ[{a, b, n, p}, x] && EqQ[a, 0]"
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         "givens": "FreeQ[{a, b, n, p}, x] && EqQ[b, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.1 Integrand simplification rules.m",
       "filename": "9.1 Integrand simplification rules.m",
            "rhs": "Int[u*(b*x^n + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_ + b_.*x_^n_. + c_.*x_^j_.)^p_., x_Symbol]",
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         "givens": "FreeQ[{a, b, c, n, p}, x] && EqQ[j, 2*n] && EqQ[a, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.1 Integrand simplification rules.m",
       "filename": "9.1 Integrand simplification rules.m",
            "rhs": "Int[u*(a + c*x^(2*n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_. + b_.*x_^n_. + c_.*x_^j_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, p}, x] && EqQ[j, 2*n] && EqQ[b, 0]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.1 Integrand simplification rules.m",
       "filename": "9.1 Integrand simplification rules.m",
            "rhs": "Int[u*(a + b*x^n)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_. + b_.*x_^n_. + c_.*x_^j_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, p}, x] && EqQ[j, 2*n] && EqQ[c, 0]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.1 Integrand simplification rules.m",
       "filename": "9.1 Integrand simplification rules.m",
            "rhs": "Int[u*((a + b)*v + w)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_.*v_ + b_.*v_ + w_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && Not[FreeQ[v, x]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.1 Integrand simplification rules.m",
       "filename": "9.1 Integrand simplification rules.m",
            "rhs": "Int[u*Px^Simplify[p], x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*Px_^p_, x_Symbol]",
        "comment": false,
         "givens": "PolyQ[Px, x] && Not[RationalQ[p]] && FreeQ[p, x] && RationalQ[Simplify[p]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.1 Integrand simplification rules.m",
       "filename": "9.1 Integrand simplification rules.m",
            "rhs": "a*x",
     "rulenumber": 0,
            "lhs": "Int[a_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[a, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.1 Integrand simplification rules.m",
       "filename": "9.1 Integrand simplification rules.m",
            "rhs": "a*(b + c*x)^2/(2*c)",
     "rulenumber": 0,
            "lhs": "Int[a_*(b_ + c_.*x_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.1 Integrand simplification rules.m",
       "filename": "9.1 Integrand simplification rules.m",
            "rhs": "Complex[Identity[0], a]*Int[u, x]",
     "rulenumber": 0,
            "lhs": "Int[Complex[0, a_]*u_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[a, x] && EqQ[a^2, 1]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.1 Integrand simplification rules.m",
       "filename": "9.1 Integrand simplification rules.m",
            "rhs": "a*Int[u, x] /; FreeQ[a, x] && Not[MatchQ[u, b_*v_",
     "rulenumber": 0,
            "lhs": "Int[a_*u_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[b, x]]]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.1 Integrand simplification rules.m",
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            "rhs": "IntSum[u, x]",
     "rulenumber": 0,
            "lhs": "Int[u_, x_Symbol] := ShowStep[\"\", \"Int[a*u + b*v + ...,x]\", \"a*Integrate[u,x] + b*Integrate[v,x] + ...\", Hold[ IntSum[u, x]]] /; SimplifyFlag && SumQ[u], Int[u_, x_Symbol]",
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         "givens": "SumQ[u]]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.1 Integrand simplification rules.m",
       "filename": "9.1 Integrand simplification rules.m",
            "rhs": "Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u] && Not[LinearQ[u, x]] && Not[MatchQ[u, a_ + b_.*v_",
     "rulenumber": 0,
            "lhs": "Int[(c_.*x_)^m_.*u_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && InverseFunctionQ[v]]]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.1 Integrand simplification rules.m",
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            "rhs": "a^IntPart[m]*(a*x^n)^FracPart[m]/x^(n*FracPart[m])* Int[u*x^(m*n), x]",
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            "lhs": "Int[u_.*(a_.*x_^n_)^m_, x_Symbol]",
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.1 Integrand simplification rules.m",
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            "rhs": "1/b^m*Int[u*(b*v)^(m + n), x]",
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            "lhs": "Int[u_.*v_^m_.*(b_*v_)^n_, x_Symbol]",
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.1 Integrand simplification rules.m",
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            "rhs": "a^(m + 1/2)*b^(n - 1/2)*Sqrt[b*v]/Sqrt[a*v]*Int[u*v^(m + n), x]",
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            "lhs": "Int[u_.*(a_.*v_)^m_*(b_.*v_)^n_, x_Symbol]",
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         "givens": "FreeQ[{a, b, m}, x] && Not[IntegerQ[m]] && IGtQ[n + 1/2, 0] && IntegerQ[m + n]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.1 Integrand simplification rules.m",
       "filename": "9.1 Integrand simplification rules.m",
            "rhs": " b^(n-1/2)*Sqrt[b*v]/(a^(n-1/2)*Sqrt[a*v])*Int[u*(a*v)^(m+n),x]",
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            "lhs": "Int[u_.*(a_.*v_)^m_*(b_.*v_)^n_,x_Symbol]",
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         "givens": " FreeQ[{a,b,m},x] && Not[IntegerQ[m]] && IGtQ[n+1/2,0] &&  Not[IntegerQ[m+n]] *)"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.1 Integrand simplification rules.m",
       "filename": "9.1 Integrand simplification rules.m",
            "rhs": "a^(m - 1/2)*b^(n + 1/2)*Sqrt[a*v]/Sqrt[b*v]*Int[u*v^(m + n), x]",
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            "lhs": "Int[u_.*(a_.*v_)^m_*(b_.*v_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m}, x] && Not[IntegerQ[m]] && ILtQ[n - 1/2, 0] && IntegerQ[m + n]"
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  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.1 Integrand simplification rules.m",
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            "rhs": " b^(n+1/2)*Sqrt[a*v]/(a^(n+1/2)*Sqrt[b*v])*Int[u*(a*v)^(m+n),x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_.*v_)^m_*(b_.*v_)^n_,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,m},x] && Not[IntegerQ[m]] && ILtQ[n-1/2,0] &&  Not[IntegerQ[m+n]] *)"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.1 Integrand simplification rules.m",
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            "lhs": "Int[u_.*(a_.*v_)^m_*(b_.*v_)^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n}, x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && IntegerQ[m + n]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.1 Integrand simplification rules.m",
       "filename": "9.1 Integrand simplification rules.m",
            "rhs": "b^IntPart[n]*(b*v)^FracPart[n]/(a^IntPart[n]*(a*v)^FracPart[n])* Int[u*(a*v)^(m + n), x]",
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            "lhs": "Int[u_.*(a_.*v_)^m_*(b_.*v_)^n_, x_Symbol]",
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         "givens": "FreeQ[{a, b, m, n}, x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && Not[IntegerQ[m + n]]"
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       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.1 Integrand simplification rules.m",
       "filename": "9.1 Integrand simplification rules.m",
            "rhs": "(b/d)^m*Int[u*(c + d*v)^(m + n), x]",
     "rulenumber": 0,
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        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && (Not[IntegerQ[n]] || SimplerQ[c + d*x, a + b*x])"
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         "givens": "FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] && GtQ[b/d, 0] && Not[IntegerQ[m] || IntegerQ[n]]"
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         "givens": "FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] && Not[IntegerQ[m] || IntegerQ[n] || GtQ[b/d, 0]]"
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            "rhs": " 1/a*Int[u*(a*v)^(m+1)*(b+c*v),x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_.*v_)^m_*(b_.*v_+c_.*v_^2),x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b,c},x] && LeQ[m,-1] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.1 Integrand simplification rules.m",
       "filename": "9.1 Integrand simplification rules.m",
            "rhs": "1/b^2*Int[u*(a + b*v)^(m + 1)*Simp[b*B - a*C + b*C*v, x], x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_ + b_.*v_)^m_*(A_. + B_.*v_ + C_.*v_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, A, B, C}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0] && LeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.1 Integrand simplification rules.m",
       "filename": "9.1 Integrand simplification rules.m",
            "rhs": "(d/a)^p*Int[u*(a + b*x^n)^(m + p)/x^(n*p), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_ + b_.*x_^n_.)^m_.*(c_ + d_.*x_^q_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && EqQ[q, -n] && IntegerQ[p] && EqQ[a*c - b*d, 0] && Not[IntegerQ[m] && NegQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.1 Integrand simplification rules.m",
       "filename": "9.1 Integrand simplification rules.m",
            "rhs": "(-b^2/d)^m*Int[u*(a - b*x^n)^(-m), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_ + b_.*x_^n_.)^m_.*(c_ + d_.*x_^j_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[j, 2*n] && EqQ[p, -m] && EqQ[b^2*c + a^2*d, 0] && GtQ[a, 0] && LtQ[d, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.1 Integrand simplification rules.m",
       "filename": "9.1 Integrand simplification rules.m",
            "rhs": "Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_ + b_.*x_ + c_.*x_^2)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.1 Integrand simplification rules.m",
       "filename": "9.1 Integrand simplification rules.m",
            "rhs": "1/c^p*Int[u*(b/2 + c*x^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_ + b_.*x_^n_ + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.2 Derivative integration rules.m",
       "filename": "9.2 Derivative integration rules.m",
            "rhs": "Derivative[n - 1][f][x]",
     "rulenumber": 0,
            "lhs": "Int[Derivative[n_][f_][x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{f, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.2 Derivative integration rules.m",
       "filename": "9.2 Derivative integration rules.m",
            "rhs": "(c*F^(a + b*x))^p*Derivative[n - 1][f][x] - b*p*Log[F]*Int[(c*F^(a + b*x))^p*Derivative[n - 1][f][x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*F_^(a_. + b_.*x_))^p_.*Derivative[n_][f_][x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, f, F, p}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.2 Derivative integration rules.m",
       "filename": "9.2 Derivative integration rules.m",
            "rhs": "(c*F^(a + b*x))^p* Derivative[n][f][x]/(b*p*Log[F]) - 1/(b*p*Log[F])*Int[(c*F^(a + b*x))^p*Derivative[n + 1][f][x], x]",
     "rulenumber": 0,
            "lhs": "Int[(c_.*F_^(a_. + b_.*x_))^p_.*Derivative[n_][f_][x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, f, F, p}, x] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.2 Derivative integration rules.m",
       "filename": "9.2 Derivative integration rules.m",
            "rhs": "Sin[a + b*x]*Derivative[n - 1][f][x] - b*Int[Cos[a + b*x]*Derivative[n - 1][f][x], x]",
     "rulenumber": 0,
            "lhs": "Int[Sin[a_. + b_.*x_]*Derivative[n_][f_][x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, f}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.2 Derivative integration rules.m",
       "filename": "9.2 Derivative integration rules.m",
            "rhs": "Cos[a + b*x]*Derivative[n - 1][f][x] + b*Int[Sin[a + b*x]*Derivative[n - 1][f][x], x]",
     "rulenumber": 0,
            "lhs": "Int[Cos[a_. + b_.*x_]*Derivative[n_][f_][x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, f}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.2 Derivative integration rules.m",
       "filename": "9.2 Derivative integration rules.m",
            "rhs": "-Cos[a + b*x]*Derivative[n][f][x]/b + 1/b*Int[Cos[a + b*x]*Derivative[n + 1][f][x], x]",
     "rulenumber": 0,
            "lhs": "Int[Sin[a_. + b_.*x_]*Derivative[n_][f_][x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, f}, x] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.2 Derivative integration rules.m",
       "filename": "9.2 Derivative integration rules.m",
            "rhs": "Sin[a + b*x]*Derivative[n][f][x]/b - 1/b*Int[Sin[a + b*x]*Derivative[n + 1][f][x], x]",
     "rulenumber": 0,
            "lhs": "Int[Cos[a_. + b_.*x_]*Derivative[n_][f_][x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, f}, x] && ILtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.2 Derivative integration rules.m",
       "filename": "9.2 Derivative integration rules.m",
            "rhs": "Subst[Int[ SimplifyIntegrand[SubstFor[Derivative[n - 1][f][x], u, x], x], x], x, Derivative[n - 1][f][x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*Derivative[n_][f_][x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{f, n}, x] && FunctionOfQ[Derivative[n - 1][f][x], u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.2 Derivative integration rules.m",
       "filename": "9.2 Derivative integration rules.m",
            "rhs": "a*Subst[Int[SimplifyIntegrand[SubstFor[f[x]*g[x], u, x], x], x], x, f[x]*g[x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*(a_.*Derivative[1][f_][x_]*g_[x_] + a_.*f_[x_]*Derivative[1][g_][x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, f, g}, x] && FunctionOfQ[f[x]*g[x], u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.2 Derivative integration rules.m",
       "filename": "9.2 Derivative integration rules.m",
            "rhs": "a*Subst[Int[ SimplifyIntegrand[SubstFor[Derivative[m - 1][f][x]*g[x], u, x], x], x], x, Derivative[m - 1][f][x]*g[x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*(a_.*Derivative[m_][f_][x_]*g_[x_] + a_.*Derivative[m1_][f_][x_]*Derivative[1][g_][x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, f, g, m}, x] && EqQ[m1, m - 1] && FunctionOfQ[Derivative[m - 1][f][x]*g[x], u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.2 Derivative integration rules.m",
       "filename": "9.2 Derivative integration rules.m",
            "rhs": "a*Subst[Int[ SimplifyIntegrand[ SubstFor[Derivative[m - 1][f][x]*Derivative[n - 1][g][x], u, x], x], x], x, Derivative[m - 1][f][x]*Derivative[n - 1][g][x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*(a_.*Derivative[m_][f_][x_]*Derivative[n1_][g_][x_] + a_.*Derivative[m1_][f_][x_]*Derivative[n_][g_][x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, f, g, m, n}, x] && EqQ[m1, m - 1] && EqQ[n1, n - 1] && FunctionOfQ[Derivative[m - 1][f][x]*Derivative[n - 1][g][x], u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.2 Derivative integration rules.m",
       "filename": "9.2 Derivative integration rules.m",
            "rhs": "b*Subst[Int[SimplifyIntegrand[SubstFor[f[x]^(p + 1)*g[x], u, x], x], x], x, f[x]^(p + 1)*g[x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*f_[x_]^ p_.*(a_.*Derivative[1][f_][x_]*g_[x_] + b_.*f_[x_]*Derivative[1][g_][x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, f, g, p}, x] && EqQ[a, b*(p + 1)] && FunctionOfQ[f[x]^(p + 1)*g[x], u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.2 Derivative integration rules.m",
       "filename": "9.2 Derivative integration rules.m",
            "rhs": "b*Subst[Int[ SimplifyIntegrand[ SubstFor[Derivative[m - 1][f][x]^(p + 1)*g[x], u, x], x], x], x, Derivative[m - 1][f][x]^(p + 1)*g[x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*Derivative[m1_][f_][x_]^p_.* (a_.*Derivative[m_][f_][x_]*g_[x_] + b_.*Derivative[m1_][f_][x_]*Derivative[1][g_][x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, f, g, m, p}, x] && EqQ[m1, m - 1] && EqQ[a, b*(p + 1)] && FunctionOfQ[Derivative[m - 1][f][x]^(p + 1)*g[x], u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.2 Derivative integration rules.m",
       "filename": "9.2 Derivative integration rules.m",
            "rhs": "a*Subst[Int[ SimplifyIntegrand[ SubstFor[Derivative[m - 1][f][x]*g[x]^(q + 1), u, x], x], x], x, Derivative[m - 1][f][x]*g[x]^(q + 1)]",
     "rulenumber": 0,
            "lhs": "Int[u_*g_[x_]^q_.* (a_.*Derivative[m_][f_][x_]*g_[x_] + b_.*Derivative[m1_][f_][x_]*Derivative[1][g_][x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, f, g, m, q}, x] && EqQ[m1, m - 1] && EqQ[a*(q + 1), b] && FunctionOfQ[Derivative[m - 1][f][x]*g[x]^(q + 1), u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.2 Derivative integration rules.m",
       "filename": "9.2 Derivative integration rules.m",
            "rhs": "b*Subst[Int[ SimplifyIntegrand[ SubstFor[ Derivative[m - 1][f][x]^(p + 1)*Derivative[n - 1][g][x], u, x], x], x], x, Derivative[m - 1][f][x]^(p + 1)*Derivative[n - 1][g][x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*Derivative[m1_][f_][x_]^p_.* (a_.*Derivative[m_][f_][x_]*Derivative[n1_][g_][x_] + b_.*Derivative[m1_][f_][x_]*Derivative[n_][g_][x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, f, g, m, n, p}, x] && EqQ[m1, m - 1] && EqQ[n1, n - 1] && EqQ[a, b*(p + 1)] && FunctionOfQ[ Derivative[m - 1][f][x]^(p + 1)*Derivative[n - 1][g][x], u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.2 Derivative integration rules.m",
       "filename": "9.2 Derivative integration rules.m",
            "rhs": "a/(p + 1)* Subst[Int[ SimplifyIntegrand[SubstFor[f[x]^(p + 1)*g[x]^(q + 1), u, x], x], x], x, f[x]^(p + 1)*g[x]^(q + 1)]",
     "rulenumber": 0,
            "lhs": "Int[u_*f_[x_]^p_.* g_[x_]^q_.*(a_.*Derivative[1][f_][x_]*g_[x_] + b_.*f_[x_]*Derivative[1][g_][x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, f, g, p, q}, x] && EqQ[a*(q + 1), b*(p + 1)] && FunctionOfQ[f[x]^(p + 1)*g[x]^(q + 1), u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.2 Derivative integration rules.m",
       "filename": "9.2 Derivative integration rules.m",
            "rhs": "a/(p + 1)* Subst[Int[ SimplifyIntegrand[ SubstFor[Derivative[m - 1][f][x]^(p + 1)*g[x]^(q + 1), u, x], x], x], x, Derivative[m - 1][f][x]^(p + 1)*g[x]^(q + 1)]",
     "rulenumber": 0,
            "lhs": "Int[u_*Derivative[m1_][f_][x_]^p_.*g_[x_]^q_.* (a_.*Derivative[m_][f_][x_]*g_[x_] + b_.*Derivative[m1_][f_][x_]*Derivative[1][g_][x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, f, g, m, p, q}, x] && EqQ[m1, m - 1] && EqQ[a*(q + 1), b*(p + 1)] && FunctionOfQ[Derivative[m - 1][f][x]^(p + 1)*g[x]^(q + 1), u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.2 Derivative integration rules.m",
       "filename": "9.2 Derivative integration rules.m",
            "rhs": "a/(p + 1)* Subst[Int[ SimplifyIntegrand[ SubstFor[ Derivative[m - 1][f][x]^(p + 1)* Derivative[n - 1][g][x]^(q + 1), u, x], x], x], x, Derivative[m - 1][f][x]^(p + 1)* Derivative[n - 1][g][x]^(q + 1)]",
     "rulenumber": 0,
            "lhs": "Int[u_*Derivative[m1_][f_][x_]^p_.*Derivative[n1_][g_][x_]^q_.* (a_.*Derivative[m_][f_][x_]*Derivative[n1_][g_][x_] + b_.*Derivative[m1_][f_][x_]*Derivative[n_][g_][x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, f, g, m, n, p, q}, x] && EqQ[m1, m - 1] && EqQ[n1, n - 1] && EqQ[a*(q + 1), b*(p + 1)] && FunctionOfQ[ Derivative[m - 1][f][x]^(p + 1)*Derivative[n - 1][g][x]^(q + 1), u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.2 Derivative integration rules.m",
       "filename": "9.2 Derivative integration rules.m",
            "rhs": "f[x]*g[x]",
     "rulenumber": 0,
            "lhs": "Int[f_'[x_]*g_[x_] + f_[x_]*g_'[x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{f, g}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.2 Derivative integration rules.m",
       "filename": "9.2 Derivative integration rules.m",
            "rhs": "f[x]/g[x]",
     "rulenumber": 0,
            "lhs": "Int[(f_'[x_]*g_[x_] - f_[x_]*g_'[x_])/g_[x_]^2, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{f, g}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.2 Derivative integration rules.m",
       "filename": "9.2 Derivative integration rules.m",
            "rhs": "Log[f[x]/g[x]]",
     "rulenumber": 0,
            "lhs": "Int[(f_'[x_]*g_[x_] - f_[x_]*g_'[x_])/(f_[x_]*g_[x_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{f, g}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.3 Piecewise linear functions.m",
       "filename": "9.3 Piecewise linear functions.m",
            "rhs": "With[{c = Simplify[D[u, x]]}, 1/c*Subst[Int[x^m, x], x, u]]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[m, x] && PiecewiseLinearQ[u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.3 Piecewise linear functions.m",
       "filename": "9.3 Piecewise linear functions.m",
            "rhs": "With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, b*x/a - (b*u - a*v)/a*Int[1/u, x] /; NeQ[b*u - a*v, 0]]",
     "rulenumber": 0,
            "lhs": "Int[v_/u_, x_Symbol]",
        "comment": false,
         "givens": "PiecewiseLinearQ[u, v, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.3 Piecewise linear functions.m",
       "filename": "9.3 Piecewise linear functions.m",
            "rhs": "With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, v^n/(a*n) - (b*u - a*v)/a*Int[v^(n - 1)/u, x] /; NeQ[b*u - a*v, 0]]",
     "rulenumber": 0,
            "lhs": "Int[v_^n_/u_, x_Symbol]",
        "comment": false,
         "givens": "PiecewiseLinearQ[u, v, x] && GtQ[n, 0] && NeQ[n, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.3 Piecewise linear functions.m",
       "filename": "9.3 Piecewise linear functions.m",
            "rhs": "With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, b/(b*u - a*v)*Int[1/v, x] - a/(b*u - a*v)*Int[1/u, x] /; NeQ[b*u - a*v, 0]]",
     "rulenumber": 0,
            "lhs": "Int[1/(u_*v_), x_Symbol]",
        "comment": false,
         "givens": "PiecewiseLinearQ[u, v, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.3 Piecewise linear functions.m",
       "filename": "9.3 Piecewise linear functions.m",
            "rhs": "With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, 2*ArcTan[ Sqrt[v]/Rt[(b*u - a*v)/a, 2]]/(a*Rt[(b*u - a*v)/a, 2]) /; NeQ[b*u - a*v, 0] && PosQ[(b*u - a*v)/a]]",
     "rulenumber": 0,
            "lhs": "Int[1/(u_*Sqrt[v_]), x_Symbol]",
        "comment": false,
         "givens": "PiecewiseLinearQ[u, v, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.3 Piecewise linear functions.m",
       "filename": "9.3 Piecewise linear functions.m",
            "rhs": "With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, -2*ArcTanh[ Sqrt[v]/Rt[-(b*u - a*v)/a, 2]]/(a* Rt[-(b*u - a*v)/a, 2]) /; NeQ[b*u - a*v, 0] && NegQ[(b*u - a*v)/a]]",
     "rulenumber": 0,
            "lhs": "Int[1/(u_*Sqrt[v_]), x_Symbol]",
        "comment": false,
         "givens": "PiecewiseLinearQ[u, v, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.3 Piecewise linear functions.m",
       "filename": "9.3 Piecewise linear functions.m",
            "rhs": "With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, v^(n + 1)/((n + 1)*(b*u - a*v)) - a*(n + 1)/((n + 1)*(b*u - a*v))*Int[v^(n + 1)/u, x] /; NeQ[b*u - a*v, 0]]",
     "rulenumber": 0,
            "lhs": "Int[v_^n_/u_, x_Symbol]",
        "comment": false,
         "givens": "PiecewiseLinearQ[u, v, x] && LtQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.3 Piecewise linear functions.m",
       "filename": "9.3 Piecewise linear functions.m",
            "rhs": "With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, v^(n + 1)/((n + 1)*(b*u - a*v))* Hypergeometric2F1[1, n + 1, n + 2, -a*v/(b*u - a*v)] /; NeQ[b*u - a*v, 0]]",
     "rulenumber": 0,
            "lhs": "Int[v_^n_/u_, x_Symbol]",
        "comment": false,
         "givens": "PiecewiseLinearQ[u, v, x] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.3 Piecewise linear functions.m",
       "filename": "9.3 Piecewise linear functions.m",
            "rhs": "With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, 2/Rt[a*b, 2]*ArcTanh[Rt[a*b, 2]*Sqrt[u]/(a*Sqrt[v])] /; NeQ[b*u - a*v, 0] && PosQ[a*b]]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[u_]*Sqrt[v_]), x_Symbol]",
        "comment": false,
         "givens": "PiecewiseLinearQ[u, v, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.3 Piecewise linear functions.m",
       "filename": "9.3 Piecewise linear functions.m",
            "rhs": "With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, 2/Rt[-a*b, 2]*ArcTan[Rt[-a*b, 2]*Sqrt[u]/(a*Sqrt[v])] /; NeQ[b*u - a*v, 0] && NegQ[a*b]]",
     "rulenumber": 0,
            "lhs": "Int[1/(Sqrt[u_]*Sqrt[v_]), x_Symbol]",
        "comment": false,
         "givens": "PiecewiseLinearQ[u, v, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.3 Piecewise linear functions.m",
       "filename": "9.3 Piecewise linear functions.m",
            "rhs": "With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, -u^(m + 1)*v^(n + 1)/((m + 1)*(b*u - a*v)) /; NeQ[b*u - a*v, 0]]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_*v_^n_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{m, n}, x] && PiecewiseLinearQ[u, v, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.3 Piecewise linear functions.m",
       "filename": "9.3 Piecewise linear functions.m",
            "rhs": "With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, u^(m + 1)*v^n/(a*(m + 1)) - b*n/(a*(m + 1))*Int[u^(m + 1)*v^(n - 1), x] /; NeQ[b*u - a*v, 0]]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_*v_^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{m, n}, x] && PiecewiseLinearQ[u, v, x] (* && NeQ[m+n+2, 0] *) && NeQ[m, -1] && ( LtQ[m, -1] && GtQ[n, 0] && Not[ILtQ[ m + n, -2] && (FractionQ[m] || GeQ[2*n + m + 1, 0])] || IGtQ[n, 0] && IGtQ[m, 0] && LeQ[n, m] || (* ILtQ[n,0] && ILtQ[m,0] && LeQ[n,m] || *) IGtQ[n, 0] && Not[IntegerQ[m]] || ILtQ[m, 0] && Not[IntegerQ[n]])"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.3 Piecewise linear functions.m",
       "filename": "9.3 Piecewise linear functions.m",
            "rhs": "With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, u^(m + 1)*v^n/(a*(m + n + 1)) - n*(b*u - a*v)/(a*(m + n + 1))*Int[u^m*v^(n - 1), x] /; NeQ[b*u - a*v, 0]]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_*v_^n_., x_Symbol]",
        "comment": false,
         "givens": "PiecewiseLinearQ[u, v, x] && NeQ[m + n + 2, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] && Not[IGtQ[m, 0] && (Not[IntegerQ[n]] || LtQ[0, m, n])] && Not[ILtQ[m + n, -2]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.3 Piecewise linear functions.m",
       "filename": "9.3 Piecewise linear functions.m",
            "rhs": "With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, u^(m + 1)*v^n/(a*(m + n + 1)) - n*(b*u - a*v)/(a*(m + n + 1))* Int[u^m*v^Simplify[n - 1], x] /; NeQ[b*u - a*v, 0]]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_*v_^n_, x_Symbol]",
        "comment": false,
         "givens": "PiecewiseLinearQ[u, v, x] && NeQ[m + n + 1, 0] && Not[RationalQ[n]] && SumSimplerQ[n, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.3 Piecewise linear functions.m",
       "filename": "9.3 Piecewise linear functions.m",
            "rhs": "With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, -u^(m + 1)*v^(n + 1)/((m + 1)*(b*u - a*v)) + b*(m + n + 2)/((m + 1)*(b*u - a*v))* Int[u^(m + 1)*v^n, x] /; NeQ[b*u - a*v, 0]]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_*v_^n_, x_Symbol]",
        "comment": false,
         "givens": "PiecewiseLinearQ[u, v, x] && NeQ[m + n + 2, 0] && LtQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.3 Piecewise linear functions.m",
       "filename": "9.3 Piecewise linear functions.m",
            "rhs": "With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, -u^(m + 1)*v^(n + 1)/((m + 1)*(b*u - a*v)) + b*(m + n + 2)/((m + 1)*(b*u - a*v))* Int[u^Simplify[m + 1]*v^n, x] /; NeQ[b*u - a*v, 0]]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_*v_^n_, x_Symbol]",
        "comment": false,
         "givens": "PiecewiseLinearQ[u, v, x] && Not[RationalQ[m]] && SumSimplerQ[m, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.3 Piecewise linear functions.m",
       "filename": "9.3 Piecewise linear functions.m",
            "rhs": "With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, u^m*v^(n + 1)/(b*(n + 1)*(b*u/(b*u - a*v))^m)* Hypergeometric2F1[-m, n + 1, n + 2, -a*v/(b*u - a*v)] /; NeQ[b*u - a*v, 0]]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_*v_^n_, x_Symbol]",
        "comment": false,
         "givens": "PiecewiseLinearQ[u, v, x] && Not[IntegerQ[m]] && Not[IntegerQ[n]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.3 Piecewise linear functions.m",
       "filename": "9.3 Piecewise linear functions.m",
            "rhs": "With[{c = Simplify[D[u, x]]}, u^n*(a + b*x)*Log[a + b*x]/b - Int[u^n, x] - c*n/b*Int[u^(n - 1)*(a + b*x)*Log[a + b*x], x]]",
     "rulenumber": 0,
            "lhs": "Int[u_^n_.*Log[a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && PiecewiseLinearQ[u, x] && Not[LinearQ[u, x]] && GtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.3 Piecewise linear functions.m",
       "filename": "9.3 Piecewise linear functions.m",
            "rhs": " With[{c=Simplify[D[u,x]]}, u^n*Log[a+b*x]^2/(2*b) - c*n/(2*b)*Int[u^(n-1)*Log[a+b*x]^2,x]]",
     "rulenumber": 0,
            "lhs": "Int[u_^n_.*Log[a_.+b_.*x_]/(a_.+b_.*x_),x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a,b},x] &&  PiecewiseLinearQ[u,x] && GtQ[n,0] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.3 Piecewise linear functions.m",
       "filename": "9.3 Piecewise linear functions.m",
            "rhs": "With[{c = Simplify[D[u, x]]}, u^n*(a + b*x)^(m + 1)*Log[a + b*x]/(b*(m + 1)) - 1/(m + 1)*Int[u^n*(a + b*x)^m, x] - c*n/(b*(m + 1))* Int[u^(n - 1)*(a + b*x)^(m + 1)*Log[a + b*x], x]]",
     "rulenumber": 0,
            "lhs": "Int[u_^n_.*(a_. + b_.*x_)^m_.*Log[a_. + b_.*x_], x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m}, x] && PiecewiseLinearQ[u, x] && Not[LinearQ[u, x]] && GtQ[n, 0] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": " c^FracPart[p]*(c*x^n)^FracPart[p]/x^(n*FracPart[p])*Int[u*x^(n*p),x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(c_.*x_^n_)^p_,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{c,n,p},x] && Not[IntegerQ[p]] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "c^IntPart[p]*(c*(a + b*x)^n)^FracPart[p]/(a + b*x)^(n*FracPart[p])* Int[u*(a + b*x)^(n*p), x] /; FreeQ[{a, b, c, n, p}, x] && Not[IntegerQ[p]] && Not[MatchQ[u, x^n1_.*v_.",
     "rulenumber": 0,
            "lhs": "Int[u_*(c_.*(a_. + b_.* x_)^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "EqQ[n, n1 + 1]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "(c*(d*(a + b*x))^p)^q/(a + b*x)^(p*q)* Int[u*(a + b*x)^(p*q), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(c_.*(d_*(a_. + b_.* x_))^p_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, p, q}, x] && Not[IntegerQ[p]] && Not[IntegerQ[q]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "(c*(d*(a + b*x)^n)^p)^q/(a + b*x)^(n*p*q)* Int[u*(a + b*x)^(n*p*q), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(c_.*(d_.*(a_. + b_.* x_)^n_)^p_)^q_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, n, p, q}, x] && Not[IntegerQ[p]] && Not[IntegerQ[q]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "2*e*g/(C*(e*f - d*g))* Subst[Int[(a + b*F[c*x])^n/x, x], x, Sqrt[d + e*x]/Sqrt[f + g*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*F_[c_.*Sqrt[d_. + e_.*x_]/Sqrt[f_. + g_.*x_]])^ n_./(A_. + B_.*x_ + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, A, B, C, F}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[B*e*g - C*(e*f + d*g), 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "2*e*g/(C*(e*f - d*g))* Subst[Int[(a + b*F[c*x])^n/x, x], x, Sqrt[d + e*x]/Sqrt[f + g*x]]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*F_[c_.*Sqrt[d_. + e_.*x_]/Sqrt[f_. + g_.*x_]])^ n_./(A_. + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, A, C, F}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[e*f + d*g, 0] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "Unintegrable[(a + b*F[c*Sqrt[d + e*x]/Sqrt[f + g*x]])^ n/(A + B*x + C*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*F_[c_.*Sqrt[d_. + e_.*x_]/Sqrt[f_. + g_.*x_]])^ n_/(A_. + B_.*x_ + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, A, B, C, F, n}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[B*e*g - C*(e*f + d*g), 0] && Not[IGtQ[n, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "Unintegrable[(a + b*F[c*Sqrt[d + e*x]/Sqrt[f + g*x]])^n/(A + C*x^2), x]",
     "rulenumber": 0,
            "lhs": "Int[(a_. + b_.*F_[c_.*Sqrt[d_. + e_.*x_]/Sqrt[f_. + g_.*x_]])^ n_/(A_ + C_.*x_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, A, C, F, n}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[e*f + d*g, 0] && Not[IGtQ[n, 0]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{q = DerivativeDivides[y, u, x]}, q*Log[RemoveContent[y, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_/y_, x_Symbol]",
        "comment": false,
         "givens": "Not[FalseQ[q]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{q = DerivativeDivides[y*w, u, x]}, q*Log[RemoveContent[y*w, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_/(y_*w_), x_Symbol]",
        "comment": false,
         "givens": "Not[FalseQ[q]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{q = DerivativeDivides[y, u, x]}, q*y^(m + 1)/(m + 1) /; Not[FalseQ[q]]]",
     "rulenumber": 0,
            "lhs": "Int[u_*y_^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[m, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{q = DerivativeDivides[y*z, u*z^(n - m), x]}, q*y^(m + 1)*z^(m + 1)/(m + 1) /; Not[FalseQ[q]]]",
     "rulenumber": 0,
            "lhs": "Int[u_*y_^m_.*z_^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{m, n}, x] && NeQ[m, -1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{v = SimplifyIntegrand[u, x]}, Int[v, x]",
     "rulenumber": 0,
            "lhs": "Int[u_, x_Symbol]",
        "comment": false,
         "givens": "SimplerIntegrandQ[v, u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "(a*e^2 - c*f^2)^m* Int[ExpandIntegrand[u*(e*Sqrt[a + b*x^n] - f*Sqrt[c + d*x^n])^(-m), x], x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(e_.*Sqrt[a_. + b_.*x_^n_.] + f_.*Sqrt[c_. + d_.*x_^n_.])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && ILtQ[m, 0] && EqQ[b*e^2 - d*f^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "(b*e^2 - d*f^2)^m* Int[ExpandIntegrand[ u*x^(m*n)*(e*Sqrt[a + b*x^n] - f*Sqrt[c + d*x^n])^(-m), x], x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(e_.*Sqrt[a_. + b_.*x_^n_.] + f_.*Sqrt[c_. + d_.*x_^n_.])^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, n}, x] && ILtQ[m, 0] && EqQ[a*e^2 - c*f^2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "Int[u^(m + n*p)*(a + u^(-n)*v)^p*w, x]",
     "rulenumber": 0,
            "lhs": "Int[u_^m_.*(a_.*u_^n_ + v_)^p_.*w_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, m, n}, x] && IntegerQ[p] && Not[GtQ[n, 0]] && Not[FreeQ[v, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{q = DerivativeDivides[y, u, x]}, q*Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, y] /; Not[FalseQ[q]]]",
     "rulenumber": 0,
            "lhs": "Int[u_*(a_. + b_.*y_)^m_.*(c_. + d_.*v_)^n_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n}, x] && EqQ[v, y]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{q = DerivativeDivides[y, u, x]}, q*Subst[Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x, y] /; Not[FalseQ[q]]]",
     "rulenumber": 0,
            "lhs": "Int[u_*(a_. + b_.*y_)^m_.*(c_. + d_.*v_)^n_.*(e_. + f_.*w_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[v, y] && EqQ[w, y]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{r = DerivativeDivides[y, u, x]}, r*Subst[Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x], x, y] /; Not[FalseQ[r]]]",
     "rulenumber": 0,
            "lhs": "Int[u_*(a_. + b_.*y_)^m_.*(c_. + d_.*v_)^n_.*(e_. + f_.*w_)^ p_.*(g_. + h_.*z_)^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, g, h, m, n, p, q}, x] && EqQ[v, y] && EqQ[w, y] && EqQ[z, y]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{q = DerivativeDivides[y, u, x]}, a*Int[u, x] + b*q*Subst[Int[x^n, x], x, y] /; Not[FalseQ[q]]]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_ + b_.*y_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{q = DerivativeDivides[y, u, x]}, q*Subst[Int[(a + b*x^n)^p, x], x, y] /; Not[FalseQ[q]]]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_. + b_.*y_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "Module[{q, r}, q*r*Subst[Int[x^m*(a + b*x^n)^p, x], x, y] /; Not[FalseQ[r = Divides[y^m, v^m, x]]] && Not[FalseQ[q = DerivativeDivides[y, u, x]]]]",
     "rulenumber": 0,
            "lhs": "Int[u_.*v_^m_.*(a_. + b_.*y_^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, n, p}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{q = DerivativeDivides[y, u, x]}, q*Subst[Int[(a + b*x^n + c*x^(2*n))^p, x], x, y] /; Not[FalseQ[q]]]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_. + b_.*y_^n_ + c_.*v_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[v, y]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{q = DerivativeDivides[y, u, x]}, q*Subst[Int[(A + B*x^n)*(a + b*x^n + c*x^(2*n))^p, x], x, y] /; Not[FalseQ[q]]]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(A_ + B_.*y_^n_) (a_. + b_.*v_^n_ + c_.*w_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, B, n, p}, x] && EqQ[n2, 2*n] && EqQ[v, y] && EqQ[w, y]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{q = DerivativeDivides[y, u, x]}, q*Subst[Int[(A + B*x^n)*(a + c*x^(2*n))^p, x], x, y] /; Not[FalseQ[q]]]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(A_ + B_.*y_^n_) (a_. + c_.*w_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, A, B, n, p}, x] && EqQ[n2, 2*n] && EqQ[w, y]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "Module[{q, r}, q*r* Subst[Int[x^m*(a + b*x^n + c*x^(2*n))^p, x], x, y] /; Not[FalseQ[r = Divides[y^m, v^m, x]]] && Not[FalseQ[q = DerivativeDivides[y, u, x]]]]",
     "rulenumber": 0,
            "lhs": "Int[u_.*v_^m_.*(a_. + b_.*y_^n_ + c_.*w_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[w, y]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "Module[{q, r}, q*r* Subst[Int[x^m*(A + B*x^n)*(a + b*x^n + c*x^(2*n))^p, x], x, y] /; Not[FalseQ[r = Divides[y^m, z^m, x]]] && Not[FalseQ[q = DerivativeDivides[y, u, x]]]]",
     "rulenumber": 0,
            "lhs": "Int[u_.*z_^m_.*(A_ + B_.*y_^n_)*(a_. + b_.*v_^n_ + c_.*w_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, A, B, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[v, y] && EqQ[w, y]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "Module[{q, r}, q*r* Subst[Int[x^m*(A + B*x^n)*(a + c*x^(2*n))^p, x], x, y] /; Not[FalseQ[r = Divides[y^m, z^m, x]]] && Not[FalseQ[q = DerivativeDivides[y, u, x]]]]",
     "rulenumber": 0,
            "lhs": "Int[u_.*z_^m_.*(A_ + B_.*y_^n_)*(a_. + c_.*w_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, c, A, B, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[w, y]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{q = DerivativeDivides[y, u, x]}, q*Subst[Int[(a + b*x^n)^m*(c + d*x^n)^p, x], x, y] /; Not[FalseQ[q]]]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_. + b_.*y_^n_)^m_.*(c_. + d_.*v_^n_)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[v, y]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{r = DerivativeDivides[y, u, x]}, r*Subst[Int[(a + b*x^n)^m*(c + d*x^n)^p*(e + f*x^n)^q, x], x, y] /; Not[FalseQ[r]]]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_. + b_.*y_^n_)^m_.*(c_. + d_.*v_^n_)^p_.*(e_. + f_.*w_^n_)^ q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, e, f, m, n, p, q}, x] && EqQ[v, y] && EqQ[w, y]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{q = DerivativeDivides[v, u, x]}, q*F^v/Log[F] /; Not[FalseQ[q]]]",
     "rulenumber": 0,
            "lhs": "Int[u_*F_^v_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[F, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{q = DerivativeDivides[v, u, x]}, q*Subst[Int[x^m*F^x, x], x, v] /; Not[FalseQ[q]]]",
     "rulenumber": 0,
            "lhs": "Int[u_*w_^m_.*F_^v_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{F, m}, x] && EqQ[w, v]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{c = Simplify[u/(w*D[v, x] + v*D[w, x])]}, c*Subst[Int[(a + b*x^p)^m, x], x, v*w] /; FreeQ[c, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*(a_ + b_.*v_^p_.*w_^p_.)^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, p}, x] && IntegerQ[p]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{c = Simplify[u/(p*w*D[v, x] + q*v*D[w, x])]}, c*p/(r + 1)* Subst[Int[(a + b*x^(p/(r + 1)))^m, x], x, v^(r + 1)*w] /; FreeQ[c, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*(a_ + b_.*v_^p_.*w_^q_.)^m_.*v_^r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, p, q, r}, x] && EqQ[p, q*(r + 1)] && NeQ[r, -1] && IntegerQ[p/(r + 1)]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{c = Simplify[u/(p*w*D[v, x] + q*v*D[w, x])]}, c*p/(r + 1)* Subst[Int[(a + b*x^(p/(r + 1)))^m, x], x, v^(r + 1)*w^(s + 1)] /; FreeQ[c, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*(a_ + b_.*v_^p_.*w_^q_.)^m_.*v_^r_.*w_^s_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, p, q, r, s}, x] && EqQ[p*(s + 1), q*(r + 1)] && NeQ[r, -1] && IntegerQ[p/(r + 1)]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{c = Simplify[u/(p*w*D[v, x] - q*v*D[w, x])]}, c*p*Subst[Int[(b + a*x^p)^m, x], x, v*w^(m*q + 1)] /; FreeQ[c, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*(a_.*v_^p_. + b_.*w_^q_.)^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, p, q}, x] && EqQ[p + q*(m*p + 1), 0] && IntegerQ[p] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": " With[{c=Simplify[u/(p*w*D[v,x]-q*v*D[w,x])]}, -c*q*Subst[Int[(a+b*x^q)^m,x],x,v^(m*p+1)*w] /; FreeQ[c,x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*(a_.*v_^p_.+b_.*w_^q_.)^m_.,x_Symbol]",
        "comment": false,
         "givens": " FreeQ[{a,b,m,p,q},x] && EqQ[p+q*(m*p+1),0] && IntegerQ[q] &&  IntegerQ[m] *)"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{c = Simplify[u/(p*w*D[v, x] - q*v*D[w, x])]}, -c*q*Subst[Int[(a + b*x^q)^m, x], x, v^(m*p + r + 1)*w] /; FreeQ[c, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*(a_.*v_^p_. + b_.*w_^q_.)^m_.*v_^r_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, p, q, r}, x] && EqQ[p + q*(m*p + r + 1), 0] && IntegerQ[q] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{c = Simplify[u/(p*w*D[v, x] - q*v*D[w, x])]}, -c*q/(s + 1)* Subst[Int[(a + b*x^(q/(s + 1)))^m, x], x, v^(m*p + 1)*w^(s + 1)] /; FreeQ[c, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*(a_.*v_^p_. + b_.*w_^q_.)^m_.*w_^s_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, p, q, s}, x] && EqQ[p*(s + 1) + q*(m*p + 1), 0] && NeQ[s, -1] && IntegerQ[q/(s + 1)] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{c = Simplify[u/(p*w*D[v, x] - q*v*D[w, x])]}, -c*q/(s + 1)* Subst[Int[(a + b*x^(q/(s + 1)))^m, x], x, v^(m*p + r + 1)*w^(s + 1)] /; FreeQ[c, x]]",
     "rulenumber": 0,
            "lhs": "Int[u_*(a_.*v_^p_. + b_.*w_^q_.)^m_.*v_^r_.*w_^s_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, p, q, r, s}, x] && EqQ[p*(s + 1) + q*(m*p + r + 1), 0] && NeQ[s, -1] && IntegerQ[q/(s + 1)] && IntegerQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "1/(m + 1)*Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)]",
     "rulenumber": 0,
            "lhs": "Int[u_*x_^m_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{lst = SubstForFractionalPowerOfLinear[u, x]}, lst[[2]]*lst[[4]]* Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])]",
     "rulenumber": 0,
            "lhs": "Int[u_, x_Symbol] := With[{lst = SubstForFractionalPowerOfLinear[u, x]}, ShowStep[\"\", \"Int[F[(a+b*x)^(1/n),x],x]\", \"n/b*Subst[Int[x^(n-1)*F[x,-a/b+x^n/b],x],x,(a+b*x)^(1/n)]\", Hold[ lst[[2]]*lst[[4]]* Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])]]] /; Not[FalseQ[lst]] && SubstForFractionalPowerQ[u, lst[[3]], x]] /; SimplifyFlag, Int[u_, x_Symbol]",
        "comment": false,
         "givens": "Not[FalseQ[lst]] && SubstForFractionalPowerQ[u, lst[[3]], x]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{lst = SubstForFractionalPowerOfQuotientOfLinears[u, x]}, lst[[2]]*lst[[4]]* Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])]",
     "rulenumber": 0,
            "lhs": "Int[u_, x_Symbol] := With[{lst = SubstForFractionalPowerOfQuotientOfLinears[u, x]}, ShowStep[\"\", \"Int[F[((a+b*x)/(c+d*x))^(1/n),x],x]\", \"n*(b*c-a*d)*Subst[Int[x^(n-1)*F[x,(-a+c*x^n)/(b-d*x^n)]/(b-d*x^ n)^2,x],x,((a+b*x)/(c+d*x))^(1/n)]\", Hold[ lst[[2]]*lst[[4]]* Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])]]] /; Not[FalseQ[lst]]] /; SimplifyFlag, Int[u_, x_Symbol]",
        "comment": false,
         "givens": "Not[FalseQ[lst]]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "a^IntPart[p]*(a*v^m*w^n*z^q)^ FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p])* z^(q*FracPart[p]))*Int[u*v^(m*p)*w^(n*p)*z^(p*q), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_.*v_^m_.*w_^n_.*z_^q_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, m, n, p, q}, x] && Not[IntegerQ[p]] && Not[FreeQ[v, x]] && Not[FreeQ[w, x]] && Not[FreeQ[z, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "a^IntPart[p]*(a*v^m*w^n)^ FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p]))* Int[u*v^(m*p)*w^(n*p), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_.*v_^m_.*w_^n_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, m, n, p}, x] && Not[IntegerQ[p]] && Not[FreeQ[v, x]] && Not[FreeQ[w, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "a^IntPart[p]*(a*v^m)^FracPart[p]/v^(m*FracPart[p])* Int[u*v^(m*p), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_.*v_^m_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, m, p}, x] && Not[IntegerQ[p]] && Not[FreeQ[v, x]] && Not[EqQ[a, 1] && EqQ[m, 1]] && Not[EqQ[v, x] && EqQ[m, 1]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "b^IntPart[p]*(a + b*x^n)^ FracPart[p]/(x^(n*FracPart[p])*(1 + a*x^(-n)/b)^FracPart[p])* Int[u*x^(n*p)*(1 + a*x^(-n)/b)^p, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_. + b_.*x_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, p}, x] && Not[IntegerQ[p]] && ILtQ[n, 0] && Not[RationalFunctionQ[u, x]] && IntegerQ[p + 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "(a + b*v^n)^ FracPart[p]/(v^(n*FracPart[p])*(b + a*v^(-n))^FracPart[p])* Int[u*v^(n*p)*(b + a*v^(-n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_. + b_.*v_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, p}, x] && Not[IntegerQ[p]] && ILtQ[n, 0] && BinomialQ[v, x] && Not[LinearQ[v, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "(a + b*x^m*v^n)^ FracPart[p]/(v^(n*FracPart[p])*(b*x^m + a*v^(-n))^FracPart[p])* Int[u*v^(n*p)*(b*x^m + a*v^(-n))^p, x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_. + b_.*x_^m_.*v_^n_)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, p}, x] && Not[IntegerQ[p]] && ILtQ[n, 0] && BinomialQ[v, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{v = (a*x^r + b*x^s)^ FracPart[ m]/(x^(r*FracPart[m])*(a + b*x^(s - r))^FracPart[m])}, v*Int[u*x^(m*r)*(a + b*x^(s - r))^m, x] /; NeQ[Simplify[v], 1]]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_.*x_^r_. + b_.*x_^s_.)^m_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, m, r, s}, x] && Not[IntegerQ[m]] && PosQ[s - r]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]]",
     "rulenumber": 0,
            "lhs": "Int[u_/(a_ + b_.*x_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "1/(4^p*c^p)*Int[u*(b + 2*c*x^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(a_. + b_.*x_^n_. + c_.*x_^n2_.)^p_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p] && Not[AlgebraicFunctionQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "(a + b*x^n + c*x^(2*n))^p/(b + 2*c*x^n)^(2*p)* Int[u*(b + 2*c*x^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(a_. + b_.*x_^n_. + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && Not[IntegerQ[p]] && Not[AlgebraicFunctionQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]]",
     "rulenumber": 0,
            "lhs": "Int[u_/(a_. + b_.*x_^n_. + c_.*x_^n2_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "Int[u*(a*x^m - b*Sqrt[c*x^n])/(a^2*x^(2*m) - b^2*c*x^n), x]",
     "rulenumber": 0,
            "lhs": "Int[u_./(a_.*x_^m_. + b_.*Sqrt[c_.*x_^n_]), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, m, n}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{lst = FunctionOfLinear[u, x]}, Dist[1/lst[[3]], Subst[Int[lst[[1]], x], x, lst[[2]] + lst[[3]]*x], x]",
     "rulenumber": 0,
            "lhs": "Int[u_, x_Symbol] := With[{lst = FunctionOfLinear[u, x]}, ShowStep[\"\", \"Int[F[a+b*x],x]\", \"Subst[Int[F[x],x],x,a+b*x]/b\", Hold[ Dist[1/lst[[3]], Subst[Int[lst[[1]], x], x, lst[[2]] + lst[[3]]*x], x]]] /; Not[FalseQ[lst]]] /; SimplifyFlag, Int[u_, x_Symbol]",
        "comment": false,
         "givens": "Not[FalseQ[lst]]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{lst = PowerVariableExpn[u, 0, x]}, 1/lst[[2]]* Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x], x], x], x, (lst[[3]]*x)^lst[[2]]] /; Not[FalseQ[lst]] && NeQ[lst[[2]], 0]]",
     "rulenumber": 0,
            "lhs": "Int[u_/x_, x_Symbol] := With[{lst = PowerVariableExpn[u, 0, x]}, ShowStep[\"\", \"Int[F[(c*x)^n]/x,x]\", \"Subst[Int[F[x]/x,x],x,(c*x)^n]/n\", Hold[ 1/lst[[2]]* Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x], x], x], x, (lst[[3]]*x)^lst[[2]]]]] /; Not[FalseQ[lst]] && NeQ[lst[[2]], 0]] /; SimplifyFlag && NonsumQ[u] && Not[RationalFunctionQ[u, x]], Int[u_/x_, x_Symbol]",
        "comment": false,
         "givens": "NonsumQ[u] && Not[RationalFunctionQ[u, x]]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{lst = PowerVariableExpn[u, m + 1, x]}, 1/lst[[2]]* Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x], x], x], x, (lst[[3]]*x)^lst[[2]]] /; Not[FalseQ[lst]] && NeQ[lst[[2]], m + 1]]",
     "rulenumber": 0,
            "lhs": "Int[u_*x_^m_., x_Symbol] := With[{lst = PowerVariableExpn[u, m + 1, x]}, ShowStep[\"If g=GCD[m+1,n]>1,\", \"Int[x^m*F[x^n],x]\", \"1/g*Subst[Int[x^((m+1)/g-1)*F[x^(n/g)],x],x,x^g]\", Hold[ 1/lst[[2]]* Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x], x], x], x, (lst[[3]]*x)^lst[[2]]]]] /; Not[FalseQ[lst]] && NeQ[lst[[2]], m + 1]] /; SimplifyFlag && IntegerQ[m] && NeQ[m, -1] && NonsumQ[u] && (GtQ[m, 0] || Not[AlgebraicFunctionQ[u, x]]), Int[u_*x_^m_., x_Symbol]",
        "comment": false,
         "givens": "IntegerQ[m] && NeQ[m, -1] && NonsumQ[u] && (GtQ[m, 0] || Not[AlgebraicFunctionQ[u, x]])]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{k = Denominator[m]}, k*Subst[Int[x^(k*(m + 1) - 1)*ReplaceAll[u, x -> x^k], x], x, x^(1/k)]]",
     "rulenumber": 0,
            "lhs": "Int[x_^m_*u_, x_Symbol]",
        "comment": false,
         "givens": "FractionQ[m]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{lst = FunctionOfSquareRootOfQuadratic[u, x]}, 2*Subst[Int[lst[[1]], x], x, lst[[2]]] /; Not[FalseQ[lst]] && EqQ[lst[[3]], 1]]",
     "rulenumber": 0,
            "lhs": "Int[u_, x_Symbol] := With[{lst = FunctionOfSquareRootOfQuadratic[u, x]}, ShowStep[\"\", \"Int[F[Sqrt[a+b*x+c*x^2],x],x]\", \"2*Subst[Int[F[(c*Sqrt[a]-b*x+Sqrt[a]*x^2)/(c-x^2),(-b+2* Sqrt[a]*x)/(c-x^2)]* (c*Sqrt[a]-b*x+Sqrt[a]*x^2)/(c-x^2)^2,x],x,(-Sqrt[a]+Sqrt[a+ b*x+c*x^2])/x]\", Hold[2*Subst[Int[lst[[1]], x], x, lst[[2]]]]] /; Not[FalseQ[lst]] && EqQ[lst[[3]], 1]] /; SimplifyFlag && EulerIntegrandQ[u, x], Int[u_, x_Symbol]",
        "comment": false,
         "givens": "EulerIntegrandQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{lst = FunctionOfSquareRootOfQuadratic[u, x]}, 2*Subst[Int[lst[[1]], x], x, lst[[2]]] /; Not[FalseQ[lst]] && EqQ[lst[[3]], 2]]",
     "rulenumber": 0,
            "lhs": "Int[u_, x_Symbol] := With[{lst = FunctionOfSquareRootOfQuadratic[u, x]}, ShowStep[\"\", \"Int[F[Sqrt[a+b*x+c*x^2],x],x]\", \"2*Subst[Int[F[(a*Sqrt[c]+b*x+Sqrt[c]*x^2)/(b+2*Sqrt[c]*x),(- a+x^2)/(b+2*Sqrt[c]*x)]* (a*Sqrt[c]+b*x+Sqrt[c]*x^2)/(b+2*Sqrt[c]*x)^2,x],x,Sqrt[c]* x+Sqrt[a+b*x+c*x^2]]\", Hold[2*Subst[Int[lst[[1]], x], x, lst[[2]]]]] /; Not[FalseQ[lst]] && EqQ[lst[[3]], 2]] /; SimplifyFlag && EulerIntegrandQ[u, x], Int[u_, x_Symbol]",
        "comment": false,
         "givens": "EulerIntegrandQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{lst = FunctionOfSquareRootOfQuadratic[u, x]}, 2*Subst[Int[lst[[1]], x], x, lst[[2]]] /; Not[FalseQ[lst]] && EqQ[lst[[3]], 3]]",
     "rulenumber": 0,
            "lhs": "Int[u_, x_Symbol] := With[{lst = FunctionOfSquareRootOfQuadratic[u, x]}, ShowStep[\"\", \"Int[F[Sqrt[a+b*x+c*x^2],x],x]\", \"-2*Sqrt[b^2-4*a*c]*Subst[Int[F[-Sqrt[b^2-4*a*c]*x/(c-x^2), (b*c+c*Sqrt[b^2-4*a*c]+(-b+Sqrt[b^2-4*a*c])*x^2)/(-2*c*(c- x^2))]*x/(c-x^2)^2,x], x,2*c*Sqrt[a+b*x+c*x^2]/(b-Sqrt[b^2-4*a*c]+2*c*x)]\", Hold[2*Subst[Int[lst[[1]], x], x, lst[[2]]]]] /; Not[FalseQ[lst]] && EqQ[lst[[3]], 3]] /; SimplifyFlag && EulerIntegrandQ[u, x], Int[u_, x_Symbol]",
        "comment": false,
         "givens": "EulerIntegrandQ[u, x]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "(*1/(2*a)*Int[Together[1/(1-Rt[-b/a,2]*v)],x] +  1/(2*a)*Int[Together[1/(1+Rt[-b/a,2]*v)],x] /; *) 1/(2*a)*Int[Together[1/(1 - v/Rt[-a/b, 2])], x] + 1/(2*a)*Int[Together[1/(1 + v/Rt[-a/b, 2])], x]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*v_^2), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "Dist[2/(a*n), Sum[Int[Together[1/(1 - v^2/((-1)^(4*k/n)*Rt[-a/b, n/2]))], x], {k, 1, n/2}], x]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*v_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IGtQ[n/2, 1]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "Dist[1/(a*n), Sum[Int[Together[1/(1 - v/((-1)^(2*k/n)*Rt[-a/b, n]))], x], {k, 1, n}], x]",
     "rulenumber": 0,
            "lhs": "Int[1/(a_ + b_.*v_^n_), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IGtQ[(n - 1)/2, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "Int[ReplaceAll[ ExpandIntegrand[PolynomialInSubst[v, u, x]/(a + b*x^n), x], x -> u], x]",
     "rulenumber": 0,
            "lhs": "Int[v_/(a_ + b_.*u_^n_.), x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b}, x] && IGtQ[n, 0] && PolynomialInQ[v, u, x]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{v = NormalizeIntegrand[u, x]}, Int[v, x]",
     "rulenumber": 0,
            "lhs": "Int[u_, x_Symbol]",
        "comment": false,
         "givens": "v =!= u]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{v = ExpandIntegrand[u, x]}, Int[v, x]",
     "rulenumber": 0,
            "lhs": "Int[u_, x_Symbol]",
        "comment": false,
         "givens": "SumQ[v]]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "(a + b*x^m)^p*(c + d*x^n)^q/x^(m*p)* Int[u*x^(m*p), x]",
     "rulenumber": 0,
            "lhs": "Int[u_.*(a_. + b_.*x_^m_.)^p_.*(c_. + d_.*x_^n_.)^q_., x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, d, m, n, p, q}, x] && EqQ[a + d, 0] && EqQ[b + c, 0] && EqQ[m + n, 0] && EqQ[p + q, 0]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "Sqrt[a + b*x^n + c*x^(2*n)]/((4*c)^(p - 1/2)*(b + 2*c*x^n))* Int[u*(b + 2*c*x^n)^(2*p), x]",
     "rulenumber": 0,
            "lhs": "Int[u_*(a_ + b_.*x_^n_. + c_.*x_^n2_.)^p_, x_Symbol]",
        "comment": false,
         "givens": "FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]"
  },
  {
       "pathname": "Rubi-4.16.1.0/Rubi/IntegrationRules/9 Miscellaneous/9.4 Miscellaneous integration rules.m",
       "filename": "9.4 Miscellaneous integration rules.m",
            "rhs": "With[{lst = SubstForFractionalPowerOfLinear[u, x]}, lst[[2]]*lst[[4]]* Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])]",
     "rulenumber": 0,
            "lhs": "Int[u_, x_Symbol] := With[{lst = SubstForFractionalPowerOfLinear[u, x]}, ShowStep[\"\", \"Int[F[(a+b*x)^(1/n),x],x]\", \"n/b*Subst[Int[x^(n-1)*F[x,-a/b+x^n/b],x],x,(a+b*x)^(1/n)]\", Hold[ lst[[2]]*lst[[4]]* Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])]]] /; Not[FalseQ[lst]]] /; SimplifyFlag, Int[u_, x_Symbol]",
        "comment": false,
         "givens": "Not[FalseQ[lst]]]]"
  }
]