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RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[x, Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], Plus[m, 1]], Power[Times[2, a, c, Plus[m, 1]], -1]], x]], Dist[Times[Plus[Times[2, m], 3], Power[Times[2, a, c, Plus[m, 1]], -1]], Int[Times[Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], Plus[m, 1]]], x], x]], And[FreeQ[List[a, b, c, d], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], NegativeIntegerQ[Plus[m, Times[3, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Power[Plus[Times[a, c], Times[b, d, Power[x, 2]]], m], x], And[FreeQ[List[a, b, c, d, m], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], Or[IntegerQ[m], And[PositiveQ[a], PositiveQ[c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[ArcCosh[Times[b, x, Power[a, -1]]], Power[b, -1]], x], And[FreeQ[List[a, b, c, d], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], PositiveQ[a], ZeroQ[Plus[a, c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], 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Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-5, 4]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 4]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[-2, Power[Times[b, Power[Plus[a, Times[b, x]], Times[1, Power[4, -1]]], Power[Plus[c, Times[d, x]], Times[1, Power[4, -1]]]], -1]], x], Dist[Times[Plus[Times[b, c], Times[-1, a, d]], Power[Times[2, b], -1]], Int[Times[1, Power[Times[Power[Plus[a, Times[b, x]], Times[5, Power[4, -1]]], Power[Plus[c, Times[d, x]], Times[5, Power[4, -1]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], PosQ[Times[b, d, Power[Times[a, c], -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-9, 4]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 4]]], Pattern[x, Blank[Symbol]]]], 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Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[a, Times[b, x]], m], Power[Plus[c, Times[d, x]], n]], x], x], And[FreeQ[List[a, b, c, d, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NegativeIntegerQ[m], IntegerQ[n], Not[And[PositiveIntegerQ[n], Less[Plus[m, n, 2], 0]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[c, Times[d, x]], n], Power[Times[b, n], -1]], x], Dist[Times[Plus[Times[b, c], Times[-1, a, d]], Power[b, -1]], Int[Times[Power[Plus[c, Times[d, x]], Plus[n, -1]], Power[Plus[a, Times[b, x]], -1]], x], x]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[n], Greater[n, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Plus[c, Times[d, x]], Plus[n, 1]], Power[Times[Plus[n, 1], Plus[Times[b, c], Times[-1, a, d]]], -1]], x]], Dist[Times[b, Plus[n, 1], Power[Times[Plus[n, 1], Plus[Times[b, c], Times[-1, a, d]]], -1]], Int[Times[Power[Plus[c, Times[d, x]], Plus[n, 1]], Power[Plus[a, Times[b, x]], -1]], x], x]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[n], Less[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 3]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Times[Plus[Times[b, c], Times[-1, a, d]], Power[b, -1]], 3]]], Plus[Times[-1, Simp[Times[Log[RemoveContent[Plus[a, Times[b, x]], x]], Power[Times[2, b, q], -1]], x]], Plus[Dist[Times[3, Power[Times[2, b], -1]], Subst[Int[Times[1, Power[Plus[Power[q, 2], Times[q, x], Power[x, 2]], -1]], x], x, Power[Plus[c, Times[d, x]], Times[1, Power[3, -1]]]], x], Times[-1, Dist[Times[3, Power[Times[2, b, q], -1]], Subst[Int[Times[1, Power[Plus[q, Times[-1, x]], -1]], x], x, Power[Plus[c, Times[d, x]], Times[1, Power[3, -1]]]], x]]]]], And[FreeQ[List[a, b, c, d], x], PosQ[Times[Plus[Times[b, c], Times[-1, a, d]], Power[b, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 3]]], Pattern[x, Blank[Symbol]]]], 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Condition[Times[-1, Simp[Times[Power[Plus[c, Times[d, x]], Plus[n, 1]], Power[Times[Plus[n, 1], Plus[Times[b, c], Times[-1, a, d]]], -1], Hypergeometric2F1[1, Plus[n, 1], Plus[n, 2], TogetherSimplify[Times[b, Plus[c, Times[d, x]], Power[Plus[Times[b, c], Times[-1, a, d]], -1]]]]], x]], And[FreeQ[List[a, b, c, d, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], n], Power[Times[b, Plus[m, 1]], -1]], x], Times[-1, Dist[Times[d, n, Power[Times[b, Plus[m, 1]], -1]], Int[Times[Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], Plus[n, -1]]], x], x]]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[m, n], Less[m, -1], Greater[n, 0], Not[And[IntegerQ[n], Not[IntegerQ[m]]]], Not[And[IntegerQ[Plus[m, n]], LessEqual[Plus[m, n, 2], 0], Or[FractionQ[m], GreaterEqual[Plus[Times[2, n], m, 1], 0]]]], IntLinearcQ[a, b, c, d, m, n, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], Plus[n, 1]], Power[Times[Plus[Times[b, c], Times[-1, a, d]], Plus[m, 1]], -1]], x], Times[-1, Dist[Times[d, Plus[m, n, 2], Power[Times[Plus[Times[b, c], Times[-1, a, d]], Plus[m, 1]], -1]], Int[Times[Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], n]], x], x]]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[m, n], Less[m, -1], Not[And[Less[n, -1], Or[ZeroQ[a], And[NonzeroQ[c], Less[m, n], IntegerQ[n]]]]], IntLinearcQ[a, b, c, d, m, n, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], n], Power[Times[b, Plus[m, n, 1]], -1]], x], Dist[Times[n, Plus[Times[b, c], Times[-1, a, d]], Power[Times[b, Plus[m, n, 1]], -1]], Int[Times[Power[Plus[a, Times[b, x]], m], Power[Plus[c, Times[d, x]], Plus[n, -1]]], x], x]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[m, n], Greater[n, 0], Unequal[Plus[m, n, 1], 0], Not[And[PositiveIntegerQ[m], Or[Not[IntegerQ[n]], Less[0, m, n]]]], Not[And[IntegerQ[Plus[m, n]], Less[Plus[m, n, 2], 0]]], IntLinearcQ[a, b, c, d, m, n, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[1, Power[Sqrt[Plus[Times[a, c], Times[-1, b, Plus[a, Times[-1, c]], x], Times[-1, Power[b, 2], Power[x, 2]]]], -1]], x], And[FreeQ[List[a, b, c, d], x], ZeroQ[Plus[b, d]], PositiveQ[Plus[a, c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[2, Power[Sqrt[b], -1]], Subst[Int[Times[1, Power[Sqrt[Plus[Times[b, c], Times[-1, a, d], Times[d, Power[x, 2]]]], -1]], x], x, Sqrt[Plus[a, Times[b, x]]]], x], And[FreeQ[List[a, b, c, d], x], PositiveQ[Plus[Times[b, c], Times[-1, a, d]]], PositiveQ[b]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[2, Power[b, -1]], Subst[Int[Times[1, Power[Sqrt[Plus[c, Times[-1, a], Power[x, 2]]], -1]], x], x, Sqrt[Plus[a, Times[b, x]]]], x], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[b, Times[-1, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[2, Subst[Int[Times[1, Power[Plus[b, Times[-1, d, Power[x, 2]]], -1]], x], x, Times[Sqrt[Plus[a, Times[b, x]]], Power[Sqrt[Plus[c, Times[d, x]]], -1]]], x], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, x]], m], Power[Plus[c, Times[d, x]], m], Power[Power[Plus[Times[a, c], Times[Plus[Times[b, c], Times[a, d]], x], Times[b, d, Power[x, 2]]], m], -1]], Int[Power[Plus[Times[a, c], Times[Plus[Times[b, c], Times[a, d]], x], Times[b, d, Power[x, 2]]], m], x], x], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[m], Less[-1, m, 0], LessEqual[3, Denominator[m], 4]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 3]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-2, 3]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Times[d, Power[b, -1]], 3]]], Plus[Times[-1, Simp[Times[Sqrt[3], q, Power[d, -1], ArcTan[Plus[Times[2, q, Power[Plus[a, Times[b, x]], Times[1, Power[3, -1]]], Power[Times[Sqrt[3], Power[Plus[c, Times[d, x]], Times[1, Power[3, -1]]]], -1]], Times[1, Power[Sqrt[3], -1]]]]], x]], Plus[Times[-1, Simp[Times[3, q, Power[Times[2, d], -1], Log[Plus[Times[q, Power[Plus[a, Times[b, x]], Times[1, Power[3, -1]]], Power[Power[Plus[c, Times[d, x]], Times[1, Power[3, -1]]], -1]], -1]]], x]], Times[-1, Simp[Times[q, Power[Times[2, d], -1], Log[Plus[c, Times[d, x]]]], x]]]]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], PosQ[Times[d, Power[b, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 3]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-2, 3]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Times[-1, d, Power[b, -1]], 3]]], Plus[Simp[Times[Sqrt[3], q, Power[d, -1], ArcTan[Plus[Times[1, Power[Sqrt[3], -1]], Times[-1, 2, q, Power[Plus[a, Times[b, x]], Times[1, Power[3, -1]]], Power[Times[Sqrt[3], Power[Plus[c, Times[d, x]], Times[1, Power[3, -1]]]], -1]]]]], x], Plus[Simp[Times[3, q, Power[Times[2, d], -1], Log[Plus[Times[q, Power[Plus[a, Times[b, x]], Times[1, Power[3, -1]]], Power[Power[Plus[c, Times[d, x]], Times[1, Power[3, -1]]], -1]], 1]]], x], Simp[Times[q, Power[Times[2, d], -1], Log[Plus[c, Times[d, x]]]], x]]]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NegQ[Times[d, Power[b, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[p, Denominator[m]]], Dist[p, Subst[Int[Times[Power[x, Plus[Times[p, Plus[m, 1]], -1]], Power[Plus[b, Times[-1, d, Power[x, p]]], -1]], x], x, Times[Power[Plus[a, Times[b, x]], Times[1, Power[p, -1]]], Power[Power[Plus[c, Times[d, x]], Times[1, Power[p, -1]]], -1]]], x]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[m, n], Less[-1, m, 0], Equal[Plus[m, n, 1], 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[p, Denominator[m]]], Dist[Times[p, Power[b, -1]], Subst[Int[Times[Power[x, Plus[Times[p, Plus[m, 1]], -1]], Power[Plus[c, Times[-1, a, d, Power[b, -1]], Times[d, Power[x, p], Power[b, -1]]], n]], x], x, Power[Plus[a, Times[b, x]], Times[1, Power[p, -1]]]], x]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[m, n], Less[-1, m, 0], Less[-1, n, 0], LessEqual[Denominator[n], Denominator[m]], IntLinearcQ[a, b, c, d, m, n, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], Plus[n, 1]], Power[Times[Plus[Times[b, c], Times[-1, a, d]], Plus[m, 1]], -1]], x], Times[-1, Dist[Times[d, Simplify[Plus[m, n, 2]], Power[Times[Plus[Times[b, c], Times[-1, a, d]], Plus[m, 1]], -1]], Int[Times[Power[Plus[a, Times[b, x]], Simplify[Plus[m, 1]]], Power[Plus[c, Times[d, x]], n]], x], x]]], And[FreeQ[List[a, b, c, d, m, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NegativeIntegerQ[Simplify[Plus[m, n, 2]]], NonzeroQ[Plus[m, 1]], Or[SumSimplerQ[m, 1], Not[SumSimplerQ[n, 1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[c, n], Power[Times[b, x], Plus[m, 1]], Power[Times[b, Plus[m, 1]], -1], Hypergeometric2F1[Times[-1, n], Plus[m, 1], Plus[m, 2], Times[-1, d, x, Power[c, -1]]]], x], And[FreeQ[List[b, c, d, m, n], x], Not[IntegerQ[m]], Or[IntegerQ[n], And[PositiveQ[c], Not[And[ZeroQ[Plus[n, Times[1, Power[2, -1]]]], ZeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], PositiveQ[Times[-1, d, Power[Times[b, c], -1]]]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[Plus[c, Times[d, x]], Plus[n, 1]], Power[Times[d, Plus[n, 1], Power[Times[-1, d, Power[Times[b, c], -1]], m]], -1], Hypergeometric2F1[Times[-1, m], Plus[n, 1], Plus[n, 2], Plus[1, Times[d, x, Power[c, -1]]]]], x], And[FreeQ[List[b, c, d, m, n], x], Not[IntegerQ[n]], Or[IntegerQ[m], PositiveQ[Times[-1, d, Power[Times[b, c], -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[c, IntPart[n]], Power[Plus[c, Times[d, x]], FracPart[n]], Power[Power[Plus[1, Times[d, x, Power[c, -1]]], FracPart[n]], -1]], Int[Times[Power[Times[b, x], m], Power[Plus[1, Times[d, x, Power[c, -1]]], n]], x], x], And[FreeQ[List[b, c, d, m, n], x], Not[IntegerQ[m]], Not[IntegerQ[n]], Not[PositiveQ[c]], Not[PositiveQ[Times[-1, d, Power[Times[b, c], -1]]]], Or[And[RationalQ[m], Not[And[ZeroQ[Plus[n, Times[1, Power[2, -1]]]], ZeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], Not[RationalQ[n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[-1, b, c, Power[d, -1]], IntPart[m]], Power[Times[b, x], FracPart[m]], Power[Power[Times[-1, d, x, Power[c, -1]], FracPart[m]], -1]], Int[Times[Power[Times[-1, d, x, Power[c, -1]], m], Power[Plus[c, Times[d, x]], n]], x], x], And[FreeQ[List[b, c, d, m, n], x], Not[IntegerQ[m]], Not[IntegerQ[n]], Not[PositiveQ[c]], Not[PositiveQ[Times[-1, d, Power[Times[b, c], -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[Plus[Times[b, c], Times[-1, a, d]], n], Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Times[Power[b, Plus[n, 1]], Plus[m, 1]], -1], Hypergeometric2F1[Times[-1, n], Plus[m, 1], Plus[m, 2], Times[-1, d, Plus[a, Times[b, x]], Power[Plus[Times[b, c], Times[-1, a, d]], -1]]]], x], And[FreeQ[List[a, b, c, d, m], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], Not[IntegerQ[m]], IntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Times[b, Plus[m, 1], Power[Times[b, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], n]], -1], Hypergeometric2F1[Times[-1, n], Plus[m, 1], Plus[m, 2], Times[-1, d, Plus[a, Times[b, x]], Power[Plus[Times[b, c], Times[-1, a, d]], -1]]]], x], And[FreeQ[List[a, b, c, d, m, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], Not[IntegerQ[m]], Not[IntegerQ[n]], PositiveQ[Times[b, Power[Plus[Times[b, c], Times[-1, a, d]], -1]]], Or[RationalQ[m], Not[And[RationalQ[n], PositiveQ[Times[-1, d, Power[Plus[Times[b, c], Times[-1, a, d]], -1]]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[c, Times[d, x]], FracPart[n]], Power[Times[Power[Times[b, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], IntPart[n]], Power[Times[b, Plus[c, Times[d, x]], Power[Plus[Times[b, c], Times[-1, a, d]], -1]], FracPart[n]]], -1]], Int[Times[Power[Plus[a, Times[b, x]], m], Power[Plus[Times[b, c, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Times[b, d, x, Power[Plus[Times[b, c], Times[-1, a, d]], -1]]], n]], x], x], And[FreeQ[List[a, b, c, d, m, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], Not[IntegerQ[m]], Not[IntegerQ[n]], Or[RationalQ[m], Not[SimplerQ[Plus[n, 1], Plus[m, 1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[u, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[u, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Coefficient[u, x, 1], -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], m], Power[Plus[c, Times[d, x]], n]], x], x, u], x], And[FreeQ[List[a, b, c, d, m, n], x], LinearQ[u, x], NonzeroQ[Coefficient[u, x, 0]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[Times[a, c], Times[b, d, Power[x, 2]]], m], Power[Plus[e, Times[f, x]], p]], x], And[FreeQ[List[a, b, c, d, e, f, m, n, p], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[m, Times[-1, n]]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[b, Power[Plus[c, Times[d, x]], Plus[n, 1]], Power[Plus[e, Times[f, x]], Plus[p, 1]], Power[Times[d, f, Plus[n, p, 2]], -1]], x], And[FreeQ[List[a, b, c, d, e, f, n, p], x], NonzeroQ[Plus[n, p, 2]], ZeroQ[Plus[Times[a, d, f, Plus[n, p, 2]], Times[-1, b, Plus[Times[d, e, Plus[n, 1]], Times[c, f, Plus[p, 1]]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[n, Blank[]]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Plus[a, Times[b, x]], Power[Times[d, x], n], Power[Plus[e, Times[f, x]], p]], x], x], And[FreeQ[List[a, b, d, e, f, n], x], PositiveIntegerQ[p], ZeroQ[Plus[Times[b, e], Times[a, f]]], Not[And[NegativeIntegerQ[Plus[n, p, 2]], Greater[Plus[n, Times[2, p]], 0]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[n, Blank[]]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Plus[a, Times[b, x]], Power[Times[d, x], n], Power[Plus[e, Times[f, x]], p]], x], x], And[FreeQ[List[a, b, d, e, f, n], x], PositiveIntegerQ[p], Or[NonzeroQ[Plus[n, 1]], Equal[p, 1]], NonzeroQ[Plus[Times[b, e], Times[a, f]]], Or[Not[IntegerQ[n]], Less[Plus[Times[9, p], Times[5, n]], 0], GreaterEqual[Plus[n, p, 1], 0], And[GreaterEqual[Plus[n, p, 2], 0], RationalQ[a, b, d, e, f]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Plus[a, Times[b, x]], Power[Plus[c, Times[d, x]], n], Power[Plus[e, Times[f, x]], p]], x], x], And[FreeQ[List[a, b, c, d, e, f, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], Or[NegativeIntegerQ[n, p], ZeroQ[Plus[p, -1]], And[PositiveIntegerQ[p], Or[Not[IntegerQ[n]], LessEqual[Plus[Times[9, p], Times[5, Plus[n, 2]]], 0], GreaterEqual[Plus[n, p, 1], 0], And[GreaterEqual[Plus[n, p, 2], 0], RationalQ[a, b, c, d, e, f]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[b, e], Times[-1, a, f]], Power[Plus[c, Times[d, x]], Plus[n, 1]], Power[Plus[e, Times[f, x]], Plus[p, 1]], Power[Times[f, Plus[p, 1], Plus[Times[c, f], Times[-1, d, e]]], -1]], x]], Dist[Times[b, Power[f, -1]], Int[Times[Power[Plus[c, Times[d, x]], n], Power[Plus[e, Times[f, x]], Plus[p, 1]]], x], x]], And[FreeQ[List[a, b, c, d, e, f, n, p], x], ZeroQ[Plus[n, p, 2]], NonzeroQ[Plus[p, 1]], Not[And[SumSimplerQ[n, 1], Not[SumSimplerQ[p, 1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[b, e], Times[-1, a, f]], Power[Plus[c, Times[d, x]], Plus[n, 1]], Power[Plus[e, Times[f, x]], Plus[p, 1]], Power[Times[f, Plus[p, 1], Plus[Times[c, f], Times[-1, d, e]]], -1]], x]], Times[-1, Dist[Times[Plus[Times[a, d, f, Plus[n, p, 2]], Times[-1, b, Plus[Times[d, e, Plus[n, 1]], Times[c, f, Plus[p, 1]]]]], Power[Times[f, Plus[p, 1], Plus[Times[c, f], Times[-1, d, e]]], -1]], Int[Times[Power[Plus[c, Times[d, x]], n], Power[Plus[e, Times[f, x]], Plus[p, 1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, n], x], NonzeroQ[Plus[n, p, 2]], RationalQ[p], Less[p, -1], Or[Not[And[RationalQ[n], Less[n, -1]]], IntegerQ[p], Not[Or[IntegerQ[n], Not[Or[ZeroQ[e], Not[Or[ZeroQ[c], Less[p, n]]]]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[b, e], Times[-1, a, f]], Power[Plus[c, Times[d, x]], Plus[n, 1]], Power[Plus[e, Times[f, x]], Plus[p, 1]], Power[Times[f, Plus[p, 1], Plus[Times[c, f], Times[-1, d, e]]], -1]], x]], Times[-1, Dist[Times[Plus[Times[a, d, f, Plus[n, p, 2]], Times[-1, b, Plus[Times[d, e, Plus[n, 1]], Times[c, f, Plus[p, 1]]]]], Power[Times[f, Plus[p, 1], Plus[Times[c, f], Times[-1, d, e]]], -1]], Int[Times[Power[Plus[c, Times[d, x]], n], Power[Plus[e, Times[f, x]], Simplify[Plus[p, 1]]]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, n, p], x], NonzeroQ[Plus[n, p, 2]], Not[RationalQ[p]], SumSimplerQ[p, 1]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, Power[Plus[c, Times[d, x]], Plus[n, 1]], Power[Plus[e, Times[f, x]], Plus[p, 1]], Power[Times[d, f, Plus[n, p, 2]], -1]], x], Dist[Times[Plus[Times[a, d, f, Plus[n, p, 2]], Times[-1, b, Plus[Times[d, e, Plus[n, 1]], Times[c, f, Plus[p, 1]]]]], Power[Times[d, f, Plus[n, p, 2]], -1]], Int[Times[Power[Plus[c, Times[d, x]], n], Power[Plus[e, Times[f, x]], p]], x], x]], And[FreeQ[List[a, b, c, d, e, f, n, p], x], NonzeroQ[Plus[n, p, 2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], 2], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[b, Power[Plus[c, Times[d, x]], Plus[n, 1]], Power[Plus[e, Times[f, x]], Plus[p, 1]], Plus[Times[2, a, d, f, Plus[n, p, 3]], Times[-1, b, Plus[Times[d, e, Plus[n, 2]], Times[c, f, Plus[p, 2]]]], Times[b, d, f, Plus[n, p, 2], x]], Power[Times[Power[d, 2], Power[f, 2], Plus[n, p, 2], Plus[n, p, 3]], -1]], x], And[FreeQ[List[a, b, c, d, e, f, n, p], x], NonzeroQ[Plus[n, p, 2]], NonzeroQ[Plus[n, p, 3]], ZeroQ[Plus[Times[d, f, Plus[n, p, 2], Plus[Times[Power[a, 2], d, f, Plus[n, p, 3]], Times[-1, b, Plus[Times[b, c, e], Times[a, Plus[Times[d, e, Plus[n, 1]], Times[c, f, Plus[p, 1]]]]]]]], Times[-1, b, Plus[Times[d, e, Plus[n, 1]], Times[c, f, Plus[p, 1]]], Plus[Times[a, d, f, Plus[n, p, 4]], Times[-1, b, Plus[Times[d, e, Plus[n, 2]], Times[c, f, Plus[p, 2]]]]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[a, Int[Times[Power[Plus[a, Times[b, x]], n], Power[Plus[c, Times[d, x]], n], Power[Times[f, x], p]], x], x], Dist[Times[b, Power[f, -1]], Int[Times[Power[Plus[a, Times[b, x]], n], Power[Plus[c, Times[d, x]], n], Power[Times[f, x], Plus[p, 1]]], x], x]], And[FreeQ[List[a, b, c, d, f, m, n, p], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[m, Times[-1, n], -1]], Not[RationalQ[p]], Not[PositiveIntegerQ[m]], NonzeroQ[Plus[m, n, p, 2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[e, Times[f, x]], p], Power[Times[Plus[a, Times[b, x]], Plus[c, Times[d, x]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f], x], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Plus[Times[b, e], Times[-1, a, f]], Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Int[Times[Power[Plus[e, Times[f, x]], Plus[p, -1]], Power[Plus[a, Times[b, x]], -1]], x], x], Times[-1, Dist[Times[Plus[Times[d, e], Times[-1, c, f]], Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Int[Times[Power[Plus[e, Times[f, x]], Plus[p, -1]], Power[Plus[c, Times[d, x]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], RationalQ[p], Less[0, p, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[f, Power[Plus[e, Times[f, x]], Plus[p, -1]], Power[Times[b, d, Plus[p, -1]], -1]], x], Dist[Times[1, Power[Times[b, d], -1]], Int[Times[Plus[Times[b, d, Power[e, 2]], Times[-1, a, c, Power[f, 2]], Times[f, Plus[Times[2, b, d, e], Times[-1, b, c, f], Times[-1, a, d, f]], x]], Power[Plus[e, Times[f, x]], Plus[p, -2]], Power[Times[Plus[a, Times[b, x]], Plus[c, Times[d, x]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], RationalQ[p], Greater[p, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[f, Power[Plus[e, Times[f, x]], Plus[p, 1]], Power[Times[Plus[p, 1], Plus[Times[b, e], Times[-1, a, f]], Plus[Times[d, e], Times[-1, c, f]]], -1]], x], Dist[Times[1, Power[Times[Plus[Times[b, e], Times[-1, a, f]], Plus[Times[d, e], Times[-1, c, f]]], -1]], Int[Times[Plus[Times[b, d, e], Times[-1, b, c, f], Times[-1, a, d, f], Times[-1, b, d, f, x]], Power[Plus[e, Times[f, x]], Plus[p, 1]], Power[Times[Plus[a, Times[b, x]], Plus[c, Times[d, x]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], RationalQ[p], Less[p, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[b, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Int[Times[Power[Plus[e, Times[f, x]], p], Power[Plus[a, Times[b, x]], -1]], x], x], Times[-1, Dist[Times[d, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Int[Times[Power[Plus[e, Times[f, x]], p], Power[Plus[c, Times[d, x]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, p], x], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Power[Plus[e, Times[f, x]], FractionalPart[p]], Times[Power[Plus[c, Times[d, x]], n], Power[Plus[e, Times[f, x]], IntegerPart[p]], Power[Plus[a, Times[b, x]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f], x], PositiveIntegerQ[n], FractionQ[p], Less[p, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[a, Times[b, x]], m], Power[Plus[c, Times[d, x]], n], Power[Plus[e, Times[f, x]], p]], x], x], And[FreeQ[List[a, b, c, d, e, f, p], x], IntegersQ[m, n], Or[IntegerQ[p], And[Greater[m, 0], GreaterEqual[n, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], 2], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[Times[b, c], Times[-1, a, d]], 2], Power[Plus[c, Times[d, x]], Plus[n, 1]], Power[Plus[e, Times[f, x]], Plus[p, 1]], Power[Times[Power[d, 2], Plus[Times[d, e], Times[-1, c, f]], Plus[n, 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[Power[d, 2], Plus[Times[d, e], Times[-1, c, f]], Plus[n, 1]], -1]], Int[Times[Power[Plus[c, Times[d, x]], Plus[n, 1]], Power[Plus[e, Times[f, x]], p], Simp[Plus[Times[Power[a, 2], Power[d, 2], f, Plus[n, p, 2]], Times[Power[b, 2], c, Plus[Times[d, e, Plus[n, 1]], Times[c, f, Plus[p, 1]]]], Times[-1, 2, a, b, d, Plus[Times[d, e, Plus[n, 1]], Times[c, f, Plus[p, 1]]]], Times[-1, Power[b, 2], d, Plus[Times[d, e], Times[-1, c, f]], Plus[n, 1], x]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, n, p], x], Or[And[RationalQ[n], Less[n, -1]], And[ZeroQ[Plus[n, p, 3]], NonzeroQ[Plus[n, 1]], Or[SumSimplerQ[n, 1], Not[SumSimplerQ[p, 1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], 2], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, Plus[a, Times[b, x]], Power[Plus[c, Times[d, x]], Plus[n, 1]], Power[Plus[e, Times[f, x]], Plus[p, 1]], Power[Times[d, f, Plus[n, p, 3]], -1]], x], Dist[Times[1, Power[Times[d, f, Plus[n, p, 3]], -1]], Int[Times[Power[Plus[c, Times[d, x]], n], Power[Plus[e, Times[f, x]], p], Simp[Plus[Times[Power[a, 2], d, f, Plus[n, p, 3]], Times[-1, b, Plus[Times[b, c, e], Times[a, Plus[Times[d, e, Plus[n, 1]], Times[c, f, Plus[p, 1]]]]]], Times[b, Plus[Times[a, d, f, Plus[n, p, 4]], Times[-1, b, Plus[Times[d, e, Plus[n, 2]], Times[c, f, Plus[p, 2]]]]], x]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, n, p], x], NonzeroQ[Plus[n, p, 3]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 3]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-2, 3]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Times[Plus[Times[d, e], Times[-1, c, f]], Power[Plus[Times[b, e], Times[-1, a, f]], -1]], 3]]], Plus[Times[-1, Simp[Times[Sqrt[3], q, ArcTan[Plus[Times[1, Power[Sqrt[3], -1]], Times[2, q, Power[Plus[a, Times[b, x]], Times[1, Power[3, -1]]], Power[Times[Sqrt[3], Power[Plus[c, Times[d, x]], Times[1, Power[3, -1]]]], -1]]]], Power[Plus[Times[d, e], Times[-1, c, f]], -1]], x]], Plus[Simp[Times[q, Log[Plus[e, Times[f, x]]], Power[Times[2, Plus[Times[d, e], Times[-1, c, f]]], -1]], x], Times[-1, Simp[Times[3, q, Log[Plus[Times[q, Power[Plus[a, Times[b, x]], Times[1, Power[3, -1]]]], Times[-1, Power[Plus[c, Times[d, x]], Times[1, Power[3, -1]]]]]], Power[Times[2, Plus[Times[d, e], Times[-1, c, f]]], -1]], x]]]]], FreeQ[List[a, b, c, d, e, f], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[b, f], Subst[Int[Times[1, Power[Plus[Times[d, Power[Plus[Times[b, e], Times[-1, a, f]], 2]], Times[b, Power[f, 2], Power[x, 2]]], -1]], x], x, Times[Sqrt[Plus[a, Times[b, x]]], Sqrt[Plus[c, Times[d, x]]]]], x], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[Times[2, b, d, e], Times[-1, f, Plus[Times[b, c], Times[a, d]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], 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Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], n], Power[Plus[e, Times[f, x]], Plus[p, 1]], Power[Times[Plus[m, 1], Plus[Times[b, e], Times[-1, a, f]]], -1]], x], Times[-1, Dist[Times[n, Plus[Times[d, e], Times[-1, c, f]], Power[Times[Plus[m, 1], Plus[Times[b, e], Times[-1, a, f]]], -1]], Int[Times[Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], Plus[n, -1]], Power[Plus[e, Times[f, x]], p]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, m, p], x], ZeroQ[Plus[m, n, p, 2]], RationalQ[n], Greater[n, 0], Not[And[SumSimplerQ[p, 1], Not[SumSimplerQ[m, 1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], 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4]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Times[-1, f, Plus[c, Times[d, x]], Power[Plus[Times[d, e], Times[-1, c, f]], -1]]], Power[Sqrt[Plus[c, Times[d, x]]], -1]], Int[Times[1, Power[Times[Plus[a, Times[b, x]], Sqrt[Plus[Times[-1, c, f, Power[Plus[Times[d, e], Times[-1, c, f]], -1]], Times[-1, d, f, x, Power[Plus[Times[d, e], Times[-1, c, f]], -1]]]], Power[Plus[e, Times[f, x]], Times[3, Power[4, -1]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f], x], Not[PositiveQ[Times[-1, f, Power[Plus[Times[d, e], Times[-1, c, f]], -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[2, Sqrt[e], Power[b, -1], Rt[Times[-1, b, Power[d, 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b, Power[d, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Plus[e, Times[f, x]]], Sqrt[Plus[1, Times[d, x, Power[c, -1]]]], Power[Times[Sqrt[Plus[c, Times[d, x]]], Sqrt[Plus[1, Times[f, x, Power[e, -1]]]]], -1]], Int[Times[Sqrt[Plus[1, Times[f, x, Power[e, -1]]]], Power[Times[Sqrt[Times[b, x]], Sqrt[Plus[1, Times[d, x, Power[c, -1]]]]], -1]], x], x], And[FreeQ[List[b, c, d, e, f], x], NonzeroQ[Plus[Times[d, e], Times[-1, c, f]]], Not[And[PositiveQ[c], PositiveQ[e]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[2, Power[b, -1], Rt[Times[-1, Plus[Times[b, e], Times[-1, a, f]], Power[d, -1]], 2], EllipticE[ArcSin[Times[Sqrt[Plus[a, Times[b, x]]], Power[Rt[Times[-1, Plus[Times[b, c], Times[-1, a, d]], Power[d, -1]], 2], -1]]], Times[f, Plus[Times[b, c], Times[-1, a, d]], Power[Times[d, Plus[Times[b, e], Times[-1, a, f]]], -1]]]], x], And[FreeQ[List[a, b, c, d, e, f], x], PositiveQ[Times[b, Power[Plus[Times[b, c], Times[-1, a, d]], -1]]], PositiveQ[Times[b, Power[Plus[Times[b, e], Times[-1, a, f]], -1]]], Not[NegativeQ[Times[-1, Plus[Times[b, c], Times[-1, a, d]], Power[d, -1]]]], Not[And[SimplerQ[Plus[c, Times[d, x]], Plus[a, Times[b, x]]], PositiveQ[Times[-1, d, Power[Plus[Times[b, c], Times[-1, a, d]], -1]]], PositiveQ[Times[d, Power[Plus[Times[d, e], Times[-1, c, f]], -1]]], Not[NegativeQ[Times[Plus[Times[b, c], Times[-1, a, d]], Power[b, -1]]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Plus[e, Times[f, x]]], Sqrt[Times[b, Plus[c, Times[d, x]], Power[Plus[Times[b, c], Times[-1, a, d]], -1]]], Power[Times[Sqrt[Plus[c, Times[d, x]]], Sqrt[Times[b, Plus[e, Times[f, x]], Power[Plus[Times[b, e], Times[-1, a, f]], -1]]]], -1]], Int[Times[Sqrt[Plus[Times[b, e, Power[Plus[Times[b, e], Times[-1, a, f]], -1]], Times[b, f, x, Power[Plus[Times[b, e], Times[-1, a, f]], -1]]]], Power[Times[Sqrt[Plus[a, Times[b, x]]], Sqrt[Plus[Times[b, c, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Times[b, d, x, Power[Plus[Times[b, c], Times[-1, a, d]], -1]]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f], x], Not[And[PositiveQ[Times[b, Power[Plus[Times[b, c], Times[-1, a, d]], -1]]], PositiveQ[Times[b, Power[Plus[Times[b, e], Times[-1, a, f]], -1]]]]], Not[NegativeQ[Times[-1, Plus[Times[b, c], Times[-1, a, d]], Power[d, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[2, Power[Times[b, Sqrt[e]], -1], Rt[Times[-1, b, Power[d, -1]], 2], EllipticF[ArcSin[Times[Sqrt[Times[b, x]], Power[Times[Sqrt[c], Rt[Times[-1, b, Power[d, -1]], 2]], -1]]], Times[c, f, Power[Times[d, e], -1]]]], x], And[FreeQ[List[b, c, d, e, f], x], PositiveQ[c], PositiveQ[e], Or[PositiveQ[Times[-1, b, Power[d, -1]]], NegativeQ[Times[-1, b, Power[f, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[2, Power[Times[b, Sqrt[e]], -1], Rt[Times[-1, b, Power[d, -1]], 2], EllipticF[ArcSin[Times[Sqrt[Times[b, x]], Power[Times[Sqrt[c], Rt[Times[-1, b, Power[d, -1]], 2]], -1]]], Times[c, f, Power[Times[d, e], -1]]]], x], And[FreeQ[List[b, c, d, e, f], x], PositiveQ[c], PositiveQ[e], Or[PosQ[Times[-1, b, Power[d, -1]]], NegQ[Times[-1, b, Power[f, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Plus[1, Times[d, x, Power[c, -1]]]], Sqrt[Plus[1, Times[f, x, Power[e, -1]]]], Power[Times[Sqrt[Plus[c, Times[d, x]]], Sqrt[Plus[e, Times[f, x]]]], -1]], Int[Times[1, Power[Times[Sqrt[Times[b, x]], Sqrt[Plus[1, Times[d, x, Power[c, -1]]]], Sqrt[Plus[1, Times[f, x, Power[e, -1]]]]], -1]], x], x], And[FreeQ[List[b, c, d, e, f], x], Not[And[PositiveQ[c], PositiveQ[e]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[2, Power[b, -1], Rt[Times[-1, Plus[Times[b, c], Times[-1, a, d]], Power[d, -1]], 2], Sqrt[Times[Power[b, 2], Power[Times[Plus[Times[b, c], Times[-1, a, d]], Plus[Times[b, e], Times[-1, a, f]]], -1]]], EllipticF[ArcSin[Times[Sqrt[Plus[a, Times[b, x]]], Power[Rt[Times[-1, Plus[Times[b, c], Times[-1, a, d]], Power[d, -1]], 2], -1]]], Times[f, Plus[Times[b, c], Times[-1, a, d]], Power[Times[d, Plus[Times[b, e], Times[-1, a, f]]], -1]]]], x], And[FreeQ[List[a, b, c, d, e, f], x], PositiveQ[Times[b, Power[Plus[Times[b, c], Times[-1, a, d]], -1]]], PositiveQ[Times[b, Power[Plus[Times[b, e], Times[-1, a, f]], -1]]], SimplerQ[Plus[a, Times[b, x]], Plus[c, Times[d, x]]], SimplerQ[Plus[a, Times[b, x]], Plus[e, Times[f, x]]], Or[PositiveQ[Times[-1, Plus[Times[b, c], Times[-1, a, d]], Power[d, -1]]], NegativeQ[Times[-1, Plus[Times[b, e], Times[-1, a, f]], Power[f, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[2, Power[b, -1], Rt[Times[-1, Plus[Times[b, c], Times[-1, a, d]], Power[d, -1]], 2], Sqrt[Times[Power[b, 2], Power[Times[Plus[Times[b, c], Times[-1, a, d]], Plus[Times[b, e], Times[-1, a, f]]], -1]]], EllipticF[ArcSin[Times[Sqrt[Plus[a, Times[b, x]]], Power[Rt[Times[-1, Plus[Times[b, c], Times[-1, a, d]], Power[d, -1]], 2], -1]]], Times[f, Plus[Times[b, c], Times[-1, a, d]], Power[Times[d, Plus[Times[b, e], Times[-1, a, f]]], -1]]]], x], And[FreeQ[List[a, b, c, d, e, f], x], PositiveQ[Times[b, Power[Plus[Times[b, c], Times[-1, a, d]], -1]]], PositiveQ[Times[b, Power[Plus[Times[b, e], Times[-1, a, f]], -1]]], SimplerQ[Plus[a, Times[b, x]], Plus[c, Times[d, x]]], SimplerQ[Plus[a, Times[b, x]], Plus[e, Times[f, x]]], Or[PosQ[Times[-1, Plus[Times[b, c], Times[-1, a, d]], Power[d, -1]]], NegQ[Times[-1, Plus[Times[b, e], Times[-1, a, f]], Power[f, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Times[b, Plus[c, Times[d, x]], Power[Plus[Times[b, c], Times[-1, a, d]], -1]]], Sqrt[Times[b, Plus[e, Times[f, x]], Power[Plus[Times[b, e], Times[-1, a, f]], -1]]], Power[Times[Sqrt[Plus[c, Times[d, x]]], Sqrt[Plus[e, Times[f, x]]]], -1]], Int[Times[1, Power[Times[Sqrt[Plus[a, Times[b, x]]], Sqrt[Plus[Times[b, c, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Times[b, d, x, Power[Plus[Times[b, c], Times[-1, a, d]], -1]]]], Sqrt[Plus[Times[b, e, Power[Plus[Times[b, e], Times[-1, a, f]], -1]], Times[b, f, x, Power[Plus[Times[b, e], Times[-1, a, f]], -1]]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f], x], Not[And[PositiveQ[Times[b, Power[Plus[Times[b, c], Times[-1, a, d]], -1]]], PositiveQ[Times[b, Power[Plus[Times[b, e], Times[-1, a, f]], -1]]]]], SimplerQ[Plus[a, Times[b, x]], Plus[c, Times[d, x]]], SimplerQ[Plus[a, Times[b, x]], Plus[e, Times[f, x]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 3]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 3]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Times[b, Plus[Times[b, e], Times[-1, a, f]], Power[Power[Plus[Times[b, c], Times[-1, a, d]], 2], -1]], 3]]], Plus[Times[-1, Simp[Times[Log[Plus[a, Times[b, x]]], Power[Times[2, q, Plus[Times[b, c], Times[-1, a, d]]], -1]], x]], Plus[Times[-1, Simp[Times[Sqrt[3], ArcTan[Plus[Times[1, Power[Sqrt[3], -1]], Times[2, q, Power[Plus[c, Times[d, x]], Times[2, Power[3, -1]]], Power[Times[Sqrt[3], Power[Plus[e, Times[f, x]], Times[1, Power[3, -1]]]], -1]]]], Power[Times[2, q, Plus[Times[b, c], Times[-1, a, d]]], -1]], x]], Simp[Times[3, Log[Plus[Times[q, Power[Plus[c, Times[d, x]], Times[2, Power[3, -1]]]], Times[-1, Power[Plus[e, Times[f, x]], Times[1, Power[3, -1]]]]]], Power[Times[4, q, Plus[Times[b, c], Times[-1, a, d]]], -1]], x]]]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[Times[2, b, d, e], Times[-1, b, c, f], Times[-1, a, d, f]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 3]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 3]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], Times[2, Power[3, -1]]], Power[Plus[e, Times[f, x]], Times[2, Power[3, -1]]], Power[Times[Plus[m, 1], Plus[Times[b, c], Times[-1, a, d]], Plus[Times[b, e], Times[-1, a, f]]], -1]], x], Dist[Times[f, Power[Times[6, Plus[m, 1], Plus[Times[b, c], Times[-1, a, d]], Plus[Times[b, e], Times[-1, a, f]]], -1]], Int[Times[Power[Plus[a, Times[b, x]], Plus[m, 1]], Plus[Times[a, d, Plus[Times[3, m], 1]], Times[-1, 3, b, c, Plus[Times[3, m], 5]], Times[-1, 2, b, d, Plus[Times[3, m], 7], x]], Power[Times[Power[Plus[c, Times[d, x]], Times[1, Power[3, -1]]], Power[Plus[e, Times[f, x]], Times[1, Power[3, -1]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[Times[2, b, d, e], Times[-1, b, c, f], Times[-1, a, d, f]]], IntegerQ[m], Less[m, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[Times[a, c], Times[b, d, Power[x, 2]]], m], Power[Times[f, x], p]], x], And[FreeQ[List[a, b, c, d, f, m, n, p], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[m, Times[-1, n]]], PositiveQ[a], PositiveQ[c]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, x]], FracPart[m]], Power[Plus[c, Times[d, x]], FracPart[m]], Power[Power[Plus[Times[a, c], Times[b, d, Power[x, 2]]], FracPart[m]], -1]], Int[Times[Power[Plus[Times[a, c], Times[b, d, Power[x, 2]]], m], Power[Times[f, x], p]], x], x], And[FreeQ[List[a, b, c, d, f, m, n, p], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[m, Times[-1, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[a, Times[b, x]], n], Power[Plus[c, Times[d, x]], n], Power[Times[f, x], p]], Power[Plus[a, Times[b, x]], Plus[m, Times[-1, n]]], x], x], And[FreeQ[List[a, b, c, d, f, m, n, p], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], PositiveIntegerQ[Plus[m, Times[-1, n]]], NonzeroQ[Plus[m, n, p, 2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[a, Times[b, x]], m], Power[Plus[c, Times[d, x]], n], Power[Plus[e, Times[f, x]], p]], x], x], And[FreeQ[List[a, b, c, d, e, f, n, p], x], Or[PositiveIntegerQ[m], NegativeIntegerQ[m, n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], Plus[n, 1]], Power[Plus[e, Times[f, x]], Plus[p, 1]], Power[Times[Plus[m, 1], Plus[Times[b, c], Times[-1, a, d]], Plus[Times[b, e], Times[-1, a, f]]], -1]], x], Dist[Times[1, Power[Times[Plus[m, 1], Plus[Times[b, c], Times[-1, a, d]], Plus[Times[b, e], Times[-1, a, f]]], -1]], Int[Times[Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], n], Power[Plus[e, Times[f, x]], p], Simp[Plus[Times[a, d, f, Plus[m, 1]], Times[-1, b, Plus[Times[d, e, Plus[m, n, 2]], Times[c, f, Plus[m, p, 2]]]], Times[-1, b, d, f, Plus[m, n, p, 3], x]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, m, n, p], x], NegativeIntegerQ[Plus[m, n, p, 2]], NonzeroQ[Plus[m, 1]], Or[SumSimplerQ[m, 1], And[Not[And[NonzeroQ[Plus[n, 1]], SumSimplerQ[n, 1]]], Not[And[NonzeroQ[Plus[p, 1]], SumSimplerQ[p, 1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[Plus[Times[b, c], Times[-1, a, d]], n], Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Times[Plus[m, 1], Power[Plus[Times[b, e], Times[-1, a, f]], Plus[n, 1]], Power[Plus[e, Times[f, x]], Plus[m, 1]]], -1], Hypergeometric2F1[Plus[m, 1], Times[-1, n], Plus[m, 2], Times[-1, Plus[Times[d, e], Times[-1, c, f]], Plus[a, Times[b, x]], Power[Times[Plus[Times[b, c], Times[-1, a, d]], Plus[e, Times[f, x]]], -1]]]], x], And[FreeQ[List[a, b, c, d, e, f, m, p], x], ZeroQ[Plus[m, n, p, 2]], NegativeIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], n], Power[Plus[e, Times[f, x]], Plus[p, 1]], Power[Times[Plus[Times[b, e], Times[-1, a, f]], Plus[m, 1]], -1], Power[Times[Plus[Times[b, e], Times[-1, a, f]], Plus[c, Times[d, x]], Power[Times[Plus[Times[b, c], Times[-1, a, d]], Plus[e, Times[f, x]]], -1]], Times[-1, n]], Hypergeometric2F1[Plus[m, 1], Times[-1, n], Plus[m, 2], Times[-1, Plus[Times[d, e], Times[-1, c, f]], Plus[a, Times[b, x]], Power[Times[Plus[Times[b, c], Times[-1, a, d]], Plus[e, Times[f, x]]], -1]]]], x], And[FreeQ[List[a, b, c, d, e, f, m, n, p], x], ZeroQ[Plus[m, n, p, 2]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[c, n], Power[e, p], Power[Times[b, x], Plus[m, 1]], Power[Times[b, Plus[m, 1]], -1], AppellF1[Plus[m, 1], Times[-1, n], Times[-1, p], Plus[m, 2], Times[-1, d, x, Power[c, -1]], Times[-1, f, x, Power[e, -1]]]], x], And[FreeQ[List[b, c, d, e, f, m, n, p], x], Not[IntegerQ[m]], Not[IntegerQ[n]], PositiveQ[c], Or[IntegerQ[p], PositiveQ[e]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[Plus[c, Times[d, x]], Plus[n, 1]], Power[Times[d, Plus[n, 1], Power[Times[-1, d, Power[Times[b, c], -1]], m], Power[Times[d, Power[Plus[Times[d, e], Times[-1, c, f]], -1]], p]], -1], AppellF1[Plus[n, 1], Times[-1, m], Times[-1, p], Plus[n, 2], Plus[1, Times[d, x, Power[c, -1]]], Times[-1, f, Plus[c, Times[d, x]], Power[Plus[Times[d, e], Times[-1, c, f]], -1]]]], x], And[FreeQ[List[b, c, d, e, f, m, n, p], x], Not[IntegerQ[m]], Not[IntegerQ[n]], PositiveQ[Times[-1, d, Power[Times[b, c], -1]]], Or[IntegerQ[p], PositiveQ[Times[d, Power[Plus[Times[d, e], Times[-1, c, f]], -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[c, IntPart[n]], Power[Plus[c, Times[d, x]], FracPart[n]], Power[Power[Plus[1, Times[d, x, Power[c, -1]]], FracPart[n]], -1]], Int[Times[Power[Times[b, x], m], Power[Plus[1, Times[d, x, Power[c, -1]]], n], Power[Plus[e, Times[f, x]], p]], x], x], And[FreeQ[List[b, c, d, e, f, m, n, p], x], Not[IntegerQ[m]], Not[IntegerQ[n]], Not[PositiveQ[c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[Plus[Times[b, e], Times[-1, a, f]], p], Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Times[Power[b, Plus[p, 1]], Plus[m, 1], Power[Times[b, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], n]], -1], AppellF1[Plus[m, 1], Times[-1, n], Times[-1, p], Plus[m, 2], Times[-1, d, Plus[a, Times[b, x]], Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Times[-1, f, Plus[a, Times[b, x]], Power[Plus[Times[b, e], Times[-1, a, f]], -1]]]], x], And[FreeQ[List[a, b, c, d, e, f, m, n], x], Not[IntegerQ[m]], Not[IntegerQ[n]], IntegerQ[p], PositiveQ[Times[b, Power[Plus[Times[b, c], Times[-1, a, d]], -1]]], Not[And[PositiveQ[Times[d, Power[Plus[Times[d, a], Times[-1, c, b]], -1]]], SimplerQ[Plus[c, Times[d, x]], Plus[a, Times[b, x]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[c, Times[d, x]], FracPart[n]], Power[Times[Power[Times[b, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], IntPart[n]], Power[Times[b, Plus[c, Times[d, x]], Power[Plus[Times[b, c], Times[-1, a, d]], -1]], FracPart[n]]], -1]], Int[Times[Power[Plus[a, Times[b, x]], m], Power[Plus[Times[b, c, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Times[b, d, x, Power[Plus[Times[b, c], Times[-1, a, d]], -1]]], n], Power[Plus[e, Times[f, x]], p]], x], x], And[FreeQ[List[a, b, c, d, e, f, m, n], x], Not[IntegerQ[m]], Not[IntegerQ[n]], IntegerQ[p], Not[PositiveQ[Times[b, Power[Plus[Times[b, c], Times[-1, a, d]], -1]]]], Not[SimplerQ[Plus[c, Times[d, x]], Plus[a, Times[b, x]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Times[b, Plus[m, 1], Power[Times[b, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], n], Power[Times[b, Power[Plus[Times[b, e], Times[-1, a, f]], -1]], p]], -1], AppellF1[Plus[m, 1], Times[-1, n], Times[-1, p], Plus[m, 2], Times[-1, d, Plus[a, Times[b, x]], Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Times[-1, f, Plus[a, Times[b, x]], Power[Plus[Times[b, e], Times[-1, a, f]], -1]]]], x], And[FreeQ[List[a, b, c, d, e, f, m, n, p], x], Not[IntegerQ[m]], Not[IntegerQ[n]], Not[IntegerQ[p]], PositiveQ[Times[b, Power[Plus[Times[b, c], Times[-1, a, d]], -1]]], PositiveQ[Times[b, Power[Plus[Times[b, e], Times[-1, a, f]], -1]]], Not[And[PositiveQ[Times[d, Power[Plus[Times[d, a], Times[-1, c, b]], -1]]], PositiveQ[Times[d, Power[Plus[Times[d, e], Times[-1, c, f]], -1]]], SimplerQ[Plus[c, Times[d, x]], Plus[a, Times[b, x]]]]], Not[And[PositiveQ[Times[f, Power[Plus[Times[f, a], Times[-1, e, b]], -1]]], PositiveQ[Times[f, Power[Plus[Times[f, c], Times[-1, e, d]], -1]]], SimplerQ[Plus[e, Times[f, x]], Plus[a, Times[b, x]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[e, Times[f, x]], FracPart[p]], Power[Times[Power[Times[b, Power[Plus[Times[b, e], Times[-1, a, f]], -1]], IntPart[p]], Power[Times[b, Plus[e, Times[f, x]], Power[Plus[Times[b, e], Times[-1, a, f]], -1]], FracPart[p]]], -1]], Int[Times[Power[Plus[a, Times[b, x]], m], Power[Plus[c, Times[d, x]], n], Power[Plus[Times[b, e, Power[Plus[Times[b, e], Times[-1, a, f]], -1]], Times[b, f, x, Power[Plus[Times[b, e], Times[-1, a, f]], -1]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, f, m, n, p], x], Not[IntegerQ[m]], Not[IntegerQ[n]], Not[IntegerQ[p]], PositiveQ[Times[b, Power[Plus[Times[b, c], Times[-1, a, d]], -1]]], Not[PositiveQ[Times[b, Power[Plus[Times[b, e], Times[-1, a, f]], -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[c, Times[d, x]], FracPart[n]], Power[Times[Power[Times[b, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], IntPart[n]], Power[Times[b, Plus[c, Times[d, x]], Power[Plus[Times[b, c], Times[-1, a, d]], -1]], FracPart[n]]], -1]], Int[Times[Power[Plus[a, Times[b, x]], m], Power[Plus[Times[b, c, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Times[b, d, x, Power[Plus[Times[b, c], Times[-1, a, d]], -1]]], n], Power[Plus[e, Times[f, x]], p]], x], x], And[FreeQ[List[a, b, c, d, e, f, m, n, p], x], Not[IntegerQ[m]], Not[IntegerQ[n]], Not[IntegerQ[p]], Not[PositiveQ[Times[b, Power[Plus[Times[b, c], Times[-1, a, d]], -1]]]], Not[SimplerQ[Plus[c, Times[d, x]], Plus[a, Times[b, x]]]], Not[SimplerQ[Plus[e, Times[f, x]], Plus[a, Times[b, x]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[u, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[u, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[u, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Coefficient[u, x, 1], -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], m], Power[Plus[c, Times[d, x]], n], Power[Plus[e, Times[f, x]], p]], x], x, u], x], And[FreeQ[List[a, b, c, d, e, f, m, n, p], x], LinearQ[u, x], NonzeroQ[Plus[u, Times[-1, x]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[a, Times[b, x]], m], Power[Plus[c, Times[d, x]], n], Plus[e, Times[f, x]], Plus[g, Times[h, x]]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, h], x], Or[PositiveIntegerQ[m], IntegersQ[m, n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[Power[b, 2], d, e, g], Times[-1, Power[a, 2], d, f, h, m], Times[-1, a, b, Plus[Times[d, Plus[Times[f, g], Times[e, h]]], Times[-1, c, f, h, Plus[m, 1]]]], Times[b, f, h, Plus[Times[b, c], Times[-1, a, d]], Plus[m, 1], x]], Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], Plus[n, 1]], Power[Times[Power[b, 2], d, Plus[Times[b, c], Times[-1, a, d]], Plus[m, 1]], -1]], x], Dist[Times[Plus[Times[a, d, f, h, m], Times[b, Plus[Times[d, Plus[Times[f, g], Times[e, h]]], Times[-1, c, f, h, Plus[m, 2]]]]], Power[Times[Power[b, 2], d], -1]], Int[Times[Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], n]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, h, m, n], x], ZeroQ[Plus[m, n, 2]], NonzeroQ[Plus[m, 1]], Not[And[SumSimplerQ[n, 1], Not[SumSimplerQ[m, 1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[Power[b, 2], c, d, e, g, Plus[n, 1]], Times[Power[a, 2], c, d, f, h, Plus[n, 1]], Times[a, b, Plus[Times[Power[d, 2], e, g, Plus[m, 1]], Times[Power[c, 2], f, h, Plus[m, 1]], Times[-1, c, d, Plus[Times[f, g], Times[e, h]], Plus[m, n, 2]]]], Times[Plus[Times[Power[a, 2], Power[d, 2], f, h, Plus[n, 1]], Times[-1, a, b, Power[d, 2], Plus[Times[f, g], Times[e, h]], Plus[n, 1]], Times[Power[b, 2], Plus[Times[Power[c, 2], f, h, Plus[m, 1]], Times[-1, c, d, Plus[Times[f, g], Times[e, h]], Plus[m, 1]], Times[Power[d, 2], e, g, Plus[m, n, 2]]]]], x]], Power[Times[b, d, Power[Plus[Times[b, c], Times[-1, a, d]], 2], Plus[m, 1], Plus[n, 1]], -1], Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], Plus[n, 1]]], x], Times[-1, Dist[Times[Plus[Times[Power[a, 2], Power[d, 2], f, h, Plus[2, Times[3, n], Power[n, 2]]], Times[a, b, d, Plus[n, 1], Plus[Times[2, c, f, h, Plus[m, 1]], Times[-1, d, Plus[Times[f, g], Times[e, h]], Plus[m, n, 3]]]], Times[Power[b, 2], Plus[Times[Power[c, 2], f, h, Plus[2, Times[3, m], Power[m, 2]]], Times[-1, c, d, Plus[Times[f, g], Times[e, h]], Plus[m, 1], Plus[m, n, 3]], Times[Power[d, 2], e, g, Plus[6, Power[m, 2], Times[5, n], Power[n, 2], Times[m, Plus[Times[2, n], 5]]]]]]], Power[Times[b, d, Power[Plus[Times[b, c], Times[-1, a, d]], 2], Plus[m, 1], Plus[n, 1]], -1]], Int[Times[Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], Plus[n, 1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g, h], x], RationalQ[m, n], Less[m, -1], Less[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[Power[b, 3], c, e, g, Plus[m, 2]], Times[-1, Power[a, 3], d, f, h, Plus[n, 2]], Times[-1, Power[a, 2], b, Plus[Times[c, f, h, m], Times[-1, d, Plus[Times[f, g], Times[e, h]], Plus[m, n, 3]]]], Times[-1, a, Power[b, 2], Plus[Times[c, Plus[Times[f, g], Times[e, h]]], Times[d, e, g, Plus[Times[2, m], n, 4]]]], Times[b, Plus[Times[Power[a, 2], d, f, h, Plus[m, Times[-1, n]]], Times[-1, a, b, Plus[Times[2, c, f, h, Plus[m, 1]], Times[-1, d, Plus[Times[f, g], Times[e, h]], Plus[n, 1]]]], Times[Power[b, 2], Plus[Times[c, Plus[Times[f, g], Times[e, h]], Plus[m, 1]], Times[-1, d, e, g, Plus[m, n, 2]]]]], x]], Power[Times[Power[b, 2], Power[Plus[Times[b, c], Times[-1, a, d]], 2], Plus[m, 1], Plus[m, 2]], -1], Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], Plus[n, 1]]], x], Dist[Plus[Times[f, h, Power[Power[b, 2], -1]], Times[-1, Times[d, Plus[m, n, 3], Plus[Times[Power[a, 2], d, f, h, Plus[m, Times[-1, n]]], Times[-1, a, b, Plus[Times[2, c, f, h, Plus[m, 1]], Times[-1, d, Plus[Times[f, g], Times[e, h]], Plus[n, 1]]]], Times[Power[b, 2], Plus[Times[c, Plus[Times[f, g], Times[e, h]], Plus[m, 1]], Times[-1, d, e, g, Plus[m, n, 2]]]]]], Power[Times[Power[b, 2], Power[Plus[Times[b, c], Times[-1, a, d]], 2], Plus[m, 1], Plus[m, 2]], -1]]], Int[Times[Power[Plus[a, Times[b, x]], Plus[m, 2]], Power[Plus[c, Times[d, x]], n]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, h, m, n], x], Or[And[RationalQ[m], Less[m, -2]], And[ZeroQ[Plus[m, n, 3]], Not[And[RationalQ[n], Less[n, -2]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[Power[a, 2], d, f, h, Plus[n, 2]], Times[Power[b, 2], d, e, g, Plus[m, n, 3]], Times[a, b, Plus[Times[c, f, h, Plus[m, 1]], Times[-1, d, Plus[Times[f, g], Times[e, h]], Plus[m, n, 3]]]], Times[b, f, h, Plus[Times[b, c], Times[-1, a, d]], Plus[m, 1], x]], Power[Times[Power[b, 2], d, Plus[Times[b, c], Times[-1, a, d]], Plus[m, 1], Plus[m, n, 3]], -1], Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], Plus[n, 1]]], x], Times[-1, Dist[Times[Plus[Times[Power[a, 2], Power[d, 2], f, h, Plus[n, 1], Plus[n, 2]], Times[a, b, d, Plus[n, 1], Plus[Times[2, c, f, h, Plus[m, 1]], Times[-1, d, Plus[Times[f, g], Times[e, h]], Plus[m, n, 3]]]], Times[Power[b, 2], Plus[Times[Power[c, 2], f, h, Plus[m, 1], Plus[m, 2]], Times[-1, c, d, Plus[Times[f, g], Times[e, h]], Plus[m, 1], Plus[m, n, 3]], Times[Power[d, 2], e, g, Plus[m, n, 2], Plus[m, n, 3]]]]], Power[Times[Power[b, 2], d, Plus[Times[b, c], Times[-1, a, d]], Plus[m, 1], Plus[m, n, 3]], -1]], Int[Times[Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], n]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g, h, m, n], x], Or[And[RationalQ[m], Inequality[-2, LessEqual, m, Less, -1]], SumSimplerQ[m, 1]], NonzeroQ[Plus[m, 1]], NonzeroQ[Plus[m, n, 3]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[a, d, f, h, Plus[n, 2]], Times[b, c, f, h, Plus[m, 2]], Times[-1, b, d, Plus[Times[f, g], Times[e, h]], Plus[m, n, 3]], Times[-1, b, d, f, h, Plus[m, n, 2], x]], Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], Plus[n, 1]], Power[Times[Power[b, 2], Power[d, 2], Plus[m, n, 2], Plus[m, n, 3]], -1]], x]], Dist[Times[Plus[Times[Power[a, 2], Power[d, 2], f, h, Plus[n, 1], Plus[n, 2]], Times[a, b, d, Plus[n, 1], Plus[Times[2, c, f, h, Plus[m, 1]], Times[-1, d, Plus[Times[f, g], Times[e, h]], Plus[m, n, 3]]]], Times[Power[b, 2], Plus[Times[Power[c, 2], f, h, Plus[m, 1], Plus[m, 2]], Times[-1, c, d, Plus[Times[f, g], Times[e, h]], Plus[m, 1], Plus[m, n, 3]], Times[Power[d, 2], e, g, Plus[m, n, 2], Plus[m, n, 3]]]]], Power[Times[Power[b, 2], Power[d, 2], Plus[m, n, 2], Plus[m, n, 3]], -1]], Int[Times[Power[Plus[a, Times[b, x]], m], Power[Plus[c, Times[d, x]], n]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, h, m, n], x], NonzeroQ[Plus[m, n, 2]], NonzeroQ[Plus[m, n, 3]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[a, Times[b, x]], m], Power[Plus[c, Times[d, x]], n], Power[Plus[e, Times[f, x]], p], Plus[g, Times[h, x]]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, h, m], x], Or[IntegersQ[m, n, p], PositiveIntegerQ[n, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[b, g], Times[-1, a, h]], Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], n], Power[Plus[e, Times[f, x]], Plus[p, 1]], Power[Times[b, Plus[Times[b, e], Times[-1, a, f]], Plus[m, 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[b, Plus[Times[b, e], Times[-1, a, f]], Plus[m, 1]], -1]], Int[Times[Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], Plus[n, -1]], Power[Plus[e, Times[f, x]], p], Simp[Plus[Times[b, c, Plus[Times[f, g], Times[-1, e, h]], Plus[m, 1]], Times[Plus[Times[b, g], Times[-1, a, h]], Plus[Times[d, e, n], Times[c, f, Plus[p, 1]]]], Times[d, Plus[Times[b, Plus[Times[f, g], Times[-1, e, h]], Plus[m, 1]], Times[f, Plus[Times[b, g], Times[-1, a, h]], Plus[n, p, 1]]], x]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g, h, p], x], RationalQ[m, n], Less[m, -1], Greater[n, 0], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[b, g], Times[-1, a, h]], Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], n], Power[Plus[e, Times[f, x]], Plus[p, 1]], Power[Times[b, Plus[Times[b, e], Times[-1, a, f]], Plus[m, 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[b, Plus[Times[b, e], Times[-1, a, f]], Plus[m, 1]], -1]], Int[Times[Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], Plus[n, -1]], Power[Plus[e, Times[f, x]], p], Simp[Plus[Times[b, c, Plus[Times[f, g], Times[-1, e, h]], Plus[m, 1]], Times[Plus[Times[b, g], Times[-1, a, h]], Plus[Times[d, e, n], Times[c, f, Plus[p, 1]]]], Times[d, Plus[Times[b, Plus[Times[f, g], Times[-1, e, h]], Plus[m, 1]], Times[f, Plus[Times[b, g], Times[-1, a, h]], Plus[n, p, 1]]], x]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g, h, p], x], RationalQ[m, n], Less[m, -1], Greater[n, 0], IntegersQ[Times[2, m], Times[2, n], Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[b, g], Times[-1, a, h]], Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], Plus[n, 1]], Power[Plus[e, Times[f, x]], Plus[p, 1]], Power[Times[Plus[m, 1], Plus[Times[b, c], Times[-1, a, d]], Plus[Times[b, e], Times[-1, a, f]]], -1]], x], Dist[Times[1, Power[Times[Plus[m, 1], Plus[Times[b, c], Times[-1, a, d]], Plus[Times[b, e], Times[-1, a, f]]], -1]], Int[Times[Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], n], Power[Plus[e, Times[f, x]], p], Simp[Plus[Times[Plus[Times[a, d, f, g], Times[-1, b, Plus[Times[d, e], Times[c, f]], g], Times[b, c, e, h]], Plus[m, 1]], Times[-1, Plus[Times[b, g], Times[-1, a, h]], Plus[Times[d, e, Plus[n, 1]], Times[c, f, Plus[p, 1]]]], Times[-1, d, f, Plus[Times[b, g], Times[-1, a, h]], Plus[m, n, p, 3], x]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, h, n, p], x], RationalQ[m], Less[m, -1], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[b, g], Times[-1, a, h]], Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], Plus[n, 1]], Power[Plus[e, Times[f, x]], Plus[p, 1]], Power[Times[Plus[m, 1], Plus[Times[b, c], Times[-1, a, d]], Plus[Times[b, e], Times[-1, a, f]]], -1]], x], Dist[Times[1, Power[Times[Plus[m, 1], Plus[Times[b, c], Times[-1, a, d]], Plus[Times[b, e], Times[-1, a, f]]], -1]], Int[Times[Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], n], Power[Plus[e, Times[f, x]], p], Simp[Plus[Times[Plus[Times[a, d, f, g], Times[-1, b, Plus[Times[d, e], Times[c, f]], g], Times[b, c, e, h]], Plus[m, 1]], Times[-1, Plus[Times[b, g], Times[-1, a, h]], Plus[Times[d, e, Plus[n, 1]], Times[c, f, Plus[p, 1]]]], Times[-1, d, f, Plus[Times[b, g], Times[-1, a, h]], Plus[m, n, p, 3], x]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, h, n, p], x], RationalQ[m], Less[m, -1], IntegersQ[Times[2, m], Times[2, n], Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[h, Power[Plus[a, Times[b, x]], m], Power[Plus[c, Times[d, x]], Plus[n, 1]], Power[Plus[e, Times[f, x]], Plus[p, 1]], Power[Times[d, f, Plus[m, n, p, 2]], -1]], x], Dist[Times[1, Power[Times[d, f, Plus[m, n, p, 2]], -1]], Int[Times[Power[Plus[a, Times[b, x]], Plus[m, -1]], Power[Plus[c, Times[d, x]], n], Power[Plus[e, Times[f, x]], p], Simp[Plus[Times[a, d, f, g, Plus[m, n, p, 2]], Times[-1, h, Plus[Times[b, c, e, m], Times[a, Plus[Times[d, e, Plus[n, 1]], Times[c, f, Plus[p, 1]]]]]], Times[Plus[Times[b, d, f, g, Plus[m, n, p, 2]], Times[h, Plus[Times[a, d, f, m], Times[-1, b, Plus[Times[d, e, Plus[m, n, 1]], Times[c, f, Plus[m, p, 1]]]]]]], x]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, h, n, p], x], RationalQ[m], Greater[m, 0], NonzeroQ[Plus[m, n, p, 2]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[h, Power[Plus[a, Times[b, x]], m], Power[Plus[c, Times[d, x]], Plus[n, 1]], Power[Plus[e, Times[f, x]], Plus[p, 1]], Power[Times[d, f, Plus[m, n, p, 2]], -1]], x], Dist[Times[1, Power[Times[d, f, Plus[m, n, p, 2]], -1]], Int[Times[Power[Plus[a, Times[b, x]], Plus[m, -1]], Power[Plus[c, Times[d, x]], n], Power[Plus[e, Times[f, x]], p], Simp[Plus[Times[a, d, f, g, Plus[m, n, p, 2]], Times[-1, h, Plus[Times[b, c, e, m], Times[a, Plus[Times[d, e, Plus[n, 1]], Times[c, f, Plus[p, 1]]]]]], Times[Plus[Times[b, d, f, g, Plus[m, n, p, 2]], Times[h, Plus[Times[a, d, f, m], Times[-1, b, Plus[Times[d, e, Plus[m, n, 1]], Times[c, f, Plus[m, p, 1]]]]]]], x]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, h, n, p], x], RationalQ[m], Greater[m, 0], NonzeroQ[Plus[m, n, p, 2]], IntegersQ[Times[2, m], Times[2, n], Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[b, g], Times[-1, a, h]], Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], Plus[n, 1]], Power[Plus[e, Times[f, x]], Plus[p, 1]], Power[Times[Plus[m, 1], Plus[Times[b, c], Times[-1, a, d]], Plus[Times[b, e], Times[-1, a, f]]], -1]], x], Dist[Times[1, Power[Times[Plus[m, 1], Plus[Times[b, c], Times[-1, a, d]], Plus[Times[b, e], Times[-1, a, f]]], -1]], Int[Times[Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Plus[c, Times[d, x]], n], Power[Plus[e, Times[f, x]], p], Simp[Plus[Times[Plus[Times[a, d, f, g], Times[-1, b, Plus[Times[d, e], Times[c, f]], g], Times[b, c, e, h]], Plus[m, 1]], Times[-1, Plus[Times[b, g], Times[-1, a, h]], Plus[Times[d, e, Plus[n, 1]], Times[c, f, Plus[p, 1]]]], Times[-1, d, f, Plus[Times[b, g], Times[-1, a, h]], Plus[m, n, p, 3], x]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, h, n, p], x], NegativeIntegerQ[Plus[m, n, p, 2]], NonzeroQ[Plus[m, 1]], Or[SumSimplerQ[m, 1], And[Not[And[NonzeroQ[Plus[n, 1]], SumSimplerQ[n, 1]]], Not[And[NonzeroQ[Plus[p, 1]], SumSimplerQ[p, 1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Plus[Times[b, g], Times[-1, a, h]], Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Int[Times[Power[Plus[e, Times[f, x]], p], Power[Plus[a, Times[b, x]], -1]], x], x], Times[-1, Dist[Times[Plus[Times[d, g], Times[-1, c, h]], Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Int[Times[Power[Plus[e, Times[f, x]], p], Power[Plus[c, Times[d, x]], -1]], x], x]]], FreeQ[List[a, b, c, d, e, f, g, h], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[h, Power[b, -1]], Int[Times[Power[Plus[c, Times[d, x]], n], Power[Plus[e, Times[f, x]], p]], x], x], Dist[Times[Plus[Times[b, g], Times[-1, a, h]], Power[b, -1]], Int[Times[Power[Plus[c, Times[d, x]], n], Power[Plus[e, Times[f, x]], p], Power[Plus[a, Times[b, x]], -1]], x], x]], FreeQ[List[a, b, c, d, e, f, g, h, n, p], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[h, Power[f, -1]], Int[Times[Sqrt[Plus[e, Times[f, x]]], Power[Times[Sqrt[Plus[a, Times[b, x]]], Sqrt[Plus[c, Times[d, x]]]], -1]], x], x], Dist[Times[Plus[Times[f, g], Times[-1, e, h]], Power[f, -1]], Int[Times[1, Power[Times[Sqrt[Plus[a, Times[b, x]]], Sqrt[Plus[c, Times[d, x]]], Sqrt[Plus[e, Times[f, x]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, h], x], SimplerQ[Plus[a, Times[b, x]], Plus[e, Times[f, x]]], SimplerQ[Plus[c, Times[d, x]], Plus[e, Times[f, x]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], 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RuleDelayed[HoldPattern[Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Plus[Times[b, Power[2, -1]], Times[c, x]], Power[Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], -1]], Int[Times[1, Power[Plus[Times[b, Power[2, -1]], Times[c, x]], -1]], x], x], And[FreeQ[List[a, b, c], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Plus[b, Times[2, c, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[2, c, Plus[Times[2, p], 1]], -1]], x], And[FreeQ[List[a, b, c, p], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], 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Condition[Int[ExpandIntegrand[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], x], x], And[FreeQ[List[a, b, c], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[p], Not[PerfectSquareQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[b, Times[2, c, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[2, c, Plus[Times[2, p], 1]], -1]], x], Times[-1, Dist[Times[p, Plus[Power[b, 2], Times[-1, 4, a, c]], Power[Times[2, c, Plus[Times[2, p], 1]], -1]], Int[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, -1]], x], x]]], And[FreeQ[List[a, b, c], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], RationalQ[p], Greater[p, 0], IntegerQ[Times[4, p]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[-2, Plus[b, Times[2, c, x]], Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]], -1]], x], And[FreeQ[List[a, b, c], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[b, Times[2, c, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], x], Times[-1, Dist[Times[2, c, Plus[Times[2, p], 3], Power[Times[Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], Int[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], x], x]]], And[FreeQ[List[a, b, c], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], RationalQ[p], Less[p, -1], Unequal[p, Times[-3, Power[2, -1]]], IntegerQ[Times[4, p]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Plus[Dist[Times[c, Power[q, -1]], Int[Times[1, Power[Simp[Plus[Times[b, Power[2, -1]], Times[-1, q, Power[2, -1]], Times[c, x]], x], -1]], x], x], Times[-1, Dist[Times[c, Power[q, -1]], Int[Times[1, Power[Simp[Plus[Times[b, Power[2, -1]], Times[q, Power[2, -1]], Times[c, x]], x], -1]], x], x]]]], And[FreeQ[List[a, b, c], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PosQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PerfectSquareQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Plus[1, Times[-1, 4, Simplify[Times[a, c, Power[Power[b, 2], -1]]]]]]], Condition[Dist[Times[-2, Power[b, -1]], Subst[Int[Times[1, Power[Plus[q, Times[-1, Power[x, 2]]], -1]], x], x, Plus[1, Times[2, c, x, Power[b, -1]]]], x], And[RationalQ[q], Or[EqQ[Power[q, 2], 1], Not[RationalQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]]]], And[FreeQ[List[a, b, c], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Pattern[x, Blank[Symbol]]]], Condition[Dist[-2, Subst[Int[Times[1, 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Rational[-1, 2]], Pattern[x, Blank[Symbol]]]], Condition[Dist[2, Subst[Int[Times[1, Power[Plus[1, Times[-1, c, Power[x, 2]]], -1]], x], x, Times[x, Power[Sqrt[Plus[Times[b, x], Times[c, Power[x, 2]]]], -1]]], x], FreeQ[List[b, c], x]]], RuleDelayed[HoldPattern[Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]]], Condition[Dist[2, Subst[Int[Times[1, Power[Plus[Times[4, c], Times[-1, Power[x, 2]]], -1]], x], x, Times[Plus[b, Times[2, c, x]], Power[Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], -1]]], x], And[FreeQ[List[a, b, c], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[Times[b, x], Times[c, Power[x, 2]]], p], Power[Power[Times[-1, c, Plus[Times[b, x], Times[c, Power[x, 2]]], Power[Power[b, 2], -1]], p], -1]], Int[Power[Plus[Times[-1, c, x, Power[b, -1]], Times[-1, Power[c, 2], Power[x, 2], Power[Power[b, 2], -1]]], p], x], x], And[FreeQ[List[b, c], x], RationalQ[p], LessEqual[3, Denominator[p], 4]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[d, Denominator[p]]], Condition[Dist[Times[d, Sqrt[Power[Plus[b, Times[2, c, x]], 2]], Power[Plus[b, Times[2, c, x]], -1]], Subst[Int[Times[Power[x, Plus[Times[d, Plus[p, 1]], -1]], Power[Sqrt[Plus[Power[b, 2], Times[-1, 4, a, c], Times[4, c, Power[x, d]]]], -1]], x], x, Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Times[1, Power[d, -1]]]], x], LessEqual[3, d, 4]]], And[FreeQ[List[a, b, c], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], RationalQ[p]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Times[-1, Simp[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[q, Plus[p, 1], Power[Times[Plus[q, Times[-1, b], Times[-1, 2, c, x]], Power[Times[2, q], -1]], Plus[p, 1]]], -1], Hypergeometric2F1[Times[-1, p], Plus[p, 1], Plus[p, 2], Times[Plus[b, q, Times[2, c, x]], Power[Times[2, q], -1]]]], x]]], And[FreeQ[List[a, b, c, p], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], Not[IntegerQ[Times[4, p]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[u, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Coefficient[u, x, 1], -1]], Subst[Int[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], x], x, u], x], And[FreeQ[List[a, b, c, p], x], LinearQ[u, x], NonzeroQ[Plus[u, Times[-1, x]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[e, m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, Times[Plus[m, 1], Power[2, -1]]]], Power[Times[Power[c, Times[Plus[m, 1], Power[2, -1]]], Plus[m, Times[2, p], 1]], -1]], x], And[FreeQ[List[a, b, c, d, e, m, p], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], IntegerQ[Times[Plus[m, 1], Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Log[RemoveContent[Plus[d, Times[e, x]], x]], Power[e, -1]], x], And[FreeQ[List[a, b, c, d, e, m, p], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], ZeroQ[Plus[m, Times[2, p], 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[e, Plus[m, Times[2, p], 1]], -1]], x], And[FreeQ[List[a, b, c, d, e, m, p], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], NonzeroQ[Plus[m, Times[2, p], 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Plus[b, Times[2, c, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[Plus[m, 1], Plus[Times[2, c, d], Times[-1, b, e]]], -1]], x]], And[FreeQ[List[a, b, c, d, e, m, p], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], ZeroQ[Plus[m, Times[2, p], 2]], NonzeroQ[Plus[m, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], Power[Plus[b, Times[2, c, x]], -1]], Int[Times[Plus[b, Times[2, c, x]], Power[Power[Plus[d, Times[e, x]], 2], -1]], x], x], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], Power[Times[e, Plus[m, 2]], -1]], x], Times[-1, Dist[Times[Plus[Times[2, c, d], Times[-1, b, e]], Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], Power[Times[e, Plus[m, 2], Plus[b, Times[2, c, x]]], -1]], Int[Power[Plus[d, Times[e, x]], m], x], x]]], And[FreeQ[List[a, b, c, d, e, m], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], NonzeroQ[Plus[m, 2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[-4, c, e, Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], Power[Times[Power[Plus[Times[2, c, d], Times[-1, b, e]], 2], Plus[d, Times[e, x]]], -1]], x], Dist[Times[2, c, Power[Plus[Times[2, c, d], Times[-1, b, e]], -1]], Int[Times[1, Power[Times[Plus[d, Times[e, x]], Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[-2, c, e, Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[Plus[Times[m, p], -1], Power[Plus[Times[2, c, d], Times[-1, b, e]], 2]], -1]], x], Times[-1, Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Plus[b, Times[2, c, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[Plus[m, 2], Plus[Times[2, c, d], Times[-1, b, e]]], -1]], x]]], And[FreeQ[List[a, b, c, d, e, m, p], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], ZeroQ[Plus[m, Times[2, p], 3]], NonzeroQ[Plus[m, 2]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[e, Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[2, c, Plus[p, 1]], -1]], x], Dist[Times[Plus[Times[2, c, d], Times[-1, b, e]], Power[Times[2, c], -1]], Int[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], x], x]], And[FreeQ[List[a, b, c, d, e, p], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], NonzeroQ[Plus[p, Times[3, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[e, Plus[m, 1]], -1]], x], Plus[Dist[Times[p, Plus[Times[2, p], -1], Plus[Times[2, c, d], Times[-1, b, e]], Power[Times[Power[e, 2], Plus[m, 1], Plus[m, Times[2, p], 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, -1]]], x], x], Times[-1, Simp[Times[p, Power[Plus[d, Times[e, x]], Plus[m, 2]], Plus[b, Times[2, c, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, -1]], Power[Times[Power[e, 2], Plus[m, 1], Plus[m, Times[2, p], 1]], -1]], x]]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], RationalQ[m, p], Greater[p, 1], Inequality[-2, LessEqual, m, Less, -1], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, 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NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], RationalQ[m, p], Greater[p, 1], Less[m, -2], IntegerQ[Times[2, p]], Not[And[NegativeIntegerQ[Plus[m, Times[2, p], 3]], Greater[Plus[m, Times[3, p], 3], 0]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[e, Plus[m, Times[2, p], 1]], -1]], x], Plus[Dist[Times[p, Plus[Times[2, p], -1], Power[Plus[Times[2, c, d], Times[-1, b, e]], 2], Power[Times[2, c, Power[e, 2], Plus[m, Times[2, p]], Plus[m, Times[2, p], 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, -1]]], x], x], Times[-1, Simp[Times[p, Plus[Times[2, c, d], Times[-1, b, e]], Power[Plus[d, Times[e, x]], Plus[m, 1]], Plus[b, Times[2, c, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, -1]], Power[Times[2, c, Power[e, 2], Plus[m, Times[2, p]], Plus[m, Times[2, p], 1]], -1]], x]]]], And[FreeQ[List[a, b, c, d, e, m], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], RationalQ[p], Greater[p, 0], NonzeroQ[Plus[m, Times[2, p]]], NonzeroQ[Plus[m, Times[2, p], 1]], Not[And[NegativeIntegerQ[Plus[m, Times[2, p], 3]], Greater[Plus[m, Times[3, p], 3], 0]]], Not[And[RationalQ[m], Less[m, -2]]], Not[And[IntegerQ[m], Less[0, m, Times[2, p]]]], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[e, Plus[m, Times[2, p], 2], Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[Plus[p, 1], Plus[Times[2, p], 1], Plus[Times[2, c, d], Times[-1, b, e]]], -1]], x]], Plus[Dist[Times[Power[e, 2], m, Plus[m, Times[2, p], 2], Power[Times[Plus[p, 1], Plus[Times[2, p], 1], Plus[Times[2, c, d], Times[-1, b, e]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]]], x], x], Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Plus[b, Times[2, c, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[Plus[Times[2, p], 1], Plus[Times[2, c, d], Times[-1, b, e]]], -1]], x]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], RationalQ[m, p], Less[p, -1], Inequality[0, Less, m, LessEqual, 1], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[e, m, Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[2, c, Plus[p, 1], Plus[Times[2, p], 1]], -1]], x]], Plus[Dist[Times[Power[e, 2], m, Plus[m, -1], Power[Times[2, c, Plus[p, 1], Plus[Times[2, p], 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -2]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]]], x], x], Simp[Times[Power[Plus[d, Times[e, x]], m], Plus[b, Times[2, c, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[2, c, Plus[Times[2, p], 1]], -1]], x]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], RationalQ[m, p], Less[p, -1], Greater[m, 1], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[-2, c, e, Plus[m, Times[2, p], 2], Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[Plus[p, 1], Plus[Times[2, p], 1], Power[Plus[Times[2, c, d], Times[-1, b, e]], 2]], -1]], x], Plus[Dist[Times[2, c, Power[e, 2], Plus[m, Times[2, p], 2], Plus[m, Times[2, p], 3], Power[Times[Plus[p, 1], Plus[Times[2, p], 1], Power[Plus[Times[2, c, d], Times[-1, b, e]], 2]], -1]], Int[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]]], x], x], Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Plus[b, Times[2, c, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[Plus[Times[2, p], 1], Plus[Times[2, c, d], Times[-1, b, e]]], -1]], x]]], And[FreeQ[List[a, b, c, d, e, m], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], RationalQ[m, p], Less[p, -1], NonzeroQ[Plus[m, p, 1]], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], m], Plus[b, Times[2, c, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[2, c, Plus[m, Times[2, p], 1]], -1]], x], Dist[Times[m, Plus[Times[2, c, d], Times[-1, b, e]], Power[Times[2, c, Plus[m, Times[2, p], 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x]], And[FreeQ[List[a, b, c, d, e, p], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], RationalQ[m], Greater[m, 0], NonzeroQ[Plus[m, Times[2, p], 1]], Or[Not[RationalQ[p]], Inequality[-1, LessEqual, p, Less, 0], And[IntegerQ[m], Less[0, m, Times[2, p]]], And[Equal[m, Times[1, Power[2, -1]]], Less[p, 0]]], Or[IntegerQ[m], IntegerQ[Times[2, p]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Plus[b, Times[2, c, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[Plus[m, 1], Plus[Times[2, c, d], Times[-1, b, e]]], -1]], x]], Dist[Times[2, c, Plus[m, Times[2, p], 2], Power[Times[Plus[m, 1], Plus[Times[2, c, d], Times[-1, b, e]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x]], And[FreeQ[List[a, b, c, d, e, p], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], RationalQ[m], Less[m, -1], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], FracPart[p]], Power[Times[Power[c, IntPart[p]], Power[Plus[Times[b, Power[2, -1]], Times[c, x]], Times[2, FracPart[p]]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[Times[b, Power[2, -1]], Times[c, x]], Times[2, p]]], x], x], And[FreeQ[List[a, b, c, d, e, m, p], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], Not[IntegerQ[p]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[d, Times[e, x]], Plus[m, p]], Power[Plus[Times[a, Power[d, -1]], Times[c, Power[e, -1], x]], p]], x], And[FreeQ[List[a, b, c, d, e, m], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[d, Times[e, x]], Plus[m, p]], Power[Plus[Times[a, Power[d, -1]], Times[c, Power[e, -1], x]], p]], x], And[FreeQ[List[a, c, d, e, m, p], x], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Or[IntegerQ[p], And[PositiveQ[a], PositiveQ[d], IntegerQ[Plus[m, p]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[e, Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, Plus[p, 1]], -1]], x], And[FreeQ[List[a, b, c, d, e, m, p], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], ZeroQ[Plus[m, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[e, Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, Plus[p, 1]], -1]], x], And[FreeQ[List[a, c, d, e, m, p], x], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], ZeroQ[Plus[m, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[e, Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[Plus[p, 1], Plus[Times[2, c, d], Times[-1, b, e]]], -1]], x], And[FreeQ[List[a, b, c, d, e, m, p], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], ZeroQ[Plus[m, Times[2, p], 2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[e, Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[2, c, d, Plus[p, 1]], -1]], x], And[FreeQ[List[a, c, d, e, m, p], x], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], ZeroQ[Plus[m, Times[2, p], 2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], 2], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[e, Plus[d, Times[e, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, Plus[p, 1]], -1]], x], Times[-1, Dist[Times[Power[e, 2], Plus[p, 2], Power[Times[c, Plus[p, 1]], -1]], Int[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], x], x]]], And[FreeQ[List[a, b, c, d, e, p], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], RationalQ[p], Less[p, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[e, Plus[d, Times[e, x]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, Plus[p, 1]], -1]], x], Times[-1, Dist[Times[Power[e, 2], Plus[p, 2], Power[Times[c, Plus[p, 1]], -1]], Int[Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], x], x]]], And[FreeQ[List[a, c, d, e, p], x], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], RationalQ[p], Less[p, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[m, p]], Power[Power[Plus[Times[a, Power[d, -1]], Times[c, x, Power[e, -1]]], m], -1]], x], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], IntegerQ[m], RationalQ[p], Or[Less[0, Times[-1, m], p], Less[p, Times[-1, m], 0]], Unequal[m, 2], Unequal[m, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[d, Times[2, m]], Power[Power[a, m], -1]], Int[Times[Power[Plus[a, Times[c, Power[x, 2]]], Plus[m, p]], Power[Power[Plus[d, Times[-1, e, x]], m], -1]], x], x], And[FreeQ[List[a, c, d, e, m, p], x], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], IntegerQ[m], RationalQ[p], Or[Less[0, Times[-1, m], p], Less[p, Times[-1, m], 0]], Unequal[m, 2], Unequal[m, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[e, Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, Plus[m, Times[2, p], 1]], -1]], x], Dist[Times[Simplify[Plus[m, p]], Plus[Times[2, c, d], Times[-1, b, e]], Power[Times[c, Plus[m, Times[2, p], 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x]], And[FreeQ[List[a, b, c, d, e, m, p], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, 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RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[e, Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[Plus[m, p, 1], Plus[Times[2, c, d], Times[-1, b, e]]], -1]], x]], Dist[Times[c, Simplify[Plus[m, Times[2, p], 2]], Power[Times[Plus[m, p, 1], Plus[Times[2, c, d], Times[-1, b, e]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x]], And[FreeQ[List[a, b, c, d, e, m, p], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], NegativeIntegerQ[Simplify[Plus[m, Times[2, p], 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[e, Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[2, c, d, Plus[m, p, 1]], -1]], x]], Dist[Times[Simplify[Plus[m, Times[2, p], 2]], Power[Times[2, d, Plus[m, p, 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x]], And[FreeQ[List[a, c, d, e, m, p], x], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], NegativeIntegerQ[Simplify[Plus[m, Times[2, p], 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[2, e], Subst[Int[Times[1, Power[Plus[Times[2, c, d], Times[-1, b, e], Times[Power[e, 2], Power[x, 2]]], -1]], x], x, Times[Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], Power[Sqrt[Plus[d, Times[e, x]]], -1]]], x], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], 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1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]]], x], x]], And[FreeQ[List[a, c, d, e], x], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], RationalQ[m, p], Less[p, -1], Inequality[0, Less, m, LessEqual, 1], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[e, Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, Plus[p, 1]], -1]], x], Times[-1, Dist[Times[Power[e, 2], Plus[m, p], Power[Times[c, Plus[p, 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -2]], 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RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[e, x], m], Power[Plus[Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[Power[x, Plus[m, p]], Power[Plus[b, Times[c, x]], p]], -1]], Int[Times[Power[x, Plus[m, p]], Power[Plus[b, Times[c, x]], p]], x], x], And[FreeQ[List[b, c, e, m], x], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[d, Times[e, x]], Plus[m, p]], Power[Plus[Times[a, Power[d, -1]], Times[c, Power[e, -1], x]], p]], x], And[FreeQ[List[a, c, d, e, m, p], x], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], PositiveQ[a], PositiveQ[d]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], FracPart[p]], Power[Times[Power[Plus[d, Times[e, x]], FracPart[p]], Power[Plus[Times[a, Power[d, -1]], Times[Times[c, x], Power[e, -1]]], FracPart[p]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, p]], Power[Plus[Times[a, Power[d, -1]], Times[c, Power[e, -1], x]], p]], x], x], And[FreeQ[List[a, b, c, d, e, m], x], 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Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[-4, b, c, Power[Times[d, Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], Int[Times[1, Power[Plus[b, Times[2, c, x]], -1]], x], x], Dist[Times[Power[b, 2], Power[Times[Power[d, 2], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], Int[Times[Plus[d, Times[e, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[2, c, d], Times[-1, b, e]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[2, c, Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[e, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], x], And[FreeQ[List[a, b, c, d, e, m, p], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], ZeroQ[Plus[m, Times[2, p], 3]], NonzeroQ[Plus[p, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, m], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], PositiveIntegerQ[p], Not[And[ZeroQ[Plus[m, -3]], Unequal[p, 1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[e, Plus[m, 1]], -1]], x], Times[-1, Dist[Times[b, p, Power[Times[d, e, Plus[m, 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 2]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, -1]]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], NonzeroQ[Plus[m, Times[2, p], 3]], RationalQ[m, p], Greater[p, 0], Less[m, -1], Not[And[EvenQ[m], Less[Plus[m, Times[2, p], 3], 0]]], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[e, Plus[m, Times[2, p], 1]], -1]], x], Times[-1, Dist[Times[d, p, Plus[Power[b, 2], Times[-1, 4, a, c]], Power[Times[b, e, Plus[m, Times[2, p], 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, -1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, m], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], NonzeroQ[Plus[m, Times[2, p], 3]], RationalQ[p], Greater[p, 0], Not[And[RationalQ[m], Less[m, -1]]], Not[And[PositiveIntegerQ[Times[Plus[m, -1], Power[2, -1]]], Or[Not[IntegerQ[p]], Less[m, Times[2, p]]]]], RationalQ[m], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[d, Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[b, Plus[p, 1]], -1]], x], Times[-1, Dist[Times[d, e, Plus[m, -1], Power[Times[b, Plus[p, 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -2]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], NonzeroQ[Plus[m, Times[2, p], 3]], RationalQ[m, p], Less[p, -1], Greater[m, 1], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[2, c, Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[e, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], x], Times[-1, Dist[Times[2, c, e, Plus[m, Times[2, p], 3], Power[Times[e, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, m], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], NonzeroQ[Plus[m, Times[2, p], 3]], RationalQ[p], Less[p, -1], Not[And[RationalQ[m], Greater[m, 1]]], RationalQ[m], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[4, c], Subst[Int[Times[1, Power[Plus[Times[Power[b, 2], e], Times[-1, 4, 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ZeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], NegativeQ[Times[c, Power[Plus[Power[b, 2], Times[-1, 4, a, c]], -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[4, Power[e, -1], Sqrt[Times[-1, c, Power[Plus[Power[b, 2], Times[-1, 4, a, c]], -1]]]], Subst[Int[Times[Power[x, 2], Power[Sqrt[Simp[Plus[1, Times[-1, Power[b, 2], Power[x, 4], Power[Times[Power[d, 2], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]]], x]], -1]], x], x, Sqrt[Plus[d, Times[e, x]]]], x], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], NegativeQ[Times[c, Power[Plus[Power[b, 2], Times[-1, 4, a, c]], -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Times[-1, c, Plus[a, Times[b, x], Times[c, Power[x, 2]]], Power[Plus[Power[b, 2], Times[-1, 4, a, c]], -1]]], Power[Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], m], Power[Sqrt[Plus[Times[-1, a, c, Power[Plus[Power[b, 2], Times[-1, 4, a, c]], -1]], Times[-1, b, c, x, Power[Plus[Power[b, 2], Times[-1, 4, a, c]], -1]], Times[-1, Power[c, 2], Power[x, 2], Power[Plus[Power[b, 2], Times[-1, 4, a, c]], -1]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], EqQ[Power[m, 2], Times[1, Power[4, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[2, d, Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[b, Plus[m, Times[2, p], 1]], -1]], x], Dist[Times[Power[d, 2], Plus[m, -1], Plus[Power[b, 2], Times[-1, 4, a, c]], Power[Times[Power[b, 2], Plus[m, Times[2, p], 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -2]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x]], And[FreeQ[List[a, b, c, d, e, p], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], NonzeroQ[Plus[m, Times[2, p], 3]], RationalQ[m], Greater[m, 1], NonzeroQ[Plus[m, Times[2, p], 1]], Or[IntegerQ[Times[2, p]], And[IntegerQ[m], RationalQ[p]], OddQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[-2, b, d, Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[Power[d, 2], Plus[m, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], x], Dist[Times[Power[b, 2], Plus[m, Times[2, p], 3], Power[Times[Power[d, 2], Plus[m, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 2]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x]], And[FreeQ[List[a, b, c, d, e, p], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], NonzeroQ[Plus[m, Times[2, p], 3]], RationalQ[m], Less[m, -1], Or[IntegerQ[Times[2, p]], And[IntegerQ[m], RationalQ[p]], IntegerQ[Times[Plus[m, Times[2, p], 3], Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[e, -1]], Subst[Int[Times[Power[x, m], Power[Plus[a, Times[-1, Power[b, 2], Power[Times[4, c], -1]], Times[Times[c, Power[x, 2]], Power[Power[e, 2], -1]]], p]], x], x, Plus[d, Times[e, x]]], x], And[FreeQ[List[a, b, c, d, e, m, p], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[2, c, d], Times[-1, b, e]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, m], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], PositiveIntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x], And[FreeQ[List[a, c, d, e, m], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], PositiveIntegerQ[p], Not[And[ZeroQ[Plus[m, -1]], Greater[p, 1]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Plus[Dist[Times[Plus[Times[c, d], Times[-1, e, Plus[Times[b, Power[2, -1]], Times[-1, q, Power[2, -1]]]]], Power[q, -1]], Int[Times[1, Power[Plus[Times[b, Power[2, -1]], Times[-1, q, Power[2, -1]], Times[c, x]], -1]], x], x], Times[-1, Dist[Times[Plus[Times[c, d], Times[-1, e, Plus[Times[b, Power[2, -1]], Times[q, Power[2, -1]]]]], Power[q, -1]], Int[Times[1, Power[Plus[Times[b, Power[2, -1]], Times[q, Power[2, -1]], Times[c, x]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], NiceSqrtQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Times[-1, a, c], 2]]], Plus[Dist[Plus[Times[e, Power[2, -1]], Times[c, d, Power[Times[2, q], -1]]], Int[Times[1, Power[Plus[Times[-1, q], Times[c, x]], -1]], x], x], Dist[Plus[Times[e, Power[2, -1]], Times[-1, c, d, Power[Times[2, q], -1]]], Int[Times[1, Power[Plus[q, Times[c, x]], -1]], x], x]]], And[FreeQ[List[a, c, d, e], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], NiceSqrtQ[Times[-1, a, c]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Plus[Times[2, c, d], Times[-1, b, e]], Power[Times[2, c], -1]], Int[Times[1, Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], -1]], x], x], Dist[Times[e, Power[Times[2, c], -1]], Int[Times[Plus[b, Times[2, c, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], Not[NiceSqrtQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[d, Int[Times[1, Power[Plus[a, Times[c, Power[x, 2]]], -1]], x], x], Dist[e, Int[Times[x, Power[Plus[a, Times[c, Power[x, 2]]], -1]], x], x]], And[FreeQ[List[a, c, d, e], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[NiceSqrtQ[Times[-1, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[2, e], Subst[Int[Times[Power[x, 2], Power[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]], Times[-1, Plus[Times[2, c, d], Times[-1, b, e]], Power[x, 2]], Times[c, Power[x, 4]]], -1]], x], x, Sqrt[Plus[d, Times[e, x]]]], x], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[2, e], Subst[Int[Times[Power[x, 2], Power[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]], Times[-1, 2, c, d, Power[x, 2]], Times[c, Power[x, 4]]], -1]], x], x, Sqrt[Plus[d, Times[e, x]]]], x], And[FreeQ[List[a, c, d, e], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Int[PolynomialDivide[Power[Plus[d, Times[e, x]], m], Plus[a, Times[b, x], Times[c, Power[x, 2]]], x], x], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], IntegerQ[m], Greater[m, 1], Or[NonzeroQ[d], Greater[m, 2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Int[PolynomialDivide[Power[Plus[d, Times[e, x]], m], Plus[a, Times[c, Power[x, 2]]], x], x], And[FreeQ[List[a, c, d, e], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], IntegerQ[m], Greater[m, 1], Or[NonzeroQ[d], Greater[m, 2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[e, Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Times[c, Plus[m, -1]], -1]], x], Dist[Times[1, Power[c, -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -2]], Simp[Plus[Times[c, Power[d, 2]], Times[-1, a, Power[e, 2]], Times[e, Plus[Times[2, c, d], Times[-1, b, e]], x]], x], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], RationalQ[m], Greater[m, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[e, Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Times[c, Plus[m, -1]], -1]], x], Dist[Times[1, Power[c, -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -2]], Simp[Plus[Times[c, Power[d, 2]], Times[-1, a, Power[e, 2]], Times[2, c, d, e, x]], x], Power[Plus[a, Times[c, Power[x, 2]]], -1]], x], x]], And[FreeQ[List[a, c, d, e], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], RationalQ[m], Greater[m, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Power[e, 2], Power[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]], -1]], Int[Times[1, Power[Plus[d, Times[e, x]], -1]], x], x], Dist[Times[1, Power[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]], -1]], Int[Times[Plus[Times[c, d], Times[-1, b, e], Times[-1, c, e, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Power[e, 2], Power[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], -1]], Int[Times[1, Power[Plus[d, Times[e, x]], -1]], x], x], Dist[Times[1, Power[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], -1]], Int[Times[Plus[Times[c, d], Times[-1, c, e, x]], Power[Plus[a, Times[c, Power[x, 2]]], -1]], x], x]], And[FreeQ[List[a, c, d, e], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[2, e], Subst[Int[Times[1, Power[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]], Times[-1, Plus[Times[2, c, d], Times[-1, b, e]], Power[x, 2]], Times[c, Power[x, 4]]], -1]], x], x, Sqrt[Plus[d, Times[e, x]]]], x], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[2, e], Subst[Int[Times[1, Power[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]], Times[-1, 2, c, d, Power[x, 2]], Times[c, Power[x, 4]]], -1]], x], x, Sqrt[Plus[d, Times[e, x]]]], x], And[FreeQ[List[a, c, d, e], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[e, Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Times[Plus[m, 1], Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], -1]], x], Dist[Times[1, Power[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Simp[Plus[Times[c, d], Times[-1, b, e], Times[-1, c, e, x]], x], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, m], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], RationalQ[m], Less[m, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[e, Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Times[Plus[m, 1], Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], -1]], x], Dist[Times[c, Power[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Plus[d, Times[-1, e, x]], Power[Plus[a, Times[c, Power[x, 2]]], -1]], x], x]], And[FreeQ[List[a, c, d, e, m], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], RationalQ[m], Less[m, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Power[Plus[d, Times[e, x]], m], Times[1, Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, m], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Power[Plus[d, Times[e, x]], m], Times[1, Power[Plus[a, Times[c, Power[x, 2]]], -1]], x], x], And[FreeQ[List[a, c, d, e, m], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[-2, Plus[Times[b, d], Times[-1, 2, a, e], Times[Plus[Times[2, c, d], Times[-1, b, e]], x]], Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]], -1]], x], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Plus[Times[-1, a, e], Times[c, d, x]], Power[Times[a, c, Sqrt[Plus[a, Times[c, Power[x, 2]]]]], -1]], x], And[FreeQ[List[a, c, d, e], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, 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RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[a, e], Times[-1, c, d, x]], Power[Times[2, a, c, Plus[p, 1]], -1], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]]], x], Dist[Times[d, Plus[Times[2, p], 3], Power[Times[2, a, Plus[p, 1]], -1]], Int[Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], x], x]], And[FreeQ[List[a, c, d, e], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], RationalQ[p], Less[p, -1], Unequal[p, Times[-3, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[e, Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[2, c, Plus[p, 1]], -1]], x], Dist[Times[Plus[Times[2, c, d], Times[-1, b, e]], Power[Times[2, c], -1]], Int[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], x], x]], And[FreeQ[List[a, b, c, d, e, p], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], Not[And[RationalQ[p], LessEqual[p, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[e, Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[2, c, 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Times[a, e]]], ZeroQ[Plus[Times[c, d], Times[b, e]]], PositiveIntegerQ[Plus[m, Times[-1, p], 1]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[d, Times[e, x]], m], Power[Times[Sqrt[Times[b, x]], Sqrt[Plus[1, Times[c, Power[b, -1], x]]]], -1]], x], And[FreeQ[List[b, c, d, e], x], NonzeroQ[Plus[Times[c, d], Times[-1, b, e]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], RationalQ[m], Equal[Power[m, 2], Times[1, Power[4, -1]]], NegativeQ[c], RationalQ[b]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[x], Sqrt[Plus[b, Times[c, x]]], Power[Sqrt[Plus[Times[b, x], Times[c, Power[x, 2]]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], m], Power[Times[Sqrt[x], Sqrt[Plus[b, Times[c, x]]]], -1]], x], x], And[FreeQ[List[b, c, d, e], x], NonzeroQ[Plus[Times[c, d], Times[-1, b, e]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], RationalQ[m], Equal[Power[m, 2], Times[1, Power[4, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[2, Subst[Int[Times[Power[x, Plus[Times[2, m], 1]], Power[Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]], -1]], x], x, Sqrt[x]], x], And[FreeQ[List[a, b, c], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Power[m, 2], Times[-1, 1, Power[4, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[e, x], m], Power[Power[x, m], -1]], Int[Times[Power[x, m], Power[Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], -1]], x], x], And[FreeQ[List[a, b, c, e], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Power[m, 2], Times[-1, 1, Power[4, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[2, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2], Power[Plus[d, Times[e, x]], m], Sqrt[Times[-1, c, Plus[a, Times[b, x], Times[c, Power[x, 2]]], Power[Plus[Power[b, 2], Times[-1, 4, a, c]], -1]]], Power[Times[c, Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], Power[Times[2, c, Plus[d, Times[e, x]], Power[Plus[Times[2, c, d], Times[-1, b, e], Times[-1, e, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], -1]], m]], -1]], Subst[Int[Times[Power[Plus[1, Times[2, e, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2], Power[x, 2], Power[Plus[Times[2, c, d], Times[-1, b, e], Times[-1, e, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], -1]]], m], Power[Sqrt[Plus[1, Times[-1, Power[x, 2]]]], -1]], x], x, Sqrt[Times[Plus[b, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2], Times[2, c, x]], Power[Times[2, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]], -1]]]], x], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], ZeroQ[Plus[Power[m, 2], Times[-1, 1, Power[4, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[2, a, Rt[Times[-1, c, Power[a, -1]], 2], Power[Plus[d, Times[e, x]], m], Sqrt[Plus[1, Times[c, Power[x, 2], Power[a, -1]]]], Power[Times[c, Sqrt[Plus[a, Times[c, Power[x, 2]]]], Power[Times[c, Plus[d, Times[e, x]], Power[Plus[Times[c, d], Times[-1, a, e, Rt[Times[-1, c, Power[a, -1]], 2]]], -1]], m]], -1]], Subst[Int[Times[Power[Plus[1, Times[2, a, e, Rt[Times[-1, c, Power[a, -1]], 2], Power[x, 2], Power[Plus[Times[c, d], Times[-1, a, e, Rt[Times[-1, c, Power[a, -1]], 2]]], -1]]], m], Power[Sqrt[Plus[1, Times[-1, Power[x, 2]]]], -1]], x], x, Sqrt[Times[Plus[1, Times[-1, Rt[Times[-1, c, Power[a, -1]], 2], x]], Power[2, -1]]]], x], And[FreeQ[List[a, c, d, e], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], ZeroQ[Plus[Power[m, 2], Times[-1, 1, Power[4, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Plus[Times[d, b], Times[-1, 2, a, e], Times[Plus[Times[2, c, d], Times[-1, b, e]], x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[2, Plus[m, 1], Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], -1]], x]], Dist[Times[p, Plus[Power[b, 2], Times[-1, 4, a, c]], Power[Times[2, Plus[m, 1], Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 2]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, -1]]], x], x]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], RationalQ[m, p], Equal[Plus[m, Times[2, p], 2], 0], Greater[p, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Plus[Times[-2, a, e], Times[Times[2, c, d], x]], Power[Plus[a, Times[c, Power[x, 2]]], p], Power[Times[2, Plus[m, 1], Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], -1]], x]], Times[-1, Dist[Times[4, a, c, p, Power[Times[2, Plus[m, 1], Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 2]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, -1]]], x], x]]], And[FreeQ[List[a, c, d, e], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], RationalQ[m, p], Equal[Plus[m, Times[2, p], 2], 0], Greater[p, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Plus[Times[d, b], Times[-1, 2, a, e], Times[Plus[Times[2, c, d], Times[-1, b, e]], x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], x], Times[-1, Dist[Times[2, Plus[Times[2, p], 3], Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]], Power[Times[Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -2]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], RationalQ[m, p], Equal[Plus[m, Times[2, p], 2], 0], Less[p, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Plus[Times[a, e], Times[-1, c, d, x]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[2, a, c, Plus[p, 1]], -1]], x], Dist[Times[Plus[Times[2, p], 3], Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], Power[Times[2, a, c, Plus[p, 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -2]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]]], x], x]], And[FreeQ[List[a, c, d, e], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], RationalQ[m, p], Equal[Plus[m, Times[2, p], 2], 0], Less[p, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[-2, Subst[Int[Times[1, Power[Plus[Times[4, c, Power[d, 2]], Times[-1, 4, b, d, e], Times[4, a, Power[e, 2]], Times[-1, Power[x, 2]]], -1]], x], x, Times[Plus[Times[2, a, e], Times[-1, b, d], Times[-1, Plus[Times[2, c, d], Times[-1, b, e]], x]], Power[Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], -1]]], x], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Subst[Int[Times[1, Power[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]], Times[-1, Power[x, 2]]], -1]], x], x, Times[Plus[Times[a, e], Times[-1, c, d, x]], Power[Sqrt[Plus[a, Times[c, Power[x, 2]]]], -1]]]], FreeQ[List[a, c, d, e], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Simp[Times[Plus[b, Times[-1, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]], Times[2, c, x]], Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[Plus[m, 1], Plus[Times[2, c, d], Times[-1, b, e], Times[e, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Power[Times[Plus[Times[2, c, d], Times[-1, b, e], Times[e, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Plus[b, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2], Times[2, c, x]], Power[Times[Plus[Times[2, c, d], Times[-1, b, e], Times[-1, e, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Plus[b, Times[-1, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]], Times[2, c, x]]], -1]], p]], -1], Hypergeometric2F1[Plus[m, 1], Times[-1, p], Plus[m, 2], Times[-4, c, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2], Plus[d, Times[e, x]], Power[Times[Plus[Times[2, c, d], Times[-1, b, e], Times[-1, e, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Plus[b, Times[-1, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]], Times[2, c, x]]], -1]]]], x]], And[FreeQ[List[a, b, c, d, e, m, p], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], Not[IntegerQ[p]], ZeroQ[Plus[m, Times[2, p], 2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Plus[Rt[Times[-1, a, c], 2], Times[-1, c, x]], Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[c, Power[x, 2]]], p], Power[Times[Plus[m, 1], Plus[Times[c, d], Times[e, Rt[Times[-1, a, c], 2]]], Power[Times[Plus[Times[c, d], Times[e, Rt[Times[-1, a, c], 2]]], Plus[Rt[Times[-1, a, c], 2], Times[c, x]], Power[Times[Plus[Times[c, d], Times[-1, e, Rt[Times[-1, a, c], 2]]], Plus[Times[-1, Rt[Times[-1, a, c], 2]], Times[c, x]]], -1]], p]], -1], Hypergeometric2F1[Plus[m, 1], Times[-1, p], Plus[m, 2], Times[2, c, Rt[Times[-1, a, c], 2], Plus[d, Times[e, x]], Power[Times[Plus[Times[c, d], Times[-1, e, Rt[Times[-1, a, c], 2]]], Plus[Rt[Times[-1, a, c], 2], Times[-1, c, x]]], -1]]]], x], And[FreeQ[List[a, c, d, e, m, p], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], ZeroQ[Plus[m, Times[2, p], 2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], m], Plus[b, Times[2, c, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], x], Dist[Times[m, Plus[Times[2, c, d], Times[-1, b, e]], Power[Times[Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]]], x], x]], And[FreeQ[List[a, b, c, d, e, m, p], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], ZeroQ[Plus[m, Times[2, p], 3]], RationalQ[p], Less[p, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Plus[d, Times[e, x]], m], Times[2, c, x], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[4, a, c, Plus[p, 1]], -1]], x]], Times[-1, Dist[Times[m, Times[2, c, d], Power[Times[4, a, c, Plus[p, 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]]], x], x]]], And[FreeQ[List[a, c, d, e, m, p], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], ZeroQ[Plus[m, Times[2, p], 3]], RationalQ[p], Less[p, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[e, Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[Plus[m, 1], Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], -1]], x], Dist[Times[Plus[Times[2, c, d], Times[-1, b, e]], Power[Times[2, Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x]], And[FreeQ[List[a, b, c, d, e, m, p], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], ZeroQ[Plus[m, Times[2, p], 3]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[e, Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[Plus[m, 1], Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], -1]], x], Dist[Times[c, d, Power[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x]], And[FreeQ[List[a, c, d, e, m, p], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], ZeroQ[Plus[m, Times[2, p], 3]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[e, Plus[m, 1]], -1]], x], Times[-1, Dist[Times[p, Power[Times[e, Plus[m, 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Plus[b, Times[2, c, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, -1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, m], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], RationalQ[p], Greater[p, 0], Or[IntegerQ[p], And[RationalQ[m], Less[m, -1]]], NonzeroQ[Plus[m, 1]], Not[NegativeIntegerQ[Plus[m, Times[2, p], 1]]], IntQuadraticQ[a, b, c, d, e, m, p, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[c, Power[x, 2]]], p], Power[Times[e, Plus[m, 1]], -1]], x], Times[-1, Dist[Times[2, c, p, Power[Times[e, Plus[m, 1]], -1]], Int[Times[x, Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, -1]]], x], x]]], And[FreeQ[List[a, c, d, e, m], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], RationalQ[p], Greater[p, 0], Or[IntegerQ[p], And[RationalQ[m], Less[m, -1]]], NonzeroQ[Plus[m, 1]], Not[NegativeIntegerQ[Plus[m, Times[2, p], 1]]], IntQuadraticQ[a, 0, c, d, e, m, p, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[e, Plus[m, Times[2, p], 1]], -1]], x], Times[-1, Dist[Times[p, Power[Times[e, Plus[m, Times[2, p], 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], m], Simp[Plus[Times[b, d], Times[-1, 2, a, e], Times[Plus[Times[2, c, d], Times[-1, b, e]], x]], x], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, -1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, m], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], RationalQ[p], Greater[p, 0], NonzeroQ[Plus[m, Times[2, p], 1]], Or[Not[RationalQ[m]], Less[m, 1]], Not[NegativeIntegerQ[Plus[m, Times[2, p]]]], IntQuadraticQ[a, b, c, d, e, m, p, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[c, Power[x, 2]]], p], Power[Times[e, Plus[m, Times[2, p], 1]], -1]], x], Dist[Times[2, p, Power[Times[e, Plus[m, Times[2, p], 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], m], Simp[Plus[Times[a, e], Times[-1, c, d, x]], x], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, -1]]], x], x]], And[FreeQ[List[a, c, d, e, m], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], RationalQ[p], Greater[p, 0], NonzeroQ[Plus[m, Times[2, p], 1]], Or[Not[RationalQ[m]], Less[m, 1]], Not[NegativeIntegerQ[Plus[m, Times[2, p]]]], IntQuadraticQ[a, 0, c, d, e, m, p, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], m], Plus[b, Times[2, c, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], x], Times[-1, Dist[Times[1, Power[Times[Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Plus[Times[b, e, m], Times[2, c, d, Plus[Times[2, p], 3]], Times[2, c, e, Plus[m, Times[2, p], 3], x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], RationalQ[m, p], Less[p, -1], Greater[m, 0], Or[Less[m, 1], And[NegativeIntegerQ[Plus[m, Times[2, p], 3]], Unequal[m, 2]]], IntQuadraticQ[a, b, c, d, e, m, p, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[x, Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[2, a, Plus[p, 1]], -1]], x]], Dist[Times[1, Power[Times[2, a, Plus[p, 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Plus[Times[d, Plus[Times[2, p], 3]], Times[e, Plus[m, Times[2, p], 3], x]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]]], x], x]], And[FreeQ[List[a, c, d, e], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], RationalQ[m, p], Less[p, -1], Greater[m, 0], Or[Less[m, 1], And[NegativeIntegerQ[Plus[m, Times[2, p], 3]], Unequal[m, 2]]], IntQuadraticQ[a, 0, c, d, e, m, p, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Plus[Times[d, b], Times[-1, 2, a, e], Times[Plus[Times[2, c, d], Times[-1, b, e]], x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], x], Dist[Times[1, Power[Times[Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -2]], Simp[Plus[Times[e, Plus[Times[2, a, e, Plus[m, -1]], Times[b, d, Plus[Times[2, p], Times[-1, m], 4]]]], Times[-1, 2, c, Power[d, 2], Plus[Times[2, p], 3]], Times[e, Plus[Times[b, e], Times[-1, 2, d, c]], Plus[m, Times[2, p], 2], x]], x], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]]], x], x]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], RationalQ[m, p], Less[p, -1], Greater[m, 1], IntQuadraticQ[a, b, c, d, e, m, p, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Plus[Times[a, e], Times[-1, c, d, x]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[2, a, c, Plus[p, 1]], -1]], x], Dist[Times[1, Power[Times[Plus[p, 1], Times[-2, a, c]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -2]], Simp[Plus[Times[a, Power[e, 2], Plus[m, -1]], Times[-1, c, Power[d, 2], Plus[Times[2, p], 3]], Times[-1, d, c, e, Plus[m, Times[2, p], 2], x]], x], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]]], x], x]], And[FreeQ[List[a, c, d, e], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], RationalQ[m, p], Less[p, -1], Greater[m, 1], IntQuadraticQ[a, 0, c, d, e, m, p, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Plus[Times[b, c, d], Times[-1, Power[b, 2], e], Times[2, a, c, e], Times[c, Plus[Times[2, c, d], Times[-1, b, e]], x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], -1]], x], Dist[Times[1, Power[Times[Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], m], Simp[Plus[Times[b, c, d, e, Plus[Times[2, p], Times[-1, m], 2]], Times[Power[b, 2], Power[e, 2], Plus[m, p, 2]], Times[-1, 2, Power[c, 2], Power[d, 2], Plus[Times[2, p], 3]], Times[-1, 2, a, c, Power[e, 2], Plus[m, Times[2, p], 3]], Times[-1, c, e, Plus[Times[2, c, d], Times[-1, b, e]], Plus[m, Times[2, p], 4], x]], x], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]]], x], x]], And[FreeQ[List[a, b, c, d, e, m], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], RationalQ[p], Less[p, -1], IntQuadraticQ[a, b, c, d, e, m, p, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Plus[Times[a, e], Times[c, d, x]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[2, a, Plus[p, 1], Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], -1]], x]], Dist[Times[1, Power[Times[2, a, Plus[p, 1], Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], m], Simp[Plus[Times[c, Power[d, 2], Plus[Times[2, p], 3]], Times[a, Power[e, 2], Plus[m, Times[2, p], 3]], Times[c, e, d, Plus[m, Times[2, p], 4], x]], x], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]]], x], x]], And[FreeQ[List[a, c, d, e, m], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], RationalQ[p], Less[p, -1], IntQuadraticQ[a, 0, c, d, e, m, p, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[e, Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, Plus[m, Times[2, p], 1]], -1]], x], Dist[Times[1, Power[Times[c, Plus[m, Times[2, p], 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -2]], Simp[Plus[Times[c, Power[d, 2], Plus[m, Times[2, p], 1]], Times[-1, e, Plus[Times[a, e, Plus[m, -1]], Times[b, d, Plus[p, 1]]]], Times[e, Plus[Times[2, c, d], Times[-1, b, e]], Plus[m, p], x]], x], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x]], And[FreeQ[List[a, b, c, d, e, m, p], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], If[RationalQ[m], Greater[m, 1], SumSimplerQ[m, -2]], NonzeroQ[Plus[m, Times[2, p], 1]], IntQuadraticQ[a, b, c, d, e, m, p, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[e, Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, Plus[m, Times[2, p], 1]], -1]], x], Dist[Times[1, Power[Times[c, Plus[m, Times[2, p], 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -2]], Simp[Plus[Times[c, Power[d, 2], Plus[m, Times[2, p], 1]], Times[-1, a, Power[e, 2], Plus[m, -1]], Times[2, c, d, e, Plus[m, p], x]], x], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x]], And[FreeQ[List[a, c, d, e, m, p], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], If[RationalQ[m], Greater[m, 1], SumSimplerQ[m, -2]], NonzeroQ[Plus[m, Times[2, p], 1]], IntQuadraticQ[a, 0, c, d, e, m, p, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[e, Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[Plus[m, 1], Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], -1]], x], Dist[Times[1, Power[Times[Plus[m, 1], Plus[Times[c, 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Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[e, Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[Plus[m, 1], Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], -1]], x], Dist[Times[c, Power[Times[Plus[m, 1], Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Simp[Plus[Times[d, Plus[m, 1]], Times[-1, e, Plus[m, Times[2, p], 3], x]], x], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x]], And[FreeQ[List[a, c, d, e, m, p], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Or[And[RationalQ[m], Less[m, -1], IntQuadraticQ[a, 0, c, d, e, m, p, x]], And[SumSimplerQ[m, 1], IntegerQ[p], NonzeroQ[Plus[m, 1]]], And[NegativeIntegerQ[Simplify[Plus[m, Times[2, p], 3]]], NonzeroQ[Plus[m, 1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 3]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Times[3, c, Power[e, 2], Plus[Times[2, c, d], Times[-1, b, e]]], 3]]], Plus[Times[-1, Simp[Times[Sqrt[3], c, e, ArcTan[Plus[Times[1, Power[Sqrt[3], -1]], Times[2, Plus[Times[c, d], Times[-1, b, e], Times[-1, c, e, x]], Power[Times[Sqrt[3], q, Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Times[1, Power[3, -1]]]], -1]]]], Power[Power[q, 2], -1]], x]], Plus[Times[-1, Simp[Times[3, c, e, Log[Plus[d, Times[e, x]]], Power[Times[2, Power[q, 2]], -1]], x]], Simp[Times[3, c, e, Log[Plus[Times[c, d], Times[-1, b, e], Times[-1, c, e, x], Times[-1, q, Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Times[1, Power[3, -1]]]]]], Power[Times[2, Power[q, 2]], -1]], x]]]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], ZeroQ[Plus[Times[Power[c, 2], Power[d, 2]], Times[-1, b, c, d, e], Times[Power[b, 2], Power[e, 2]], Times[-1, 3, a, c, Power[e, 2]]]], PosQ[Times[c, Power[e, 2], Plus[Times[2, c, d], Times[-1, b, e]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 3]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Times[6, Power[c, 2], Power[e, 2], Power[Power[d, 2], -1]], 3]]], Plus[Times[-1, Simp[Times[Sqrt[3], c, e, ArcTan[Plus[Times[1, Power[Sqrt[3], -1]], Times[2, c, Plus[d, Times[-1, e, x]], Power[Times[Sqrt[3], d, q, Power[Plus[a, Times[c, Power[x, 2]]], Times[1, Power[3, -1]]]], -1]]]], Power[Times[Power[d, 2], Power[q, 2]], -1]], x]], Plus[Times[-1, Simp[Times[3, c, e, Log[Plus[d, Times[e, x]]], Power[Times[2, Power[d, 2], Power[q, 2]], -1]], x]], Simp[Times[3, c, e, Log[Plus[Times[c, d], Times[-1, c, e, x], Times[-1, d, q, Power[Plus[a, Times[c, Power[x, 2]]], Times[1, Power[3, -1]]]]]], Power[Times[2, Power[d, 2], Power[q, 2]], -1]], x]]]], And[FreeQ[List[a, c, d, e], x], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, 3, a, Power[e, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 3]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Times[-3, c, Power[e, 2], Plus[Times[2, c, d], Times[-1, b, e]]], 3]]], Plus[Times[-1, Simp[Times[Sqrt[3], c, e, ArcTan[Plus[Times[1, Power[Sqrt[3], -1]], Times[-1, 2, Plus[Times[c, d], Times[-1, b, e], Times[-1, c, e, x]], Power[Times[Sqrt[3], q, Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Times[1, Power[3, -1]]]], -1]]]], Power[Power[q, 2], -1]], x]], Plus[Times[-1, Simp[Times[3, c, e, Log[Plus[d, Times[e, x]]], Power[Times[2, Power[q, 2]], -1]], x]], Simp[Times[3, c, e, Log[Plus[Times[c, d], Times[-1, b, e], Times[-1, c, e, x], Times[q, Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Times[1, Power[3, -1]]]]]], Power[Times[2, Power[q, 2]], -1]], x]]]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], ZeroQ[Plus[Times[Power[c, 2], Power[d, 2]], Times[-1, b, c, d, e], Times[Power[b, 2], Power[e, 2]], Times[-1, 3, a, c, Power[e, 2]]]], NegQ[Times[c, Power[e, 2], Plus[Times[2, c, d], Times[-1, b, e]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 3]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Dist[Times[Power[Plus[b, q, Times[2, c, x]], Times[1, Power[3, -1]]], Power[Plus[b, Times[-1, q], Times[2, c, x]], Times[1, Power[3, -1]]], Power[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Times[1, Power[3, -1]]], -1]], Int[Times[1, Power[Times[Plus[d, Times[e, x]], Power[Plus[b, q, Times[2, c, x]], Times[1, Power[3, -1]]], Power[Plus[b, Times[-1, q], Times[2, c, x]], Times[1, Power[3, -1]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[Power[c, 2], Power[d, 2]], Times[-1, b, c, d, e], Times[-1, 2, Power[b, 2], Power[e, 2]], Times[9, a, c, Power[e, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 4]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[d, Int[Times[1, Power[Times[Plus[Power[d, 2], Times[-1, Power[e, 2], Power[x, 2]]], Power[Plus[a, Times[c, Power[x, 2]]], Times[1, Power[4, -1]]]], -1]], x], x], Times[-1, Dist[e, Int[Times[x, Power[Times[Plus[Power[d, 2], Times[-1, Power[e, 2], Power[x, 2]]], Power[Plus[a, Times[c, Power[x, 2]]], Times[1, Power[4, -1]]]], -1]], x], x]]], And[FreeQ[List[a, c, d, e], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 4]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[d, Int[Times[1, Power[Times[Plus[Power[d, 2], Times[-1, Power[e, 2], Power[x, 2]]], Power[Plus[a, Times[c, Power[x, 2]]], Times[3, Power[4, -1]]]], -1]], x], x], Times[-1, Dist[e, Int[Times[x, Power[Times[Plus[Power[d, 2], Times[-1, Power[e, 2], Power[x, 2]]], Power[Plus[a, Times[c, Power[x, 2]]], Times[3, Power[4, -1]]]], -1]], x], x]]], And[FreeQ[List[a, c, d, e], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Power[Times[-4, c, Power[Plus[Power[b, 2], Times[-1, 4, a, c]], -1]], p], -1]], Subst[Int[Times[Power[Simp[Plus[1, Times[-1, Power[x, 2], Power[Plus[Power[b, 2], Times[-1, 4, a, c]], -1]]], x], p], Power[Simp[Plus[Times[2, c, d], Times[-1, b, e], Times[e, x]], x], -1]], x], x, Plus[b, Times[2, c, x]]], x], And[FreeQ[List[a, b, c, d, e, p], x], PositiveQ[Plus[Times[4, a], Times[-1, Power[b, 2], Power[c, -1]]]], IntegerQ[Times[4, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Power[Times[-1, c, Plus[a, Times[b, x], Times[c, Power[x, 2]]], Power[Plus[Power[b, 2], Times[-1, 4, a, c]], -1]], p], -1]], Int[Times[Power[Plus[Times[-1, a, c, Power[Plus[Power[b, 2], Times[-1, 4, a, c]], -1]], Times[-1, b, c, x, Power[Plus[Power[b, 2], Times[-1, 4, a, c]], -1]], Times[-1, Power[c, 2], Power[x, 2], Power[Plus[Power[b, 2], Times[-1, 4, a, c]], -1]]], p], Power[Plus[d, Times[e, x]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, p], x], Not[PositiveQ[Plus[Times[4, a], Times[-1, Power[b, 2], Power[c, -1]]]]], IntegerQ[Times[4, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[Rt[a, 2], Times[Rt[Times[-1, c], 2], x]], p], Power[Plus[Rt[a, 2], Times[-1, Rt[Times[-1, c], 2], x]], p]], x], And[FreeQ[List[a, c, d, e, m, p], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], PositiveQ[a], NegativeQ[c]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Times[-1, Dist[Times[Power[Times[1, Power[Plus[d, Times[e, x]], -1]], Times[2, p]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[e, Power[Times[e, Plus[b, Times[-1, q], Times[2, c, x]], Power[Times[2, c, Plus[d, Times[e, x]]], -1]], p], Power[Times[e, Plus[b, q, Times[2, c, x]], Power[Times[2, c, Plus[d, Times[e, x]]], -1]], p]], -1]], Subst[Int[Times[Power[x, Plus[Times[-1, m], Times[-1, 2, Plus[p, 1]]]], Power[Simp[Plus[1, Times[-1, Plus[d, Times[-1, e, Plus[b, Times[-1, q]], Power[Times[2, c], -1]]], x]], x], p], Power[Simp[Plus[1, Times[-1, Plus[d, Times[-1, e, Plus[b, q], Power[Times[2, c], -1]]], x]], x], p]], x], x, Times[1, Power[Plus[d, Times[e, x]], -1]]], x]]], And[FreeQ[List[a, b, c, d, e, p], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], Not[IntegerQ[p]], NegativeIntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Times[-1, a, c], 2]]], Times[-1, Dist[Times[Power[Times[1, Power[Plus[d, Times[e, x]], -1]], Times[2, p]], Power[Plus[a, Times[c, Power[x, 2]]], p], Power[Times[e, Power[Times[-1, e, Plus[q, Times[-1, c, x]], Power[Times[c, Plus[d, Times[e, x]]], -1]], p], Power[Times[e, Plus[q, Times[c, x]], Power[Times[c, Plus[d, Times[e, x]]], -1]], p]], -1]], Subst[Int[Times[Power[x, Plus[Times[-1, m], Times[-1, 2, Plus[p, 1]]]], Power[Simp[Plus[1, Times[-1, Plus[d, Times[e, q, Power[c, -1]]], x]], x], p], Power[Simp[Plus[1, Times[-1, Plus[d, Times[-1, e, q, Power[c, -1]]], x]], x], p]], x], x, Times[1, Power[Plus[d, Times[e, x]], -1]]], x]]], And[FreeQ[List[a, c, d, e, p], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], NegativeIntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Dist[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[e, Power[Plus[1, Times[-1, Plus[d, Times[e, x]], Power[Plus[d, Times[-1, e, Plus[b, Times[-1, q]], Power[Times[2, c], -1]]], -1]]], p], Power[Plus[1, Times[-1, Plus[d, Times[e, x]], Power[Plus[d, Times[-1, e, Plus[b, q], Power[Times[2, c], -1]]], -1]]], p]], -1]], Subst[Int[Times[Power[x, m], Power[Simp[Plus[1, Times[-1, x, Power[Plus[d, Times[-1, e, Plus[b, Times[-1, q]], Power[Times[2, c], -1]]], -1]]], x], p], Power[Simp[Plus[1, Times[-1, x, Power[Plus[d, Times[-1, e, Plus[b, q], Power[Times[2, c], -1]]], -1]]], x], p]], x], x, Plus[d, Times[e, x]]], x]], And[FreeQ[List[a, b, c, d, e, m, p], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Times[-1, a, c], 2]]], Dist[Times[Power[Plus[a, Times[c, Power[x, 2]]], p], Power[Times[e, Power[Plus[1, Times[-1, Plus[d, Times[e, x]], Power[Plus[d, Times[e, q, Power[c, -1]]], -1]]], p], Power[Plus[1, Times[-1, Plus[d, Times[e, x]], Power[Plus[d, Times[-1, e, q, Power[c, -1]]], -1]]], p]], -1]], Subst[Int[Times[Power[x, m], Power[Simp[Plus[1, Times[-1, x, Power[Plus[d, Times[e, q, Power[c, -1]]], -1]]], x], p], Power[Simp[Plus[1, Times[-1, x, Power[Plus[d, Times[-1, e, q, Power[c, -1]]], -1]]], x], p]], x], x, Plus[d, Times[e, x]]], x]], And[FreeQ[List[a, c, d, e, m, p], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[u, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[u, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Coefficient[u, x, 1], -1]], Subst[Int[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x, u], x], And[FreeQ[List[a, b, c, d, e, m, p], x], LinearQ[u, x], NonzeroQ[Plus[u, Times[-1, x]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[u, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Coefficient[u, x, 1], -1]], Subst[Int[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x, u], x], And[FreeQ[List[a, c, d, e, m, p], x], LinearQ[u, x], NonzeroQ[Plus[u, Times[-1, x]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[d, Int[Times[Power[x, n], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x], Dist[e, Int[Times[Power[x, Plus[n, 1]], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x]], And[FreeQ[List[a, c, d, e, p], x], IntegerQ[n], Not[IntegerQ[Times[2, p]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Plus[f, Times[g, x]], Power[Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], -1]], Int[Power[Plus[d, Times[e, x]], m], x], x], And[FreeQ[List[a, b, c, d, e, f, g, m], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[2, c, f], Times[-1, b, g]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Simp[Times[f, g, Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[b, Plus[p, 1], Plus[Times[e, f], Times[-1, d, g]]], -1]], x]], And[FreeQ[List[a, b, c, d, e, f, g, m, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[2, c, f], Times[-1, b, g]]], Not[IntegerQ[p]], ZeroQ[Plus[m, Times[2, p], 3]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[g, Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[2, c, Plus[p, 1]], -1]], x], Times[-1, Dist[Times[e, g, m, Power[Times[2, c, Plus[p, 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[2, c, f], Times[-1, b, g]]], Not[IntegerQ[p]], RationalQ[m, p], Less[p, -1], Greater[m, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[f, g, Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[b, Plus[p, 1], Plus[Times[e, f], Times[-1, d, g]]], -1]], x]], Dist[Times[e, f, g, Plus[m, Times[2, p], 3], Power[Times[b, Plus[p, 1], Plus[Times[e, f], Times[-1, d, g]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, m], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[2, c, f], Times[-1, b, g]]], Not[IntegerQ[p]], RationalQ[p], Less[p, -1], Not[And[RationalQ[m], Greater[m, 0]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Plus[f, Times[g, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[e, Plus[m, 1]], -1]], x], Times[-1, Dist[Times[g, Plus[Times[2, p], 1], Power[Times[e, Plus[m, 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[2, c, f], Times[-1, b, g]]], Not[IntegerQ[p]], RationalQ[m], Less[m, -1], NonzeroQ[Plus[Times[2, p], 1]], Or[Not[RationalQ[p]], And[Greater[p, 0], Or[Not[IntegerQ[m]], GreaterEqual[m, Plus[Times[-2, p], -2]], Less[m, Times[-4, Plus[p, 1]]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[2, f, g, Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[b, Plus[m, 1], Plus[Times[e, f], Times[-1, d, g]]], -1]], x], Times[-1, Dist[Times[g, Plus[m, Times[2, p], 3], Power[Times[Plus[m, 1], Plus[Times[e, f], Times[-1, d, g]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Plus[f, Times[g, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[2, c, f], Times[-1, b, g]]], Not[IntegerQ[p]], RationalQ[m], Less[m, -1], NonzeroQ[Plus[m, Times[2, p], 2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[g, Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, Plus[m, Times[2, p], 2]], -1]], x], Times[-1, Dist[Times[b, m, Plus[Times[e, f], Times[-1, d, g]], Power[Times[2, c, f, Plus[m, Times[2, p], 2]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Plus[f, Times[g, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[2, c, f], Times[-1, b, g]]], Not[IntegerQ[p]], PositiveIntegerQ[m], NonzeroQ[Plus[m, Times[2, p], 2]], Or[Not[RationalQ[p]], Less[m, Plus[Times[2, p], 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Plus[f, Times[g, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[e, Plus[m, Times[2, p], 2]], -1]], x], Dist[Times[Plus[Times[e, f], Times[-1, d, g]], Plus[Times[2, p], 1], Power[Times[e, Plus[m, Times[2, p], 2]], -1]], Int[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, m, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[2, c, f], Times[-1, b, g]]], Not[IntegerQ[p]], NonzeroQ[Plus[m, Times[2, p], 2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Plus[Times[2, c, f], Times[-1, b, g]], Power[Plus[Times[2, c, d], Times[-1, b, e]], -1]], Int[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], x], x], Times[-1, Dist[Times[Plus[Times[e, f], Times[-1, d, g]], Power[Plus[Times[2, c, d], Times[-1, b, e]], -1]], Int[Times[Plus[b, Times[2, c, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Plus[d, Times[e, x]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[2, c, f], Times[-1, b, g]]], Not[IntegerQ[p]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], RationalQ[p], Less[p, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Plus[Times[e, f], Times[-1, d, g]], Power[e, -1]], Int[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x], Dist[Times[g, Power[e, -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, m, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[2, c, f], Times[-1, b, g]]], Not[IntegerQ[p]], Or[And[ZeroQ[Plus[m, Times[2, p], 2]], NonzeroQ[Plus[m, 1]]], And[ZeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], NonzeroQ[Plus[m, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Plus[Times[2, c, f], Times[-1, b, g]], Power[Plus[Times[2, c, d], Times[-1, b, e]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x], Times[-1, Dist[Times[Plus[Times[e, f], Times[-1, d, g]], Power[Plus[Times[2, c, d], Times[-1, b, e]], -1]], Int[Times[Power[Plus[d, Times[e, x]], m], Plus[b, Times[2, c, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g, m, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[2, c, f], Times[-1, b, g]]], Not[IntegerQ[p]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], ZeroQ[Plus[m, Times[2, p], 3]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[e, f], Times[-1, d, g]], Power[Plus[d, Times[e, x]], Plus[m, 1]], Plus[b, Times[2, c, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[e, Plus[m, 1], Plus[Times[2, c, d], Times[-1, b, e]]], -1]], x]], Dist[Times[Plus[Times[2, c, e, f, Plus[m, Times[2, p], 2]], Times[-1, g, Plus[Times[2, c, d, Plus[Times[2, p], 1]], Times[b, e, Plus[m, 1]]]]], Power[Times[e, Plus[m, 1], Plus[Times[2, c, d], Times[-1, b, e]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[2, c, f], Times[-1, b, g]]], Not[IntegerQ[p]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], NonzeroQ[Plus[m, Times[2, p], 2]], NonzeroQ[Plus[m, Times[2, p], 3]], RationalQ[m], Less[m, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[g, Power[Plus[d, Times[e, x]], Plus[m, 1]], Plus[b, Times[2, c, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[2, c, e, Plus[m, Times[2, p], 2]], -1]], x], Dist[Times[Plus[Times[2, c, e, f, Plus[m, Times[2, p], 2]], Times[-1, g, Plus[Times[b, e, Plus[m, 1]], Times[2, c, Plus[d, Times[2, d, p]]]]]], Power[Times[2, c, e, Plus[m, Times[2, p], 2]], -1]], Int[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, m, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[2, c, f], Times[-1, b, g]]], Not[IntegerQ[p]], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]], NonzeroQ[Plus[m, Times[2, p], 2]], NonzeroQ[Plus[m, Times[2, p], 3]], Not[And[RationalQ[m], Less[m, -1]]], Not[And[ZeroQ[Plus[m, -1]], SimplerQ[Plus[f, Times[g, x]], Plus[d, Times[e, x]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], FracPart[p]], Power[Times[Power[c, IntPart[p]], Power[Plus[Times[b, Power[2, -1]], Times[c, x]], Times[2, FracPart[p]]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], n], Power[Plus[Times[b, Power[2, -1]], Times[c, x]], Times[2, p]]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, m, n], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[d, Times[e, x]], Plus[m, p]], Power[Plus[f, Times[g, x]], n], Power[Plus[Times[a, Power[d, -1]], Times[c, Power[e, -1], x]], p]], x], And[FreeQ[List[a, b, c, d, e, f, g, m, n], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[d, Times[e, x]], Plus[m, p]], Power[Plus[f, Times[g, x]], n], Power[Plus[Times[a, Power[d, -1]], Times[c, Power[e, -1], x]], p]], x], And[FreeQ[List[a, c, d, e, f, g, m, n], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Or[IntegerQ[p], And[PositiveQ[a], PositiveQ[d], ZeroQ[Plus[m, p]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[d, m], Power[e, m]], Int[Times[Power[Plus[Times[a, e], Times[c, d, x]], Times[-1, m]], Power[Plus[f, Times[g, x]], n], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[m, p]]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, n, p], x], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], NegativeIntegerQ[m], Not[IntegerQ[Times[2, p]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[d, m], Power[e, m]], Int[Times[Power[Plus[Times[a, e], Times[c, d, x]], Times[-1, m]], Power[Plus[f, Times[g, x]], n], Power[Plus[a, Times[c, Power[x, 2]]], Plus[m, p]]], x], x], And[FreeQ[List[a, c, d, e, f, g, n, p], x], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], NegativeIntegerQ[m], Not[IntegerQ[Times[2, p]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[g, Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, Plus[m, Times[2, p], 2]], -1]], x], And[FreeQ[List[a, b, c, d, e, f, g, m, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], ZeroQ[Plus[Times[m, Plus[Times[g, Plus[Times[c, d], Times[-1, b, e]]], Times[c, e, f]]], Times[e, Plus[p, 1], Plus[Times[2, c, f], Times[-1, b, g]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[g, Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, Plus[m, Times[2, p], 2]], -1]], x], And[FreeQ[List[a, c, d, e, f, g, m, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], ZeroQ[Plus[Times[m, Plus[Times[d, g], Times[e, f]]], Times[2, e, f, Plus[p, 1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[g, Plus[Times[c, d], Times[-1, b, e]]], Times[c, e, f]], Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, Plus[p, 1], Plus[Times[2, c, d], Times[-1, b, e]]], -1]], x], Times[-1, Dist[Times[e, Plus[Times[m, Plus[Times[g, Plus[Times[c, d], Times[-1, b, e]]], Times[c, e, f]]], Times[e, Plus[p, 1], Plus[Times[2, c, f], Times[-1, b, g]]]], Power[Times[c, Plus[p, 1], Plus[Times[2, c, d], Times[-1, b, e]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], RationalQ[m, p], Less[p, -1], Greater[m, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[d, g], Times[e, f]], Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[2, c, d, Plus[p, 1]], -1]], x], Times[-1, Dist[Times[e, Plus[Times[m, Plus[Times[d, g], Times[e, f]]], Times[2, e, f, Plus[p, 1]]], Power[Times[2, c, d, Plus[p, 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]]], x], x]]], And[FreeQ[List[a, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], RationalQ[m, p], Less[p, -1], Greater[m, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[g, Plus[Times[c, d], Times[-1, b, e]]], Times[c, e, f]], Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, Plus[p, 1], Plus[Times[2, c, d], Times[-1, b, e]]], -1]], x], Times[-1, Dist[Times[e, Plus[Times[m, Plus[Times[g, Plus[Times[c, d], Times[-1, b, e]]], Times[c, e, f]]], Times[e, Plus[p, 1], Plus[Times[2, c, f], Times[-1, b, g]]]], Power[Times[c, Plus[p, 1], Plus[Times[2, c, d], Times[-1, b, e]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Simplify[Plus[m, -1]]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Simplify[Plus[p, 1]]]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g, m, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], SumSimplerQ[p, 1], SumSimplerQ[m, -1], NonzeroQ[Plus[p, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[d, g], Times[e, f]], Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[2, c, d, Plus[p, 1]], -1]], x], Times[-1, Dist[Times[e, Plus[Times[m, Plus[Times[d, g], Times[e, f]]], Times[2, e, f, Plus[p, 1]]], Power[Times[2, c, d, Plus[p, 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Simplify[Plus[m, -1]]], Power[Plus[a, Times[c, Power[x, 2]]], Simplify[Plus[p, 1]]]], x], x]]], And[FreeQ[List[a, c, d, e, f, g, m, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], SumSimplerQ[p, 1], SumSimplerQ[m, -1], NonzeroQ[Plus[p, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[d, g], Times[-1, e, f]], Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[Plus[Times[2, c, d], Times[-1, b, e]], Plus[m, p, 1]], -1]], x], Dist[Times[Plus[Times[m, Plus[Times[g, Plus[Times[c, d], Times[-1, b, e]]], Times[c, e, f]]], Times[e, Plus[p, 1], Plus[Times[2, c, f], Times[-1, b, g]]]], Power[Times[e, Plus[Times[2, c, d], Times[-1, b, e]], Plus[m, p, 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, m, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Or[And[RationalQ[m], Less[m, -1], Not[PositiveIntegerQ[Plus[m, p, 1]]]], And[RationalQ[m, p], Less[m, 0], Less[p, -1]], ZeroQ[Plus[m, Times[2, p], 2]]], NonzeroQ[Plus[m, p, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[d, g], Times[-1, e, f]], Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[2, c, d, Plus[m, p, 1]], -1]], x], Dist[Times[Plus[Times[m, Plus[Times[g, c, d], Times[c, e, f]]], Times[2, e, c, f, Plus[p, 1]]], Power[Times[e, Times[2, c, d], Plus[m, p, 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x]], And[FreeQ[List[a, c, d, e, f, g, m, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Or[And[RationalQ[m], Less[m, -1], Not[PositiveIntegerQ[Plus[m, p, 1]]]], And[RationalQ[m, p], Less[m, 0], Less[p, -1]], ZeroQ[Plus[m, Times[2, p], 2]]], NonzeroQ[Plus[m, p, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[g, Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, Plus[m, Times[2, p], 2]], -1]], x], Dist[Times[Plus[Times[m, Plus[Times[g, Plus[Times[c, d], Times[-1, b, e]]], Times[c, e, f]]], Times[e, Plus[p, 1], Plus[Times[2, c, f], Times[-1, b, g]]]], Power[Times[c, e, Plus[m, Times[2, p], 2]], -1]], Int[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, m, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], NonzeroQ[Plus[m, Times[2, p], 2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[g, Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, Plus[m, Times[2, p], 2]], -1]], x], Dist[Times[Plus[Times[m, Plus[Times[d, g], Times[e, f]]], Times[2, e, f, Plus[p, 1]]], Power[Times[e, Plus[m, Times[2, p], 2]], -1]], Int[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x]], And[FreeQ[List[a, c, d, e, f, g, m, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], NonzeroQ[Plus[m, Times[2, p], 2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], 2], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, 2], Plus[Times[a, g], Times[-1, c, f, x]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[2, a, c, Plus[p, 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[2, a, c, Plus[p, 1]], -1]], Int[Times[x, Simp[Plus[Times[2, a, g], Times[-1, c, f, Plus[Times[2, p], 5], x]], x], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]]], x], x]]], And[FreeQ[List[a, c, f, g], x], ZeroQ[Plus[Times[a, Power[g, 2]], Times[Power[f, 2], c]]], RationalQ[p], Less[p, -2]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], 2], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[c, -1]], Int[Times[Plus[f, Times[g, x]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]]], x], x], Times[-1, Dist[Times[Power[f, 2], Power[c, -1]], Int[Times[Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[f, Times[-1, g, x]], -1]], x], x]]], And[FreeQ[List[a, c, f, g, p], x], ZeroQ[Plus[Times[a, Power[g, 2]], Times[Power[f, 2], c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[f, Times[g, x]], n], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[m, p]], Power[Power[Plus[Times[a, Power[d, -1]], Times[c, x, Power[e, -1]]], m], -1]], x], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], IntegersQ[m, n], RationalQ[p], Or[Less[0, Times[-1, m], Plus[p, 1]], Less[p, Times[-1, m], 0]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[d, Times[2, m]], Power[Power[a, m], -1]], Int[Times[Power[Plus[f, Times[g, x]], n], Power[Plus[a, Times[c, Power[x, 2]]], Plus[m, p]], Power[Power[Plus[d, Times[-1, e, x]], m], -1]], x], x], And[FreeQ[List[a, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], IntegersQ[m, n], RationalQ[p], Or[Less[0, Times[-1, m], Plus[p, 1]], Less[p, Times[-1, m], 0]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Plus[f, Times[g, x]], n], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Plus[Times[a, Plus[Times[2, c, d], Times[-1, b, e]]], Times[c, Plus[Times[b, d], Times[-1, 2, a, e]], x]], Power[Times[d, e, p, Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], x]], Times[-1, Dist[Times[1, Power[Times[d, e, p, Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], Int[Times[Power[Plus[f, Times[g, x]], Plus[n, -1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Simp[Plus[Times[b, Plus[Times[a, e, g, n], Times[-1, c, d, f, Plus[Times[2, p], 1]]]], Times[-1, 2, a, c, Plus[Times[d, g, n], Times[-1, e, f, Plus[Times[2, p], 1]]]], Times[-1, c, g, Plus[Times[b, d], Times[-1, 2, a, e]], Plus[n, Times[2, p], 1], x]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], PositiveIntegerQ[n], NegativeIntegerQ[Plus[n, Times[2, p]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[f, Times[g, x]], n], Power[Plus[a, Times[c, Power[x, 2]]], p], Plus[d, Times[-1, e, x]], Power[Times[2, d, e, p], -1]], x], Times[-1, Dist[Times[1, Power[Times[2, d, e, p], -1]], Int[Times[Power[Plus[f, Times[g, x]], Plus[n, -1]], Power[Plus[a, Times[c, Power[x, 2]]], p], Simp[Plus[Times[d, g, n], Times[-1, e, f, Plus[Times[2, p], 1]], Times[-1, e, g, Plus[n, Times[2, p], 1], x]], x]], x], x]]], And[FreeQ[List[a, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], PositiveIntegerQ[n], NegativeIntegerQ[Plus[n, Times[2, p]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Plus[f, Times[g, x]], Plus[n, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Plus[Times[a, c, d, Plus[Times[2, c, f], Times[-1, b, g]]], Times[-1, a, e, Plus[Times[b, c, f], Times[-1, Power[b, 2], g], Times[2, a, c, g]]], Times[c, Plus[Times[c, d, Plus[Times[b, f], Times[-1, 2, a, g]]], Times[-1, a, e, Plus[Times[2, c, f], Times[-1, b, g]]]], x]], Power[Times[d, e, p, Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[Times[c, Power[f, 2]], Times[-1, b, f, g], Times[a, Power[g, 2]]]], -1]], x]], Times[-1, Dist[Times[1, Power[Times[d, e, p, Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[Times[c, Power[f, 2]], Times[-1, b, f, g], Times[a, Power[g, 2]]]], -1]], Int[Times[Power[Plus[f, Times[g, x]], n], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Simp[Plus[Times[Power[b, 2], g, Plus[Times[c, d, f, p], Times[-1, a, e, g, Plus[n, p, 1]]]], Times[b, c, Plus[Times[a, g, Plus[Times[d, g, Plus[n, 1]], Times[e, f, Plus[n, Times[-1, 2, p]]]]], Times[-1, c, d, Power[f, 2], Plus[Times[2, p], 1]]]], Times[2, a, c, Plus[Times[a, e, Power[g, 2], Plus[n, Times[2, p], 1]], Times[c, f, Plus[Times[e, f], Times[-1, d, g, n], Times[2, e, f, p]]]]], Times[c, g, Plus[Times[2, a, c, Plus[Times[e, f], Times[d, g]]], Times[-1, b, Plus[Times[c, d, f], Times[a, e, g]]]], Plus[n, Times[2, p], 2], x]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], NegativeIntegerQ[n], NegativeIntegerQ[Plus[n, Times[2, p]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[f, Times[g, x]], Plus[n, 1]], Power[Plus[a, Times[c, Power[x, 2]]], p], Plus[Times[c, d, f], Times[-1, a, e, g], Times[-1, c, Plus[Times[e, f], Times[d, g]], x]], Power[Times[2, d, e, p, Plus[Times[c, Power[f, 2]], Times[a, Power[g, 2]]]], -1]], x], Dist[Times[1, Power[Times[2, d, e, p, Plus[Times[c, Power[f, 2]], Times[a, Power[g, 2]]]], -1]], Int[Times[Power[Plus[f, Times[g, x]], n], Power[Plus[a, Times[c, Power[x, 2]]], p], Simp[Plus[Plus[Times[a, e, Power[g, 2], Plus[n, Times[2, p], 1]], Times[-1, c, f, Plus[Times[d, g, n], Times[-1, e, Plus[f, Times[2, f, p]]]]]], Times[c, g, Plus[Times[e, f], Times[d, g]], Plus[n, Times[2, p], 2], x]], x]], x], x]], And[FreeQ[List[a, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], NegativeIntegerQ[n], NegativeIntegerQ[Plus[n, Times[2, p]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Simp[Times[e, Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], n], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, Plus[m, Times[-1, n], -1]], -1]], x]], And[FreeQ[List[a, b, c, d, e, f, g, m, n, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], ZeroQ[Plus[m, p]], ZeroQ[Plus[Times[c, e, f], Times[c, d, g], Times[-1, b, e, g]]], NonzeroQ[Plus[m, Times[-1, n], -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Simp[Times[e, Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], n], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, Plus[m, Times[-1, n], -1]], -1]], x]], And[FreeQ[List[a, c, d, e, f, g, m, n, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], ZeroQ[Plus[m, p]], ZeroQ[Plus[Times[e, f], Times[d, g]]], NonzeroQ[Plus[m, Times[-1, n], -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Simp[Times[Power[e, 2], Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], Plus[n, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[Plus[n, 1], Plus[Times[c, e, f], Times[c, d, g], Times[-1, b, e, g]]], -1]], x]], And[FreeQ[List[a, b, c, d, e, f, g, m, n, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], ZeroQ[Plus[m, p]], ZeroQ[Plus[m, Times[-1, n], -2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Simp[Times[Power[e, 2], Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], Plus[n, 1]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, Plus[n, 1], Plus[Times[e, f], Times[d, g]]], -1]], x]], And[FreeQ[List[a, c, d, e, f, g, m, n, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], ZeroQ[Plus[m, p]], ZeroQ[Plus[m, Times[-1, n], -2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], Plus[n, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[g, Plus[n, 1]], -1]], x], Dist[Times[c, m, Power[Times[e, g, Plus[n, 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[f, Times[g, x]], Plus[n, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, -1]]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], ZeroQ[Plus[m, p]], RationalQ[n, p], Greater[p, 0], Less[n, -1], Not[And[IntegerQ[Plus[n, p]], LessEqual[Plus[n, p, 2], 0]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], Plus[n, 1]], Power[Plus[a, Times[c, Power[x, 2]]], p], Power[Times[g, Plus[n, 1]], -1]], x], Dist[Times[c, m, Power[Times[e, g, Plus[n, 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[f, Times[g, x]], Plus[n, 1]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, -1]]], x], x]], And[FreeQ[List[a, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], ZeroQ[Plus[m, p]], RationalQ[n, p], Greater[p, 0], Less[n, -1], Not[And[IntegerQ[Plus[n, p]], LessEqual[Plus[n, p, 2], 0]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], Plus[n, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[g, Plus[m, Times[-1, n], -1]], -1]], x]], Times[-1, Dist[Times[m, Plus[Times[c, e, f], Times[c, d, g], Times[-1, b, e, g]], Power[Times[Power[e, 2], g, Plus[m, Times[-1, n], -1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[f, Times[g, x]], n], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, -1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g, n], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], ZeroQ[Plus[m, p]], RationalQ[n, p], Greater[p, 0], NonzeroQ[Plus[m, Times[-1, n], -1]], Not[PositiveIntegerQ[n]], Not[And[IntegerQ[Plus[n, p]], Less[Plus[n, p, 2], 0]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], Plus[n, 1]], Power[Plus[a, Times[c, Power[x, 2]]], p], Power[Times[g, Plus[m, Times[-1, n], -1]], -1]], x]], Times[-1, Dist[Times[c, m, Plus[Times[e, f], Times[d, g]], Power[Times[Power[e, 2], g, Plus[m, Times[-1, n], -1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[f, Times[g, x]], n], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, -1]]], x], x]]], And[FreeQ[List[a, c, d, e, f, g, n], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], ZeroQ[Plus[m, p]], RationalQ[n, p], Greater[p, 0], NonzeroQ[Plus[m, Times[-1, n], -1]], Not[PositiveIntegerQ[n]], Not[And[IntegerQ[Plus[n, p]], Less[Plus[n, p, 2], 0]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[e, Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], n], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, Plus[p, 1]], -1]], x], Times[-1, Dist[Times[e, g, n, Power[Times[c, Plus[p, 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], Plus[n, -1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], ZeroQ[Plus[m, p]], RationalQ[n, p], Less[p, -1], Greater[n, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[e, Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], n], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, Plus[p, 1]], -1]], x], Times[-1, Dist[Times[e, g, n, Power[Times[c, Plus[p, 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], Plus[n, -1]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]]], x], x]]], And[FreeQ[List[a, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], ZeroQ[Plus[m, p]], RationalQ[n, p], Less[p, -1], Greater[n, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[e, 2], Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], Plus[n, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[Plus[p, 1], Plus[Times[c, e, f], Times[c, d, g], Times[-1, b, e, g]]], -1]], x], Dist[Times[Power[e, 2], g, Plus[m, Times[-1, n], -2], Power[Times[Plus[p, 1], Plus[Times[c, e, f], Times[c, d, g], Times[-1, b, e, g]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], n], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, n], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], ZeroQ[Plus[m, p]], RationalQ[n, p], Less[p, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[e, 2], Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], Plus[n, 1]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, Plus[p, 1], Plus[Times[e, f], Times[d, g]]], -1]], x], Dist[Times[Power[e, 2], g, Plus[m, Times[-1, n], -2], Power[Times[c, Plus[p, 1], Plus[Times[e, f], Times[d, g]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], n], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]]], x], x]], And[FreeQ[List[a, c, d, e, f, g, n], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], ZeroQ[Plus[m, p]], RationalQ[n, p], Less[p, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[e, Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], n], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, Plus[m, Times[-1, n], -1]], -1]], x]], Times[-1, Dist[Times[n, Plus[Times[c, e, f], Times[c, d, g], Times[-1, b, e, g]], Power[Times[c, e, Plus[m, Times[-1, n], -1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], Plus[n, -1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g, m, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], ZeroQ[Plus[m, p]], RationalQ[n], Greater[n, 0], NonzeroQ[Plus[m, Times[-1, n], -1]], Or[IntegerQ[Times[2, p]], IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[e, Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], n], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, Plus[m, Times[-1, n], -1]], -1]], x]], Times[-1, Dist[Times[n, Plus[Times[e, f], Times[d, g]], Power[Times[e, Plus[m, Times[-1, n], -1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], Plus[n, -1]], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x]]], And[FreeQ[List[a, c, d, e, f, g, m, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], ZeroQ[Plus[m, p]], RationalQ[n], Greater[n, 0], NonzeroQ[Plus[m, Times[-1, n], -1]], Or[IntegerQ[Times[2, p]], IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[e, 2], Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], Plus[n, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[Plus[n, 1], Plus[Times[c, e, f], Times[c, d, g], Times[-1, b, e, g]]], -1]], x]], Times[-1, Dist[Times[c, e, Plus[m, Times[-1, n], -2], Power[Times[Plus[n, 1], Plus[Times[c, e, f], Times[c, d, g], Times[-1, b, e, g]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], Plus[n, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g, m, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], ZeroQ[Plus[m, p]], RationalQ[n], Less[n, -1], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[e, 2], Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], Plus[n, 1]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[Plus[n, 1], Plus[Times[c, e, f], Times[c, d, g]]], -1]], x]], Times[-1, Dist[Times[e, Plus[m, Times[-1, n], -2], Power[Times[Plus[n, 1], Plus[Times[e, f], Times[d, g]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], Plus[n, 1]], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x]]], And[FreeQ[List[a, c, d, e, f, g, m, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], ZeroQ[Plus[m, p]], RationalQ[n], Less[n, -1], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, 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Blank[]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[2, Power[e, 2]], Subst[Int[Times[1, Power[Plus[Times[c, Plus[Times[e, f], Times[d, g]]], Times[Power[e, 2], g, Power[x, 2]]], -1]], x], x, Times[Sqrt[Plus[a, Times[c, Power[x, 2]]]], Power[Sqrt[Plus[d, Times[e, x]]], -1]]], x], And[FreeQ[List[a, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[e, 2], Power[Plus[d, Times[e, x]], Plus[m, -2]], Power[Plus[f, Times[g, x]], Plus[n, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, g, Plus[n, p, 2]], -1]], x], And[FreeQ[List[a, b, c, d, e, f, g, m, n, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], ZeroQ[Plus[m, p, -1]], ZeroQ[Plus[Times[b, e, g, Plus[n, 1]], Times[c, e, f, Plus[p, 1]], Times[-1, c, d, g, Plus[Times[2, n], p, 3]]]], NonzeroQ[Plus[n, p, 2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], 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x]]], And[FreeQ[List[a, b, c, d, e, f, g, m, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], ZeroQ[Plus[m, p, -1]], RationalQ[n], Less[n, -1], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[e, 2], Plus[Times[e, f], Times[-1, d, g]], Power[Plus[d, Times[e, x]], Plus[m, -2]], Power[Plus[f, Times[g, x]], Plus[n, 1]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, g, Plus[n, 1], Plus[Times[e, f], Times[d, g]]], -1]], x], Times[-1, Dist[Times[e, Plus[Times[e, f, Plus[p, 1]], Times[-1, d, g, Plus[Times[2, n], p, 3]]], Power[Times[g, Plus[n, 1], Plus[Times[e, f], Times[d, g]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], Plus[n, 1]], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x]]], And[FreeQ[List[a, c, d, e, f, g, m, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], ZeroQ[Plus[m, p, -1]], RationalQ[n], Less[n, -1], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[e, 2], Power[Plus[d, Times[e, x]], Plus[m, -2]], Power[Plus[f, Times[g, x]], Plus[n, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, g, Plus[n, p, 2]], -1]], x], Times[-1, Dist[Times[Plus[Times[b, e, g, Plus[n, 1]], Times[c, e, f, Plus[p, 1]], Times[-1, c, d, g, Plus[Times[2, n], p, 3]]], Power[Times[c, g, Plus[n, p, 2]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], n], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g, m, n, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], ZeroQ[Plus[m, p, -1]], Not[And[RationalQ[n], Less[n, -1]]], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[e, 2], Power[Plus[d, Times[e, x]], Plus[m, -2]], Power[Plus[f, Times[g, x]], Plus[n, 1]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, g, Plus[n, p, 2]], -1]], x], Times[-1, Dist[Times[Plus[Times[e, f, Plus[p, 1]], Times[-1, d, g, Plus[Times[2, n], p, 3]]], Power[Times[g, Plus[n, p, 2]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], n], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x]]], And[FreeQ[List[a, c, d, e, f, g, m, n, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], ZeroQ[Plus[m, p, -1]], Not[And[RationalQ[n], Less[n, -1]]], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], n], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, n, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], Or[PositiveIntegerQ[m], IntegersQ[m, n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[1, Power[Sqrt[Plus[a, Times[c, Power[x, 2]]]], -1]], Times[Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], n], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, Times[1, Power[2, -1]]]]], x], x], And[FreeQ[List[a, c, d, e, f, g, n, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], IntegerQ[Plus[p, Times[-1, 1, 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Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Power[d, 2], Power[Power[e, 2], -1]], Int[Times[Power[Plus[a, Times[c, Power[x, 2]]], p], Power[Plus[d, Times[e, x]], -1]], x], x], Times[-1, Dist[Times[1, Power[Power[e, 2], -1]], Int[Times[Plus[d, Times[-1, e, x]], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x]]], And[FreeQ[List[a, c, d, e, p], x], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], 2], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], 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x], x]]], And[FreeQ[List[a, c, d, e, f, g, m, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[p]], NonzeroQ[Plus[m, Times[2, p], 3]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[e, x], m], Power[Plus[Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[Power[x, Plus[m, p]], Power[Plus[b, Times[c, x]], p]], -1]], Int[Times[Power[x, Plus[m, p]], Power[Plus[f, Times[g, x]], n], Power[Plus[b, Times[c, x]], p]], x], x], And[FreeQ[List[b, c, e, f, g, m, n], x], Not[IntegerQ[p]]]]], 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RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[d, Times[e, x]], m], Plus[f, Times[g, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, m], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], PositiveIntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, 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Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Simp[Times[Plus[Times[b, e, g, Plus[p, 2]], Times[-1, c, Plus[Times[e, f], Times[d, g]], Plus[Times[2, p], 3]], Times[-1, 2, c, e, g, Plus[p, 1], x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[2, Power[c, 2], Plus[p, 1], Plus[Times[2, p], 3]], -1]], x]], And[FreeQ[List[a, b, c, d, e, f, g, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], ZeroQ[Plus[Times[Power[b, 2], e, g, Plus[p, 2]], Times[-1, 2, a, c, e, g], Times[c, Plus[Times[2, c, d, f], Times[-1, b, Plus[Times[e, f], Times[d, g]]]], Plus[Times[2, p], 3]]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Plus[Times[Plus[Times[e, f], Times[d, g]], Plus[Times[2, p], 3]], Times[2, e, g, Plus[p, 1], x]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[2, c, Plus[p, 1], Plus[Times[2, p], 3]], -1]], x], And[FreeQ[List[a, c, d, e, f, g, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], ZeroQ[Plus[Times[a, e, g], Times[-1, c, d, f, Plus[Times[2, p], 3]]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[2, a, c, Plus[Times[e, f], Times[d, g]]], Times[-1, b, Plus[Times[c, d, f], Times[a, e, g]]], Times[-1, Plus[Times[Power[b, 2], e, g], Times[-1, b, c, Plus[Times[e, f], Times[d, g]]], Times[2, c, Plus[Times[c, d, f], Times[-1, a, e, g]]]], x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], x]], Times[-1, Dist[Times[Plus[Times[Power[b, 2], e, g, Plus[p, 2]], Times[-1, 2, a, c, e, g], Times[c, Plus[Times[2, c, d, f], Times[-1, b, Plus[Times[e, f], Times[d, g]]]], Plus[Times[2, p], 3]]], Power[Times[c, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], Int[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], RationalQ[p], Less[p, -1]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[c, d, f, x], Times[-1, a, Plus[Times[d, g], Times[e, Plus[f, Times[g, x]]]]]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[2, a, c, Plus[p, 1]], -1]], x]], Times[-1, Dist[Times[Plus[Times[a, e, g], Times[-1, c, d, f, Plus[Times[2, p], 3]]], Power[Times[2, a, c, Plus[p, 1]], -1]], Int[Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], x], x]]], And[FreeQ[List[a, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], RationalQ[p], Less[p, -1]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[b, e, g, Plus[p, 2]], Times[-1, c, Plus[Times[e, f], Times[d, g]], Plus[Times[2, p], 3]], Times[-1, 2, c, e, g, Plus[p, 1], x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[2, Power[c, 2], Plus[p, 1], Plus[Times[2, p], 3]], -1]], x]], Dist[Times[Plus[Times[Power[b, 2], e, g, Plus[p, 2]], Times[-1, 2, a, c, e, g], Times[c, Plus[Times[2, c, d, f], Times[-1, b, Plus[Times[e, f], Times[d, g]]]], Plus[Times[2, p], 3]]], Power[Times[2, Power[c, 2], Plus[Times[2, p], 3]], -1]], Int[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Not[And[RationalQ[p], LessEqual[p, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[Plus[Times[e, f], Times[d, g]], Plus[Times[2, p], 3]], Times[2, e, g, Plus[p, 1], x]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[2, c, Plus[p, 1], Plus[Times[2, p], 3]], -1]], x], Times[-1, Dist[Times[Plus[Times[a, e, g], Times[-1, c, d, f, Plus[Times[2, p], 3]]], Power[Times[c, Plus[Times[2, p], 3]], -1]], Int[Power[Plus[a, Times[c, Power[x, 2]]], p], x], x]]], And[FreeQ[List[a, c, d, e, f, g, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[And[RationalQ[p], LessEqual[p, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[f, Int[Times[Power[Times[e, x], m], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x], Dist[Times[g, Power[e, -1]], Int[Times[Power[Times[e, x], Plus[m, 1]], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x]], And[FreeQ[List[a, c, e, f, g, p], x], Not[RationalQ[m]], Not[PositiveIntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[d, Times[e, x]], FracPart[p]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], FracPart[p]], Power[Power[Plus[Times[a, d], Times[c, e, Power[x, 3]]], FracPart[p]], -1]], Int[Times[Plus[f, Times[g, x]], Power[Plus[Times[a, d], Times[c, e, Power[x, 3]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, m, p], x], ZeroQ[Plus[m, Times[-1, p]]], ZeroQ[Plus[Times[b, d], Times[a, e]]], ZeroQ[Plus[Times[c, d], Times[b, e]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[Power[e, 2], Plus[m, 1], Plus[m, 2], Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], -1], Plus[Times[Plus[Times[d, g], Times[-1, e, f, Plus[m, 2]]], Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Times[-1, d, p, Plus[Times[2, c, d], Times[-1, b, e]], Plus[Times[e, f], Times[-1, d, g]]], Times[-1, e, Plus[Times[g, Plus[m, 1], Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Times[p, Plus[Times[2, c, d], Times[-1, b, e]], Plus[Times[e, f], Times[-1, d, g]]]], x]]], x]], Times[-1, Dist[Times[p, Power[Times[Power[e, 2], Plus[m, 1], Plus[m, 2], Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 2]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, -1]], Simp[Plus[Times[2, a, c, e, Plus[Times[e, f], Times[-1, d, g]], Plus[m, 2]], Times[Power[b, 2], e, Plus[Times[d, g, Plus[p, 1]], Times[-1, e, f, Plus[m, p, 2]]]], Times[b, Plus[Times[a, Power[e, 2], g, Plus[m, 1]], Times[-1, c, d, Plus[Times[d, g, Plus[Times[2, p], 1]], Times[-1, e, f, Plus[m, Times[2, p], 2]]]]]], Times[-1, c, Plus[Times[2, c, d, Plus[Times[d, g, Plus[Times[2, p], 1]], Times[-1, e, f, Plus[m, Times[2, p], 2]]]], Times[-1, e, Plus[Times[2, a, e, g, Plus[m, 1]], Times[-1, b, Plus[Times[d, g, Plus[m, Times[-1, 2, p]]], Times[e, f, Plus[m, Times[2, p], 2]]]]]]], x]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], RationalQ[m, p], Greater[p, 0], Less[m, -2], Less[Plus[m, Times[2, p]], 0], Not[NegativeIntegerQ[Plus[m, Times[2, p], 3]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[c, Power[x, 2]]], p], Power[Times[Power[e, 2], Plus[m, 1], Plus[m, 2], Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], -1], Plus[Times[Plus[Times[d, g], Times[-1, e, f, Plus[m, 2]]], Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Times[-1, 2, c, Power[d, 2], p, Plus[Times[e, f], Times[-1, d, g]]], Times[-1, e, Plus[Times[g, Plus[m, 1], Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Times[2, c, d, p, Plus[Times[e, f], Times[-1, d, g]]]], x]]], x]], Times[-1, Dist[Times[p, Power[Times[Power[e, 2], Plus[m, 1], Plus[m, 2], Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 2]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, -1]], Simp[Plus[Times[2, a, c, e, Plus[Times[e, f], Times[-1, d, g]], Plus[m, 2]], Times[-1, c, Plus[Times[2, c, d, Plus[Times[d, g, Plus[Times[2, p], 1]], Times[-1, e, f, Plus[m, Times[2, p], 2]]]], Times[-1, 2, a, Power[e, 2], g, Plus[m, 1]]], x]], x]], x], x]]], And[FreeQ[List[a, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], RationalQ[m, p], Greater[p, 0], Less[m, -2], Less[Plus[m, Times[2, p]], 0], Not[NegativeIntegerQ[Plus[m, Times[2, p], 3]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Plus[Times[e, f, Plus[m, Times[2, p], 2]], Times[-1, d, g, Plus[Times[2, p], 1]], Times[e, g, Plus[m, 1], x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[Power[e, 2], Plus[m, 1], Plus[m, Times[2, p], 2]], -1]], x], Dist[Times[p, Power[Times[Power[e, 2], Plus[m, 1], Plus[m, Times[2, p], 2]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, -1]], Simp[Plus[Times[g, Plus[Times[b, d], Times[2, a, e], Times[2, a, e, m], Times[2, b, d, p]]], Times[-1, f, b, e, Plus[m, Times[2, p], 2]], Times[Plus[Times[g, Plus[Times[2, c, d], Times[b, e], Times[b, e, m], Times[4, c, d, p]]], Times[-1, 2, c, e, f, Plus[m, Times[2, p], 2]]], x]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, m], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], RationalQ[p], Greater[p, 0], Or[And[RationalQ[m], Less[m, -1]], Equal[p, 1], And[IntegerQ[p], Not[RationalQ[m]]]], NonzeroQ[Plus[m, 1]], Not[NegativeIntegerQ[Plus[m, Times[2, p], 1]]], Or[IntegerQ[m], IntegerQ[p], IntegersQ[Times[2, m], Times[2, p]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Plus[Times[e, f, Plus[m, Times[2, p], 2]], Times[-1, d, g, Plus[Times[2, p], 1]], Times[e, g, Plus[m, 1], x]], Power[Plus[a, Times[c, Power[x, 2]]], p], Power[Times[Power[e, 2], Plus[m, 1], Plus[m, Times[2, p], 2]], -1]], x], Dist[Times[p, Power[Times[Power[e, 2], Plus[m, 1], Plus[m, Times[2, p], 2]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, -1]], Simp[Plus[Times[g, Plus[Times[2, a, e], Times[2, a, e, m]]], Times[Plus[Times[g, Plus[Times[2, c, d], Times[4, c, d, p]]], Times[-1, 2, c, e, f, Plus[m, Times[2, p], 2]]], x]], x]], x], x]], And[FreeQ[List[a, c, d, e, f, g, m], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], RationalQ[p], Greater[p, 0], Or[And[RationalQ[m], Less[m, -1]], Equal[p, 1], And[IntegerQ[p], Not[RationalQ[m]]]], NonzeroQ[Plus[m, 1]], Not[NegativeIntegerQ[Plus[m, Times[2, p], 1]]], Or[IntegerQ[m], IntegerQ[p], IntegersQ[Times[2, m], Times[2, p]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Plus[Times[c, e, f, Plus[m, Times[2, p], 2]], Times[-1, g, Plus[Times[c, d], Times[2, c, d, p], Times[-1, b, e, p]]], Times[g, c, e, Plus[m, Times[2, p], 1], x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Times[c, Power[e, 2], Plus[m, Times[2, p], 1], Plus[m, Times[2, p], 2]], -1]], x], Times[-1, Dist[Times[p, Power[Times[c, Power[e, 2], Plus[m, Times[2, p], 1], Plus[m, Times[2, p], 2]], -1]], Int[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, -1]], Simp[Plus[Times[c, e, f, Plus[Times[b, d], Times[-1, 2, a, e]], Plus[m, Times[2, p], 2]], Times[g, Plus[Times[a, e, Plus[Times[b, e], Times[-1, 2, c, d, m], Times[b, e, m]]], Times[b, d, Plus[Times[b, e, p], Times[-1, c, d], Times[-1, 2, c, d, p]]]]], Times[Plus[Times[c, e, f, Plus[Times[2, c, d], Times[-1, b, e]], Plus[m, Times[2, p], 2]], Times[g, Plus[Times[Power[b, 2], Power[e, 2], Plus[p, m, 1]], Times[-1, 2, Power[c, 2], Power[d, 2], Plus[1, Times[2, p]]], Times[-1, c, e, Plus[Times[b, d, Plus[m, Times[-1, 2, p]]], Times[2, a, e, Plus[m, Times[2, p], 1]]]]]]], x]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g, m], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], RationalQ[p], Greater[p, 0], Or[IntegerQ[p], Not[RationalQ[m]], Inequality[-1, LessEqual, m, Less, 0]], Not[NegativeIntegerQ[Plus[m, Times[2, p]]]], Or[IntegerQ[m], IntegerQ[p], IntegersQ[Times[2, m], Times[2, p]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, 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Times[a, c, Power[e, 2], Plus[m, Times[2, p], 1]]]]], x]], x]], x], x]], And[FreeQ[List[a, c, d, e, f, g, m], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], RationalQ[p], Greater[p, 0], Or[IntegerQ[p], Not[RationalQ[m]], Inequality[-1, LessEqual, m, Less, 0]], Not[NegativeIntegerQ[Plus[m, Times[2, p]]]], Or[IntegerQ[m], IntegerQ[p], IntegersQ[Times[2, m], Times[2, p]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 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Times[2, Power[c, 2], d, f, Plus[m, Times[2, p], 2]], Times[-1, c, Plus[Times[2, a, e, g, m], Times[b, Plus[Times[e, f], Times[d, g]], Plus[m, Times[2, p], 2]]]]], x]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], RationalQ[m, p], Less[p, -1], Greater[m, 1], Or[And[Equal[m, 2], Equal[p, -3], RationalQ[a, b, c, d, e, f, g]], Not[NegativeIntegerQ[Plus[m, Times[2, p], 3]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], 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RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Plus[Times[f, b], Times[-1, 2, a, g], Times[Plus[Times[2, c, f], Times[-1, b, g]], x]], Power[Times[Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], x], Dist[Times[1, Power[Times[Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Simp[Plus[Times[g, Plus[Times[2, a, e, m], Times[b, d, Plus[Times[2, p], 3]]]], Times[-1, f, Plus[Times[b, e, m], Times[2, c, d, Plus[Times[2, p], 3]]]], Times[-1, e, Plus[Times[2, c, f], Times[-1, b, g]], Plus[m, Times[2, p], 3], x]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], RationalQ[m, p], Less[p, -1], Greater[m, 0], Not[And[Equal[m, 1], SimplerQ[Plus[d, Times[e, x]], Plus[f, Times[g, x]]]]], Or[IntegerQ[m], IntegerQ[p], IntegersQ[Times[2, m], Times[2, p]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Plus[Times[a, g], Times[-1, c, f, x]], Power[Times[2, a, c, Plus[p, 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[2, a, c, Plus[p, 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Simp[Plus[Times[a, e, g, m], Times[-1, c, d, f, Plus[Times[2, p], 3]], Times[-1, c, e, f, Plus[m, Times[2, p], 3], x]], x]], x], x]]], And[FreeQ[List[a, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], RationalQ[m, p], Less[p, -1], Greater[m, 0], Not[And[Equal[m, 1], SimplerQ[Plus[d, Times[e, x]], Plus[f, Times[g, x]]]]], Or[IntegerQ[m], IntegerQ[p], IntegersQ[Times[2, m], Times[2, p]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Plus[Times[f, Plus[Times[b, c, d], Times[-1, Power[b, 2], e], Times[2, a, c, e]]], Times[-1, a, g, Plus[Times[2, c, d], Times[-1, b, e]]], Times[c, Plus[Times[f, Plus[Times[2, c, d], Times[-1, b, e]]], Times[-1, g, Plus[Times[b, d], Times[-1, 2, a, e]]]], x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], -1]], x], Dist[Times[1, Power[Times[Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]], 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Times[c, Power[x, 2]]], Plus[p, 1]], Simp[Plus[Times[f, Plus[Times[Power[c, 2], Power[d, 2], Plus[Times[2, p], 3]], Times[a, c, Power[e, 2], Plus[m, Times[2, p], 3]]]], Times[-1, a, c, d, e, g, m], Times[c, e, Plus[Times[c, d, f], Times[a, e, g]], Plus[m, Times[2, p], 4], x]], x]], x], x]], And[FreeQ[List[a, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], RationalQ[p], Less[p, -1], Or[IntegerQ[m], IntegerQ[p], IntegersQ[Times[2, m], Times[2, p]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, 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NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[g, Power[Plus[d, Times[e, x]], m], Power[Times[c, m], -1]], x], Dist[Times[1, Power[c, -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Simp[Plus[Times[c, d, f], Times[-1, a, e, g], Times[Plus[Times[g, c, d], Times[-1, b, e, g], Times[c, e, f]], x]], x], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], FractionQ[m], Greater[m, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[g, Power[Plus[d, Times[e, x]], m], Power[Times[c, m], -1]], x], Dist[Times[1, Power[c, -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Simp[Plus[Times[c, d, f], Times[-1, a, e, g], Times[Plus[Times[g, c, d], Times[c, e, f]], x]], x], Power[Plus[a, Times[c, Power[x, 2]]], -1]], x], x]], And[FreeQ[List[a, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, 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NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[2, Subst[Int[Times[Plus[Times[e, f], Times[-1, d, g], Times[g, Power[x, 2]]], Power[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]], Times[-1, 2, c, d, Power[x, 2]], Times[c, Power[x, 4]]], -1]], x], x, Sqrt[Plus[d, Times[e, x]]]], x], And[FreeQ[List[a, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, 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Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[g, Power[e, -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x], Dist[Times[Plus[Times[e, f], Times[-1, d, g]], Power[e, -1]], Int[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x]], And[FreeQ[List[a, c, d, e, f, g, m, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], n], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], IntegersQ[m, n, p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], n], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x], And[FreeQ[List[a, c, d, e, f, g], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], IntegersQ[m, n, p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]], Power[Times[e, Plus[Times[e, f], Times[-1, d, g]]], -1]], Int[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, -1]], Power[Plus[d, Times[e, x]], -1]], x], x], Times[-1, Dist[Times[1, Power[Times[e, Plus[Times[e, f], Times[-1, d, g]]], -1]], Int[Times[Simp[Plus[Times[c, d, f], Times[-1, b, e, f], Times[a, e, g], Times[-1, c, Plus[Times[e, f], Times[-1, d, g]], x]], x], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, -1]], Power[Plus[f, Times[g, x]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], FractionQ[p], Greater[p, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], Power[Times[e, Plus[Times[e, f], Times[-1, d, g]]], -1]], Int[Times[Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, -1]], Power[Plus[d, Times[e, x]], -1]], x], x], Times[-1, Dist[Times[1, Power[Times[e, Plus[Times[e, f], Times[-1, d, g]]], -1]], Int[Times[Simp[Plus[Times[c, d, f], Times[a, e, g], Times[-1, c, Plus[Times[e, f], Times[-1, d, g]], x]], x], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, -1]], Power[Plus[f, Times[g, x]], -1]], x], x]]], And[FreeQ[List[a, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], FractionQ[p], Greater[p, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Denominator[m]]], Dist[Times[q, Power[e, -1]], Subst[Int[Times[Power[x, Plus[Times[q, Plus[m, 1]], -1]], Power[Plus[Times[Plus[Times[e, f], Times[-1, d, g]], Power[e, -1]], Times[g, Power[x, q], Power[e, -1]]], n], Power[Plus[Times[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]], Power[Power[e, 2], -1]], Times[-1, Plus[Times[2, c, d], Times[-1, b, e]], Power[x, q], Power[Power[e, 2], -1]], Times[c, Power[x, Times[2, q]], Power[Power[e, 2], -1]]], p]], x], x, Power[Plus[d, Times[e, x]], Times[1, Power[q, -1]]]], x]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], IntegersQ[n, p], FractionQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Denominator[m]]], Dist[Times[q, Power[e, -1]], Subst[Int[Times[Power[x, Plus[Times[q, Plus[m, 1]], -1]], Power[Plus[Times[Plus[Times[e, f], Times[-1, d, g]], Power[e, -1]], Times[g, Power[x, q], Power[e, -1]]], n], Power[Plus[Times[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], Power[Power[e, 2], -1]], Times[-1, 2, c, d, Power[x, q], Power[Power[e, 2], -1]], Times[c, Power[x, Times[2, q]], Power[Power[e, 2], -1]]], p]], x], x, Power[Plus[d, Times[e, x]], Times[1, Power[q, -1]]]], x]], And[FreeQ[List[a, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], IntegersQ[n, p], FractionQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[Times[d, f], Times[e, g, Power[x, 2]]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], And[FreeQ[List[a, b, c, d, e, f, g, m, n, p], x], ZeroQ[Plus[m, Times[-1, n]]], ZeroQ[Plus[Times[e, f], Times[d, g]]], Or[IntegerQ[m], And[PositiveQ[d], PositiveQ[f]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[Times[d, f], Times[e, g, Power[x, 2]]], m], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], And[FreeQ[List[a, c, d, e, f, g, m, n, p], x], ZeroQ[Plus[m, Times[-1, n]]], ZeroQ[Plus[Times[e, f], Times[d, g]]], Or[IntegerQ[m], And[PositiveQ[d], PositiveQ[f]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[d, Times[e, x]], FracPart[m]], Power[Plus[f, Times[g, x]], FracPart[m]], Power[Power[Plus[Times[d, f], Times[e, g, Power[x, 2]]], FracPart[m]], -1]], Int[Times[Power[Plus[Times[d, f], Times[e, g, Power[x, 2]]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, m, n, p], x], ZeroQ[Plus[m, Times[-1, n]]], ZeroQ[Plus[Times[e, f], Times[d, g]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[d, Times[e, x]], FracPart[m]], Power[Plus[f, Times[g, x]], FracPart[m]], Power[Power[Plus[Times[d, f], Times[e, g, Power[x, 2]]], FracPart[m]], -1]], Int[Times[Power[Plus[Times[d, f], Times[e, g, Power[x, 2]]], m], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x], And[FreeQ[List[a, c, d, e, f, g, m, n, p], x], ZeroQ[Plus[m, Times[-1, n]]], ZeroQ[Plus[Times[e, f], Times[d, g]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Int[Times[Plus[a, Times[b, x]], Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], n]], x], Dist[c, Int[Times[Power[x, 2], Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], n]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], RationalQ[m, n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[a, Int[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], n]], x], x], Dist[c, Int[Times[Power[x, 2], Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], n]], x], x]], And[FreeQ[List[a, c, d, e, f, g], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], RationalQ[m, n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[g, Power[Power[c, 2], -1]], Int[Times[Simp[Plus[Times[2, c, e, f], Times[c, d, g], Times[-1, b, e, g], Times[c, e, g, x]], x], Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], Plus[n, -2]]], x], x], Dist[Times[1, Power[Power[c, 2], -1]], Int[Times[Simp[Plus[Times[Power[c, 2], d, Power[f, 2]], Times[-1, 2, a, c, e, f, g], Times[-1, a, c, d, Power[g, 2]], Times[a, b, e, Power[g, 2]], Times[Plus[Times[Power[c, 2], e, Power[f, 2]], Times[2, Power[c, 2], d, f, g], Times[-1, 2, b, c, e, f, g], Times[-1, b, c, d, Power[g, 2]], Times[Power[b, 2], e, Power[g, 2]], Times[-1, a, c, e, Power[g, 2]]], x]], x], Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], Plus[n, -2]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Not[IntegerQ[m]], Not[IntegerQ[n]], RationalQ[m, n], Greater[m, 0], Greater[n, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[g, Power[c, -1]], Int[Times[Simp[Plus[Times[2, e, f], Times[d, g], Times[e, g, x]], x], Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], Plus[n, -2]]], x], x], Dist[Times[1, Power[c, -1]], Int[Times[Simp[Plus[Times[c, d, Power[f, 2]], Times[-1, 2, a, e, f, g], Times[-1, a, d, Power[g, 2]], Times[Plus[Times[c, e, Power[f, 2]], Times[2, c, d, f, g], Times[-1, a, e, Power[g, 2]]], x]], x], Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], Plus[n, -2]], Power[Plus[a, Times[c, Power[x, 2]]], -1]], x], x]], And[FreeQ[List[a, c, d, e, f, g], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[m]], Not[IntegerQ[n]], RationalQ[m, n], Greater[m, 0], Greater[n, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[e, g, Power[c, -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], Plus[n, -1]]], x], x], Dist[Times[1, Power[c, -1]], Int[Times[Simp[Plus[Times[c, d, f], Times[-1, a, e, g], Times[Plus[Times[c, e, f], Times[c, d, g], Times[-1, b, e, g]], x]], x], Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], Plus[n, -1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Not[IntegerQ[m]], Not[IntegerQ[n]], RationalQ[m, n], Greater[m, 0], Greater[n, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[e, g, Power[c, -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], Plus[n, -1]]], x], x], Dist[Times[1, Power[c, -1]], Int[Times[Simp[Plus[Times[c, d, f], Times[-1, a, e, g], Times[Plus[Times[c, e, f], Times[c, d, g]], x]], x], Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], Plus[n, -1]], Power[Plus[a, Times[c, Power[x, 2]]], -1]], x], x]], And[FreeQ[List[a, c, d, e, f, g], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[m]], Not[IntegerQ[n]], RationalQ[m, n], Greater[m, 0], Greater[n, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Dist[Times[g, Plus[Times[e, f], Times[-1, d, g]], Power[Plus[Times[c, Power[f, 2]], Times[-1, b, f, g], Times[a, Power[g, 2]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], n]], x], x]], Dist[Times[1, Power[Plus[Times[c, Power[f, 2]], Times[-1, b, f, g], Times[a, Power[g, 2]]], -1]], Int[Times[Simp[Plus[Times[c, d, f], Times[-1, b, d, g], Times[a, e, g], Times[c, Plus[Times[e, f], Times[-1, d, g]], x]], x], Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], Plus[n, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Not[IntegerQ[m]], Not[IntegerQ[n]], RationalQ[m, n], Greater[m, 0], Less[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Dist[Times[g, Plus[Times[e, f], Times[-1, d, g]], Power[Plus[Times[c, Power[f, 2]], Times[a, Power[g, 2]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], n]], x], x]], Dist[Times[1, Power[Plus[Times[c, Power[f, 2]], Times[a, Power[g, 2]]], -1]], Int[Times[Simp[Plus[Times[c, d, f], Times[a, e, g], Times[c, Plus[Times[e, f], Times[-1, d, g]], x]], x], Power[Plus[d, Times[e, x]], Plus[m, -1]], Power[Plus[f, Times[g, x]], Plus[n, 1]], Power[Plus[a, Times[c, Power[x, 2]]], -1]], x], x]], And[FreeQ[List[a, c, d, e, f, g], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[m]], Not[IntegerQ[n]], RationalQ[m, n], Greater[m, 0], Less[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, 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RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], n], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Or[IntegerQ[p], IntegersQ[m, n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, 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2]]], Simp[Times[-2, Sqrt[Times[g, Plus[q, Times[-1, c, x]], Power[Plus[Times[c, f], Times[g, q]], -1]]], Sqrt[Times[-1, g, Plus[q, Times[c, x]], Power[Plus[Times[c, f], Times[-1, g, q]], -1]]], Power[Times[Plus[Times[e, f], Times[-1, d, g]], Sqrt[Times[c, Power[Plus[Times[c, f], Times[g, q]], -1]]], Sqrt[Plus[a, Times[c, Power[x, 2]]]]], -1], EllipticPi[Times[e, Plus[Times[c, f], Times[g, q]], Power[Times[c, Plus[Times[e, f], Times[-1, d, g]]], -1]], ArcSin[Times[Sqrt[Times[c, Power[Plus[Times[c, f], Times[g, q]], -1]]], Sqrt[Plus[f, Times[g, x]]]]], Times[Plus[Times[c, f], Times[g, q]], Power[Plus[Times[c, f], Times[-1, g, q]], -1]]]], x]], And[FreeQ[List[a, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], 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Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[-2, Plus[d, Times[e, x]], Sqrt[Times[Power[Plus[Times[e, f], Times[-1, d, g]], 2], Plus[a, Times[b, x], Times[c, Power[x, 2]]], Power[Times[Plus[Times[c, Power[f, 2]], Times[-1, b, f, g], Times[a, Power[g, 2]]], Power[Plus[d, Times[e, x]], 2]], -1]]], Power[Times[Plus[Times[e, f], Times[-1, d, g]], Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]], -1]], Subst[Int[Times[1, Power[Sqrt[Plus[1, Times[-1, Plus[Times[2, c, d, f], Times[-1, b, e, f], Times[-1, b, d, g], Times[2, a, e, g]], Power[x, 2], Power[Plus[Times[c, Power[f, 2]], Times[-1, b, f, g], Times[a, Power[g, 2]]], -1]], Times[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]], Power[x, 4], Power[Plus[Times[c, Power[f, 2]], Times[-1, b, f, g], Times[a, Power[g, 2]]], -1]]]], -1]], x], x, Times[Sqrt[Plus[f, Times[g, x]]], 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Power[x, 2]]]]], -1]], Subst[Int[Times[1, Power[Sqrt[Plus[1, Times[-1, Plus[Times[2, c, d, f], Times[2, a, e, g]], Power[x, 2], Power[Plus[Times[c, Power[f, 2]], Times[a, Power[g, 2]]], -1]], Times[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], Power[x, 4], Power[Plus[Times[c, Power[f, 2]], Times[a, Power[g, 2]]], -1]]]], -1]], x], x, Times[Sqrt[Plus[f, Times[g, x]]], Power[Sqrt[Plus[d, Times[e, x]]], -1]]], x], And[FreeQ[List[a, c, d, e, f, g], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[2, f, g, Power[e, -1]], Int[Times[Power[Times[e, x], Plus[m, 1]], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x], Int[Times[Power[Times[e, x], m], Plus[Power[f, 2], Times[Power[g, 2], Power[x, 2]]], Power[Plus[a, Times[c, Power[x, 2]]], p]], x]], FreeQ[List[a, c, e, f, g, m, p], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], 3], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[f, Int[Times[Power[Times[e, x], m], Plus[Power[f, 2], Times[3, Power[g, 2], Power[x, 2]]], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x], Dist[Times[g, Power[e, -1]], Int[Times[Power[Times[e, x], Plus[m, 1]], Plus[Times[3, Power[f, 2]], Times[Power[g, 2], Power[x, 2]]], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x]], FreeQ[List[a, c, e, f, g, m, p], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[g, Power[e, -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[f, Times[g, x]], Plus[n, -1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x], Dist[Times[Plus[Times[e, f], Times[-1, d, g]], Power[e, -1]], Int[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], Plus[n, -1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, m, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[g, Power[e, -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Power[Plus[f, Times[g, x]], Plus[n, -1]], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x], Dist[Times[Plus[Times[e, f], Times[-1, d, g]], Power[e, -1]], Int[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], Plus[n, -1]], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x]], And[FreeQ[List[a, c, d, e, f, g, m, p], x], NonzeroQ[Plus[Times[e, f], Times[-1, d, g]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], n], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], FreeQ[List[a, b, c, d, e, f, g, m, n, p], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], n], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], FreeQ[List[a, c, d, e, f, g, m, n, p], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[u, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[u, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[u, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Coefficient[u, x, 1], -1]], Subst[Int[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], n], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x, u], x], And[FreeQ[List[a, b, c, d, e, f, g, m, n, p], x], LinearQ[u, x], NonzeroQ[Plus[u, Times[-1, x]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[u, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[u, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Coefficient[u, x, 1], -1]], Subst[Int[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], n], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x, u], x], And[FreeQ[List[a, c, d, e, f, g, m, n, p], x], LinearQ[u, x], NonzeroQ[Plus[u, Times[-1, x]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[Times[c, Power[f, -1]], p], Int[Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[p, q]], x], x], And[FreeQ[List[a, b, c, d, e, f, p, q], x], ZeroQ[Plus[Times[c, d], Times[-1, a, f]]], ZeroQ[Plus[Times[b, d], Times[-1, a, e]]], Or[IntegerQ[p], PositiveQ[Times[c, Power[f, -1]]]], Or[Not[IntegerQ[q]], LessEqual[LeafCount[Plus[d, Times[e, x], Times[f, Power[x, 2]]]], LeafCount[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[a, IntPart[p]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], FracPart[p]], Power[Times[Power[d, IntPart[p]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], FracPart[p]]], -1]], Int[Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[p, q]], x], x], And[FreeQ[List[a, b, c, d, e, f, p, q], x], ZeroQ[Plus[Times[c, d], Times[-1, a, f]]], ZeroQ[Plus[Times[b, d], Times[-1, a, e]]], Not[IntegerQ[p]], Not[IntegerQ[q]], Not[PositiveQ[Times[c, Power[f, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], FracPart[p]], Power[Times[Power[Times[4, c], IntPart[p]], Power[Plus[b, Times[2, c, x]], Times[2, FracPart[p]]]], -1]], Int[Times[Power[Plus[b, Times[2, c, x]], Times[2, p]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q]], x], x], And[FreeQ[List[a, b, c, d, e, f, p, q], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], FracPart[p]], Power[Times[Power[Times[4, c], IntPart[p]], Power[Plus[b, Times[2, c, x]], Times[2, FracPart[p]]]], -1]], Int[Times[Power[Plus[b, Times[2, c, x]], Times[2, p]], Power[Plus[d, Times[f, Power[x, 2]]], q]], x], x], And[FreeQ[List[a, b, c, d, f, p, q], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Plus[Times[b, f, Plus[Times[2, q], 3]], Times[-1, c, e, Plus[q, 2]], Times[2, c, f, Plus[q, 1], x]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[2, Power[f, 2], Plus[q, 1], Plus[Times[2, q], 3]], -1]], x], And[FreeQ[List[a, b, c, d, e, f, q], x], ZeroQ[Plus[Times[f, Plus[Times[2, a, f], Times[-1, b, e]], Plus[Times[2, q], 3]], Times[c, Plus[Times[Power[e, 2], Plus[q, 2]], Times[-1, 2, d, f]]]]], NonzeroQ[Plus[q, 1]], NonzeroQ[Plus[Times[2, q], 3]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Plus[Times[-1, c, e, Plus[q, 2]], Times[2, c, f, Plus[q, 1], x]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[2, Power[f, 2], Plus[q, 1], Plus[Times[2, q], 3]], -1]], x], And[FreeQ[List[a, c, d, e, f, q], x], ZeroQ[Plus[Times[2, a, Power[f, 2], Plus[Times[2, q], 3]], Times[c, Plus[Times[Power[e, 2], Plus[q, 2]], Times[-1, 2, d, f]]]]], NonzeroQ[Plus[q, 1]], NonzeroQ[Plus[Times[2, q], 3]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Plus[Times[b, d], Times[2, a, f, Plus[q, 1], x]], Power[Plus[d, Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[2, d, f, Plus[q, 1]], -1]], x], And[FreeQ[List[a, b, c, d, f, q], x], NonzeroQ[Plus[q, 1]], ZeroQ[Plus[Times[3, a, f], Times[-1, c, d], Times[2, a, f, q]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[b, Int[Times[x, Power[Plus[d, Times[f, Power[x, 2]]], q]], x], x], Int[Times[Plus[a, Times[c, Power[x, 2]]], Power[Plus[d, Times[f, Power[x, 2]]], q]], x]], And[FreeQ[List[a, b, c, d, f, q], x], PositiveIntegerQ[Plus[q, 2]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q]], x], x], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], PositiveIntegerQ[Plus[q, 2]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Plus[a, Times[c, Power[x, 2]]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q]], x], x], And[FreeQ[List[a, c, d, e, f], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], PositiveIntegerQ[Plus[q, 2]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[a, e, f], Times[-1, 2, b, d, f], Times[c, d, e], Times[Plus[Times[f, Plus[Times[2, a, f], Times[-1, b, e]]], Times[c, Plus[Power[e, 2], Times[-1, 2, d, f]]]], x]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[f, Plus[q, 1], Plus[Power[e, 2], Times[-1, 4, d, f]]], -1]], x], Times[-1, Dist[Times[Plus[Times[f, Plus[Times[2, a, f], Times[-1, b, e]], Plus[Times[2, q], 3]], Times[c, Plus[Times[Power[e, 2], Plus[q, 2]], Times[-1, 2, d, f]]]], Power[Times[f, Plus[q, 1], Plus[Power[e, 2], Times[-1, 4, d, f]]], -1]], Int[Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[q, 1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[q], Less[q, -1], NonzeroQ[Plus[Times[f, Plus[Times[2, a, f], Times[-1, b, e]], Plus[Times[2, q], 3]], Times[c, Plus[Times[Power[e, 2], Plus[q, 2]], Times[-1, 2, d, f]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[a, e, f], Times[c, d, e], Times[Plus[Times[2, a, Power[f, 2]], Times[c, Plus[Power[e, 2], Times[-1, 2, d, f]]]], x]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[f, Plus[q, 1], Plus[Power[e, 2], Times[-1, 4, d, f]]], -1]], x], Times[-1, Dist[Times[Plus[Times[2, a, Power[f, 2], Plus[Times[2, q], 3]], Times[c, Plus[Times[Power[e, 2], Plus[q, 2]], Times[-1, 2, d, f]]]], Power[Times[f, Plus[q, 1], Plus[Power[e, 2], Times[-1, 4, d, f]]], -1]], Int[Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[q, 1]], x], x]]], And[FreeQ[List[a, c, d, e, f], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[q], Less[q, -1], NonzeroQ[Plus[Times[2, a, Power[f, 2], Plus[Times[2, q], 3]], Times[c, Plus[Times[Power[e, 2], Plus[q, 2]], Times[-1, 2, d, f]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[b, d], Times[-1, Plus[Times[a, f], Times[-1, c, d]], x]], Power[Plus[d, Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[2, d, f, Plus[q, 1]], -1]], x], Dist[Times[Plus[Times[3, a, f], Times[-1, c, d], Times[2, a, f, q]], Power[Times[2, d, f, Plus[q, 1]], -1]], Int[Power[Plus[d, Times[f, Power[x, 2]]], Plus[q, 1]], x], x]], And[FreeQ[List[a, b, c, d, f], x], RationalQ[q], Less[q, -1], NonzeroQ[Plus[Times[3, a, f], Times[-1, c, d], Times[2, a, f, q]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[b, f, Plus[Times[2, q], 3]], Times[-1, c, e, Plus[q, 2]], Times[2, c, f, Plus[q, 1], x]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[2, Power[f, 2], Plus[q, 1], Plus[Times[2, q], 3]], -1]], x], Dist[Times[Plus[Times[f, Plus[Times[2, a, f], Times[-1, b, e]], Plus[Times[2, q], 3]], Times[c, Plus[Times[Power[e, 2], Plus[q, 2]], Times[-1, 2, d, f]]]], Power[Times[2, Power[f, 2], Plus[Times[2, q], 3]], -1]], Int[Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], x], x]], And[FreeQ[List[d, e, f, a, b, c, q], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], Not[PositiveIntegerQ[q]], Not[And[RationalQ[q], LessEqual[q, -1]]], NonzeroQ[Plus[Times[f, Plus[Times[2, a, f], Times[-1, b, e]], Plus[Times[2, q], 3]], Times[c, Plus[Times[Power[e, 2], Plus[q, 2]], Times[-1, 2, d, f]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[-1, c, e, Plus[q, 2]], Times[2, c, f, Plus[q, 1], x]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[2, Power[f, 2], Plus[q, 1], Plus[Times[2, q], 3]], -1]], x], Dist[Times[Plus[Times[2, a, Power[f, 2], Plus[Times[2, q], 3]], Times[c, Plus[Times[Power[e, 2], Plus[q, 2]], Times[-1, 2, d, f]]]], Power[Times[2, Power[f, 2], Plus[Times[2, q], 3]], -1]], Int[Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], x], x]], And[FreeQ[List[d, e, f, a, c, q], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], Not[PositiveIntegerQ[q]], Not[And[RationalQ[q], LessEqual[q, -1]]], NonzeroQ[Plus[Times[2, a, Power[f, 2], Plus[Times[2, q], 3]], Times[c, Plus[Times[Power[e, 2], Plus[q, 2]], Times[-1, 2, d, f]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[b, f, Plus[Times[2, q], 3]], Times[2, c, f, Plus[q, 1], x]], Power[Plus[d, Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[2, Power[f, 2], Plus[q, 1], Plus[Times[2, q], 3]], -1]], x], Dist[Times[Plus[Times[3, a, f], Times[-1, c, d], Times[2, a, f, q]], Power[Times[f, Plus[Times[2, q], 3]], -1]], Int[Power[Plus[d, Times[f, Power[x, 2]]], q], x], x]], And[FreeQ[List[d, f, a, b, c, q], x], Not[PositiveIntegerQ[q]], Not[And[RationalQ[q], LessEqual[q, -1]]], NonzeroQ[Plus[Times[3, a, f], Times[-1, c, d], Times[2, a, f, q]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[b, Times[2, c, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[p, 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[q, -1]], Simp[Plus[Times[2, c, d, Plus[Times[2, p], 3]], Times[b, e, q], Times[Plus[Times[2, b, f, q], Times[2, c, e, Plus[Times[2, p], q, 3]]], x], Times[2, c, f, Plus[Times[2, p], Times[2, q], 3], Power[x, 2]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[p, q], Less[p, -1], Greater[q, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[b, Times[2, c, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[f, Power[x, 2]]], q], Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[p, 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[f, Power[x, 2]]], Plus[q, -1]], Simp[Plus[Times[2, c, d, Plus[Times[2, p], 3]], Times[Times[2, b, f, q], x], Times[2, c, f, Plus[Times[2, p], Times[2, q], 3], Power[x, 2]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, f], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], RationalQ[p, q], Less[p, -1], Greater[q, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Times[2, c, x], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], Power[Times[Times[-4, a, c], Plus[p, 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[Times[-4, a, c], Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[q, -1]], Simp[Plus[Times[2, c, d, Plus[Times[2, p], 3]], Times[Times[2, c, e, Plus[Times[2, p], q, 3]], x], Times[2, c, f, Plus[Times[2, p], Times[2, q], 3], Power[x, 2]]], x]], x], x]]], And[FreeQ[List[a, c, d, e, f], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[p, q], Less[p, -1], Greater[q, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[2, a, Power[c, 2], e], Times[-1, Power[b, 2], c, e], Times[Power[b, 3], f], Times[b, c, Plus[Times[c, d], Times[-1, 3, a, f]]], Times[c, Plus[Times[2, Power[c, 2], d], Times[Power[b, 2], f], Times[-1, c, Plus[Times[b, e], Times[2, a, f]]]], x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[Power[Plus[Times[c, d], Times[-1, a, f]], 2], Times[-1, Plus[Times[b, d], Times[-1, a, e]], Plus[Times[c, e], Times[-1, b, f]]]], Plus[p, 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[Power[Plus[Times[c, d], Times[-1, a, f]], 2], Times[-1, Plus[Times[b, d], Times[-1, a, e]], Plus[Times[c, e], Times[-1, b, f]]]], Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], Simp[Plus[Times[2, c, Plus[Power[Plus[Times[c, d], Times[-1, a, f]], 2], Times[-1, Plus[Times[b, d], Times[-1, a, e]], Plus[Times[c, e], Times[-1, b, f]]]], Plus[p, 1]], Times[-1, Plus[Times[2, Power[c, 2], d], Times[Power[b, 2], f], Times[-1, c, Plus[Times[b, e], Times[2, a, f]]]], Plus[Times[a, f, Plus[p, 1]], Times[-1, c, d, Plus[p, 2]]]], Times[-1, e, Plus[Times[Power[b, 2], c, e], Times[-1, 2, a, Power[c, 2], e], Times[-1, Power[b, 3], f], Times[-1, b, c, Plus[Times[c, d], Times[-1, 3, a, f]]]], Plus[p, q, 2]], Times[Plus[Times[2, f, Plus[Times[2, a, Power[c, 2], e], Times[-1, Power[b, 2], c, e], Times[Power[b, 3], f], Times[b, c, Plus[Times[c, d], Times[-1, 3, a, f]]]], Plus[p, q, 2]], Times[-1, Plus[Times[2, Power[c, 2], d], Times[Power[b, 2], f], Times[-1, c, Plus[Times[b, e], Times[2, a, f]]]], Plus[Times[b, f, Plus[p, 1]], Times[-1, c, e, Plus[Times[2, p], q, 4]]]]], x], Times[c, f, Plus[Times[2, Power[c, 2], d], Times[Power[b, 2], f], Times[-1, c, Plus[Times[b, e], Times[2, a, f]]]], Plus[Times[2, p], Times[2, q], 5], Power[x, 2]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, q], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[p], Less[p, -1], NonzeroQ[Plus[Power[Plus[Times[c, d], Times[-1, a, f]], 2], Times[-1, Plus[Times[b, d], Times[-1, a, e]], Plus[Times[c, e], Times[-1, b, f]]]]], Not[And[Not[IntegerQ[p]], IntegerQ[q], Less[q, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[Power[b, 3], f], Times[b, c, Plus[Times[c, d], Times[-1, 3, a, f]]], Times[c, Plus[Times[2, Power[c, 2], d], Times[Power[b, 2], f], Times[-1, c, Times[2, a, f]]], x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[Times[Power[b, 2], d, f], Power[Plus[Times[c, d], Times[-1, a, f]], 2]], Plus[p, 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[Times[Power[b, 2], d, f], Power[Plus[Times[c, d], Times[-1, a, f]], 2]], Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[f, Power[x, 2]]], q], Simp[Plus[Times[2, c, Plus[Times[Power[b, 2], d, f], Power[Plus[Times[c, d], Times[-1, a, f]], 2]], Plus[p, 1]], Times[-1, Plus[Times[2, Power[c, 2], d], Times[Power[b, 2], f], Times[-1, c, Times[2, a, f]]], Plus[Times[a, f, Plus[p, 1]], Times[-1, c, d, Plus[p, 2]]]], Times[Plus[Times[2, f, Plus[Times[Power[b, 3], f], Times[b, c, Plus[Times[c, d], Times[-1, 3, a, f]]]], Plus[p, q, 2]], Times[-1, Plus[Times[2, Power[c, 2], d], Times[Power[b, 2], f], Times[-1, c, Times[2, a, f]]], Times[b, f, Plus[p, 1]]]], x], Times[c, f, Plus[Times[2, Power[c, 2], d], Times[Power[b, 2], f], Times[-1, c, Times[2, a, f]]], Plus[Times[2, p], Times[2, q], 5], Power[x, 2]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, f, q], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], RationalQ[p], Less[p, -1], NonzeroQ[Plus[Times[Power[b, 2], d, f], Power[Plus[Times[c, d], Times[-1, a, f]], 2]]], Not[And[Not[IntegerQ[p]], IntegerQ[q], Less[q, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[2, a, Power[c, 2], e], Times[c, Plus[Times[2, Power[c, 2], d], Times[-1, c, Times[2, a, f]]], x]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[Times[-4, a, c], Plus[Times[a, c, Power[e, 2]], Power[Plus[Times[c, d], Times[-1, a, f]], 2]], Plus[p, 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[Times[-4, a, c], Plus[Times[a, c, Power[e, 2]], Power[Plus[Times[c, d], Times[-1, a, f]], 2]], Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], Simp[Plus[Times[2, c, Plus[Power[Plus[Times[c, d], Times[-1, a, f]], 2], Times[-1, Times[-1, a, e], Times[c, e]]], Plus[p, 1]], Times[-1, Plus[Times[2, Power[c, 2], d], Times[-1, c, Times[2, a, f]]], Plus[Times[a, f, Plus[p, 1]], Times[-1, c, d, Plus[p, 2]]]], Times[-1, e, Times[-2, a, Power[c, 2], e], Plus[p, q, 2]], Times[Plus[Times[2, f, Times[2, a, Power[c, 2], e], Plus[p, q, 2]], Times[-1, Plus[Times[2, Power[c, 2], d], Times[-1, c, Times[2, a, f]]], Times[-1, c, e, Plus[Times[2, p], q, 4]]]], x], Times[c, f, Plus[Times[2, Power[c, 2], d], Times[-1, c, Times[2, a, f]]], Plus[Times[2, p], Times[2, q], 5], Power[x, 2]]], x]], x], x]]], And[FreeQ[List[a, c, d, e, f, q], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[p], Less[p, -1], NonzeroQ[Plus[Times[a, c, Power[e, 2]], Power[Plus[Times[c, d], Times[-1, a, f]], 2]]], Not[And[Not[IntegerQ[p]], IntegerQ[q], Less[q, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[b, f, Plus[Times[3, p], Times[2, q]]], Times[-1, c, e, Plus[Times[2, p], q]], Times[2, c, f, Plus[p, q], x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, -1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[2, Power[f, 2], Plus[p, q], Plus[Times[2, p], Times[2, q], 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[2, Power[f, 2], Plus[p, q], Plus[Times[2, p], Times[2, q], 1]], -1]], Int[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, -2]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], Simp[Plus[Times[Plus[Times[b, d], Times[-1, a, e]], Plus[Times[c, e], Times[-1, b, f]], Plus[1, Times[-1, p]], Plus[Times[2, p], q]], Times[-1, Plus[p, q], Plus[Times[Power[b, 2], d, f, Plus[1, Times[-1, p]]], Times[-1, a, Plus[Times[f, Plus[Times[b, e], Times[-1, 2, a, f]], Plus[Times[2, p], Times[2, q], 1]], Times[c, Plus[Times[2, d, f], Times[-1, Power[e, 2], Plus[Times[2, p], q]]]]]]]], Times[Plus[Times[2, Plus[Times[c, d], Times[-1, a, f]], Plus[Times[c, e], Times[-1, b, f]], Plus[1, Times[-1, p]], Plus[Times[2, p], q]], Times[-1, Plus[p, q], Plus[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], e, f, Plus[1, Times[-1, p]]], Times[b, Plus[Times[c, Plus[Power[e, 2], Times[-1, 4, d, f]], Plus[Times[2, p], q]], Times[f, Plus[Times[2, c, d], Times[-1, b, e], Times[2, a, f]], Plus[Times[2, p], Times[2, q], 1]]]]]]], x], Times[Plus[Times[Power[Plus[Times[c, e], Times[-1, b, f]], 2], Plus[1, Times[-1, p]], p], Times[c, Plus[p, q], Plus[Times[f, Plus[Times[b, e], Times[-1, 2, a, f]], Plus[Times[4, p], Times[2, q], -1]], Times[-1, c, Plus[Times[2, d, f, Plus[1, Times[-1, 2, p]]], Times[Power[e, 2], Plus[Times[3, p], q, -1]]]]]]], Power[x, 2]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, q], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[p], Greater[p, 1], NonzeroQ[Plus[p, q]], NonzeroQ[Plus[Times[2, p], Times[2, q], 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[b, Plus[Times[3, p], Times[2, q]]], Times[2, c, Plus[p, q], x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, -1]], Power[Plus[d, Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[2, f, Plus[p, q], Plus[Times[2, p], Times[2, q], 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[2, f, Plus[p, q], Plus[Times[2, p], Times[2, q], 1]], -1]], Int[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, -2]], Power[Plus[d, Times[f, Power[x, 2]]], q], Simp[Plus[Times[Power[b, 2], d, Plus[p, -1], Plus[Times[2, p], q]], Times[-1, Plus[p, q], Plus[Times[Power[b, 2], d, Plus[1, Times[-1, p]]], Times[-1, 2, a, Plus[Times[c, d], Times[-1, a, f, Plus[Times[2, p], Times[2, q], 1]]]]]], Times[-1, Plus[Times[2, b, Plus[Times[c, d], Times[-1, a, f]], Plus[1, Times[-1, p]], Plus[Times[2, p], q]], Times[-1, 2, Plus[p, q], b, Plus[Times[2, c, d, Plus[Times[2, p], q]], Times[-1, Plus[Times[c, d], Times[a, f]], Plus[Times[2, p], Times[2, q], 1]]]]], x], Times[Plus[Times[Power[b, 2], f, p, Plus[1, Times[-1, p]]], Times[2, c, Plus[p, q], Plus[Times[c, d, Plus[Times[2, p], -1]], Times[-1, a, f, Plus[Times[4, p], Times[2, q], -1]]]]], Power[x, 2]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, f, q], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], RationalQ[p], Greater[p, 1], NonzeroQ[Plus[p, q]], NonzeroQ[Plus[Times[2, p], Times[2, q], 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[c, Plus[Times[e, Plus[Times[2, p], q]], Times[-1, 2, f, Plus[p, q], x]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, -1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[2, Power[f, 2], Plus[p, q], Plus[Times[2, p], Times[2, q], 1]], -1]], x]], Times[-1, Dist[Times[1, Power[Times[2, Power[f, 2], Plus[p, q], Plus[Times[2, p], Times[2, q], 1]], -1]], Int[Times[Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, -2]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], Simp[Plus[Times[-1, a, c, Power[e, 2], Plus[1, Times[-1, p]], Plus[Times[2, p], q]], Times[a, Plus[p, q], Plus[Times[-2, a, Power[f, 2], Plus[Times[2, p], Times[2, q], 1]], Times[c, Plus[Times[2, d, f], Times[-1, Power[e, 2], Plus[Times[2, p], q]]]]]], Times[Plus[Times[2, Plus[Times[c, d], Times[-1, a, f]], Times[c, e], Plus[1, Times[-1, p]], Plus[Times[2, p], q]], Times[4, a, c, e, f, Plus[1, Times[-1, p]], Plus[p, q]]], x], Times[Plus[Times[p, Power[c, 2], Power[e, 2], Plus[1, Times[-1, p]]], Times[-1, c, Plus[p, q], Plus[Times[2, a, Power[f, 2], Plus[Times[4, p], Times[2, q], -1]], Times[c, Plus[Times[2, d, f, Plus[1, Times[-1, 2, p]]], Times[Power[e, 2], Plus[Times[3, p], q, -1]]]]]]], Power[x, 2]]], x]], x], x]]], And[FreeQ[List[a, c, d, e, f, q], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[p], Greater[p, 1], NonzeroQ[Plus[p, q]], NonzeroQ[Plus[Times[2, p], Times[2, q], 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Plus[Times[Power[c, 2], Power[d, 2]], Times[-1, b, c, d, e], Times[a, c, Power[e, 2]], Times[Power[b, 2], d, f], Times[-1, 2, a, c, d, f], Times[-1, a, b, e, f], Times[Power[a, 2], Power[f, 2]]]]], Condition[Plus[Dist[Times[1, Power[q, -1]], Int[Times[Plus[Times[Power[c, 2], d], Times[-1, b, c, e], Times[Power[b, 2], f], Times[-1, a, c, f], Times[-1, Plus[Times[Power[c, 2], e], Times[-1, b, c, f]], x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], -1]], x], x], Dist[Times[1, Power[q, -1]], Int[Times[Plus[Times[c, Power[e, 2]], Times[-1, c, d, f], Times[-1, b, e, f], Times[a, Power[f, 2]], Times[Plus[Times[c, e, f], Times[-1, b, Power[f, 2]]], x]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], -1]], x], x]], NonzeroQ[q]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Plus[Times[Power[c, 2], Power[d, 2]], Times[Power[b, 2], d, f], Times[-1, 2, a, c, d, f], Times[Power[a, 2], Power[f, 2]]]]], Condition[Plus[Dist[Times[1, Power[q, -1]], Int[Times[Plus[Times[Power[c, 2], d], Times[Power[b, 2], f], Times[-1, a, c, f], Times[b, c, f, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], -1]], x], x], Times[-1, Dist[Times[1, Power[q, -1]], Int[Times[Plus[Times[c, d, f], Times[-1, a, Power[f, 2]], Times[b, Power[f, 2], x]], Power[Plus[d, Times[f, Power[x, 2]]], -1]], x], x]]], NonzeroQ[q]]], And[FreeQ[List[a, b, c, d, f], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[-2, e], Subst[Int[Times[1, Power[Plus[Times[e, Plus[Times[b, e], Times[-1, 4, a, f]]], Times[-1, Plus[Times[b, d], Times[-1, a, e]], Power[x, 2]]], -1]], x], x, Times[Plus[e, Times[2, f, x]], Power[Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]], -1]]], x], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], ZeroQ[Plus[Times[c, e], Times[-1, b, f]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Plus[Dist[Times[2, c, Power[q, -1]], Int[Times[1, Power[Times[Plus[b, Times[-1, q], Times[2, c, x]], Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]]], -1]], x], x], Times[-1, Dist[Times[2, c, Power[q, -1]], Int[Times[1, Power[Times[Plus[b, q, Times[2, c, x]], Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], NonzeroQ[Plus[Times[c, e], Times[-1, b, f]]], PosQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[2, -1]], Int[Times[1, Power[Times[Plus[a, Times[-1, Rt[Times[-1, a, c], 2], x]], Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]]], -1]], x], x], Dist[Times[1, Power[2, -1]], Int[Times[1, Power[Times[Plus[a, Times[Rt[Times[-1, a, c], 2], x]], Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]]], -1]], x], x]], And[FreeQ[List[a, c, d, e, f], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], PosQ[Times[-1, a, c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Plus[Dist[Times[2, c, Power[q, -1]], Int[Times[1, Power[Times[Plus[b, Times[-1, q], Times[2, c, x]], Sqrt[Plus[d, Times[f, Power[x, 2]]]]], -1]], x], x], Times[-1, Dist[Times[2, c, Power[q, -1]], Int[Times[1, Power[Times[Plus[b, q, Times[2, c, x]], Sqrt[Plus[d, Times[f, Power[x, 2]]]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d, f], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PosQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[Plus[Times[c, d], Times[-1, a, f]], 2], Times[-1, Plus[Times[b, d], Times[-1, a, e]], Plus[Times[c, e], Times[-1, b, f]]]], 2]]], Plus[Dist[Times[1, Power[Times[2, q], -1]], Int[Times[Plus[Times[c, d], Times[-1, a, f], q, Times[Plus[Times[c, e], Times[-1, b, f]], x]], Power[Times[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]]], -1]], x], x], Times[-1, Dist[Times[1, Power[Times[2, q], -1]], Int[Times[Plus[Times[c, d], Times[-1, a, f], Times[-1, q], Times[Plus[Times[c, e], Times[-1, b, f]], x]], Power[Times[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], NonzeroQ[Plus[Times[c, e], Times[-1, b, f]]], NegQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[Plus[Times[c, d], Times[-1, a, f]], 2], Times[a, c, Power[e, 2]]], 2]]], Plus[Dist[Times[1, Power[Times[2, q], -1]], Int[Times[Plus[Times[c, d], Times[-1, a, f], q, Times[c, e, x]], Power[Times[Plus[a, Times[c, Power[x, 2]]], Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]]], -1]], x], x], Times[-1, Dist[Times[1, Power[Times[2, q], -1]], Int[Times[Plus[Times[c, d], Times[-1, a, f], Times[-1, q], Times[c, e, x]], Power[Times[Plus[a, Times[c, Power[x, 2]]], Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]]], -1]], x], x]]]], And[FreeQ[List[a, c, d, e, f], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], NegQ[Times[-1, a, c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[Plus[Times[c, d], Times[-1, a, f]], 2], Times[Power[b, 2], d, f]], 2]]], Plus[Dist[Times[1, Power[Times[2, q], -1]], Int[Times[Plus[Times[c, d], Times[-1, a, f], q, Times[Times[-1, b, f], x]], Power[Times[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Sqrt[Plus[d, Times[f, Power[x, 2]]]]], -1]], x], x], Times[-1, Dist[Times[1, Power[Times[2, q], -1]], Int[Times[Plus[Times[c, d], Times[-1, a, f], Times[-1, q], Times[Times[-1, b, f], x]], Power[Times[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Sqrt[Plus[d, Times[f, Power[x, 2]]]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d, f], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NegQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[c, Power[f, -1]], Int[Times[1, Power[Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], -1]], x], x], Times[-1, Dist[Times[1, Power[f, -1]], Int[Times[Plus[Times[c, d], Times[-1, a, f], Times[Plus[Times[c, e], Times[-1, b, f]], x]], Power[Times[Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], Plus[d, Times[e, x], Times[f, Power[x, 2]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[c, Power[f, -1]], Int[Times[1, Power[Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], -1]], x], x], Times[-1, Dist[Times[1, Power[f, -1]], Int[Times[Plus[Times[c, d], Times[-1, a, f], Times[-1, b, f, x]], Power[Times[Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], Plus[d, Times[f, Power[x, 2]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, f], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[c, Power[f, -1]], Int[Times[1, Power[Sqrt[Plus[a, Times[c, Power[x, 2]]]], -1]], x], x], Times[-1, Dist[Times[1, Power[f, -1]], Int[Times[Plus[Times[c, d], Times[-1, a, f], Times[c, e, x]], Power[Times[Sqrt[Plus[a, Times[c, Power[x, 2]]]], Plus[d, Times[e, x], Times[f, Power[x, 2]]]], -1]], x], x]]], And[FreeQ[List[a, c, d, e, f], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[r, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Dist[Times[Sqrt[Plus[b, r, Times[2, c, x]]], Sqrt[Plus[Times[2, a], Times[Plus[b, r], x]]], Power[Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], -1]], Int[Times[1, Power[Times[Sqrt[Plus[b, r, Times[2, c, x]]], Sqrt[Plus[Times[2, a], Times[Plus[b, r], x]]], Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[r, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Dist[Times[Sqrt[Plus[b, r, Times[2, c, x]]], Sqrt[Plus[Times[2, a], Times[Plus[b, r], x]]], Power[Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], -1]], Int[Times[1, Power[Times[Sqrt[Plus[b, r, Times[2, c, x]]], Sqrt[Plus[Times[2, a], Times[Plus[b, r], x]]], Sqrt[Plus[d, Times[f, Power[x, 2]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, f], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q]], x], FreeQ[List[a, b, c, d, e, f, p, q], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Plus[a, Times[c, Power[x, 2]]], p], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q]], x], FreeQ[List[a, c, d, e, f, p, q], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[u, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[u, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Coefficient[u, x, 1], -1]], Subst[Int[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q]], x], x, u], x], And[FreeQ[List[a, b, c, d, e, f, p, q], x], LinearQ[u, x], NonzeroQ[Plus[u, Times[-1, x]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[u, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Coefficient[u, x, 1], -1]], Subst[Int[Times[Power[Plus[a, Times[c, Power[x, 2]]], p], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q]], x], x, u], x], And[FreeQ[List[a, c, d, e, f, p, q], x], LinearQ[u, x], NonzeroQ[Plus[u, Times[-1, x]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[Times[c, Power[f, -1]], p], Int[Times[Power[Plus[g, Times[h, x]], m], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[p, q]]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, h, p, q], x], ZeroQ[Plus[Times[c, d], Times[-1, a, f]]], ZeroQ[Plus[Times[b, d], Times[-1, a, e]]], Or[IntegerQ[p], PositiveQ[Times[c, Power[f, -1]]]], Or[Not[IntegerQ[q]], LessEqual[LeafCount[Plus[d, Times[e, x], Times[f, Power[x, 2]]]], LeafCount[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[a, IntPart[p]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], FracPart[p]], Power[Times[Power[d, IntPart[p]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], FracPart[p]]], -1]], Int[Times[Power[Plus[g, Times[h, x]], m], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[p, q]]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, h, p, q], x], ZeroQ[Plus[Times[c, d], Times[-1, a, f]]], ZeroQ[Plus[Times[b, d], Times[-1, a, e]]], Not[IntegerQ[p]], Not[IntegerQ[q]], Not[PositiveQ[Times[c, Power[f, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], FracPart[p]], Power[Times[Power[Times[4, c], IntPart[p]], Power[Plus[b, Times[2, c, x]], Times[2, FracPart[p]]]], -1]], Int[Times[Power[Plus[g, Times[h, x]], m], Power[Plus[b, Times[2, c, x]], Times[2, p]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, h, m, p, q], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], FracPart[p]], Power[Times[Power[Times[4, c], IntPart[p]], Power[Plus[b, Times[2, c, x]], Times[2, FracPart[p]]]], -1]], Int[Times[Power[Plus[g, Times[h, x]], m], Power[Plus[b, Times[2, c, x]], Times[2, p]], Power[Plus[d, Times[f, Power[x, 2]]], q]], x], x], And[FreeQ[List[a, b, c, d, f, g, h, m, p, q], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[Times[d, g, Power[a, -1]], Times[f, h, x, Power[c, -1]]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[m, p]]], x], And[FreeQ[List[a, b, c, d, e, f, g, h, p], x], ZeroQ[Plus[Times[c, Power[g, 2]], Times[-1, b, g, h], Times[a, Power[h, 2]]]], ZeroQ[Plus[Times[Power[c, 2], d, Power[g, 2]], Times[-1, a, c, e, g, h], Times[Power[a, 2], f, Power[h, 2]]]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[Times[d, g, Power[a, -1]], Times[f, h, x, Power[c, -1]]], m], Power[Plus[a, Times[c, Power[x, 2]]], Plus[m, p]]], x], And[FreeQ[List[a, c, d, e, f, g, h, p], x], ZeroQ[Plus[Times[c, Power[g, 2]], Times[a, Power[h, 2]]]], ZeroQ[Plus[Times[Power[c, 2], d, Power[g, 2]], Times[-1, a, c, e, g, h], Times[Power[a, 2], f, Power[h, 2]]]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[Times[d, g, Power[a, -1]], Times[f, h, x, Power[c, -1]]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[m, p]]], x], And[FreeQ[List[a, b, c, d, f, g, h, p], x], ZeroQ[Plus[Times[c, Power[g, 2]], Times[-1, b, g, h], Times[a, Power[h, 2]]]], ZeroQ[Plus[Times[Power[c, 2], d, Power[g, 2]], Times[Power[a, 2], f, Power[h, 2]]]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[Times[d, g, Power[a, -1]], Times[f, h, x, Power[c, -1]]], m], Power[Plus[a, Times[c, Power[x, 2]]], Plus[m, p]]], x], And[FreeQ[List[a, c, d, f, g, h, p], x], ZeroQ[Plus[Times[c, Power[g, 2]], Times[a, Power[h, 2]]]], ZeroQ[Plus[Times[Power[c, 2], d, Power[g, 2]], Times[Power[a, 2], f, Power[h, 2]]]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[Times[a, Power[e, -1]], Times[c, Power[f, -1], x]], p], Power[Plus[Times[e, x], Times[f, Power[x, 2]]], Plus[p, q]]], x], And[FreeQ[List[a, b, c, e, f, q], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[e, 2]], Times[-1, b, e, f], Times[a, Power[f, 2]]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[Times[a, Power[e, -1]], Times[c, Power[f, -1], x]], p], Power[Plus[Times[e, x], Times[f, Power[x, 2]]], Plus[p, q]]], x], And[FreeQ[List[a, c, e, f, q], x], ZeroQ[Plus[Times[c, Power[e, 2]], Times[a, Power[f, 2]]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[f, Power[Plus[g, Times[h, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, h, Plus[m, Times[2, p], 3]], -1]], x], And[FreeQ[List[a, b, c, d, e, f, g, h, m, p], x], ZeroQ[Plus[Times[b, f, h, Plus[m, p, 2]], Times[c, Plus[Times[2, f, g, Plus[p, 1]], Times[-1, e, h, Plus[m, Times[2, p], 3]]]]]], ZeroQ[Plus[Times[b, f, g, Plus[p, 1]], Times[h, Plus[Times[a, f, Plus[m, 1]], Times[-1, c, d, Plus[m, Times[2, p], 3]]]]]], NonzeroQ[Plus[m, Times[2, p], 3]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[f, Power[Plus[g, Times[h, x]], Plus[m, 1]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, h, Plus[m, Times[2, p], 3]], -1]], x], And[FreeQ[List[a, c, d, e, f, g, h, m, p], x], ZeroQ[Times[c, Plus[Times[2, f, g, Plus[p, 1]], Times[-1, e, h, Plus[m, Times[2, p], 3]]]]], ZeroQ[Times[h, Plus[Times[a, f, Plus[m, 1]], Times[-1, c, d, Plus[m, Times[2, p], 3]]]]], NonzeroQ[Plus[m, Times[2, p], 3]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[f, Power[Plus[g, Times[h, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, h, Plus[m, Times[2, p], 3]], -1]], x], And[FreeQ[List[a, b, c, d, f, g, h, m, p], x], ZeroQ[Plus[Times[b, f, h, Plus[m, p, 2]], Times[c, Times[2, f, g, Plus[p, 1]]]]], ZeroQ[Plus[Times[b, f, g, Plus[p, 1]], Times[h, Plus[Times[a, f, Plus[m, 1]], Times[-1, c, d, Plus[m, Times[2, p], 3]]]]]], NonzeroQ[Plus[m, Times[2, p], 3]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[g, Times[h, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Plus[d, Times[e, x], Times[f, Power[x, 2]]]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, h, m], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], Or[IntegersQ[m, p], PositiveIntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[g, Times[h, x]], m], Power[Plus[a, Times[c, Power[x, 2]]], p], Plus[d, Times[e, x], Times[f, Power[x, 2]]]], x], x], And[FreeQ[List[a, c, d, e, f, g, h, m], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], Or[IntegersQ[m, p], PositiveIntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[g, Times[h, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Plus[d, Times[f, Power[x, 2]]]], x], x], And[FreeQ[List[a, b, c, d, f, g, h, m], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], Or[IntegersQ[m, p], PositiveIntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[g, Times[h, x]], m], Power[Plus[a, Times[c, Power[x, 2]]], p], Plus[d, Times[f, Power[x, 2]]]], x], x], And[FreeQ[List[a, c, d, f, g, h, m], x], Or[IntegersQ[m, p], PositiveIntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[f, Power[g, 2]], Times[-1, e, g, h], Times[d, Power[h, 2]]], Power[Plus[g, Times[h, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[h, Plus[m, 1], Plus[Times[c, Power[g, 2]], Times[-1, b, g, h], Times[a, Power[h, 2]]]], -1]], x], Dist[Times[1, Power[Times[h, Plus[m, 1], Plus[Times[c, Power[g, 2]], Times[-1, b, g, h], Times[a, Power[h, 2]]]], -1]], Int[Times[Power[Plus[g, Times[h, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Simp[Plus[Times[h, Plus[Times[c, d, g], Times[-1, a, f, g], Times[a, e, h]], Plus[m, 1]], Times[-1, b, Plus[Times[f, Power[g, 2], Plus[p, 1]], Times[-1, h, Plus[Times[e, g, Plus[p, 1]], Times[-1, d, h, Plus[m, p, 2]]]]]], Times[-1, Plus[Times[f, h, Plus[Times[b, g], Times[-1, a, h]], Plus[m, 1]], Times[c, Plus[Times[2, f, Power[g, 2], Plus[p, 1]], Times[-1, h, Plus[Times[e, g], Times[-1, d, h]], Plus[m, Times[2, p], 3]]]]], x]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, h, p], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[m], Less[m, -1], NonzeroQ[Plus[Times[c, Power[g, 2]], Times[-1, b, g, h], Times[a, Power[h, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[f, Power[g, 2]], Times[-1, e, g, h], Times[d, Power[h, 2]]], Power[Plus[g, Times[h, x]], Plus[m, 1]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[h, Plus[m, 1], Plus[Times[c, Power[g, 2]], Times[a, Power[h, 2]]]], -1]], x], Dist[Times[1, Power[Times[h, Plus[m, 1], Plus[Times[c, Power[g, 2]], Times[a, Power[h, 2]]]], -1]], Int[Times[Power[Plus[g, Times[h, x]], Plus[m, 1]], Power[Plus[a, Times[c, Power[x, 2]]], p], Simp[Plus[Times[h, Plus[Times[c, d, g], Times[-1, a, f, g], Times[a, e, h]], Plus[m, 1]], Times[Plus[Times[a, f, Power[h, 2], Plus[m, 1]], Times[-1, c, Plus[Times[2, f, Power[g, 2], Plus[p, 1]], Times[-1, h, Plus[Times[e, g], Times[-1, d, h]], Plus[m, Times[2, p], 3]]]]], x]], x]], x], x]], And[FreeQ[List[a, c, d, e, f, g, h, p], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[m], Less[m, -1], NonzeroQ[Plus[Times[c, Power[g, 2]], Times[a, Power[h, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[f, Power[g, 2]], Times[d, Power[h, 2]]], Power[Plus[g, Times[h, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[h, Plus[m, 1], Plus[Times[c, Power[g, 2]], Times[-1, b, g, h], Times[a, Power[h, 2]]]], -1]], x], Dist[Times[1, Power[Times[h, Plus[m, 1], Plus[Times[c, Power[g, 2]], Times[-1, b, g, h], Times[a, Power[h, 2]]]], -1]], Int[Times[Power[Plus[g, Times[h, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Simp[Plus[Times[h, Plus[Times[c, d, g], Times[-1, a, f, g]], Plus[m, 1]], Times[-1, b, Plus[Times[f, Power[g, 2], Plus[p, 1]], Times[d, Power[h, 2], Plus[m, p, 2]]]], Times[-1, Plus[Times[f, h, Plus[Times[b, g], Times[-1, a, h]], Plus[m, 1]], Times[c, Plus[Times[2, f, Power[g, 2], Plus[p, 1]], Times[d, Power[h, 2], Plus[m, Times[2, p], 3]]]]], x]], x]], x], x]], And[FreeQ[List[a, b, c, d, f, g, h, p], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], RationalQ[m], Less[m, -1], NonzeroQ[Plus[Times[c, Power[g, 2]], Times[-1, b, g, h], Times[a, Power[h, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[f, Power[g, 2]], Times[d, Power[h, 2]]], Power[Plus[g, Times[h, x]], Plus[m, 1]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[h, Plus[m, 1], Plus[Times[c, Power[g, 2]], Times[a, Power[h, 2]]]], -1]], x], Dist[Times[1, Power[Times[h, Plus[m, 1], Plus[Times[c, Power[g, 2]], Times[a, Power[h, 2]]]], -1]], Int[Times[Power[Plus[g, Times[h, x]], Plus[m, 1]], Power[Plus[a, Times[c, Power[x, 2]]], p], Simp[Plus[Times[h, Plus[Times[c, d, g], Times[-1, a, f, g]], Plus[m, 1]], Times[Plus[Times[a, f, Power[h, 2], Plus[m, 1]], Times[-1, c, Plus[Times[2, f, Power[g, 2], Plus[p, 1]], Times[d, Power[h, 2], Plus[m, Times[2, p], 3]]]]], x]], x]], x], x]], And[FreeQ[List[a, c, d, f, g, h, p], x], RationalQ[m], Less[m, -1], NonzeroQ[Plus[Times[c, Power[g, 2]], Times[a, Power[h, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Plus[Times[f, Power[g, 2]], Times[-1, e, g, h], Times[d, Power[h, 2]]], Power[Plus[Times[c, Power[g, 2]], Times[-1, b, g, h], Times[a, Power[h, 2]]], -1]], Int[Times[1, Power[Times[Plus[g, Times[h, x]], Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]], -1]], x], x], Dist[Times[1, Power[Plus[Times[c, Power[g, 2]], Times[-1, b, g, h], Times[a, Power[h, 2]]], -1]], Int[Times[Plus[Times[c, d, g], Times[-1, a, f, g], Times[-1, b, d, h], Times[a, e, h], Times[Plus[Times[c, e, g], Times[-1, b, f, g], Times[-1, c, d, h], Times[a, f, h]], x]], Power[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Times[3, Power[2, -1]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, h], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], NonzeroQ[Plus[Times[c, Power[g, 2]], Times[-1, b, g, h], Times[a, Power[h, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Plus[Times[f, Power[g, 2]], Times[-1, e, g, h], Times[d, Power[h, 2]]], Power[Plus[Times[c, Power[g, 2]], Times[a, Power[h, 2]]], -1]], Int[Times[1, Power[Times[Plus[g, Times[h, x]], Sqrt[Plus[a, Times[c, Power[x, 2]]]]], -1]], x], x], Dist[Times[1, Power[Plus[Times[c, Power[g, 2]], Times[a, Power[h, 2]]], -1]], Int[Times[Plus[Times[c, d, g], Times[-1, a, f, g], Times[a, e, h], Times[Plus[Times[c, e, g], Times[-1, c, d, h], Times[a, f, h]], x]], Power[Power[Plus[a, Times[c, Power[x, 2]]], Times[3, Power[2, -1]]], -1]], x], x]], And[FreeQ[List[a, c, d, e, f, g, h], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], NonzeroQ[Plus[Times[c, Power[g, 2]], Times[a, Power[h, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Plus[Times[f, Power[g, 2]], Times[d, Power[h, 2]]], Power[Plus[Times[c, Power[g, 2]], Times[-1, b, g, h], Times[a, Power[h, 2]]], -1]], Int[Times[1, Power[Times[Plus[g, Times[h, x]], Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]], -1]], x], x], Dist[Times[1, Power[Plus[Times[c, Power[g, 2]], Times[-1, b, g, h], Times[a, Power[h, 2]]], -1]], Int[Times[Plus[Times[c, d, g], Times[-1, a, f, g], Times[-1, b, d, h], Times[-1, Plus[Times[b, f, g], Times[c, d, h], Times[-1, a, f, h]], x]], Power[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Times[3, Power[2, -1]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, f, g, h], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[g, 2]], Times[-1, b, g, h], Times[a, Power[h, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Plus[Times[f, Power[g, 2]], Times[d, Power[h, 2]]], Power[Plus[Times[c, Power[g, 2]], Times[a, Power[h, 2]]], -1]], Int[Times[1, Power[Times[Plus[g, Times[h, x]], Sqrt[Plus[a, Times[c, Power[x, 2]]]]], -1]], x], x], Dist[Times[1, Power[Plus[Times[c, Power[g, 2]], Times[a, Power[h, 2]]], -1]], Int[Times[Plus[Times[c, d, g], Times[-1, a, f, g], Times[-1, Plus[Times[c, d, h], Times[-1, a, f, h]], x]], Power[Power[Plus[a, Times[c, Power[x, 2]]], Times[3, Power[2, -1]]], -1]], x], x]], And[FreeQ[List[a, c, d, f, g, h], x], NonzeroQ[Plus[Times[c, Power[g, 2]], Times[a, Power[h, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[b, c, d], Times[-1, 2, a, c, e], Times[a, b, f], Times[Plus[Times[c, Plus[Times[2, c, d], Times[-1, b, e]]], Times[f, Plus[Power[b, 2], Times[-1, 2, a, c]]]], x]], Power[Plus[g, Times[h, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[p, 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[c, Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[p, 1]], -1]], Int[Times[Power[Plus[g, Times[h, x]], Plus[m, -1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Simp[Plus[Times[h, m, Plus[Times[b, c, d], Times[-1, 2, a, c, e], Times[a, b, f]]], Times[g, Plus[Times[c, Plus[Times[2, c, d], Times[-1, b, e]], Plus[Times[2, p], 3]], Times[-1, f, Plus[Times[2, a, c], Times[-1, Power[b, 2], Plus[p, 2]]]]]], Times[h, Plus[Times[c, Plus[Times[2, c, d], Times[-1, b, e]], Plus[m, Times[2, p], 3]], Times[-1, f, Plus[Times[2, a, c, Plus[m, 1]], Times[-1, Power[b, 2], Plus[m, p, 2]]]]], x]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g, h], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[m, p], Less[p, -1], Greater[m, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[a, e], Times[-1, Plus[Times[c, d], Times[-1, a, f]], x]], Power[Plus[g, Times[h, x]], m], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[2, a, c, Plus[p, 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[2, a, c, Plus[p, 1]], -1]], Int[Times[Power[Plus[g, Times[h, x]], Plus[m, -1]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Simp[Plus[Times[a, Plus[Times[f, g], Times[e, h, m]]], Times[-1, c, d, g, Plus[Times[2, p], 3]], Times[h, Plus[Times[a, f, Plus[m, 1]], Times[-1, c, d, Plus[m, Times[2, p], 3]]], x]], x]], x], x]]], And[FreeQ[List[a, c, d, e, f, g, h], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[m, p], Less[p, -1], Greater[m, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[b, c, d], Times[a, b, f], Times[Plus[Times[2, Power[c, 2], d], Times[f, Plus[Power[b, 2], Times[-1, 2, a, c]]]], x]], Power[Plus[g, Times[h, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[p, 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[c, Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[p, 1]], -1]], Int[Times[Power[Plus[g, Times[h, x]], Plus[m, -1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Simp[Plus[Times[h, m, Plus[Times[b, c, d], Times[a, b, f]]], Times[g, Plus[Times[c, Times[2, c, d], Plus[Times[2, p], 3]], Times[-1, f, Plus[Times[2, a, c], Times[-1, Power[b, 2], Plus[p, 2]]]]]], Times[h, Plus[Times[2, Power[c, 2], d, Plus[m, Times[2, p], 3]], Times[-1, f, Plus[Times[2, a, c, Plus[m, 1]], Times[-1, Power[b, 2], Plus[m, p, 2]]]]], x]], x]], x], x]]], And[FreeQ[List[a, b, c, d, f, g, h], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], RationalQ[m, p], Less[p, -1], Greater[m, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[c, d], Times[-1, a, f]], x, Power[Plus[g, Times[h, x]], m], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[2, a, c, Plus[p, 1]], -1]], x]], Times[-1, Dist[Times[1, Power[Times[2, a, c, Plus[p, 1]], -1]], Int[Times[Power[Plus[g, Times[h, x]], Plus[m, -1]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Simp[Plus[Times[a, f, g], Times[-1, c, d, g, Plus[Times[2, p], 3]], Times[h, Plus[Times[a, f, Plus[m, 1]], Times[-1, c, d, Plus[m, Times[2, p], 3]]], x]], x]], x], x]]], And[FreeQ[List[a, c, d, f, g, h], x], RationalQ[m, p], Less[p, -1], Greater[m, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[Plus[Times[c, d], Times[-1, a, f]], Plus[Times[b, g], Times[-1, 2, a, h]]], Times[-1, Plus[Times[b, d], Times[-1, a, e]], Plus[Times[2, c, g], Times[-1, b, h]]], Times[-1, Plus[Times[2, Power[c, 2], d, g], Times[b, f, Plus[Times[b, g], Times[-1, a, h]]], Times[-1, c, Plus[Times[b, e, g], Times[2, a, f, g], Times[b, d, h], Times[-1, 2, a, e, h]]]], x]], Power[Plus[g, Times[h, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[Times[c, Power[g, 2]], Times[-1, h, Plus[Times[b, g], Times[-1, a, h]]]], Plus[p, 1]], -1]], x]], Times[-1, Dist[Times[1, Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[Times[c, Power[g, 2]], Times[-1, h, Plus[Times[b, g], Times[-1, a, h]]]], Plus[p, 1]], -1]], Int[Times[Power[Plus[g, Times[h, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Simp[Plus[Times[Plus[Times[2, c, d], Times[-1, b, e], Times[2, a, f]], Plus[Times[c, Power[g, 2]], Times[-1, h, Plus[Times[b, g], Times[-1, a, h]]]], Plus[p, 1]], Times[g, Plus[Times[Power[b, 2], f, g], Times[-1, b, Plus[Times[c, e, g], Times[c, d, h], Times[a, f, h]]], Times[2, Plus[Times[Power[c, 2], d, g], Times[-1, a, Plus[Times[c, f, g], Times[-1, c, e, h]]]]]], Plus[p, 2]], Times[-1, h, Plus[Times[Plus[Times[c, d], Times[-1, a, f]], Plus[Times[b, g], Times[-1, 2, a, h]]], Times[-1, Plus[Times[b, d], Times[-1, a, e]], Plus[Times[2, c, g], Times[-1, b, h]]]], Plus[p, m, 2]], Times[h, Plus[Times[Power[b, 2], f, g], Times[-1, b, Plus[Times[c, e, g], Times[c, d, h], Times[a, f, h]]], Times[2, Plus[Times[Power[c, 2], d, g], Times[-1, a, Plus[Times[c, f, g], Times[-1, c, e, h]]]]]], Plus[Times[2, p], m, 4], x]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g, h, m], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[p], Less[p, -1], NonzeroQ[Plus[Times[c, Power[g, 2]], Times[-1, h, Plus[Times[b, g], Times[-1, a, h]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[a, c, e, g], Times[-1, a, h, Plus[Times[c, d], Times[-1, a, f]]], Times[-1, c, Plus[Times[c, d, g], Times[-1, Plus[Times[a, f, g], Times[-1, a, e, h]]]], x]], Power[Plus[g, Times[h, x]], Plus[m, 1]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[2, a, c, Plus[Times[c, Power[g, 2]], Times[a, Power[h, 2]]], Plus[p, 1]], -1]], x], Dist[Times[1, Power[Times[2, a, c, Plus[Times[c, Power[g, 2]], Times[a, Power[h, 2]]], Plus[p, 1]], -1]], Int[Times[Power[Plus[g, Times[h, x]], m], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Simp[Plus[Times[Plus[Times[c, d], Times[a, f]], Plus[Times[c, Power[g, 2]], Times[a, Power[h, 2]]], Plus[p, 1]], Times[g, Plus[Times[Power[c, 2], d, g], Times[-1, a, Plus[Times[c, f, g], Times[-1, c, e, h]]]], Plus[p, 2]], Times[-1, h, Plus[Times[a, c, e, g], Times[-1, a, h, Plus[Times[c, d], Times[-1, a, f]]]], Plus[p, m, 2]], Times[h, Plus[Times[Power[c, 2], d, g], Times[-1, a, Plus[Times[c, f, g], Times[-1, c, e, h]]]], Plus[Times[2, p], m, 4], x]], x]], x], x]], And[FreeQ[List[a, c, d, e, f, g, h, m], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[p], Less[p, -1], NonzeroQ[Plus[Times[c, Power[g, 2]], Times[a, Power[h, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[Plus[Times[c, d], Times[-1, a, f]], Plus[Times[b, g], Times[-1, 2, a, h]]], Times[-1, b, d, Plus[Times[2, c, g], Times[-1, b, h]]], Times[-1, Plus[Times[2, Power[c, 2], d, g], Times[b, f, Plus[Times[b, g], Times[-1, a, h]]], Times[-1, c, Plus[Times[2, a, f, g], Times[b, d, h]]]], x]], Power[Plus[g, Times[h, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[Times[c, Power[g, 2]], Times[-1, h, Plus[Times[b, g], Times[-1, a, h]]]], Plus[p, 1]], -1]], x]], Times[-1, Dist[Times[1, Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[Times[c, Power[g, 2]], Times[-1, h, Plus[Times[b, g], Times[-1, a, h]]]], Plus[p, 1]], -1]], Int[Times[Power[Plus[g, Times[h, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Simp[Plus[Times[2, Plus[Times[c, d], Times[a, f]], Plus[Times[c, Power[g, 2]], Times[-1, h, Plus[Times[b, g], Times[-1, a, h]]]], Plus[p, 1]], Times[g, Plus[Times[Power[b, 2], f, g], Times[-1, b, Plus[Times[c, d, h], Times[a, f, h]]], Times[2, Plus[Times[Power[c, 2], d, g], Times[-1, a, c, f, g]]]], Plus[p, 2]], Times[-1, h, Plus[Times[Plus[Times[c, d], Times[-1, a, f]], Plus[Times[b, g], Times[-1, 2, a, h]]], Times[-1, b, d, Plus[Times[2, c, g], Times[-1, b, h]]]], Plus[p, m, 2]], Times[h, Plus[Times[Power[b, 2], f, g], Times[-1, b, Plus[Times[c, d, h], Times[a, f, h]]], Times[2, Plus[Times[Power[c, 2], d, g], Times[-1, a, c, f, g]]]], Plus[Times[2, p], m, 4], x]], x]], x], x]]], And[FreeQ[List[a, b, c, d, f, g, h, m], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], RationalQ[p], Less[p, -1], NonzeroQ[Plus[Times[c, Power[g, 2]], Times[-1, h, Plus[Times[b, g], Times[-1, a, h]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[a, h, Plus[Times[c, d], Times[-1, a, f]]], Times[c, Plus[Times[c, d, g], Times[-1, a, f, g]], x]], Power[Plus[g, Times[h, x]], Plus[m, 1]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[2, a, c, Plus[Times[c, Power[g, 2]], Times[a, Power[h, 2]]], Plus[p, 1]], -1]], x]], Dist[Times[1, Power[Times[2, a, c, Plus[Times[c, Power[g, 2]], Times[a, Power[h, 2]]], Plus[p, 1]], -1]], Int[Times[Power[Plus[g, Times[h, x]], m], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Simp[Plus[Times[Plus[Times[c, d], Times[a, f]], Plus[Times[c, Power[g, 2]], Times[a, Power[h, 2]]], Plus[p, 1]], Times[g, Plus[Times[Power[c, 2], d, g], Times[-1, a, c, f, g]], Plus[p, 2]], Times[-1, h, Times[-1, a, h, Plus[Times[c, d], Times[-1, a, f]]], Plus[p, m, 2]], Times[h, Plus[Times[Power[c, 2], d, g], Times[-1, a, Times[c, f, g]]], Plus[Times[2, p], m, 4], x]], x]], x], x]], And[FreeQ[List[a, c, d, f, g, h, m], x], RationalQ[p], Less[p, -1], NonzeroQ[Plus[Times[c, Power[g, 2]], Times[a, Power[h, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[f, Power[Power[h, 2], -1]], Int[Times[Power[Plus[g, Times[h, x]], Plus[m, 2]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x], Times[-1, Dist[Times[1, Power[Power[h, 2], -1]], Int[Times[Power[Plus[g, Times[h, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Plus[Times[f, Power[g, 2]], Times[-1, d, Power[h, 2]], Times[h, Plus[Times[2, f, g], Times[-1, e, h]], x]]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g, h, m, p], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], ZeroQ[Plus[m, Times[2, p], 3]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[f, Power[Power[h, 2], -1]], Int[Times[Power[Plus[g, Times[h, x]], Plus[m, 2]], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x], Times[-1, Dist[Times[1, Power[Power[h, 2], -1]], Int[Times[Power[Plus[g, Times[h, x]], m], Power[Plus[a, Times[c, Power[x, 2]]], p], Plus[Times[f, Power[g, 2]], Times[-1, d, Power[h, 2]], Times[h, Plus[Times[2, f, g], Times[-1, e, h]], x]]], x], x]]], And[FreeQ[List[a, c, d, e, f, g, h, m, p], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], ZeroQ[Plus[m, Times[2, p], 3]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[f, Power[Power[h, 2], -1]], Int[Times[Power[Plus[g, Times[h, x]], Plus[m, 2]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x], Times[-1, Dist[Times[1, Power[Power[h, 2], -1]], Int[Times[Power[Plus[g, Times[h, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Plus[Times[f, Power[g, 2]], Times[-1, d, Power[h, 2]], Times[2, f, g, h, x]]], x], x]]], And[FreeQ[List[a, b, c, d, f, g, h, m, p], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[m, Times[2, p], 3]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[f, Power[Power[h, 2], -1]], Int[Times[Power[Plus[g, Times[h, x]], Plus[m, 2]], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x], Times[-1, Dist[Times[1, Power[Power[h, 2], -1]], Int[Times[Power[Plus[g, Times[h, x]], m], Power[Plus[a, Times[c, Power[x, 2]]], p], Plus[Times[f, Power[g, 2]], Times[-1, d, Power[h, 2]], Times[2, f, g, h, x]]], x], x]]], And[FreeQ[List[a, c, d, f, g, h, m, p], x], ZeroQ[Plus[m, Times[2, p], 3]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[f, Power[Plus[g, Times[h, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, h, Plus[m, Times[2, p], 3]], -1]], x], Times[-1, Dist[Times[1, Power[Times[c, h, Plus[m, Times[2, p], 3]], -1]], Int[Times[Power[Plus[g, Times[h, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Simp[Plus[Times[b, f, g, Plus[p, 1]], Times[h, Plus[Times[a, f, Plus[m, 1]], Times[-1, c, d, Plus[m, Times[2, p], 3]]]], Times[Plus[Times[b, f, h, Plus[m, p, 2]], Times[c, Plus[Times[2, f, g, Plus[p, 1]], Times[-1, e, h, Plus[m, Times[2, p], 3]]]]], x]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g, h, m, p], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], NonzeroQ[Plus[m, Times[2, p], 3]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[f, Power[Plus[g, Times[h, x]], Plus[m, 1]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, h, Plus[m, Times[2, p], 3]], -1]], x], Times[-1, Dist[Times[1, Power[Times[c, h, Plus[m, Times[2, p], 3]], -1]], Int[Times[Power[Plus[g, Times[h, x]], m], Power[Plus[a, Times[c, Power[x, 2]]], p], Simp[Plus[Times[h, Plus[Times[a, f, Plus[m, 1]], Times[-1, c, d, Plus[m, Times[2, p], 3]]]], Times[Times[c, Plus[Times[2, f, g, Plus[p, 1]], Times[-1, e, h, Plus[m, Times[2, p], 3]]]], x]], x]], x], x]]], And[FreeQ[List[a, c, d, e, f, g, h, m, p], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], NonzeroQ[Plus[m, Times[2, p], 3]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[f, Power[Plus[g, Times[h, x]], Plus[m, 1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, h, Plus[m, Times[2, p], 3]], -1]], x], Times[-1, Dist[Times[1, Power[Times[c, h, Plus[m, Times[2, p], 3]], -1]], Int[Times[Power[Plus[g, Times[h, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Simp[Plus[Times[b, f, g, Plus[p, 1]], Times[h, Plus[Times[a, f, Plus[m, 1]], Times[-1, c, d, Plus[m, Times[2, p], 3]]]], Times[f, Plus[Times[b, h, Plus[m, p, 2]], Times[2, c, g, Plus[p, 1]]], x]], x]], x], x]]], And[FreeQ[List[a, b, c, d, f, g, h, m, p], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[m, Times[2, p], 3]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[f, Power[Plus[g, Times[h, x]], Plus[m, 1]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Times[c, h, Plus[m, Times[2, p], 3]], -1]], x], Times[-1, Dist[Times[1, Power[Times[c, h, Plus[m, Times[2, p], 3]], -1]], Int[Times[Power[Plus[g, Times[h, x]], m], Power[Plus[a, Times[c, Power[x, 2]]], p], Simp[Plus[Times[h, Plus[Times[a, f, Plus[m, 1]], Times[-1, c, d, Plus[m, Times[2, p], 3]]]], Times[2, c, f, g, Plus[p, 1], x]], x]], x], x]]], And[FreeQ[List[a, c, d, f, g, h, m, p], x], NonzeroQ[Plus[m, Times[2, p], 3]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], Plus[g, Times[h, x]]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, h], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], IntegersQ[p, q], Greater[p, 0]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[a, Times[c, Power[x, 2]]], p], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], Plus[g, Times[h, x]]], x], x], And[FreeQ[List[a, c, d, e, f, g, h], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], IntegersQ[p, q], Or[Greater[p, 0], Greater[q, 0]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[g, b], Times[-1, 2, a, h], Times[-1, Plus[Times[b, h], Times[-1, 2, g, c]], x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[p, 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[q, -1]], Simp[Plus[Times[e, q, Plus[Times[g, b], Times[-1, 2, a, h]]], Times[-1, d, Plus[Times[b, h], Times[-1, 2, g, c]], Plus[Times[2, p], 3]], Times[Plus[Times[2, f, q, Plus[Times[g, b], Times[-1, 2, a, h]]], Times[-1, e, Plus[Times[b, h], Times[-1, 2, g, c]], Plus[Times[2, p], q, 3]]], x], Times[-1, f, Plus[Times[b, h], Times[-1, 2, g, c]], Plus[Times[2, p], Times[2, q], 3], Power[x, 2]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g, h], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[p, q], Less[p, -1], Greater[q, 0]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[a, h], Times[-1, g, c, x]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], Power[Times[2, a, c, Plus[p, 1]], -1]], x], Dist[Times[2, Power[Times[4, a, c, Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[q, -1]], Simp[Plus[Times[g, c, d, Plus[Times[2, p], 3]], Times[-1, a, Times[h, e, q]], Times[Plus[Times[g, c, e, Plus[Times[2, p], q, 3]], Times[-1, a, Times[2, h, f, q]]], x], Times[g, c, f, Plus[Times[2, p], Times[2, q], 3], Power[x, 2]]], x]], x], x]], And[FreeQ[List[a, c, d, e, f, g, h], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[p, q], Less[p, -1], Greater[q, 0]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[g, b], Times[-1, 2, a, h], Times[-1, Plus[Times[b, h], Times[-1, 2, g, c]], x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[f, Power[x, 2]]], q], Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[p, 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[f, Power[x, 2]]], Plus[q, 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Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[Power[Plus[Times[c, d], Times[-1, a, f]], 2], Times[-1, Plus[Times[b, d], Times[-1, a, e]], Plus[Times[c, e], Times[-1, b, f]]]], Plus[p, 1]], -1], Plus[Times[g, c, Plus[Times[2, a, c, e], Times[-1, b, Plus[Times[c, d], Times[a, f]]]]], Times[Plus[Times[g, b], Times[-1, a, h]], Plus[Times[2, Power[c, 2], d], Times[Power[b, 2], f], Times[-1, c, Plus[Times[b, e], Times[2, a, f]]]]], Times[c, Plus[Times[g, Plus[Times[2, Power[c, 2], d], Times[Power[b, 2], f], Times[-1, c, Plus[Times[b, e], Times[2, a, f]]]]], Times[-1, h, Plus[Times[b, c, d], Times[-1, 2, a, c, e], Times[a, b, f]]]], x]]], x], Dist[Times[1, Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[Power[Plus[Times[c, d], Times[-1, a, f]], 2], Times[-1, Plus[Times[b, d], Times[-1, a, e]], Plus[Times[c, e], Times[-1, b, f]]]], Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], Simp[Plus[Times[Plus[Times[b, h], Times[-1, 2, g, c]], Plus[Power[Plus[Times[c, d], Times[-1, a, f]], 2], Times[-1, Plus[Times[b, d], Times[-1, a, e]], Plus[Times[c, e], Times[-1, b, f]]]], Plus[p, 1]], Times[Plus[Times[Power[b, 2], Times[g, f]], Times[-1, b, Plus[Times[h, c, d], Times[g, c, e], Times[a, h, f]]], Times[2, Plus[Times[g, c, Plus[Times[c, d], Times[-1, a, f]]], Times[-1, a, Times[-1, h, c, e]]]]], Plus[Times[a, f, Plus[p, 1]], Times[-1, c, d, Plus[p, 2]]]], Times[-1, e, Plus[Times[Times[g, c], Plus[Times[2, a, c, e], Times[-1, b, Plus[Times[c, d], Times[a, f]]]]], Times[Plus[Times[g, b], Times[-1, a, h]], Plus[Times[2, Power[c, 2], d], Times[Power[b, 2], f], Times[-1, c, Plus[Times[b, e], Times[2, a, f]]]]]], Plus[p, q, 2]], Times[-1, Plus[Times[2, f, Plus[Times[Times[g, c], Plus[Times[2, a, c, e], Times[-1, b, Plus[Times[c, d], Times[a, f]]]]], Times[Plus[Times[g, b], Times[-1, a, h]], Plus[Times[2, Power[c, 2], d], Times[Power[b, 2], f], Times[-1, c, Plus[Times[b, e], Times[2, a, f]]]]]], Plus[p, q, 2]], Times[-1, Plus[Times[Power[b, 2], g, f], Times[-1, b, Plus[Times[h, c, d], Times[g, c, e], Times[a, h, f]]], Times[2, Plus[Times[g, c, Plus[Times[c, d], Times[-1, a, f]]], Times[-1, a, Times[-1, h, c, e]]]]], Plus[Times[b, f, Plus[p, 1]], Times[-1, c, e, Plus[Times[2, p], q, 4]]]]], x], Times[-1, c, f, Plus[Times[Power[b, 2], Times[g, f]], Times[-1, b, Plus[Times[h, c, d], Times[g, c, e], Times[a, h, f]]], Times[2, Plus[Times[g, c, Plus[Times[c, d], Times[-1, a, f]]], Times[a, h, c, e]]]], Plus[Times[2, p], Times[2, q], 5], Power[x, 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, h, q], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[p], Less[p, -1], NonzeroQ[Plus[Power[Plus[Times[c, d], Times[-1, a, f]], 2], Times[-1, Plus[Times[b, d], Times[-1, a, e]], Plus[Times[c, e], Times[-1, b, f]]]]], Not[And[Not[IntegerQ[p]], IntegerQ[q], Less[q, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[Times[-4, a, c], Plus[Times[a, c, Power[e, 2]], Power[Plus[Times[c, d], Times[-1, a, f]], 2]], Plus[p, 1]], -1], Plus[Times[g, c, Times[2, a, c, e]], Times[Times[-1, a, h], Plus[Times[2, Power[c, 2], d], Times[-1, c, Times[2, a, f]]]], Times[c, Plus[Times[g, Plus[Times[2, Power[c, 2], d], Times[-1, c, Times[2, a, f]]]], Times[-1, h, Times[-2, a, c, e]]], x]]], x], Dist[Times[1, Power[Times[Times[-4, a, c], Plus[Times[a, c, Power[e, 2]], Power[Plus[Times[c, d], Times[-1, a, f]], 2]], Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], Simp[Plus[Times[Times[-2, g, c], Plus[Power[Plus[Times[c, d], Times[-1, a, f]], 2], Times[-1, Times[-1, a, e], Times[c, e]]], Plus[p, 1]], Times[Times[2, Plus[Times[g, c, Plus[Times[c, d], Times[-1, a, f]]], Times[-1, a, Times[-1, h, c, e]]]], Plus[Times[a, f, Plus[p, 1]], Times[-1, c, d, Plus[p, 2]]]], Times[-1, e, Plus[Times[Times[g, c], Times[2, a, c, e]], Times[Times[-1, a, h], Plus[Times[2, Power[c, 2], d], Times[-1, c, Times[Plus[2], a, f]]]]], Plus[p, q, 2]], Times[-1, Plus[Times[2, f, Plus[Times[Times[g, c], Times[2, a, c, e]], Times[Times[-1, a, h], Plus[Times[2, Power[c, 2], d], Times[-1, c, Times[Plus[2], a, f]]]]], Plus[p, q, 2]], Times[-1, Times[2, Plus[Times[g, c, Plus[Times[c, d], Times[-1, a, f]]], Times[-1, a, Times[-1, h, c, e]]]], Times[-1, c, e, Plus[Times[2, p], q, 4]]]], x], Times[-1, c, f, Times[2, Plus[Times[g, c, Plus[Times[c, d], Times[-1, a, f]]], Times[-1, a, Times[-1, h, c, e]]]], Plus[Times[2, p], Times[2, q], 5], Power[x, 2]]], x]], x], x]], And[FreeQ[List[a, c, d, e, f, g, h, q], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[p], Less[p, -1], NonzeroQ[Plus[Times[a, c, Power[e, 2]], Power[Plus[Times[c, d], Times[-1, a, f]], 2]]], Not[And[Not[IntegerQ[p]], IntegerQ[q], Less[q, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[Times[Power[b, 2], d, f], Power[Plus[Times[c, d], Times[-1, a, f]], 2]], Plus[p, 1]], -1], Plus[Times[Times[g, c], Times[-1, b, Plus[Times[c, d], Times[a, f]]]], Times[Plus[Times[g, b], Times[-1, a, h]], Plus[Times[2, Power[c, 2], d], Times[Power[b, 2], f], Times[-1, c, Times[2, a, f]]]], Times[c, Plus[Times[g, Plus[Times[2, Power[c, 2], d], Times[Power[b, 2], f], Times[-1, c, Times[2, a, f]]]], Times[-1, h, Plus[Times[b, c, d], Times[a, b, f]]]], x]]], x], Dist[Times[1, Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[Times[Power[b, 2], d, f], Power[Plus[Times[c, d], Times[-1, a, f]], 2]], Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[f, Power[x, 2]]], q], Simp[Plus[Times[Plus[Times[b, h], Times[-1, 2, g, c]], Plus[Power[Plus[Times[c, d], Times[-1, a, f]], 2], Times[-1, Times[b, d], Times[-1, b, f]]], Plus[p, 1]], Times[Plus[Times[Power[b, 2], Times[g, f]], Times[-1, b, Plus[Times[h, c, d], Times[a, h, f]]], Times[2, Times[g, c, Plus[Times[c, d], Times[-1, a, f]]]]], Plus[Times[a, f, Plus[p, 1]], Times[-1, c, d, Plus[p, 2]]]], Times[-1, Plus[Times[2, f, Plus[Times[Times[g, c], Times[-1, b, Plus[Times[c, d], Times[a, f]]]], Times[Plus[Times[g, b], Times[-1, a, h]], Plus[Times[2, Power[c, 2], d], Times[Power[b, 2], f], Times[-1, c, Times[2, a, f]]]]], Plus[p, q, 2]], Times[-1, Plus[Times[Power[b, 2], Times[g, f]], Times[-1, b, Plus[Times[h, c, d], Times[a, h, f]]], Times[2, Times[g, c, Plus[Times[c, d], Times[-1, a, f]]]]], Times[b, f, Plus[p, 1]]]], x], Times[-1, c, f, Plus[Times[Power[b, 2], Times[g, f]], Times[-1, b, Plus[Times[h, c, d], Times[a, h, f]]], Times[2, Times[g, c, Plus[Times[c, d], Times[-1, a, f]]]]], Plus[Times[2, p], Times[2, q], 5], Power[x, 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, f, g, h, q], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], RationalQ[p], Less[p, -1], NonzeroQ[Plus[Times[Power[b, 2], d, f], Power[Plus[Times[c, d], Times[-1, a, f]], 2]]], Not[And[Not[IntegerQ[p]], IntegerQ[q], Less[q, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[h, Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Plus[d, 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Power[x, 2]]], x]], x], x]], And[FreeQ[List[a, c, d, e, f, g, h, q], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[p], Greater[p, 0], NonzeroQ[Plus[p, q, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[h, Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Plus[d, Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[2, f, Plus[p, q, 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[2, f, Plus[p, q, 1]], -1]], Int[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, -1]], Power[Plus[d, Times[f, Power[x, 2]]], q], Simp[Plus[Times[h, p, Times[b, d]], Times[a, Times[-2, g, f], Plus[p, q, 1]], Times[Plus[Times[2, h, p, Plus[Times[c, d], Times[-1, a, f]]], Times[b, Times[-2, g, f], Plus[p, q, 1]]], x], Times[Plus[Times[h, p, Times[-1, b, f]], Times[c, Times[-2, g, f], Plus[p, q, 1]]], Power[x, 2]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, f, g, h, q], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], RationalQ[p], Greater[p, 0], NonzeroQ[Plus[p, q, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Simplify[Plus[Times[Power[c, 2], Power[d, 2]], Times[-1, b, c, d, e], Times[a, c, Power[e, 2]], Times[Power[b, 2], d, f], Times[-1, 2, a, c, d, f], Times[-1, a, b, e, f], Times[Power[a, 2], Power[f, 2]]]]]], Condition[Plus[Dist[Times[1, Power[q, -1]], Int[Times[Simp[Plus[Times[g, Power[c, 2], d], Times[-1, g, b, c, e], Times[a, h, c, e], Times[g, Power[b, 2], f], Times[-1, a, b, h, f], Times[-1, a, g, c, f], Times[c, Plus[Times[h, c, d], Times[-1, g, c, e], Times[g, b, f], Times[-1, a, h, f]], x]], x], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], -1]], x], x], Dist[Times[1, Power[q, -1]], Int[Times[Simp[Plus[Times[-1, h, c, d, e], Times[g, c, Power[e, 2]], Times[b, h, d, f], Times[-1, g, c, d, f], Times[-1, g, b, e, f], Times[a, g, Power[f, 2]], Times[-1, f, Plus[Times[h, c, d], Times[-1, g, c, e], Times[g, b, f], Times[-1, a, h, f]], x]], x], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], -1]], x], x]], NonzeroQ[q]]], And[FreeQ[List[a, b, c, d, e, f, g, h], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Simplify[Plus[Times[Power[c, 2], Power[d, 2]], Times[Power[b, 2], d, f], Times[-1, 2, a, c, d, f], Times[Power[a, 2], Power[f, 2]]]]]], Condition[Plus[Dist[Times[1, Power[q, -1]], Int[Times[Simp[Plus[Times[g, Power[c, 2], d], Times[g, Power[b, 2], f], Times[-1, a, b, h, f], Times[-1, a, g, c, f], Times[c, Plus[Times[h, c, d], Times[g, b, f], Times[-1, a, h, f]], x]], x], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], -1]], x], x], Dist[Times[1, Power[q, -1]], Int[Times[Simp[Plus[Times[b, h, d, f], Times[-1, g, c, d, f], Times[a, g, Power[f, 2]], Times[-1, f, Plus[Times[h, c, d], Times[g, b, f], Times[-1, a, h, f]], x]], x], Power[Plus[d, Times[f, Power[x, 2]]], -1]], x], x]], NonzeroQ[q]]], And[FreeQ[List[a, b, c, d, f, g, h], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[g, Int[Times[1, Power[Times[Plus[a, Times[c, Power[x, 2]]], Sqrt[Plus[d, Times[f, Power[x, 2]]]]], -1]], x], x], Dist[h, Int[Times[x, Power[Times[Plus[a, Times[c, Power[x, 2]]], Sqrt[Plus[d, Times[f, Power[x, 2]]]]], -1]], x], x]], And[FreeQ[List[a, c, d, f, g, h], x], PositiveQ[Times[a, c]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Times[-1, a, c], 2]]], Plus[Times[-1, Dist[Times[Plus[Times[c, g], Times[h, q]], Power[Times[2, q], -1]], Int[Times[1, Power[Times[Plus[q, Times[-1, c, x]], Sqrt[Plus[d, Times[f, Power[x, 2]]]]], -1]], x], x]], Times[-1, Dist[Times[Plus[Times[c, g], Times[-1, h, q]], Power[Times[2, q], -1]], Int[Times[1, Power[Times[Plus[q, Times[c, x]], Sqrt[Plus[d, Times[f, Power[x, 2]]]]], -1]], x], x]]]], And[FreeQ[List[a, c, d, f, g, h], x], Not[PositiveQ[Times[a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[-2, g], Subst[Int[Times[1, Power[Plus[Times[b, d], Times[-1, a, e], Times[-1, b, Power[x, 2]]], -1]], x], x, Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]]], x], And[FreeQ[List[a, b, c, d, e, f, g, h], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], ZeroQ[Plus[Times[c, e], Times[-1, b, f]]], ZeroQ[Plus[Times[h, e], Times[-1, 2, g, f]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Dist[Times[Plus[Times[h, e], Times[-1, 2, g, f]], Power[Times[2, f], -1]], Int[Times[1, Power[Times[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]]], -1]], x], x]], Dist[Times[h, Power[Times[2, f], -1]], Int[Times[Plus[e, Times[2, f, x]], Power[Times[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, h], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], ZeroQ[Plus[Times[c, e], Times[-1, b, f]]], NonzeroQ[Plus[Times[h, e], Times[-1, 2, g, f]]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[x, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[-2, e], Subst[Int[Times[Plus[1, Times[-1, d, Power[x, 2]]], Power[Plus[Times[c, e], Times[-1, b, f], Times[-1, e, Plus[Times[2, c, d], Times[-1, b, e], Times[2, a, f]], Power[x, 2]], Times[Power[d, 2], Plus[Times[c, e], Times[-1, b, f]], Power[x, 4]]], -1]], x], x, Times[Plus[1, Times[Plus[e, Sqrt[Plus[Power[e, 2], Times[-1, 4, d, f]]]], x, Power[Times[2, d], -1]]], Power[Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]], -1]]], x], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], ZeroQ[Plus[Times[b, d], Times[-1, a, e]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[g, Subst[Int[Times[1, Power[Plus[a, Times[Plus[Times[c, d], Times[-1, a, f]], Power[x, 2]]], -1]], x], x, Times[x, Power[Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]], -1]]], x], And[FreeQ[List[a, b, c, d, e, f, g, h], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], ZeroQ[Plus[Times[b, d], Times[-1, a, e]]], ZeroQ[Plus[Times[2, h, d], Times[-1, g, e]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Dist[Times[Plus[Times[2, h, d], Times[-1, g, e]], Power[e, -1]], Int[Times[1, Power[Times[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]]], -1]], x], x]], Dist[Times[h, Power[e, -1]], Int[Times[Plus[Times[2, d], Times[e, x]], Power[Times[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, h], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], ZeroQ[Plus[Times[b, d], Times[-1, a, e]]], NonzeroQ[Plus[Times[2, h, d], Times[-1, g, e]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[-2, g, Plus[Times[g, b], Times[-1, 2, a, h]]], Subst[Int[Times[1, Power[Simp[Plus[Times[g, Plus[Times[g, b], Times[-1, 2, a, h]], Plus[Power[b, 2], Times[-1, 4, a, c]]], Times[-1, Plus[Times[b, d], Times[-1, a, e]], Power[x, 2]]], x], -1]], x], x, Times[Simp[Plus[Times[g, b], Times[-1, 2, a, h], Times[-1, Plus[Times[b, h], Times[-1, 2, g, c]], x]], x], Power[Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]], -1]]], x], And[FreeQ[List[a, b, c, d, e, f, g, h], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], NonzeroQ[Plus[Times[b, d], Times[-1, a, e]]], ZeroQ[Plus[Times[Power[h, 2], Plus[Times[b, d], Times[-1, a, e]]], Times[-1, 2, g, h, Plus[Times[c, d], Times[-1, a, f]]], Times[Power[g, 2], Plus[Times[c, e], Times[-1, b, f]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[-2, a, g, h], Subst[Int[Times[1, Power[Simp[Plus[Times[2, Power[a, 2], g, h, c], Times[a, e, Power[x, 2]]], x], -1]], x], x, Times[Simp[Plus[Times[a, h], Times[-1, g, c, x]], x], Power[Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]], -1]]], x], And[FreeQ[List[a, c, d, e, f, g, h], x], ZeroQ[Plus[Times[a, Power[h, 2], e], Times[2, g, h, Plus[Times[c, d], Times[-1, a, f]]], Times[-1, Power[g, 2], c, e]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[-2, g, Plus[Times[g, b], Times[-1, 2, a, h]]], Subst[Int[Times[1, Power[Simp[Plus[Times[g, Plus[Times[g, b], Times[-1, 2, a, h]], Plus[Power[b, 2], Times[-1, 4, a, c]]], Times[-1, b, d, Power[x, 2]]], x], -1]], x], x, Times[Simp[Plus[Times[g, b], Times[-1, 2, a, h], Times[-1, Plus[Times[b, h], Times[-1, 2, g, c]], x]], x], Power[Sqrt[Plus[d, Times[f, Power[x, 2]]]], -1]]], x], And[FreeQ[List[a, b, c, d, f, g, h], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[b, Power[h, 2], d], Times[-1, 2, g, h, Plus[Times[c, d], Times[-1, a, f]]], Times[-1, Power[g, 2], b, f]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Plus[Dist[Times[Plus[Times[2, c, g], Times[-1, h, Plus[b, Times[-1, q]]]], Power[q, -1]], Int[Times[1, Power[Times[Plus[b, Times[-1, q], Times[2, c, x]], Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]]], -1]], x], x], Times[-1, Dist[Times[Plus[Times[2, c, g], Times[-1, h, Plus[b, q]]], Power[q, -1]], Int[Times[1, Power[Times[Plus[b, q, Times[2, c, x]], Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d, e, f, g, h], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], PosQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Times[-1, a, c], 2]]], Plus[Dist[Plus[Times[h, Power[2, -1]], Times[c, g, Power[Times[2, q], -1]]], Int[Times[1, Power[Times[Plus[Times[-1, q], Times[c, x]], Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]]], -1]], x], x], Dist[Plus[Times[h, Power[2, -1]], Times[-1, c, g, Power[Times[2, q], -1]]], Int[Times[1, Power[Times[Plus[q, Times[c, x]], Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]]], -1]], x], x]]], And[FreeQ[List[a, c, d, e, f, g, h], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], PosQ[Times[-1, a, c]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Plus[Dist[Times[Plus[Times[2, c, g], Times[-1, h, Plus[b, Times[-1, q]]]], Power[q, -1]], Int[Times[1, Power[Times[Plus[b, Times[-1, q], Times[2, c, x]], Sqrt[Plus[d, Times[f, Power[x, 2]]]]], -1]], x], x], Times[-1, Dist[Times[Plus[Times[2, c, g], Times[-1, h, Plus[b, q]]], Power[q, -1]], Int[Times[1, Power[Times[Plus[b, q, Times[2, c, x]], Sqrt[Plus[d, Times[f, Power[x, 2]]]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d, f, g, h], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PosQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[Plus[Times[c, d], Times[-1, a, f]], 2], Times[-1, Plus[Times[b, d], Times[-1, a, e]], Plus[Times[c, e], Times[-1, b, f]]]], 2]]], Plus[Dist[Times[1, Power[Times[2, q], -1]], Int[Times[Simp[Plus[Times[h, Plus[Times[b, d], Times[-1, a, e]]], Times[-1, g, Plus[Times[c, d], Times[-1, a, f], Times[-1, q]]], Times[-1, Plus[Times[g, Plus[Times[c, e], Times[-1, b, f]]], Times[-1, h, Plus[Times[c, d], Times[-1, a, f], q]]], x]], x], Power[Times[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]]], -1]], x], x], Times[-1, Dist[Times[1, Power[Times[2, q], -1]], Int[Times[Simp[Plus[Times[h, Plus[Times[b, d], Times[-1, a, e]]], Times[-1, g, Plus[Times[c, d], Times[-1, a, f], q]], Times[-1, Plus[Times[g, Plus[Times[c, e], Times[-1, b, f]]], Times[-1, h, Plus[Times[c, d], Times[-1, a, f], Times[-1, q]]]], x]], x], Power[Times[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d, e, f, g, h], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], NonzeroQ[Plus[Times[b, d], Times[-1, a, e]]], NegQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[Plus[Times[c, d], Times[-1, a, f]], 2], Times[a, c, Power[e, 2]]], 2]]], Plus[Dist[Times[1, Power[Times[2, q], -1]], Int[Times[Simp[Plus[Times[-1, a, h, e], Times[-1, g, Plus[Times[c, d], Times[-1, a, f], Times[-1, q]]], Times[Plus[Times[h, Plus[Times[c, d], Times[-1, a, f], q]], Times[-1, g, c, e]], x]], x], Power[Times[Plus[a, Times[c, Power[x, 2]]], Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]]], -1]], x], x], Times[-1, Dist[Times[1, Power[Times[2, q], -1]], Int[Times[Simp[Plus[Times[-1, a, h, e], Times[-1, g, Plus[Times[c, d], Times[-1, a, f], q]], Times[Plus[Times[h, Plus[Times[c, d], Times[-1, a, f], Times[-1, q]]], Times[-1, g, c, e]], x]], x], Power[Times[Plus[a, Times[c, Power[x, 2]]], Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]]], -1]], x], x]]]], And[FreeQ[List[a, c, d, e, f, g, h], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], NegQ[Times[-1, a, c]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[Plus[Times[c, d], Times[-1, a, f]], 2], Times[Power[b, 2], d, f]], 2]]], Plus[Dist[Times[1, Power[Times[2, q], -1]], Int[Times[Simp[Plus[Times[h, b, d], Times[-1, g, Plus[Times[c, d], Times[-1, a, f], Times[-1, q]]], Times[Plus[Times[h, Plus[Times[c, d], Times[-1, a, f], q]], Times[g, b, f]], x]], x], Power[Times[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Sqrt[Plus[d, Times[f, Power[x, 2]]]]], -1]], x], x], Times[-1, Dist[Times[1, Power[Times[2, q], -1]], Int[Times[Simp[Plus[Times[h, b, d], Times[-1, g, Plus[Times[c, d], Times[-1, a, f], q]], Times[Plus[Times[h, Plus[Times[c, d], Times[-1, a, f], Times[-1, q]]], Times[g, b, f]], x]], x], Power[Times[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Sqrt[Plus[d, Times[f, Power[x, 2]]]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d, f, g, h], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NegQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[s, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]], Set[t, Rt[Plus[Power[e, 2], Times[-1, 4, d, f]], 2]]], Dist[Times[Sqrt[Plus[b, s, Times[2, c, x]]], Sqrt[Plus[Times[2, a], Times[Plus[b, s], x]]], Sqrt[Plus[e, t, Times[2, f, x]]], Sqrt[Plus[Times[2, d], Times[Plus[e, t], x]]], Power[Times[Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]]], -1]], Int[Times[Plus[g, Times[h, x]], Power[Times[Sqrt[Plus[b, s, Times[2, c, x]]], Sqrt[Plus[Times[2, a], Times[Plus[b, s], x]]], Sqrt[Plus[e, t, Times[2, f, x]]], Sqrt[Plus[Times[2, d], Times[Plus[e, t], x]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, h], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, 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Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 3]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Power[Times[-9, c, Power[h, 2], Power[Power[Plus[Times[2, c, g], Times[-1, b, h]], 2], -1]], Times[1, Power[3, -1]]]]], Plus[Simp[Times[Sqrt[3], h, q, ArcTan[Plus[Times[1, Power[Sqrt[3], -1]], Times[-1, Power[2, Times[2, Power[3, -1]]], Power[Plus[1, Times[-1, Times[3, h, Plus[b, Times[2, c, x]]], Power[Plus[Times[2, c, g], Times[-1, b, h]], -1]]], Times[2, Power[3, -1]]], Power[Times[Sqrt[3], Power[Plus[1, Times[Times[3, h, Plus[b, Times[2, c, x]]], Power[Plus[Times[2, c, g], Times[-1, b, h]], -1]]], Times[1, Power[3, -1]]]], -1]]]], Power[f, -1]], x], Plus[Times[-1, Simp[Times[3, h, q, Log[Plus[Power[Plus[1, Times[-1, 3, h, Plus[b, Times[2, c, x]], Power[Plus[Times[2, c, g], Times[-1, b, h]], -1]]], Times[2, Power[3, -1]]], Times[Power[2, Times[1, Power[3, -1]]], Power[Plus[1, Times[3, h, Plus[b, Times[2, c, x]], Power[Plus[Times[2, c, g], Times[-1, b, h]], -1]]], Times[1, Power[3, -1]]]]]], Power[Times[2, f], -1]], x]], Simp[Times[h, q, Log[Plus[d, Times[e, x], Times[f, Power[x, 2]]]], Power[Times[2, f], -1]], x]]]], And[FreeQ[List[a, b, c, d, e, f, g, h], x], ZeroQ[Plus[Times[c, e], Times[-1, b, f]]], ZeroQ[Plus[Times[Power[c, 2], d], Times[-1, f, Plus[Power[b, 2], Times[-1, 3, a, c]]]]], ZeroQ[Plus[Times[Power[c, 2], Power[g, 2]], Times[-1, b, c, g, h], Times[-1, 2, Power[b, 2], Power[h, 2]], Times[9, a, c, Power[h, 2]]]], PositiveQ[Times[-9, c, Power[h, 2], Power[Power[Plus[Times[2, c, g], Times[-1, b, h]], 2], -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 3]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Sqrt[3], h, ArcTan[Plus[Times[1, Power[Sqrt[3], -1]], Times[-1, Power[2, Times[2, Power[3, -1]]], Power[Plus[1, Times[-1, 3, h, x, Power[g, -1]]], Times[2, Power[3, -1]]], Power[Times[Sqrt[3], Power[Plus[1, Times[3, h, x, Power[g, -1]]], Times[1, Power[3, -1]]]], -1]]]], Power[Times[Power[2, Times[2, Power[3, -1]]], Power[a, Times[1, Power[3, -1]]], f], -1]], x], Plus[Times[-1, Simp[Times[3, h, Log[Plus[Power[Plus[1, Times[-1, 3, h, x, Power[g, -1]]], Times[2, Power[3, -1]]], Times[Power[2, Times[1, Power[3, -1]]], Power[Plus[1, Times[3, h, x, Power[g, -1]]], Times[1, Power[3, -1]]]]]], Power[Times[Power[2, Times[5, Power[3, -1]]], Power[a, Times[1, Power[3, -1]]], f], -1]], x]], Simp[Times[h, Log[Plus[d, Times[f, Power[x, 2]]]], Power[Times[Power[2, Times[5, Power[3, -1]]], Power[a, Times[1, Power[3, -1]]], f], -1]], x]]], And[FreeQ[List[a, c, d, f, g, h], x], ZeroQ[Plus[Times[c, d], Times[3, a, f]]], ZeroQ[Plus[Times[c, Power[g, 2]], Times[9, a, Power[h, 2]]]], PositiveQ[a]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 3]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Times[-1, c, Power[Plus[Power[b, 2], Times[-1, 4, a, c]], -1]]]], Dist[Times[Power[Times[q, Plus[a, Times[b, x], Times[c, Power[x, 2]]]], Times[1, Power[3, -1]]], Power[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Times[1, Power[3, -1]]], -1]], Int[Times[Plus[g, Times[h, x]], Power[Times[Power[Plus[Times[q, a], Times[b, q, x], Times[c, q, Power[x, 2]]], Times[1, Power[3, -1]]], Plus[d, Times[e, x], Times[f, Power[x, 2]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, h], x], ZeroQ[Plus[Times[c, e], Times[-1, b, f]]], ZeroQ[Plus[Times[Power[c, 2], d], Times[-1, f, Plus[Power[b, 2], Times[-1, 3, a, c]]]]], ZeroQ[Plus[Times[Power[c, 2], Power[g, 2]], Times[-1, b, c, g, h], Times[-1, 2, Power[b, 2], Power[h, 2]], Times[9, a, c, Power[h, 2]]]], Not[PositiveQ[Plus[Times[4, a], Times[-1, Power[b, 2], Power[c, -1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 3]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[1, Times[c, Power[x, 2], Power[a, -1]]], Times[1, Power[3, -1]]], Power[Power[Plus[a, Times[c, Power[x, 2]]], Times[1, Power[3, -1]]], -1]], Int[Times[Plus[g, Times[h, x]], Power[Times[Power[Plus[1, Times[c, Power[x, 2], Power[a, -1]]], Times[1, Power[3, -1]]], Plus[d, Times[f, Power[x, 2]]]], -1]], x], x], And[FreeQ[List[a, c, d, f, g, h], x], ZeroQ[Plus[Times[c, d], Times[3, a, f]]], ZeroQ[Plus[Times[c, Power[g, 2]], Times[9, a, Power[h, 2]]]], Not[PositiveQ[a]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Plus[g, Times[h, x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q]], x], FreeQ[List[a, b, c, d, e, f, g, h, p, q], x]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Plus[g, Times[h, x]], Power[Plus[a, Times[c, Power[x, 2]]], p], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q]], x], FreeQ[List[a, c, d, e, f, g, h, p, q], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[u, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[u, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[u, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Coefficient[u, x, 1], -1]], Subst[Int[Times[Power[Plus[g, Times[h, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q]], x], x, u], x], And[FreeQ[List[a, b, c, d, e, f, g, h, m, p, q], x], LinearQ[u, x], NonzeroQ[Plus[u, Times[-1, x]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[u, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[u, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[u, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Coefficient[u, x, 1], -1]], Subst[Int[Times[Power[Plus[g, Times[h, x]], m], Power[Plus[a, Times[c, Power[x, 2]]], p], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q]], x], x, u], x], And[FreeQ[List[a, c, d, e, f, g, h, m, p, q], x], LinearQ[u, x], NonzeroQ[Plus[u, Times[-1, x]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Pattern[v, Blank[]], Optional[Pattern[q, Blank[]]]], Power[Pattern[z, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[ExpandToSum[z, x], m], Power[ExpandToSum[u, x], p], Power[ExpandToSum[v, x], q]], x], And[FreeQ[List[m, p, q], x], LinearQ[z, x], QuadraticQ[List[u, v], x], Not[And[LinearMatchQ[z, x], QuadraticMatchQ[List[u, v], x]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[h, Blank[]]], Times[Optional[Pattern[i, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[h, Times[i, x]], q], Power[Plus[Times[d, f], Times[e, g, Power[x, 2]]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], And[FreeQ[List[a, b, c, d, e, f, g, h, i, m, n, p, q], x], ZeroQ[Plus[Times[e, f], Times[d, g]]], ZeroQ[Plus[m, Times[-1, n]]], Or[IntegerQ[m], And[PositiveQ[d], PositiveQ[f]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[h, Blank[]]], Times[Optional[Pattern[i, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], n], Power[Plus[h, Times[i, x]], q], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, h, i, m, n, p, q], x], PositiveIntegerQ[p], NegativeIntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[h, Blank[]]], Times[Optional[Pattern[i, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[d, Times[e, x]], FracPart[m]], Power[Plus[f, Times[g, x]], FracPart[m]], Power[Power[Plus[Times[d, f], Times[e, g, Power[x, 2]]], FracPart[m]], -1]], Int[Times[Power[Plus[h, Times[i, x]], q], Power[Plus[Times[d, f], Times[e, g, Power[x, 2]]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, h, i, m, n, p, q], x], ZeroQ[Plus[Times[e, f], Times[d, g]]], ZeroQ[Plus[m, Times[-1, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[Times[c, Power[f, -1]], p], Int[Times[Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[p, q]], Plus[A, Times[B, x], Times[C, Power[x, 2]]]], x], x], And[FreeQ[List[a, b, c, d, e, f, A, B, C, p, q], x], ZeroQ[Plus[Times[c, d], Times[-1, a, f]]], ZeroQ[Plus[Times[b, d], Times[-1, a, e]]], Or[IntegerQ[p], PositiveQ[Times[c, Power[f, -1]]]], Or[Not[IntegerQ[q]], LessEqual[LeafCount[Plus[d, Times[e, x], Times[f, Power[x, 2]]]], LeafCount[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[Times[c, Power[f, -1]], p], Int[Times[Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[p, q]], Plus[A, Times[C, Power[x, 2]]]], x], x], And[FreeQ[List[a, b, c, d, e, f, A, C, p, q], x], ZeroQ[Plus[Times[c, d], Times[-1, a, f]]], ZeroQ[Plus[Times[b, d], Times[-1, a, e]]], Or[IntegerQ[p], PositiveQ[Times[c, Power[f, -1]]]], Or[Not[IntegerQ[q]], LessEqual[LeafCount[Plus[d, Times[e, x], Times[f, Power[x, 2]]]], LeafCount[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[a, IntPart[p]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], FracPart[p]], Power[Times[Power[d, IntPart[p]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], FracPart[p]]], -1]], Int[Times[Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[p, q]], Plus[A, Times[B, x], Times[C, Power[x, 2]]]], x], x], And[FreeQ[List[a, b, c, d, e, f, A, B, C, p, q], x], ZeroQ[Plus[Times[c, d], Times[-1, a, f]]], ZeroQ[Plus[Times[b, d], Times[-1, a, e]]], Not[IntegerQ[p]], Not[IntegerQ[q]], Not[PositiveQ[Times[c, Power[f, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[a, IntPart[p]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], FracPart[p]], Power[Times[Power[d, IntPart[p]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], FracPart[p]]], -1]], Int[Times[Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[p, q]], Plus[A, Times[C, Power[x, 2]]]], x], x], And[FreeQ[List[a, b, c, d, e, f, A, C, p, q], x], ZeroQ[Plus[Times[c, d], Times[-1, a, f]]], ZeroQ[Plus[Times[b, d], Times[-1, a, e]]], Not[IntegerQ[p]], Not[IntegerQ[q]], Not[PositiveQ[Times[c, Power[f, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], FracPart[p]], Power[Times[Power[Times[4, c], IntPart[p]], Power[Plus[b, Times[2, c, x]], Times[2, FracPart[p]]]], -1]], Int[Times[Power[Plus[b, Times[2, c, x]], Times[2, p]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], Plus[A, Times[B, x], Times[C, Power[x, 2]]]], x], x], And[FreeQ[List[a, b, c, d, e, f, A, B, C, p, q], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], FracPart[p]], Power[Times[Power[Times[4, c], IntPart[p]], Power[Plus[b, Times[2, c, x]], Times[2, FracPart[p]]]], -1]], Int[Times[Power[Plus[b, Times[2, c, x]], Times[2, p]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], Plus[A, Times[C, Power[x, 2]]]], x], x], And[FreeQ[List[a, b, c, d, e, f, A, C, p, q], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], FracPart[p]], Power[Times[Power[Times[4, c], IntPart[p]], Power[Plus[b, Times[2, c, x]], Times[2, FracPart[p]]]], -1]], Int[Times[Power[Plus[b, Times[2, c, x]], Times[2, p]], Power[Plus[d, Times[f, Power[x, 2]]], q], Plus[A, Times[B, x], Times[C, Power[x, 2]]]], x], x], And[FreeQ[List[a, b, c, d, f, A, B, C, p, q], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], FracPart[p]], Power[Times[Power[Times[4, c], IntPart[p]], Power[Plus[b, Times[2, c, x]], Times[2, FracPart[p]]]], -1]], Int[Times[Power[Plus[b, Times[2, c, x]], Times[2, p]], Power[Plus[d, Times[f, Power[x, 2]]], q], Plus[A, Times[C, Power[x, 2]]]], x], x], And[FreeQ[List[a, b, c, d, f, A, C, p, q], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], Plus[A, Times[B, x], Times[C, Power[x, 2]]]], x], x], And[FreeQ[List[a, b, c, d, e, f, A, B, C], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], IntegersQ[p, q], Greater[p, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], Plus[A, Times[C, Power[x, 2]]]], x], x], And[FreeQ[List[a, b, c, d, e, f, A, C], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], IntegersQ[p, q], Greater[p, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[a, Times[c, Power[x, 2]]], p], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], Plus[A, Times[B, x], Times[C, Power[x, 2]]]], x], x], And[FreeQ[List[a, c, d, e, f, A, B, C], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], IntegersQ[p, q], Or[Greater[p, 0], Greater[q, 0]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[a, Times[c, Power[x, 2]]], p], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], Plus[A, Times[C, Power[x, 2]]]], x], x], And[FreeQ[List[a, c, d, e, f, A, C], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], IntegersQ[p, q], Or[Greater[p, 0], Greater[q, 0]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[A, b, c], Times[-1, 2, a, B, c], Times[a, b, C], Times[-1, Plus[Times[c, Plus[Times[b, B], Times[-1, 2, A, c]]], Times[-1, C, Plus[Power[b, 2], Times[-1, 2, a, c]]]], x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], Power[Times[c, Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[p, 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[c, Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[q, -1]], Simp[Plus[Times[e, q, Plus[Times[A, b, c], Times[-1, 2, a, B, c], Times[a, b, C]]], Times[-1, d, Plus[Times[c, Plus[Times[b, B], Times[-1, 2, A, c]], Plus[Times[2, p], 3]], Times[C, Plus[Times[2, a, c], Times[-1, Power[b, 2], Plus[p, 2]]]]]], Times[Plus[Times[2, f, q, Plus[Times[A, b, c], Times[-1, 2, a, B, c], Times[a, b, C]]], Times[-1, e, Plus[Times[c, Plus[Times[b, B], Times[-1, 2, A, c]], Plus[Times[2, p], q, 3]], Times[C, Plus[Times[2, a, c, Plus[q, 1]], Times[-1, Power[b, 2], Plus[p, q, 2]]]]]]], x], Times[-1, f, Plus[Times[c, Plus[Times[b, B], Times[-1, 2, A, c]], Plus[Times[2, p], Times[2, q], 3]], Times[C, Plus[Times[2, a, c, Plus[Times[2, q], 1]], Times[-1, Power[b, 2], Plus[p, Times[2, q], 2]]]]], Power[x, 2]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, A, B, C], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[p, q], Less[p, -1], Greater[q, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[A, b, c], Times[a, b, C], Times[Plus[Times[2, A, Power[c, 2]], Times[C, Plus[Power[b, 2], Times[-1, 2, a, c]]]], x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], Power[Times[c, Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[p, 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[c, Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[q, -1]], Simp[Plus[Times[A, c, Plus[Times[2, c, d, Plus[Times[2, p], 3]], Times[b, e, q]]], Times[-1, C, Plus[Times[2, a, c, d], Times[-1, Power[b, 2], d, Plus[p, 2]], Times[-1, a, b, e, q]]], Times[Plus[Times[C, Plus[Times[2, a, b, f, q], Times[-1, 2, a, c, e, Plus[q, 1]], Times[Power[b, 2], e, Plus[p, q, 2]]]], Times[2, A, c, Plus[Times[b, f, q], Times[c, e, Plus[Times[2, p], q, 3]]]]], x], Times[-1, f, Plus[Times[-2, A, Power[c, 2], Plus[Times[2, p], Times[2, q], 3]], Times[C, Plus[Times[2, a, c, Plus[Times[2, q], 1]], Times[-1, Power[b, 2], Plus[p, Times[2, q], 2]]]]], Power[x, 2]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, A, C], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[p, q], Less[p, -1], Greater[q, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[a, B], Times[-1, Plus[Times[A, c], Times[-1, a, C]], x]], Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], Power[Times[2, a, c, Plus[p, 1]], -1]], x], Times[-1, Dist[Times[2, Power[Times[Times[-4, a, c], Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[q, -1]], Simp[Plus[Times[A, c, d, Plus[Times[2, p], 3]], Times[-1, a, Plus[Times[C, d], Times[B, e, q]]], Times[Plus[Times[A, c, e, Plus[Times[2, p], q, 3]], Times[-1, a, Plus[Times[2, B, f, q], Times[C, e, Plus[q, 1]]]]], x], Times[-1, f, Plus[Times[a, C, Plus[Times[2, q], 1]], Times[-1, A, c, Plus[Times[2, p], Times[2, q], 3]]], Power[x, 2]]], x]], x], x]]], And[FreeQ[List[a, c, d, e, f, A, B, C], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[p, q], Less[p, -1], Greater[q, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[A, c], Times[-1, a, C]], x, Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], Power[Times[2, a, c, Plus[p, 1]], -1]], x]], Dist[Times[2, Power[Times[4, a, c, Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[q, -1]], Simp[Plus[Times[A, c, d, Plus[Times[2, p], 3]], Times[-1, a, C, d], Times[Plus[Times[A, c, e, Plus[Times[2, p], q, 3]], Times[-1, a, C, e, Plus[q, 1]]], x], Times[-1, f, Plus[Times[a, C, Plus[Times[2, q], 1]], Times[-1, A, c, Plus[Times[2, p], Times[2, q], 3]]], Power[x, 2]]], x]], x], x]], And[FreeQ[List[a, c, d, e, f, A, C], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[p, q], Less[p, -1], Greater[q, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[A, b, c], Times[-1, 2, a, B, c], Times[a, b, C], Times[-1, Plus[Times[c, Plus[Times[b, B], Times[-1, 2, A, c]]], Times[-1, C, Plus[Power[b, 2], Times[-1, 2, a, c]]]], x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[f, Power[x, 2]]], q], Power[Times[c, Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[p, 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[c, Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[f, Power[x, 2]]], Plus[q, -1]], Simp[Plus[Times[-1, d, Plus[Times[c, Plus[Times[b, B], Times[-1, 2, A, c]], Plus[Times[2, p], 3]], Times[C, Plus[Times[2, a, c], Times[-1, Power[b, 2], Plus[p, 2]]]]]], Times[Times[2, f, q, Plus[Times[A, b, c], Times[-1, 2, a, B, c], Times[a, b, C]]], x], Times[-1, f, Plus[Times[c, Plus[Times[b, B], Times[-1, 2, A, c]], Plus[Times[2, p], Times[2, q], 3]], Times[C, Plus[Times[2, a, c, Plus[Times[2, q], 1]], Times[-1, Power[b, 2], Plus[p, Times[2, q], 2]]]]], Power[x, 2]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, f, A, B, C], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], RationalQ[p, q], Less[p, -1], Greater[q, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[A, b, c], Times[a, b, C], Times[Plus[Times[2, A, Power[c, 2]], Times[C, Plus[Power[b, 2], Times[-1, 2, a, c]]]], x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[f, Power[x, 2]]], q], Power[Times[c, Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[p, 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[c, Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[f, Power[x, 2]]], Plus[q, -1]], Simp[Plus[Times[A, c, Times[2, c, d, Plus[Times[2, p], 3]]], Times[-1, C, Plus[Times[2, a, c, d], Times[-1, Power[b, 2], d, Plus[p, 2]]]], Times[Plus[Times[C, Times[2, a, b, f, q]], Times[2, A, c, Times[b, f, q]]], x], Times[-1, f, Plus[Times[-2, A, Power[c, 2], Plus[Times[2, p], Times[2, q], 3]], Times[C, Plus[Times[2, a, c, Plus[Times[2, q], 1]], Times[-1, Power[b, 2], Plus[p, Times[2, q], 2]]]]], Power[x, 2]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, f, A, C], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], RationalQ[p, q], Less[p, -1], Greater[q, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[Power[Plus[Times[c, d], Times[-1, a, f]], 2], Times[-1, Plus[Times[b, d], Times[-1, a, e]], Plus[Times[c, e], Times[-1, b, f]]]], Plus[p, 1]], -1], Plus[Times[Plus[Times[A, c], Times[-1, a, C]], Plus[Times[2, a, c, e], Times[-1, b, Plus[Times[c, d], Times[a, f]]]]], Times[Plus[Times[A, b], Times[-1, a, B]], Plus[Times[2, Power[c, 2], d], Times[Power[b, 2], f], Times[-1, c, Plus[Times[b, e], Times[2, a, f]]]]], Times[c, Plus[Times[A, Plus[Times[2, Power[c, 2], d], Times[Power[b, 2], f], Times[-1, c, Plus[Times[b, e], Times[2, a, f]]]]], Times[-1, B, Plus[Times[b, c, d], Times[-1, 2, a, c, e], Times[a, b, f]]], Times[C, Plus[Times[Power[b, 2], d], Times[-1, a, b, e], Times[-1, 2, a, Plus[Times[c, d], Times[-1, a, f]]]]]], x]]], x], Dist[Times[1, Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[Power[Plus[Times[c, d], Times[-1, a, f]], 2], Times[-1, Plus[Times[b, d], Times[-1, a, e]], Plus[Times[c, e], Times[-1, b, f]]]], Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], Simp[Plus[Times[Plus[Times[b, B], Times[-1, 2, A, c], Times[-1, 2, a, C]], Plus[Power[Plus[Times[c, d], Times[-1, a, f]], 2], Times[-1, Plus[Times[b, d], Times[-1, a, e]], Plus[Times[c, e], Times[-1, b, f]]]], Plus[p, 1]], Times[Plus[Times[Power[b, 2], Plus[Times[C, d], Times[A, f]]], Times[-1, b, Plus[Times[B, c, d], Times[A, c, e], Times[a, C, e], Times[a, B, f]]], Times[2, Plus[Times[A, c, Plus[Times[c, d], Times[-1, a, f]]], Times[-1, a, Plus[Times[c, C, d], Times[-1, B, c, e], Times[-1, a, C, f]]]]]], Plus[Times[a, f, Plus[p, 1]], Times[-1, c, d, Plus[p, 2]]]], Times[-1, e, Plus[Times[Plus[Times[A, c], Times[-1, a, C]], Plus[Times[2, a, c, e], Times[-1, b, Plus[Times[c, d], Times[a, f]]]]], Times[Plus[Times[A, b], Times[-1, a, B]], Plus[Times[2, Power[c, 2], d], Times[Power[b, 2], f], Times[-1, c, Plus[Times[b, e], Times[2, a, f]]]]]], Plus[p, q, 2]], Times[-1, Plus[Times[2, f, Plus[Times[Plus[Times[A, c], Times[-1, a, C]], Plus[Times[2, a, c, e], Times[-1, b, Plus[Times[c, d], Times[a, f]]]]], Times[Plus[Times[A, b], Times[-1, a, B]], Plus[Times[2, Power[c, 2], d], Times[Power[b, 2], f], Times[-1, c, Plus[Times[b, e], Times[2, a, f]]]]]], Plus[p, q, 2]], Times[-1, Plus[Times[Power[b, 2], Plus[Times[C, d], Times[A, f]]], Times[-1, b, Plus[Times[B, c, d], Times[A, c, e], Times[a, C, e], Times[a, B, f]]], Times[2, Plus[Times[A, c, Plus[Times[c, d], Times[-1, a, f]]], Times[-1, a, Plus[Times[c, C, d], Times[-1, B, c, e], Times[-1, a, C, f]]]]]], Plus[Times[b, f, Plus[p, 1]], Times[-1, c, e, Plus[Times[2, p], q, 4]]]]], x], Times[-1, c, f, Plus[Times[Power[b, 2], Plus[Times[C, d], Times[A, f]]], Times[-1, b, Plus[Times[B, c, d], Times[A, c, e], Times[a, C, e], Times[a, B, f]]], Times[2, Plus[Times[A, c, Plus[Times[c, d], Times[-1, a, f]]], Times[-1, a, Plus[Times[c, C, d], Times[-1, B, c, e], Times[-1, a, C, f]]]]]], Plus[Times[2, p], Times[2, q], 5], Power[x, 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B, C, q], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[p], Less[p, -1], NonzeroQ[Plus[Power[Plus[Times[c, d], Times[-1, a, f]], 2], Times[-1, Plus[Times[b, d], Times[-1, a, e]], Plus[Times[c, e], Times[-1, b, f]]]]], Not[And[Not[IntegerQ[p]], IntegerQ[q], Less[q, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[Power[Plus[Times[c, d], Times[-1, a, f]], 2], Times[-1, Plus[Times[b, d], Times[-1, a, e]], Plus[Times[c, e], Times[-1, b, f]]]], Plus[p, 1]], -1], Plus[Times[Plus[Times[A, c], Times[-1, a, C]], Plus[Times[2, a, c, e], Times[-1, b, Plus[Times[c, d], Times[a, f]]]]], Times[Times[A, b], Plus[Times[2, Power[c, 2], d], Times[Power[b, 2], f], Times[-1, c, Plus[Times[b, e], Times[2, a, f]]]]], Times[c, Plus[Times[A, Plus[Times[2, Power[c, 2], d], Times[Power[b, 2], f], Times[-1, c, Plus[Times[b, e], Times[2, a, f]]]]], Times[C, Plus[Times[Power[b, 2], d], Times[-1, a, b, e], Times[-1, 2, a, Plus[Times[c, d], Times[-1, a, f]]]]]], x]]], x], Dist[Times[1, Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[Power[Plus[Times[c, d], Times[-1, a, f]], 2], Times[-1, Plus[Times[b, d], Times[-1, a, e]], Plus[Times[c, e], Times[-1, b, f]]]], Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], Simp[Plus[Times[Plus[Times[-2, A, c], Times[-1, 2, a, C]], Plus[Power[Plus[Times[c, d], Times[-1, a, f]], 2], Times[-1, Plus[Times[b, d], Times[-1, a, e]], Plus[Times[c, e], Times[-1, b, f]]]], Plus[p, 1]], Times[Plus[Times[Power[b, 2], Plus[Times[C, d], Times[A, f]]], Times[-1, b, Plus[Times[Plus[A], c, e], Times[a, C, e]]], Times[2, Plus[Times[A, c, Plus[Times[c, d], Times[-1, a, f]]], Times[-1, a, Plus[Times[c, C, d], Times[-1, a, C, f]]]]]], Plus[Times[a, f, Plus[p, 1]], Times[-1, c, d, Plus[p, 2]]]], Times[-1, e, Plus[Times[Plus[Times[A, c], Times[-1, a, C]], Plus[Times[2, a, c, e], Times[-1, b, Plus[Times[c, d], Times[a, f]]]]], Times[Times[A, b], Plus[Times[2, Power[c, 2], d], Times[Power[b, 2], f], Times[-1, c, Plus[Times[b, e], Times[2, a, f]]]]]], Plus[p, q, 2]], Times[-1, Plus[Times[2, f, Plus[Times[Plus[Times[A, c], Times[-1, a, C]], Plus[Times[2, a, c, e], Times[-1, b, Plus[Times[c, d], Times[a, f]]]]], Times[Times[A, b], Plus[Times[2, Power[c, 2], d], Times[Power[b, 2], f], Times[-1, c, Plus[Times[b, e], Times[2, a, f]]]]]], Plus[p, q, 2]], Times[-1, Plus[Times[Power[b, 2], Plus[Times[C, d], Times[A, f]]], Times[-1, b, Plus[Times[A, c, e], Times[a, C, e]]], Times[2, Plus[Times[A, c, Plus[Times[c, d], Times[-1, a, f]]], Times[-1, a, Plus[Times[c, C, d], Times[-1, a, C, f]]]]]], Plus[Times[b, f, Plus[p, 1]], Times[-1, c, e, Plus[Times[2, p], q, 4]]]]], x], Times[-1, c, f, Plus[Times[Power[b, 2], Plus[Times[C, d], Times[A, f]]], Times[-1, b, Plus[Times[A, c, e], Times[a, C, e]]], Times[2, Plus[Times[A, c, Plus[Times[c, d], Times[-1, a, f]]], Times[-1, a, Plus[Times[c, C, d], Times[-1, a, C, f]]]]]], Plus[Times[2, p], Times[2, q], 5], Power[x, 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, C, q], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[p], Less[p, -1], NonzeroQ[Plus[Power[Plus[Times[c, d], Times[-1, a, f]], 2], Times[-1, Plus[Times[b, d], Times[-1, a, e]], Plus[Times[c, e], Times[-1, b, f]]]]], Not[And[Not[IntegerQ[p]], IntegerQ[q], Less[q, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[Times[-4, a, c], Plus[Times[a, c, Power[e, 2]], Power[Plus[Times[c, d], Times[-1, a, f]], 2]], Plus[p, 1]], -1], Plus[Times[Plus[Times[A, c], Times[-1, a, C]], Times[2, a, c, e]], Times[Times[-1, a, B], Plus[Times[2, Power[c, 2], d], Times[-1, c, Times[2, a, f]]]], Times[c, Plus[Times[A, Plus[Times[2, Power[c, 2], d], Times[-1, c, Times[2, a, f]]]], Times[-1, B, Times[-2, a, c, e]], Times[C, Times[-2, a, Plus[Times[c, d], Times[-1, a, f]]]]], x]]], x], Dist[Times[1, Power[Times[Times[-4, a, c], Plus[Times[a, c, Power[e, 2]], Power[Plus[Times[c, d], Times[-1, a, f]], 2]], Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], Simp[Plus[Times[Plus[Times[-2, A, c], Times[-1, 2, a, C]], Plus[Power[Plus[Times[c, d], Times[-1, a, f]], 2], Times[-1, Times[-1, a, e], Times[c, e]]], Plus[p, 1]], Times[Times[2, Plus[Times[A, c, Plus[Times[c, d], Times[-1, a, f]]], Times[-1, a, Plus[Times[c, C, d], Times[-1, B, c, e], Times[-1, a, C, f]]]]], Plus[Times[a, f, Plus[p, 1]], Times[-1, c, d, Plus[p, 2]]]], Times[-1, e, Plus[Times[Plus[Times[A, c], Times[-1, a, C]], Times[2, a, c, e]], Times[Times[-1, a, B], Plus[Times[2, Power[c, 2], d], Times[-1, c, Times[Plus[2], a, f]]]]], Plus[p, q, 2]], Times[-1, Plus[Times[2, f, Plus[Times[Plus[Times[A, c], Times[-1, a, C]], Times[2, a, c, e]], Times[Times[-1, a, B], Plus[Times[2, Power[c, 2], d], Times[-1, c, Times[Plus[2], a, f]]]]], Plus[p, q, 2]], Times[-1, Times[2, Plus[Times[A, c, Plus[Times[c, d], Times[-1, a, f]]], Times[-1, a, Plus[Times[c, C, d], Times[-1, B, c, e], Times[-1, a, C, f]]]]], Times[-1, c, e, Plus[Times[2, p], q, 4]]]], x], Times[-1, c, f, Times[2, Plus[Times[A, c, Plus[Times[c, d], Times[-1, a, f]]], Times[-1, a, Plus[Times[c, C, d], Times[-1, B, c, e], Times[-1, a, C, f]]]]], Plus[Times[2, p], Times[2, q], 5], Power[x, 2]]], x]], x], x]], And[FreeQ[List[a, c, d, e, f, A, B, C, q], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[p], Less[p, -1], NonzeroQ[Plus[Times[a, c, Power[e, 2]], Power[Plus[Times[c, d], Times[-1, a, f]], 2]]], Not[And[Not[IntegerQ[p]], IntegerQ[q], Less[q, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[Times[-4, a, c], Plus[Times[a, c, Power[e, 2]], Power[Plus[Times[c, d], Times[-1, a, f]], 2]], Plus[p, 1]], -1], Plus[Times[Plus[Times[A, c], Times[-1, a, C]], Times[2, a, c, e]], Times[c, Plus[Times[A, Plus[Times[2, Power[c, 2], d], Times[-1, c, Times[2, a, f]]]], Times[C, Times[-2, a, Plus[Times[c, d], Times[-1, a, f]]]]], x]]], x], Dist[Times[1, Power[Times[Times[-4, a, c], Plus[Times[a, c, Power[e, 2]], Power[Plus[Times[c, d], Times[-1, a, f]], 2]], Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], Simp[Plus[Times[Plus[Times[-2, A, c], Times[-1, 2, a, C]], Plus[Power[Plus[Times[c, d], Times[-1, a, f]], 2], Times[-1, Times[-1, a, e], Times[c, e]]], Plus[p, 1]], Times[Times[2, Plus[Times[A, c, Plus[Times[c, d], Times[-1, a, f]]], Times[-1, a, Plus[Times[c, C, d], Times[-1, a, C, f]]]]], Plus[Times[a, f, Plus[p, 1]], Times[-1, c, d, Plus[p, 2]]]], Times[-1, e, Times[Plus[Times[A, c], Times[-1, a, C]], Times[2, a, c, e]], Plus[p, q, 2]], Times[-1, Plus[Times[2, f, Times[Plus[Times[A, c], Times[-1, a, C]], Times[2, a, c, e]], Plus[p, q, 2]], Times[-1, Times[2, Plus[Times[A, c, Plus[Times[c, d], Times[-1, a, f]]], Times[-1, a, Plus[Times[c, C, d], Times[-1, a, C, f]]]]], Times[-1, c, e, Plus[Times[2, p], q, 4]]]], x], Times[-1, c, f, Times[2, Plus[Times[A, c, Plus[Times[c, d], Times[-1, a, f]]], Times[-1, a, Plus[Times[c, C, d], Times[-1, a, C, f]]]]], Plus[Times[2, p], Times[2, q], 5], Power[x, 2]]], x]], x], x]], And[FreeQ[List[a, c, d, e, f, A, C, q], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[p], Less[p, -1], NonzeroQ[Plus[Times[a, c, Power[e, 2]], Power[Plus[Times[c, d], Times[-1, a, f]], 2]]], Not[And[Not[IntegerQ[p]], IntegerQ[q], Less[q, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[Times[Power[b, 2], d, f], Power[Plus[Times[c, d], Times[-1, a, f]], 2]], Plus[p, 1]], -1], Plus[Times[Plus[Times[A, c], Times[-1, a, C]], Times[-1, b, Plus[Times[c, d], Times[a, f]]]], Times[Plus[Times[A, b], Times[-1, a, B]], Plus[Times[2, Power[c, 2], d], Times[Power[b, 2], f], Times[-1, c, Times[2, a, f]]]], Times[c, Plus[Times[A, Plus[Times[2, Power[c, 2], d], Times[Power[b, 2], f], Times[-1, c, Times[2, a, f]]]], Times[-1, B, Plus[Times[b, c, d], Times[a, b, f]]], Times[C, Plus[Times[Power[b, 2], d], Times[-1, 2, a, Plus[Times[c, d], Times[-1, a, f]]]]]], x]]], x], Dist[Times[1, Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[Times[Power[b, 2], d, f], Power[Plus[Times[c, d], Times[-1, a, f]], 2]], Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[f, Power[x, 2]]], q], Simp[Plus[Times[Plus[Times[b, B], Times[-1, 2, A, c], Times[-1, 2, a, C]], Plus[Power[Plus[Times[c, d], Times[-1, a, f]], 2], Times[-1, Times[b, d], Times[-1, b, f]]], Plus[p, 1]], Times[Plus[Times[Power[b, 2], Plus[Times[C, d], Times[A, f]]], Times[-1, b, Plus[Times[B, c, d], Times[a, B, f]]], Times[2, Plus[Times[A, c, Plus[Times[c, d], Times[-1, a, f]]], Times[-1, a, Plus[Times[c, C, d], Times[-1, a, C, f]]]]]], Plus[Times[a, f, Plus[p, 1]], Times[-1, c, d, Plus[p, 2]]]], Times[-1, Plus[Times[2, f, Plus[Times[Plus[Times[A, c], Times[-1, a, C]], Times[-1, b, Plus[Times[c, d], Times[a, f]]]], Times[Plus[Times[A, b], Times[-1, a, B]], Plus[Times[2, Power[c, 2], d], Times[Power[b, 2], f], Times[-1, c, Times[2, a, f]]]]], Plus[p, q, 2]], Times[-1, Plus[Times[Power[b, 2], Plus[Times[C, d], Times[A, f]]], Times[-1, b, Plus[Times[B, c, d], Times[a, B, f]]], Times[2, Plus[Times[A, c, Plus[Times[c, d], Times[-1, a, f]]], Times[-1, a, Plus[Times[c, C, d], Times[-1, a, C, f]]]]]], Times[b, f, Plus[p, 1]]]], x], Times[-1, c, f, Plus[Times[Power[b, 2], Plus[Times[C, d], Times[A, f]]], Times[-1, b, Plus[Times[B, c, d], Times[a, B, f]]], Times[2, Plus[Times[A, c, Plus[Times[c, d], Times[-1, a, f]]], Times[-1, a, Plus[Times[c, C, d], Times[-1, a, C, f]]]]]], Plus[Times[2, p], Times[2, q], 5], Power[x, 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, f, A, B, C, q], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], RationalQ[p], Less[p, -1], NonzeroQ[Plus[Times[Power[b, 2], d, f], Power[Plus[Times[c, d], Times[-1, a, f]], 2]]], Not[And[Not[IntegerQ[p]], IntegerQ[q], Less[q, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[Times[Power[b, 2], d, f], Power[Plus[Times[c, d], Times[-1, a, f]], 2]], Plus[p, 1]], -1], Plus[Times[Plus[Times[A, c], Times[-1, a, C]], Times[-1, b, Plus[Times[c, d], Times[a, f]]]], Times[Times[A, b], Plus[Times[2, Power[c, 2], d], Times[Power[b, 2], f], Times[-1, c, Times[2, a, f]]]], Times[c, Plus[Times[A, Plus[Times[2, Power[c, 2], d], Times[Power[b, 2], f], Times[-1, c, Times[2, a, f]]]], Times[C, Plus[Times[Power[b, 2], d], Times[-1, 2, a, Plus[Times[c, d], Times[-1, a, f]]]]]], x]]], x], Dist[Times[1, Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[Times[Power[b, 2], d, f], Power[Plus[Times[c, d], Times[-1, a, f]], 2]], Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, 1]], Power[Plus[d, Times[f, Power[x, 2]]], q], Simp[Plus[Times[Plus[Times[-2, A, c], Times[-1, 2, a, C]], Plus[Power[Plus[Times[c, d], Times[-1, a, f]], 2], Times[-1, Times[b, d], Times[-1, b, f]]], Plus[p, 1]], Times[Plus[Times[Power[b, 2], Plus[Times[C, d], Times[A, f]]], Times[2, Plus[Times[A, c, Plus[Times[c, d], Times[-1, a, f]]], Times[-1, a, Plus[Times[c, C, d], Times[-1, a, C, f]]]]]], Plus[Times[a, f, Plus[p, 1]], Times[-1, c, d, Plus[p, 2]]]], Times[-1, Plus[Times[2, f, Plus[Times[Plus[Times[A, c], Times[-1, a, C]], Times[-1, b, Plus[Times[c, d], Times[a, f]]]], Times[Times[A, b], Plus[Times[2, Power[c, 2], d], Times[Power[b, 2], f], Times[-1, c, Times[2, a, f]]]]], Plus[p, q, 2]], Times[-1, Plus[Times[Power[b, 2], Plus[Times[C, d], Times[A, f]]], Times[2, Plus[Times[A, c, Plus[Times[c, d], Times[-1, a, f]]], Times[-1, a, Plus[Times[c, C, d], Times[-1, a, C, f]]]]]], Times[b, f, Plus[p, 1]]]], x], Times[-1, c, f, Plus[Times[Power[b, 2], Plus[Times[C, d], Times[A, f]]], Times[2, Plus[Times[A, c, Plus[Times[c, d], Times[-1, a, f]]], Times[-1, a, Plus[Times[c, C, d], Times[-1, a, C, f]]]]]], Plus[Times[2, p], Times[2, q], 5], Power[x, 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, f, A, C, q], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], RationalQ[p], Less[p, -1], NonzeroQ[Plus[Times[Power[b, 2], d, f], Power[Plus[Times[c, d], Times[-1, a, f]], 2]]], Not[And[Not[IntegerQ[p]], IntegerQ[q], Less[q, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[B, c, f, Plus[Times[2, p], Times[2, q], 3]], Times[C, Plus[Times[b, f, p], Times[-1, c, e, Plus[Times[2, p], q, 2]]]], Times[2, c, C, f, Plus[p, q, 1], x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[2, c, Power[f, 2], Plus[p, q, 1], Plus[Times[2, p], Times[2, q], 3]], -1]], x], Times[-1, Dist[Times[1, Power[Times[2, c, Power[f, 2], Plus[p, q, 1], Plus[Times[2, p], Times[2, q], 3]], -1]], Int[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, -1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], Simp[Plus[Times[p, Plus[Times[b, d], Times[-1, a, e]], Plus[Times[C, Plus[Times[c, e], Times[-1, b, f]], Plus[q, 1]], Times[-1, c, Plus[Times[C, e], Times[-1, B, f]], Plus[Times[2, p], Times[2, q], 3]]]], Times[Plus[p, q, 1], Plus[Times[Power[b, 2], C, d, f, p], Times[a, c, Plus[Times[C, Plus[Times[2, d, f], Times[-1, Power[e, 2], Plus[Times[2, p], q, 2]]]], Times[f, Plus[Times[B, e], Times[-1, 2, A, f]], Plus[Times[2, p], Times[2, q], 3]]]]]], Times[Plus[Times[2, p, Plus[Times[c, d], Times[-1, a, f]], Plus[Times[C, Plus[Times[c, e], Times[-1, b, f]], Plus[q, 1]], Times[-1, c, Plus[Times[C, e], Times[-1, B, f]], Plus[Times[2, p], Times[2, q], 3]]]], Times[Plus[p, q, 1], Plus[Times[C, e, f, p, Plus[Power[b, 2], Times[-1, 4, a, c]]], Times[-1, b, c, Plus[Times[C, Plus[Power[e, 2], Times[-1, 4, d, f]], Plus[Times[2, p], q, 2]], Times[f, Plus[Times[2, C, d], Times[-1, B, e], Times[2, A, f]], Plus[Times[2, p], Times[2, q], 3]]]]]]], x], Times[Plus[Times[p, Plus[Times[c, e], Times[-1, b, f]], Plus[Times[C, Plus[Times[c, e], Times[-1, b, f]], Plus[q, 1]], Times[-1, c, Plus[Times[C, e], Times[-1, B, f]], Plus[Times[2, p], Times[2, q], 3]]]], Times[Plus[p, q, 1], Plus[Times[C, Power[f, 2], p, Plus[Power[b, 2], Times[-1, 4, a, c]]], Times[-1, Power[c, 2], Plus[Times[C, Plus[Power[e, 2], Times[-1, 4, d, f]], Plus[Times[2, p], q, 2]], Times[f, Plus[Times[2, C, d], Times[-1, B, e], Times[2, A, f]], Plus[Times[2, p], Times[2, q], 3]]]]]]], Power[x, 2]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, A, B, C, q], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[p], Greater[p, 0], NonzeroQ[Plus[p, q, 1]], NonzeroQ[Plus[Times[2, p], Times[2, q], 3]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[C, Plus[Times[b, f, p], Times[-1, c, e, Plus[Times[2, p], q, 2]]]], Times[2, c, C, f, Plus[p, q, 1], x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[2, c, Power[f, 2], Plus[p, q, 1], Plus[Times[2, p], Times[2, q], 3]], -1]], x], Times[-1, Dist[Times[1, Power[Times[2, c, Power[f, 2], Plus[p, q, 1], Plus[Times[2, p], Times[2, q], 3]], -1]], Int[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, -1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], Simp[Plus[Times[p, Plus[Times[b, d], Times[-1, a, e]], Plus[Times[C, Plus[Times[c, e], Times[-1, b, f]], Plus[q, 1]], Times[-1, c, Times[C, e], Plus[Times[2, p], Times[2, q], 3]]]], Times[Plus[p, q, 1], Plus[Times[Power[b, 2], C, d, f, p], Times[a, c, Plus[Times[C, Plus[Times[2, d, f], Times[-1, Power[e, 2], Plus[Times[2, p], q, 2]]]], Times[f, Times[-2, A, f], Plus[Times[2, p], Times[2, q], 3]]]]]], Times[Plus[Times[2, p, Plus[Times[c, d], Times[-1, a, f]], Plus[Times[C, Plus[Times[c, e], Times[-1, b, f]], Plus[q, 1]], Times[-1, c, Times[C, e], Plus[Times[2, p], Times[2, q], 3]]]], Times[Plus[p, q, 1], Plus[Times[C, e, f, p, Plus[Power[b, 2], Times[-1, 4, a, c]]], Times[-1, b, c, Plus[Times[C, Plus[Power[e, 2], Times[-1, 4, d, f]], Plus[Times[2, p], q, 2]], Times[f, Plus[Times[2, C, d], Times[2, A, f]], Plus[Times[2, p], Times[2, q], 3]]]]]]], x], Times[Plus[Times[p, Plus[Times[c, e], Times[-1, b, f]], Plus[Times[C, Plus[Times[c, e], Times[-1, b, f]], Plus[q, 1]], Times[-1, c, Times[C, e], Plus[Times[2, p], Times[2, q], 3]]]], Times[Plus[p, q, 1], Plus[Times[C, Power[f, 2], p, Plus[Power[b, 2], Times[-1, 4, a, c]]], Times[-1, Power[c, 2], Plus[Times[C, Plus[Power[e, 2], Times[-1, 4, d, f]], Plus[Times[2, p], q, 2]], Times[f, Plus[Times[2, C, d], Times[2, A, f]], Plus[Times[2, p], Times[2, q], 3]]]]]]], Power[x, 2]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, A, C, q], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[p], Greater[p, 0], NonzeroQ[Plus[p, q, 1]], NonzeroQ[Plus[Times[2, p], Times[2, q], 3]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[B, c, f, Plus[Times[2, p], Times[2, q], 3]], Times[C, Times[-1, c, e, Plus[Times[2, p], q, 2]]], Times[2, c, C, f, Plus[p, q, 1], x]], Power[Plus[a, Times[c, Power[x, 2]]], p], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[2, c, Power[f, 2], Plus[p, q, 1], Plus[Times[2, p], Times[2, q], 3]], -1]], x], Times[-1, Dist[Times[1, Power[Times[2, c, Power[f, 2], Plus[p, q, 1], Plus[Times[2, p], Times[2, q], 3]], -1]], Int[Times[Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, -1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], Simp[Plus[Times[p, Times[-1, a, e], Plus[Times[C, Times[c, e], Plus[q, 1]], Times[-1, c, Plus[Times[C, e], Times[-1, B, f]], Plus[Times[2, p], Times[2, q], 3]]]], Times[Plus[p, q, 1], Times[a, c, Plus[Times[C, Plus[Times[2, d, f], Times[-1, Power[e, 2], Plus[Times[2, p], q, 2]]]], Times[f, Plus[Times[B, e], Times[-1, 2, A, f]], Plus[Times[2, p], Times[2, q], 3]]]]], Times[Plus[Times[2, p, Plus[Times[c, d], Times[-1, a, f]], Plus[Times[C, Times[c, e], Plus[q, 1]], Times[-1, c, Plus[Times[C, e], Times[-1, B, f]], Plus[Times[2, p], Times[2, q], 3]]]], Times[Plus[p, q, 1], Times[C, e, f, p, Times[-4, a, c]]]], x], Times[Plus[Times[p, Times[c, e], Plus[Times[C, Times[c, e], Plus[q, 1]], Times[-1, c, Plus[Times[C, e], Times[-1, B, f]], Plus[Times[2, p], Times[2, q], 3]]]], Times[Plus[p, q, 1], Plus[Times[C, Power[f, 2], p, Times[-4, a, c]], Times[-1, Power[c, 2], Plus[Times[C, Plus[Power[e, 2], Times[-1, 4, d, f]], Plus[Times[2, p], q, 2]], Times[f, Plus[Times[2, C, d], Times[-1, B, e], Times[2, A, f]], Plus[Times[2, p], Times[2, q], 3]]]]]]], Power[x, 2]]], x]], x], x]]], And[FreeQ[List[a, c, d, e, f, A, B, C, q], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[p], Greater[p, 0], NonzeroQ[Plus[p, q, 1]], NonzeroQ[Plus[Times[2, p], Times[2, q], 3]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[C, Times[-1, c, e, Plus[Times[2, p], q, 2]]], Times[2, c, C, f, Plus[p, q, 1], x]], Power[Plus[a, Times[c, Power[x, 2]]], p], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[2, c, Power[f, 2], Plus[p, q, 1], Plus[Times[2, p], Times[2, q], 3]], -1]], x], Times[-1, Dist[Times[1, Power[Times[2, c, Power[f, 2], Plus[p, q, 1], Plus[Times[2, p], Times[2, q], 3]], -1]], Int[Times[Power[Plus[a, Times[c, Power[x, 2]]], Plus[p, -1]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q], Simp[Plus[Times[p, Times[-1, a, e], Plus[Times[C, Times[c, e], Plus[q, 1]], Times[-1, c, Times[C, e], Plus[Times[2, p], Times[2, q], 3]]]], Times[Plus[p, q, 1], Times[a, c, Plus[Times[C, Plus[Times[2, d, f], Times[-1, Power[e, 2], Plus[Times[2, p], q, 2]]]], Times[f, Times[-2, A, f], Plus[Times[2, p], Times[2, q], 3]]]]], Times[Plus[Times[2, p, Plus[Times[c, d], Times[-1, a, f]], Plus[Times[C, Times[c, e], Plus[q, 1]], Times[-1, c, Times[C, e], Plus[Times[2, p], Times[2, q], 3]]]], Times[Plus[p, q, 1], Times[C, e, f, p, Times[-4, a, c]]]], x], Times[Plus[Times[p, Times[c, e], Plus[Times[C, Times[c, e], Plus[q, 1]], Times[-1, c, Times[C, e], Plus[Times[2, p], Times[2, q], 3]]]], Times[Plus[p, q, 1], Plus[Times[C, Power[f, 2], p, Times[-4, a, c]], Times[-1, Power[c, 2], Plus[Times[C, Plus[Power[e, 2], Times[-1, 4, d, f]], Plus[Times[2, p], q, 2]], Times[f, Plus[Times[2, C, d], Times[2, A, f]], Plus[Times[2, p], Times[2, q], 3]]]]]]], Power[x, 2]]], x]], x], x]]], And[FreeQ[List[a, c, d, e, f, A, C, q], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]], RationalQ[p], Greater[p, 0], NonzeroQ[Plus[p, q, 1]], NonzeroQ[Plus[Times[2, p], Times[2, q], 3]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[B, c, f, Plus[Times[2, p], Times[2, q], 3]], Times[C, Times[b, f, p]], Times[2, c, C, f, Plus[p, q, 1], x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Plus[d, Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[2, c, Power[f, 2], Plus[p, q, 1], Plus[Times[2, p], Times[2, q], 3]], -1]], x], Times[-1, Dist[Times[1, Power[Times[2, c, Power[f, 2], Plus[p, q, 1], Plus[Times[2, p], Times[2, q], 3]], -1]], Int[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, -1]], Power[Plus[d, Times[f, Power[x, 2]]], q], Simp[Plus[Times[p, Times[b, d], Plus[Times[C, Times[-1, b, f], Plus[q, 1]], Times[-1, c, Times[-1, B, f], Plus[Times[2, p], Times[2, q], 3]]]], Times[Plus[p, q, 1], Plus[Times[Power[b, 2], C, d, f, p], Times[a, c, Plus[Times[C, Times[2, d, f]], Times[f, Times[-2, A, f], Plus[Times[2, p], Times[2, q], 3]]]]]], Times[Plus[Times[2, p, Plus[Times[c, d], Times[-1, a, f]], Plus[Times[C, Times[-1, b, f], Plus[q, 1]], Times[-1, c, Times[-1, B, f], Plus[Times[2, p], Times[2, q], 3]]]], Times[Plus[p, q, 1], Times[-1, b, c, Plus[Times[C, Times[-4, d, f], Plus[Times[2, p], q, 2]], Times[f, Plus[Times[2, C, d], Times[2, A, f]], Plus[Times[2, p], Times[2, q], 3]]]]]], x], Times[Plus[Times[p, Times[-1, b, f], Plus[Times[C, Times[-1, b, f], Plus[q, 1]], Times[-1, c, Times[-1, B, f], Plus[Times[2, p], Times[2, q], 3]]]], Times[Plus[p, q, 1], Plus[Times[C, Power[f, 2], p, Plus[Power[b, 2], Times[-1, 4, a, c]]], Times[-1, Power[c, 2], Plus[Times[C, Times[-4, d, f], Plus[Times[2, p], q, 2]], Times[f, Plus[Times[2, C, d], Times[2, A, f]], Plus[Times[2, p], Times[2, q], 3]]]]]]], Power[x, 2]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, f, A, B, C, q], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], RationalQ[p], Greater[p, 0], NonzeroQ[Plus[p, q, 1]], NonzeroQ[Plus[Times[2, p], Times[2, q], 3]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[C, Times[b, f, p]], Times[2, c, C, f, Plus[p, q, 1], x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Power[Plus[d, Times[f, Power[x, 2]]], Plus[q, 1]], Power[Times[2, c, Power[f, 2], Plus[p, q, 1], Plus[Times[2, p], Times[2, q], 3]], -1]], x], Times[-1, Dist[Times[1, Power[Times[2, c, Power[f, 2], Plus[p, q, 1], Plus[Times[2, p], Times[2, q], 3]], -1]], Int[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, -1]], Power[Plus[d, Times[f, Power[x, 2]]], q], Simp[Plus[Times[p, Times[b, d], Times[C, Times[-1, b, f], Plus[q, 1]]], Times[Plus[p, q, 1], Plus[Times[Power[b, 2], C, d, f, p], Times[a, c, Plus[Times[C, Times[2, d, f]], Times[f, Times[-2, A, f], Plus[Times[2, p], Times[2, q], 3]]]]]], Times[Plus[Times[2, p, Plus[Times[c, d], Times[-1, a, f]], Times[C, Times[-1, b, f], Plus[q, 1]]], Times[Plus[p, q, 1], Times[-1, b, c, Plus[Times[C, Times[-4, d, f], Plus[Times[2, p], q, 2]], Times[f, Plus[Times[2, C, d], Times[2, A, f]], Plus[Times[2, p], Times[2, q], 3]]]]]], x], Times[Plus[Times[p, Times[-1, b, f], Times[C, Times[-1, b, f], Plus[q, 1]]], Times[Plus[p, q, 1], Plus[Times[C, Power[f, 2], p, Plus[Power[b, 2], Times[-1, 4, a, c]]], Times[-1, Power[c, 2], Plus[Times[C, Times[-4, d, f], Plus[Times[2, p], q, 2]], Times[f, Plus[Times[2, C, d], Times[2, A, f]], Plus[Times[2, p], Times[2, q], 3]]]]]]], Power[x, 2]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, f, A, C, q], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], RationalQ[p], Greater[p, 0], NonzeroQ[Plus[p, q, 1]], NonzeroQ[Plus[Times[2, p], Times[2, q], 3]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Plus[Times[Power[c, 2], Power[d, 2]], Times[-1, b, c, d, e], Times[a, c, Power[e, 2]], Times[Power[b, 2], d, f], Times[-1, 2, a, c, d, f], Times[-1, a, b, e, f], Times[Power[a, 2], Power[f, 2]]]]], Condition[Plus[Dist[Times[1, Power[q, -1]], Int[Times[Plus[Times[A, Power[c, 2], d], Times[-1, a, c, C, d], Times[-1, A, b, c, e], Times[a, B, c, e], Times[A, Power[b, 2], f], Times[-1, a, b, B, f], Times[-1, a, A, c, f], Times[Power[a, 2], C, f], Times[c, Plus[Times[B, c, d], Times[-1, b, C, d], Times[-1, A, c, e], Times[a, C, e], Times[A, b, f], Times[-1, a, B, f]], x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], -1]], x], x], Dist[Times[1, Power[q, -1]], Int[Times[Plus[Times[c, C, Power[d, 2]], Times[-1, B, c, d, e], Times[A, c, Power[e, 2]], Times[b, B, d, f], Times[-1, A, c, d, f], Times[-1, a, C, d, f], Times[-1, A, b, e, f], Times[a, A, Power[f, 2]], Times[-1, f, Plus[Times[B, c, d], Times[-1, b, C, d], Times[-1, A, c, e], Times[a, C, e], Times[A, b, f], Times[-1, a, B, f]], x]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], -1]], x], x]], NonzeroQ[q]]], And[FreeQ[List[a, b, c, d, e, f, A, B, C], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Plus[Times[Power[c, 2], Power[d, 2]], Times[-1, b, c, d, e], Times[a, c, Power[e, 2]], Times[Power[b, 2], d, f], Times[-1, 2, a, c, d, f], Times[-1, a, b, e, f], Times[Power[a, 2], Power[f, 2]]]]], Condition[Plus[Dist[Times[1, Power[q, -1]], Int[Times[Plus[Times[A, Power[c, 2], d], Times[-1, a, c, C, d], Times[-1, A, b, c, e], Times[A, Power[b, 2], f], Times[-1, a, A, c, f], Times[Power[a, 2], C, f], Times[c, Plus[Times[-1, b, C, d], Times[-1, A, c, e], Times[a, C, e], Times[A, b, f]], x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], -1]], x], x], Dist[Times[1, Power[q, -1]], Int[Times[Plus[Times[c, C, Power[d, 2]], Times[A, c, Power[e, 2]], Times[-1, A, c, d, f], Times[-1, a, C, d, f], Times[-1, A, b, e, f], Times[a, A, Power[f, 2]], Times[-1, f, Plus[Times[-1, b, C, d], Times[-1, A, c, e], Times[a, C, e], Times[A, b, f]], x]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], -1]], x], x]], NonzeroQ[q]]], And[FreeQ[List[a, b, c, d, e, f, A, C], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Plus[Times[Power[c, 2], Power[d, 2]], Times[Power[b, 2], d, f], Times[-1, 2, a, c, d, f], Times[Power[a, 2], Power[f, 2]]]]], Condition[Plus[Dist[Times[1, Power[q, -1]], Int[Times[Plus[Times[A, Power[c, 2], d], Times[-1, a, c, C, d], Times[A, Power[b, 2], f], Times[-1, a, b, B, f], Times[-1, a, A, c, f], Times[Power[a, 2], C, f], Times[c, Plus[Times[B, c, d], Times[-1, b, C, d], Times[A, b, f], Times[-1, a, B, f]], x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], -1]], x], x], Dist[Times[1, Power[q, -1]], Int[Times[Plus[Times[c, C, Power[d, 2]], Times[b, B, d, f], Times[-1, A, c, d, f], Times[-1, a, C, d, f], Times[a, A, Power[f, 2]], Times[-1, f, Plus[Times[B, c, d], Times[-1, b, C, d], Times[A, b, f], Times[-1, a, B, f]], x]], Power[Plus[d, Times[f, Power[x, 2]]], -1]], x], x]], NonzeroQ[q]]], And[FreeQ[List[a, b, c, d, f, A, B, C], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Plus[Times[Power[c, 2], Power[d, 2]], Times[Power[b, 2], d, f], Times[-1, 2, a, c, d, f], Times[Power[a, 2], Power[f, 2]]]]], Condition[Plus[Dist[Times[1, Power[q, -1]], Int[Times[Plus[Times[A, Power[c, 2], d], Times[-1, a, c, C, d], Times[A, Power[b, 2], f], Times[-1, a, A, c, f], Times[Power[a, 2], C, f], Times[c, Plus[Times[-1, b, C, d], Times[A, b, f]], x]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], -1]], x], x], Dist[Times[1, Power[q, -1]], Int[Times[Plus[Times[c, C, Power[d, 2]], Times[-1, A, c, d, f], Times[-1, a, C, d, f], Times[a, A, Power[f, 2]], Times[-1, f, Plus[Times[-1, b, C, d], Times[A, b, f]], x]], Power[Plus[d, Times[f, Power[x, 2]]], -1]], x], x]], NonzeroQ[q]]], And[FreeQ[List[a, b, c, d, f, A, C], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[C, Power[c, -1]], Int[Times[1, Power[Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]], -1]], x], x], Dist[Times[1, Power[c, -1]], Int[Times[Plus[Times[A, c], Times[-1, a, C], Times[Plus[Times[B, c], Times[-1, b, C]], x]], Power[Times[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B, C], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[C, Power[c, -1]], Int[Times[1, Power[Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]], -1]], x], x], Dist[Times[1, Power[c, -1]], Int[Times[Plus[Times[A, c], Times[-1, a, C], Times[-1, b, C, x]], Power[Times[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, C], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[C, Power[c, -1]], Int[Times[1, Power[Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]], -1]], x], x], Dist[Times[1, Power[c, -1]], Int[Times[Plus[Times[A, c], Times[-1, a, C], Times[B, c, x]], Power[Times[Plus[a, Times[c, Power[x, 2]]], Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]]], -1]], x], x]], And[FreeQ[List[a, c, d, e, f, A, B, C], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[C, Power[c, -1]], Int[Times[1, Power[Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]], -1]], x], x], Dist[Times[Plus[Times[A, c], Times[-1, a, C]], Power[c, -1]], Int[Times[1, Power[Times[Plus[a, Times[c, Power[x, 2]]], Sqrt[Plus[d, Times[e, x], Times[f, Power[x, 2]]]]], -1]], x], x]], And[FreeQ[List[a, c, d, e, f, A, C], x], NonzeroQ[Plus[Power[e, 2], Times[-1, 4, d, f]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[C, Power[c, -1]], Int[Times[1, Power[Sqrt[Plus[d, Times[f, Power[x, 2]]]], -1]], x], x], Dist[Times[1, Power[c, -1]], Int[Times[Plus[Times[A, c], Times[-1, a, C], Times[Plus[Times[B, c], Times[-1, b, C]], x]], Power[Times[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Sqrt[Plus[d, Times[f, Power[x, 2]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, f, A, B, C], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[C, Power[c, -1]], Int[Times[1, Power[Sqrt[Plus[d, Times[f, Power[x, 2]]]], -1]], x], x], Dist[Times[1, Power[c, -1]], Int[Times[Plus[Times[A, c], Times[-1, a, C], Times[-1, b, C, x]], Power[Times[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Sqrt[Plus[d, Times[f, Power[x, 2]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, f, A, C], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p], Plus[A, Times[B, x], Times[C, Power[x, 2]]], Power[Plus[d, Times[e, x], Times[f, Power[x, 2]]], q]], x], FreeQ[List[a, b, c, d, e, f, A, B, C, p, q], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, 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Int[Times[Plus[Times[r, Cos[Times[Plus[Times[2, k], -1], m, Pi, Power[n, -1]]]], Times[-1, s, Cos[Times[Plus[Times[2, k], -1], Plus[m, 1], Pi, Power[n, -1]]], x]], Power[Plus[Power[r, 2], Times[-1, 2, r, s, Cos[Times[Plus[Times[2, k], -1], Pi, Power[n, -1]]], x], Times[Power[s, 2], Power[x, 2]]], -1]], x]], Plus[Times[-1, Power[Times[-1, r], Plus[m, 1]], Power[Times[a, n, Power[s, m]], -1], Int[Times[1, Power[Plus[r, Times[s, x]], -1]], x]], Dist[Times[2, Power[r, Plus[m, 1]], Power[Times[a, n, Power[s, m]], -1]], Sum[u, List[k, 1, Times[Plus[n, -1], Power[2, -1]]]], x]]], x]], And[FreeQ[List[a, b], x], PositiveIntegerQ[Times[Plus[n, -1], Power[2, -1]]], PositiveIntegerQ[m], Less[m, Plus[n, -1]], PosQ[Times[a, Power[b, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]]], 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Times[2, n, Plus[p, 1]], 1], Power[Times[2, a1, a2, n, Plus[p, 1]], -1]], Int[Times[Power[Times[c, x], m], Power[Plus[a1, Times[b1, Power[x, n]]], Plus[p, 1]], Power[Plus[a2, Times[b2, Power[x, n]]], Plus[p, 1]]], x], x]], And[FreeQ[List[a1, b1, a2, b2, c, m, n], x], ZeroQ[Plus[Times[a2, b1], Times[a1, b2]]], IntegerQ[Plus[p, Simplify[Times[Plus[m, 1], Power[n, -1]]]]], RationalQ[p], Less[p, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[mn, Simplify[Plus[m, Times[-1, n]]]]], Plus[Simp[Times[Power[x, Plus[mn, 1]], Power[Times[b, Plus[mn, 1]], -1]], x], Times[-1, Dist[Times[a, Power[b, -1]], Int[Times[Power[x, mn], Power[Plus[a, Times[b, Power[x, n]]], -1]], x], x]]]], And[FreeQ[List[a, b, m, n], x], FractionQ[Simplify[Times[Plus[m, 1], Power[n, -1]]]], 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Power[Plus[a, Times[b, Power[x, n]]], -1]], x], x], And[FreeQ[List[a, b, c, m, n], x], FractionQ[Simplify[Times[Plus[m, 1], Power[n, -1]]]], Or[SumSimplerQ[m, n], SumSimplerQ[m, Times[-1, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[a, p], Power[Times[c, x], Plus[m, 1]], Power[Times[c, Plus[m, 1]], -1], Hypergeometric2F1[Times[-1, p], Times[Plus[m, 1], Power[n, -1]], Plus[Times[Plus[m, 1], Power[n, -1]], 1], Times[-1, b, Power[x, n], Power[a, -1]]]], x], And[FreeQ[List[a, b, c, m, n, p], x], Not[PositiveIntegerQ[p]], Or[NegativeIntegerQ[p], PositiveQ[a]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[a, IntPart[p]], Power[Plus[a, Times[b, Power[x, n]]], FracPart[p]], Power[Power[Plus[1, Times[b, Power[x, n], Power[a, -1]]], FracPart[p]], -1]], Int[Times[Power[Times[c, x], m], Power[Plus[1, Times[b, Power[x, n], Power[a, -1]]], p]], x], x], And[FreeQ[List[a, b, c, m, n, p], x], Not[PositiveIntegerQ[p]], Not[Or[NegativeIntegerQ[p], PositiveQ[a]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Power[Coefficient[v, x, 1], Plus[m, 1]], -1]], Subst[Int[SimplifyIntegrand[Times[Power[Plus[x, Times[-1, Coefficient[v, x, 0]]], m], Power[Plus[a, Times[b, Power[x, n]]], p]], x], x], x, v], x], And[FreeQ[List[a, b, n, p], x], LinearQ[v, x], IntegerQ[m], NonzeroQ[Plus[v, Times[-1, x]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[u, m], Power[Times[Coefficient[v, x, 1], Power[v, m]], -1]], Subst[Int[Times[Power[x, m], Power[Plus[a, Times[b, Power[x, n]]], p]], x], x, v], x], And[FreeQ[List[a, b, m, n, p], x], LinearPairQ[u, v, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a1, Times[b1, Power[x, n]]], FracPart[p]], Power[Plus[a2, Times[b2, Power[x, n]]], FracPart[p]], Power[Power[Plus[Times[a1, a2], Times[b1, b2, Power[x, Times[2, n]]]], FracPart[p]], -1]], Int[Times[Power[Times[c, x], m], Power[Plus[Times[a1, a2], Times[b1, b2, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a1, b1, a2, b2, c, m, n, p], x], ZeroQ[Plus[Times[a2, b1], Times[a1, b2]]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], q]], x], x], And[FreeQ[List[a, b, c, d, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], PositiveIntegerQ[p, q]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[x, Times[n, Plus[p, q]]], Power[Plus[b, Times[a, Power[x, Times[-1, n]]]], p], Power[Plus[d, Times[c, Power[x, Times[-1, n]]]], q]], x], And[FreeQ[List[a, b, c, d, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], IntegersQ[p, q], NegQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Subst[Int[Times[Power[Plus[a, Times[b, Power[x, Times[-1, n]]]], p], Power[Plus[c, Times[d, Power[x, Times[-1, n]]]], q], Power[Power[x, 2], -1]], x], x, Times[1, Power[x, -1]]]], And[FreeQ[List[a, b, c, d, p, q], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NegativeIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[g, Denominator[n]]], Dist[g, Subst[Int[Times[Power[x, Plus[g, -1]], Power[Plus[a, Times[b, Power[x, Times[g, n]]]], p], Power[Plus[c, Times[d, Power[x, Times[g, n]]]], q]], x], x, Power[x, Times[1, Power[g, -1]]]], x]], And[FreeQ[List[a, b, c, d, p, q], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], FractionQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Subst[Int[Times[1, Power[Plus[c, Times[-1, Plus[Times[b, c], Times[-1, a, d]], Power[x, n]]], -1]], x], x, Times[x, Power[Power[Plus[a, Times[b, Power[x, n]]], Times[1, Power[n, -1]]], -1]]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Times[n, p], 1]], IntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[x, Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Power[Plus[c, Times[d, Power[x, n]]], q], Power[Times[a, n, Plus[p, 1]], -1]], x]], Times[-1, Dist[Times[c, q, Power[Times[a, Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Power[Plus[c, Times[d, Power[x, n]]], Plus[q, -1]]], x], x]]], And[FreeQ[List[a, b, c, d, n, p], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Times[n, Plus[p, q, 1]], 1]], RationalQ[q], Greater[q, 0], NonzeroQ[Plus[p, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[a, p], x, Power[Times[Power[c, Plus[p, 1]], Power[Plus[c, Times[d, Power[x, n]]], Times[1, Power[n, -1]]]], -1], Hypergeometric2F1[Times[1, Power[n, -1]], Times[-1, p], Plus[1, Times[1, Power[n, -1]]], Times[-1, Plus[Times[b, c], Times[-1, a, d]], Power[x, n], Power[Times[a, Plus[c, Times[d, Power[x, n]]]], -1]]]], x], And[FreeQ[List[a, b, c, d, n, q], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Times[n, Plus[p, q, 1]], 1]], NegativeIntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[x, Power[Plus[a, Times[b, Power[x, n]]], p], Power[Times[c, Power[Times[c, Plus[a, Times[b, Power[x, n]]], Power[Times[a, Plus[c, Times[d, Power[x, n]]]], -1]], p], Power[Plus[c, Times[d, Power[x, n]]], Plus[Times[1, Power[n, -1]], p]]], -1], Hypergeometric2F1[Times[1, Power[n, -1]], Times[-1, p], Plus[1, Times[1, Power[n, -1]]], Times[-1, Plus[Times[b, c], Times[-1, a, d]], Power[x, n], Power[Times[a, Plus[c, Times[d, Power[x, n]]]], -1]]]], x], And[FreeQ[List[a, b, c, d, n, p, q], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Times[n, Plus[p, q, 1]], 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[x, Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Power[Plus[c, Times[d, Power[x, n]]], Plus[q, 1]], Power[Times[a, c], -1]], x], And[FreeQ[List[a, b, c, d, n, p, q], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Times[n, Plus[p, q, 2]], 1]], ZeroQ[Plus[Times[a, d, Plus[p, 1]], Times[b, c, Plus[q, 1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[b, x, Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Power[Plus[c, Times[d, Power[x, n]]], Plus[q, 1]], Power[Times[a, n, Plus[p, 1], Plus[Times[b, c], Times[-1, a, d]]], -1]], x]], Dist[Times[Plus[Times[b, c], Times[n, Plus[p, 1], Plus[Times[b, c], Times[-1, a, d]]]], Power[Times[a, n, Plus[p, 1], Plus[Times[b, c], Times[-1, a, d]]], -1]], Int[Times[Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Power[Plus[c, Times[d, Power[x, n]]], q]], x], x]], And[FreeQ[List[a, b, c, d, n, q], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Times[n, Plus[p, q, 2]], 1]], Or[And[RationalQ[p], Less[p, -1]], Not[And[RationalQ[q], Less[q, -1]]]], NonzeroQ[Plus[p, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[c, x, Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Power[a, -1]], x], And[FreeQ[List[a, b, c, d, n, p], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Times[a, d], Times[-1, b, c, Plus[Times[n, Plus[p, 1]], 1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[b, c], Times[-1, a, d]], x, Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Power[Times[a, b, n, Plus[p, 1]], -1]], x]], Times[-1, Dist[Times[Plus[Times[a, d], Times[-1, b, c, Plus[Times[n, Plus[p, 1]], 1]]], Power[Times[a, b, n, Plus[p, 1]], -1]], Int[Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], x], x]]], And[FreeQ[List[a, b, c, d, n, p], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], Or[And[RationalQ[p], Less[p, -1]], NegativeIntegerQ[Plus[Times[1, Power[n, -1]], p]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[c, x, Power[a, -1]], x], Times[-1, Dist[Times[Plus[Times[b, c], Times[-1, a, d]], Power[a, -1]], Int[Times[1, Power[Plus[b, Times[a, Power[x, Times[-1, n]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[n], Less[n, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[d, x, Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Power[Times[b, Plus[Times[n, Plus[p, 1]], 1]], -1]], x], Times[-1, Dist[Times[Plus[Times[a, d], Times[-1, b, c, Plus[Times[n, Plus[p, 1]], 1]]], Power[Times[b, Plus[Times[n, Plus[p, 1]], 1]], -1]], Int[Power[Plus[a, Times[b, Power[x, n]]], p], x], x]]], And[FreeQ[List[a, b, c, d, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Times[n, Plus[p, 1]], 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[PolynomialDivide[Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], Times[-1, q]], x], x], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], PositiveIntegerQ[n, p], NegativeIntegerQ[q], GreaterEqual[p, Times[-1, q]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[b, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Int[Times[1, Power[Plus[a, Times[b, Power[x, n]]], -1]], x], x], Times[-1, Dist[Times[d, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Int[Times[1, Power[Plus[c, Times[d, Power[x, n]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 3]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Sqrt[3], Power[Times[2, c], -1]], Int[Times[1, Power[Times[Power[Plus[a, Times[b, Power[x, 2]]], Times[1, Power[3, -1]]], Plus[Sqrt[3], Times[-1, Rt[Times[b, Power[a, -1]], 2], x]]], -1]], x], x], Dist[Times[Sqrt[3], Power[Times[2, c], -1]], Int[Times[1, Power[Times[Power[Plus[a, Times[b, Power[x, 2]]], Times[1, Power[3, -1]]], Plus[Sqrt[3], Times[Rt[Times[b, Power[a, -1]], 2], x]]], -1]], x], x]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Times[b, c], Times[3, a, d]]], PosQ[Times[b, Power[a, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 3]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[6, -1]], Int[Times[Plus[3, Times[Rt[Times[-1, b, Power[a, -1]], 2], x]], Power[Times[Power[Plus[a, Times[b, Power[x, 2]]], Times[1, Power[3, -1]]], Plus[c, Times[d, Power[x, 2]]]], -1]], x], x], Dist[Times[1, Power[6, -1]], Int[Times[Plus[3, Times[-1, Rt[Times[-1, b, Power[a, -1]], 2], x]], Power[Times[Power[Plus[a, Times[b, Power[x, 2]]], Times[1, Power[3, -1]]], Plus[c, Times[d, Power[x, 2]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Times[b, c], Times[3, a, d]]], NegQ[Times[b, Power[a, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[2, 3]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[b, Power[d, -1]], Int[Times[1, Power[Power[Plus[a, Times[b, Power[x, 2]]], Times[1, Power[3, -1]]], -1]], x], x], Times[-1, Dist[Times[Plus[Times[b, c], Times[-1, a, d]], Power[d, -1]], Int[Times[1, Power[Times[Power[Plus[a, Times[b, Power[x, 2]]], Times[1, Power[3, -1]]], Plus[c, Times[d, Power[x, 2]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Times[b, c], Times[3, a, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 4]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Times[-1, b, Power[x, 2], Power[a, -1]]], Power[Times[2, x], -1]], Subst[Int[Times[1, Power[Times[Sqrt[Times[-1, b, x, Power[a, -1]]], Power[Plus[a, Times[b, x]], Times[1, Power[4, -1]]], Plus[c, Times[d, x]]], -1]], x], x, Power[x, 2]], x], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 4]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Times[-1, b, Power[x, 2], Power[a, -1]]], Power[Times[2, x], -1]], Subst[Int[Times[1, Power[Times[Sqrt[Times[-1, b, x, Power[a, -1]]], Power[Plus[a, Times[b, x]], Times[3, Power[4, -1]]], Plus[c, Times[d, x]]], -1]], x], x, Power[x, 2]], x], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[b, Power[d, -1]], Int[Power[Plus[a, Times[b, Power[x, 2]]], Plus[p, -1]], x], x], Times[-1, Dist[Times[Plus[Times[b, c], Times[-1, a, d]], Power[d, -1]], Int[Times[Power[Plus[a, Times[b, Power[x, 2]]], Plus[p, -1]], Power[Plus[c, Times[d, Power[x, 2]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[p], Greater[p, 0], Or[Equal[p, Times[1, Power[2, -1]]], Equal[Denominator[p], 4]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[b, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Int[Power[Plus[a, Times[b, Power[x, 2]]], p], x], x], Times[-1, Dist[Times[d, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Int[Times[Power[Plus[a, Times[b, Power[x, 2]]], Plus[p, 1]], Power[Plus[c, Times[d, Power[x, 2]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[p], Less[p, -1], Equal[Denominator[p], 4], Or[Equal[p, Times[-5, Power[4, -1]]], Equal[p, Times[-7, Power[4, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[a, Power[c, -1]], Subst[Int[Times[1, Power[Plus[1, Times[-1, 4, a, b, Power[x, 4]]], -1]], x], x, Times[x, Power[Sqrt[Plus[a, Times[b, Power[x, 4]]]], -1]]], x], And[FreeQ[List[a, b, c, d], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], PosQ[Times[a, b]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Times[-1, a, b], 4]]], Plus[Simp[Times[a, Power[Times[2, c, q], -1], ArcTan[Times[q, x, Plus[a, Times[Power[q, 2], Power[x, 2]]], Power[Times[a, Sqrt[Plus[a, Times[b, Power[x, 4]]]]], -1]]]], x], Simp[Times[a, Power[Times[2, c, q], -1], ArcTanh[Times[q, x, Plus[a, Times[-1, Power[q, 2], Power[x, 2]]], Power[Times[a, Sqrt[Plus[a, Times[b, Power[x, 4]]]]], -1]]]], x]]], And[FreeQ[List[a, b, c, d], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], NegQ[Times[a, b]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[b, Power[d, -1]], Int[Times[1, Power[Sqrt[Plus[a, Times[b, Power[x, 4]]]], -1]], x], x], Times[-1, Dist[Times[Plus[Times[b, c], Times[-1, a, d]], Power[d, -1]], Int[Times[1, Power[Times[Sqrt[Plus[a, Times[b, Power[x, 4]]]], Plus[c, Times[d, Power[x, 4]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[1, 4]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Plus[a, Times[b, Power[x, 4]]]], Sqrt[Times[a, Power[Plus[a, Times[b, Power[x, 4]]], -1]]]], Subst[Int[Times[1, Power[Times[Sqrt[Plus[1, Times[-1, b, Power[x, 4]]]], Plus[c, Times[-1, Plus[Times[b, c], Times[-1, a, d]], Power[x, 4]]]], -1]], x], x, Times[x, Power[Power[Plus[a, Times[b, Power[x, 4]]], Times[1, Power[4, -1]]], -1]]], x], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[b, Power[d, -1]], Int[Power[Plus[a, Times[b, Power[x, 4]]], Plus[p, -1]], x], x], Times[-1, Dist[Times[Plus[Times[b, c], Times[-1, a, d]], Power[d, -1]], Int[Times[Power[Plus[a, Times[b, Power[x, 4]]], Plus[p, -1]], Power[Plus[c, Times[d, Power[x, 4]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[p], Or[Equal[p, Times[3, Power[4, -1]]], Equal[p, Times[5, Power[4, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[Times[2, c], -1]], Int[Times[1, Power[Times[Sqrt[Plus[a, Times[b, Power[x, 4]]]], Plus[1, Times[-1, Rt[Times[-1, d, Power[c, -1]], 2], Power[x, 2]]]], -1]], x], x], Dist[Times[1, Power[Times[2, c], -1]], Int[Times[1, Power[Times[Sqrt[Plus[a, Times[b, Power[x, 4]]]], Plus[1, Times[Rt[Times[-1, d, Power[c, -1]], 2], Power[x, 2]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-3, 4]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[b, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Int[Times[1, Power[Power[Plus[a, Times[b, Power[x, 4]]], Times[3, Power[4, -1]]], -1]], x], x], Times[-1, Dist[Times[d, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Int[Times[Power[Plus[a, Times[b, Power[x, 4]]], Times[1, Power[4, -1]]], Power[Plus[c, Times[d, Power[x, 4]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Sqrt[Plus[a, Times[b, Power[x, 2]]]], Power[Times[c, Rt[Times[d, Power[c, -1]], 2], Sqrt[Plus[c, Times[d, Power[x, 2]]]], Sqrt[Times[c, Plus[a, Times[b, Power[x, 2]]], Power[Times[a, Plus[c, Times[d, Power[x, 2]]]], -1]]]], -1], EllipticE[ArcTan[Times[Rt[Times[d, Power[c, -1]], 2], x]], Plus[1, Times[-1, b, c, Power[Times[a, d], -1]]]]], x], And[FreeQ[List[a, b, c, d], x], PosQ[Times[b, Power[a, -1]]], PosQ[Times[d, Power[c, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[x, Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Power[Plus[c, Times[d, Power[x, n]]], q], Power[Times[a, n, Plus[p, 1]], -1]], x]], Dist[Times[1, Power[Times[a, n, Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Power[Plus[c, Times[d, Power[x, n]]], Plus[q, -1]], Simp[Plus[Times[c, Plus[Times[n, Plus[p, 1]], 1]], Times[d, Plus[Times[n, Plus[p, q, 1]], 1], Power[x, n]]], x]], x], x]], And[FreeQ[List[a, b, c, d, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[p, q], Less[p, -1], Less[0, q, 1], IntBinomialQ[a, b, c, d, n, p, q, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[a, d], Times[-1, c, b]], x, Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Power[Plus[c, Times[d, Power[x, n]]], Plus[q, -1]], Power[Times[a, b, n, Plus[p, 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[a, b, n, Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Power[Plus[c, Times[d, Power[x, n]]], Plus[q, -2]], Simp[Plus[Times[c, Plus[Times[a, d], Times[-1, c, b, Plus[Times[n, Plus[p, 1]], 1]]]], Times[d, Plus[Times[a, d, Plus[Times[n, Plus[q, -1]], 1]], Times[-1, b, c, Plus[Times[n, Plus[p, q]], 1]]], Power[x, n]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[p, q], Less[p, -1], Greater[q, 1], IntBinomialQ[a, b, c, d, n, p, q, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[b, x, Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Power[Plus[c, Times[d, Power[x, n]]], Plus[q, 1]], Power[Times[a, n, Plus[p, 1], Plus[Times[b, c], Times[-1, a, d]]], -1]], x]], Dist[Times[1, Power[Times[a, n, Plus[p, 1], Plus[Times[b, c], Times[-1, a, d]]], -1]], Int[Times[Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Power[Plus[c, Times[d, Power[x, n]]], q], Simp[Plus[Times[b, c], Times[n, Plus[p, 1], Plus[Times[b, c], Times[-1, a, d]]], Times[d, b, Plus[Times[n, Plus[p, q, 2]], 1], Power[x, n]]], x]], x], x]], And[FreeQ[List[a, b, c, d, n, q], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[p], Less[p, -1], Not[And[Not[IntegerQ[p]], IntegerQ[q], Less[q, -1]]], IntBinomialQ[a, b, c, d, n, p, q, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], q]], x], x], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], PositiveIntegerQ[n], IntegersQ[p, q], Greater[Plus[p, q], 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[d, x, Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Power[Plus[c, Times[d, Power[x, n]]], Plus[q, -1]], Power[Times[b, Plus[Times[n, Plus[p, q]], 1]], -1]], x], Dist[Times[1, Power[Times[b, Plus[Times[n, Plus[p, q]], 1]], -1]], Int[Times[Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], Plus[q, -2]], Simp[Plus[Times[c, Plus[Times[b, c, Plus[Times[n, Plus[p, q]], 1]], Times[-1, a, d]]], Times[d, Plus[Times[b, c, Plus[Times[n, Plus[p, Times[2, q], -1]], 1]], Times[-1, a, d, Plus[Times[n, Plus[q, -1]], 1]]], Power[x, n]]], x]], x], x]], And[FreeQ[List[a, b, c, d, n, p], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[q], Greater[q, 1], NonzeroQ[Plus[Times[n, Plus[p, q]], 1]], Not[And[IntegerQ[p], Greater[p, 1]]], IntBinomialQ[a, b, c, d, n, p, q, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[x, Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], q], Power[Plus[Times[n, Plus[p, q]], 1], -1]], x], Dist[Times[n, Power[Plus[Times[n, Plus[p, q]], 1], -1]], Int[Times[Power[Plus[a, Times[b, Power[x, n]]], Plus[p, -1]], Power[Plus[c, Times[d, Power[x, n]]], Plus[q, -1]], Simp[Plus[Times[a, c, Plus[p, q]], Times[Plus[Times[q, Plus[Times[b, c], Times[-1, a, d]]], Times[a, d, Plus[p, q]]], Power[x, n]]], x]], x], x]], And[FreeQ[List[a, b, c, d, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[p, q], Greater[q, 0], Greater[p, 0], IntBinomialQ[a, b, c, d, n, p, q, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Sqrt[Plus[a, Times[b, Power[x, 2]]]], Power[Times[a, Rt[Times[d, Power[c, -1]], 2], Sqrt[Plus[c, Times[d, Power[x, 2]]]], Sqrt[Times[c, Plus[a, Times[b, Power[x, 2]]], Power[Times[a, Plus[c, Times[d, Power[x, 2]]]], -1]]]], -1], EllipticF[ArcTan[Times[Rt[Times[d, Power[c, -1]], 2], x]], Plus[1, Times[-1, b, c, Power[Times[a, d], -1]]]]], x], And[FreeQ[List[a, b, c, d], x], PosQ[Times[d, Power[c, -1]]], PosQ[Times[b, Power[a, -1]]], Not[SimplerSqrtQ[Times[b, Power[a, -1]], Times[d, Power[c, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, 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x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], PositiveIntegerQ[n], RationalQ[p, q], Less[p, -1], Less[0, q, 1], IntBinomialQ[a, b, c, d, e, m, n, p, q, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[a, Power[e, Plus[Times[2, n], -1]], Power[Times[e, x], Plus[m, Times[-1, 2, n], 1]], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Power[Plus[c, Times[d, Power[x, n]]], Plus[q, 1]], Power[Times[b, n, Plus[Times[b, c], Times[-1, a, d]], Plus[p, 1]], -1]], x]], Dist[Times[Power[e, Times[2, n]], Power[Times[b, n, Plus[Times[b, c], Times[-1, a, d]], 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Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[e, Plus[Times[2, n], -1]], Power[Times[e, x], Plus[m, Times[-1, 2, n], 1]], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Power[Plus[c, Times[d, Power[x, n]]], Plus[q, 1]], Power[Times[b, d, Plus[m, Times[n, Plus[p, q]], 1]], -1]], x], Times[-1, Dist[Times[Power[e, Times[2, n]], Power[Times[b, d, Plus[m, Times[n, Plus[p, q]], 1]], -1]], Int[Times[Power[Times[e, x], Plus[m, Times[-1, 2, n]]], Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], q], Simp[Plus[Times[a, c, Plus[m, Times[-1, 2, n], 1]], Times[Plus[Times[a, d, Plus[m, Times[n, Plus[q, -1]], 1]], Times[b, c, Plus[m, Times[n, Plus[p, -1]], 1]]], Power[x, n]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, p, q], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], 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Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[e, IntPart[m]], Power[Times[e, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], q]], x], x], And[FreeQ[List[a, b, c, d, e, m, p, q], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], FractionQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Plus[m, 1], -1]], Subst[Int[Times[Power[Plus[a, Times[b, Power[x, Simplify[Times[n, Power[Plus[m, 1], -1]]]]]], p], Power[Plus[c, Times[d, Power[x, Simplify[Times[n, Power[Plus[m, 1], -1]]]]]], q]], x], x, Power[x, Plus[m, 1]]], x], And[FreeQ[List[a, b, c, d, m, n, p, q], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], IntegerQ[Simplify[Times[n, Power[Plus[m, 1], -1]]]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[e, IntPart[m]], Power[Times[e, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], q]], x], x], And[FreeQ[List[a, b, c, d, e, m, n, p, q], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], IntegerQ[Simplify[Times[n, Power[Plus[m, 1], -1]]]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[c, b], Times[-1, a, d]], Power[Times[e, x], Plus[m, 1]], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Power[Plus[c, Times[d, Power[x, n]]], Plus[q, -1]], Power[Times[a, b, e, n, Plus[p, 1]], -1]], x]], Dist[Times[1, Power[Times[a, b, n, Plus[p, 1]], -1]], Int[Times[Power[Times[e, x], m], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Power[Plus[c, Times[d, Power[x, n]]], Plus[q, -2]], Simp[Plus[Times[c, Plus[Times[c, b, n, Plus[p, 1]], Times[Plus[Times[c, b], Times[-1, a, d]], Plus[m, 1]]]], Times[d, Plus[Times[c, b, n, Plus[p, 1]], Times[Plus[Times[c, b], Times[-1, a, d]], Plus[m, Times[n, Plus[q, -1]], 1]]], Power[x, n]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, m, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[p, q], Less[p, -1], Greater[q, 1], IntBinomialQ[a, b, c, d, e, m, n, p, q, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Times[e, x], Plus[m, 1]], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Power[Plus[c, Times[d, Power[x, n]]], q], Power[Times[a, e, n, Plus[p, 1]], -1]], x]], Dist[Times[1, Power[Times[a, n, Plus[p, 1]], -1]], Int[Times[Power[Times[e, x], m], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Power[Plus[c, Times[d, Power[x, n]]], Plus[q, -1]], Simp[Plus[Times[c, Plus[m, Times[n, Plus[p, 1]], 1]], Times[d, Plus[m, Times[n, Plus[p, q, 1]], 1], Power[x, n]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, m, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[p, q], Less[p, -1], Less[0, q, 1], IntBinomialQ[a, b, c, d, e, m, n, p, q, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[b, Power[Times[e, x], Plus[m, 1]], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Power[Plus[c, Times[d, Power[x, n]]], Plus[q, 1]], Power[Times[a, e, n, Plus[Times[b, c], Times[-1, a, d]], Plus[p, 1]], -1]], x]], Dist[Times[1, Power[Times[a, n, Plus[Times[b, c], Times[-1, a, d]], Plus[p, 1]], -1]], Int[Times[Power[Times[e, x], m], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Power[Plus[c, Times[d, Power[x, n]]], q], Simp[Plus[Times[c, b, Plus[m, 1]], Times[n, Plus[Times[b, c], Times[-1, a, d]], Plus[p, 1]], Times[d, b, Plus[m, Times[n, Plus[p, q, 2]], 1], Power[x, n]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, m, n, q], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[p], Less[p, -1], IntBinomialQ[a, b, c, d, e, m, n, p, q, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[e, x], Plus[m, 1]], Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], q], Power[Times[e, Plus[m, Times[n, Plus[p, q]], 1]], -1]], x], Dist[Times[n, Power[Plus[m, Times[n, Plus[p, q]], 1], -1]], Int[Times[Power[Times[e, x], m], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, -1]], Power[Plus[c, Times[d, Power[x, n]]], Plus[q, -1]], Simp[Plus[Times[a, c, Plus[p, q]], Times[Plus[Times[q, Plus[Times[b, c], Times[-1, a, d]]], Times[a, d, Plus[p, q]]], Power[x, n]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, m, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[p, q], Greater[q, 0], Greater[p, 0], IntBinomialQ[a, b, c, d, e, m, n, p, q, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[d, Power[Times[e, x], Plus[m, 1]], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Power[Plus[c, Times[d, Power[x, n]]], Plus[q, -1]], Power[Times[b, e, Plus[m, Times[n, Plus[p, q]], 1]], -1]], x], Dist[Times[1, Power[Times[b, Plus[m, Times[n, Plus[p, q]], 1]], -1]], Int[Times[Power[Times[e, x], m], Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], Plus[q, -2]], Simp[Plus[Times[c, Plus[Times[Plus[Times[c, b], Times[-1, a, d]], Plus[m, 1]], Times[c, b, n, Plus[p, q]]]], Times[Plus[Times[d, Plus[Times[c, b], Times[-1, a, d]], Plus[m, 1]], Times[d, n, Plus[q, -1], Plus[Times[b, c], Times[-1, a, d]]], Times[c, b, d, n, Plus[p, q]]], Power[x, n]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, m, n, p], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[q], Greater[q, 1], IntBinomialQ[a, b, c, d, e, m, n, p, q, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, 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q], GreaterEqual[p, -2], Or[GreaterEqual[q, -2], And[Equal[q, -3], IntegerQ[Times[Plus[m, -1], Power[2, -1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[x, Plus[m, Times[-1, n, q]]], Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[d, Times[c, Power[x, n]]], q]], x], And[FreeQ[List[a, b, c, d, m, n, p], x], EqQ[mn, Times[-1, n]], IntegerQ[q], Or[PosQ[n], Not[IntegerQ[p]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, 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Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[e, IntPart[m]], Power[Times[e, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, Times[-1, n]]]], q]], x], x], And[FreeQ[List[a, b, c, d, e, m, n, p, q], x], EqQ[mn, Times[-1, n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[a, p], Power[c, q], Power[Times[e, x], Plus[m, 1]], Power[Times[e, Plus[m, 1]], -1], AppellF1[Times[Plus[m, 1], Power[n, -1]], Times[-1, p], Times[-1, q], Plus[1, Times[Plus[m, 1], Power[n, -1]]], Times[-1, b, Power[x, n], Power[a, -1]], Times[-1, d, Power[x, n], Power[c, -1]]]], x], And[FreeQ[List[a, b, c, d, e, m, n, p, q], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[m, 1]], NonzeroQ[Plus[m, Times[-1, n], 1]], PositiveQ[a], PositiveQ[c]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[a, IntPart[p]], Power[Plus[a, Times[b, Power[x, n]]], FracPart[p]], Power[Power[Plus[1, Times[b, Power[x, n], 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Power[Plus[c, Times[d, Power[x, n]]], q]], x], x], x, v], x], And[FreeQ[List[a, b, c, d, n, p, q], x], LinearQ[v, x], IntegerQ[m], NonzeroQ[Plus[v, Times[-1, x]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[u, m], Power[Times[Coefficient[v, x, 1], Power[v, m]], -1]], Subst[Int[Times[Power[x, m], Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], q]], x], x, v], x], And[FreeQ[List[a, b, c, d, m, n, p, q], x], LinearPairQ[u, v, x]]]], RuleDelayed[HoldPattern[Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], 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Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a1, Times[b1, Power[x, Times[n, Power[2, -1]]]]], FracPart[p]], Power[Plus[a2, Times[b2, Power[x, Times[n, Power[2, -1]]]]], FracPart[p]], Power[Power[Plus[Times[a1, a2], Times[b1, b2, Power[x, n]]], FracPart[p]], -1]], Int[Times[u, Power[Plus[Times[a1, a2], Times[b1, b2, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], q]], x], x], And[FreeQ[List[a1, b1, a2, b2, c, d, n, p, q], x], ZeroQ[Plus[non2, Times[-1, n, Power[2, -1]]]], ZeroQ[Plus[Times[a2, b1], Times[a1, b2]]]]]], 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e, f], x], NegQ[Times[Plus[Times[d, e], Times[-1, c, f]], Power[c, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[b, Power[f, -1]], Int[Times[Sqrt[Plus[c, Times[d, Power[x, 2]]]], Power[Times[Sqrt[Plus[a, Times[b, Power[x, 2]]]], Sqrt[Plus[e, Times[f, Power[x, 2]]]]], -1]], x], x], Times[-1, Dist[Times[Plus[Times[b, e], Times[-1, a, f]], Power[f, -1]], Int[Times[Sqrt[Plus[c, Times[d, Power[x, 2]]]], Power[Times[Sqrt[Plus[a, Times[b, Power[x, 2]]]], Power[Plus[e, Times[f, Power[x, 2]]], Times[3, Power[2, -1]]]], -1]], x], x]]], FreeQ[List[a, b, c, d, e, f], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[r, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, ExpandIntegrand[Times[Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], q], Power[Plus[e, Times[f, Power[x, n]]], r]], x]]], Condition[Int[u, x], SumQ[u]]], And[FreeQ[List[a, b, c, d, e, f, p, q, r], x], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[r, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Subst[Int[Times[Power[Plus[a, Times[b, Power[x, Times[-1, n]]]], p], Power[Plus[c, Times[d, Power[x, Times[-1, n]]]], q], Power[Plus[e, Times[f, Power[x, Times[-1, n]]]], r], Power[Power[x, 2], -1]], x], x, Times[1, Power[x, -1]]]], And[FreeQ[List[a, b, c, d, e, f, p, q, r], x], NegativeIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], q], Power[Plus[e, Times[f, Power[x, n]]], r]], x], FreeQ[List[a, b, c, d, e, f, n, p, q, r], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[w, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Coefficient[u, x, 1], -1]], Subst[Int[Times[Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], q], Power[Plus[e, Times[f, Power[x, n]]], r]], x], x, u], x], And[FreeQ[List[a, b, c, d, e, f, p, n, q, r], x], ZeroQ[Plus[u, Times[-1, v]]], ZeroQ[Plus[u, Times[-1, w]]], LinearQ[u, x], NonzeroQ[Plus[u, Times[-1, x]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[d, Times[c, Power[x, n]]], q], Power[Plus[e, Times[f, Power[x, n]]], r], Power[Power[x, Times[n, q]], -1]], x], And[FreeQ[List[a, b, c, d, e, f, n, p, r], x], EqQ[mn, Times[-1, n]], IntegerQ[q]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[x, Times[n, Plus[p, r]]], Power[Plus[b, Times[a, Power[x, Times[-1, n]]]], p], Power[Plus[c, Times[d, Power[x, Times[-1, n]]]], q], Power[Plus[f, Times[e, Power[x, Times[-1, n]]]], r]], x], And[FreeQ[List[a, b, c, d, e, f, n, q], x], EqQ[mn, Times[-1, n]], IntegerQ[p], IntegerQ[r]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Pattern[q, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[x, Times[n, FracPart[q]]], Power[Plus[c, Times[d, Power[x, Times[-1, n]]]], FracPart[q]], Power[Power[Plus[d, Times[c, Power[x, n]]], FracPart[q]], -1]], Int[Times[Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[d, Times[c, Power[x, n]]], q], Power[Plus[e, Times[f, Power[x, n]]], r], Power[Power[x, Times[n, q]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, n, p, q, r], x], EqQ[mn, Times[-1, n]], Not[IntegerQ[q]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[e1, Blank[]], Times[Optional[Pattern[f1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[r, Blank[]]]], Power[Plus[Pattern[e2, Blank[]], Times[Optional[Pattern[f2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[r, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], q], Power[Plus[Times[e1, e2], Times[f1, f2, Power[x, n]]], r]], x], And[FreeQ[List[a, b, c, d, e1, f1, e2, f2, n, p, q, r], x], ZeroQ[Plus[n2, Times[-1, n, Power[2, -1]]]], ZeroQ[Plus[Times[e2, f1], Times[e1, f2]]], Or[IntegerQ[r], And[PositiveQ[e1], PositiveQ[e2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[e1, Blank[]], Times[Optional[Pattern[f1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[r, Blank[]]]], Power[Plus[Pattern[e2, Blank[]], Times[Optional[Pattern[f2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[r, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[e1, Times[f1, Power[x, Times[n, Power[2, -1]]]]], FracPart[r]], Power[Plus[e2, Times[f2, Power[x, Times[n, Power[2, -1]]]]], FracPart[r]], Power[Power[Plus[Times[e1, e2], Times[f1, f2, Power[x, n]]], FracPart[r]], -1]], Int[Times[Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], q], Power[Plus[Times[e1, e2], Times[f1, f2, Power[x, n]]], r]], x], x], And[FreeQ[List[a, b, c, d, e1, f1, e2, f2, n, p, q, r], x], ZeroQ[Plus[n2, Times[-1, n, Power[2, -1]]]], ZeroQ[Plus[Times[e2, f1], Times[e1, f2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[g, m], Power[Times[n, Power[b, Plus[Simplify[Times[Plus[m, 1], Power[n, -1]]], -1]]], -1]], Subst[Int[Times[Power[Times[b, x], Plus[p, Simplify[Times[Plus[m, 1], Power[n, -1]]], -1]], Power[Plus[c, Times[d, x]], q], Power[Plus[e, Times[f, x]], r]], x], x, Power[x, n]], x], And[FreeQ[List[b, c, d, e, f, g, m, n, p, q, r], x], Or[IntegerQ[m], PositiveQ[g]], IntegerQ[Simplify[Times[Plus[m, 1], Power[n, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[g, m], Power[b, IntPart[p]], Power[Times[b, Power[x, n]], FracPart[p]], Power[Power[x, Times[n, FracPart[p]]], -1]], Int[Times[Power[x, Plus[m, Times[n, p]]], Power[Plus[c, Times[d, Power[x, n]]], q], Power[Plus[e, Times[f, Power[x, n]]], r]], x], x], And[FreeQ[List[b, c, d, e, f, g, m, n, p, q, r], x], Or[IntegerQ[m], PositiveQ[g]], Not[IntegerQ[Simplify[Times[Plus[m, 1], Power[n, -1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[g, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[g, IntPart[m]], Power[Times[g, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Times[b, Power[x, n]], p], Power[Plus[c, Times[d, Power[x, n]]], q], Power[Plus[e, Times[f, Power[x, n]]], r]], x], x], And[FreeQ[List[b, c, d, e, f, g, m, n, p, q, r], x], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Times[g, x], m], Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], q], Power[Plus[e, Times[f, Power[x, n]]], r]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, m, n], x], PositiveIntegerQ[Plus[p, 2], q, r]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[n, -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], p], Power[Plus[c, Times[d, x]], q], Power[Plus[e, Times[f, x]], r]], x], x, Power[x, n]], x], And[FreeQ[List[a, b, c, d, e, f, m, n, p, q, r], x], ZeroQ[Plus[m, Times[-1, n], 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[x, Plus[m, Times[n, Plus[p, q, r]]]], Power[Plus[b, Times[a, Power[x, Times[-1, n]]]], p], Power[Plus[d, Times[c, Power[x, Times[-1, n]]]], q], Power[Plus[f, Times[e, Power[x, Times[-1, n]]]], r]], x], And[FreeQ[List[a, b, c, d, e, f, m, n], x], IntegersQ[p, q, r], NegQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[n, -1]], Subst[Int[Times[Power[x, Plus[Simplify[Times[Plus[m, 1], Power[n, -1]]], -1]], Power[Plus[a, Times[b, x]], p], Power[Plus[c, Times[d, x]], q], Power[Plus[e, Times[f, x]], r]], x], x, Power[x, n]], x], And[FreeQ[List[a, b, c, d, e, f, m, n, p, q, r], x], IntegerQ[Simplify[Times[Plus[m, 1], Power[n, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[g, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[g, IntPart[m]], Power[Times[g, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], q], Power[Plus[e, Times[f, Power[x, n]]], r]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, m, n, p, q, r], x], IntegerQ[Simplify[Times[Plus[m, 1], Power[n, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[k, GCD[Plus[m, 1], n]]], Condition[Dist[Times[1, Power[k, -1]], Subst[Int[Times[Power[x, Plus[Times[Plus[m, 1], Power[k, -1]], -1]], Power[Plus[a, Times[b, Power[x, Times[n, Power[k, -1]]]]], p], Power[Plus[c, Times[d, Power[x, Times[n, Power[k, -1]]]]], q], Power[Plus[e, Times[f, Power[x, Times[n, Power[k, -1]]]]], r]], x], x, Power[x, k]], x], Unequal[k, 1]]], And[FreeQ[List[a, b, c, d, e, f, p, q, r], x], PositiveIntegerQ[n], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[r, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[k, Denominator[m]]], Dist[Times[k, Power[g, -1]], Subst[Int[Times[Power[x, Plus[Times[k, Plus[m, 1]], -1]], Power[Plus[a, Times[b, Power[x, Times[k, n]], Power[Power[g, n], -1]]], p], Power[Plus[c, Times[d, Power[x, Times[k, n]], Power[Power[g, n], -1]]], q], Power[Plus[e, Times[f, Power[x, Times[k, n]], Power[Power[g, n], -1]]], r]], x], x, Power[Times[g, x], Times[1, Power[k, -1]]]], x]], 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Blank[Symbol]]]], Condition[Plus[Simp[Times[f, Power[g, Plus[n, -1]], Power[Times[g, x], Plus[m, Times[-1, n], 1]], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Power[Plus[c, Times[d, Power[x, n]]], Plus[q, 1]], Power[Times[b, d, Plus[m, Times[n, Plus[p, q, 1]], 1]], -1]], x], Times[-1, Dist[Times[Power[g, n], Power[Times[b, d, Plus[m, Times[n, Plus[p, q, 1]], 1]], -1]], Int[Times[Power[Times[g, x], Plus[m, Times[-1, n]]], Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], q], Simp[Plus[Times[a, f, c, Plus[m, Times[-1, n], 1]], Times[Plus[Times[a, f, d, Plus[m, Times[n, q], 1]], Times[b, Plus[Times[f, c, Plus[m, Times[n, p], 1]], Times[-1, e, d, Plus[m, Times[n, Plus[p, q, 1]], 1]]]]], Power[x, n]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g, p, q], x], PositiveIntegerQ[n], RationalQ[m], Greater[m, Plus[n, -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[e, Power[Times[g, x], Plus[m, 1]], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Power[Plus[c, Times[d, Power[x, n]]], Plus[q, 1]], Power[Times[a, c, g, Plus[m, 1]], -1]], x], Dist[Times[1, Power[Times[a, c, Power[g, n], Plus[m, 1]], -1]], Int[Times[Power[Times[g, x], Plus[m, n]], Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], q], Simp[Plus[Times[a, f, c, Plus[m, 1]], Times[-1, e, Plus[Times[b, c], Times[a, d]], Plus[m, n, 1]], Times[-1, e, n, Plus[Times[b, c, p], Times[a, d, q]]], Times[-1, b, e, d, Plus[m, Times[n, Plus[p, q, 2]], 1], Power[x, n]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, p, q], x], PositiveIntegerQ[n], RationalQ[m], Less[m, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Times[g, x], m], Power[Plus[a, Times[b, Power[x, n]]], p], Plus[e, Times[f, Power[x, n]]], Power[Plus[c, Times[d, Power[x, n]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, m, p], x], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[e, Int[Times[Power[Times[g, x], m], Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], q]], x], x], Dist[Times[f, Power[Power[e, n], -1]], Int[Times[Power[Times[g, x], Plus[m, n]], Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], q]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, m, p, q], x], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[e, Int[Times[Power[Times[g, x], m], Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], q], Power[Plus[e, Times[f, Power[x, n]]], Plus[r, -1]]], x], x], Dist[Times[f, Power[Power[e, n], -1]], Int[Times[Power[Times[g, x], Plus[m, n]], Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], q], Power[Plus[e, 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And[FreeQ[List[a, b, c, d, e, f, p, q, r], x], NegativeIntegerQ[n], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[k, Denominator[m]]], Times[-1, Dist[Times[k, Power[g, -1]], Subst[Int[Times[Power[Plus[a, Times[b, Power[g, Times[-1, n]], Power[x, Times[-1, k, n]]]], p], Power[Plus[c, Times[d, Power[g, Times[-1, n]], Power[x, Times[-1, k, n]]]], q], Power[Plus[e, Times[f, Power[g, Times[-1, n]], 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n]]]], p], Power[Plus[c, Times[d, Power[x, Times[-1, n]]]], q], Power[Plus[e, Times[f, Power[x, Times[-1, n]]]], r], Power[Power[x, Plus[m, 2]], -1]], x], x, Times[1, Power[x, -1]]], x]], And[FreeQ[List[a, b, c, d, e, f, g, m, p, q, r], x], NegativeIntegerQ[n], Not[RationalQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[k, Denominator[n]]], Dist[k, Subst[Int[Times[Power[x, Plus[Times[k, Plus[m, 1]], -1]], Power[Plus[a, Times[b, 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p, q, r], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[u, m], Power[Times[Coefficient[v, x, 1], Power[v, m]], -1]], Subst[Int[Times[Power[x, m], Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], q], Power[Plus[e, Times[f, Power[x, n]]], r]], x], x, v], x], And[FreeQ[List[a, b, c, d, e, f, m, n, p, q, r], x], LinearPairQ[u, v, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[e1, Blank[]], Times[Optional[Pattern[f1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[r, Blank[]]]], Power[Plus[Pattern[e2, Blank[]], Times[Optional[Pattern[f2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[r, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Times[g, x], m], Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], q], Power[Plus[Times[e1, e2], Times[f1, f2, Power[x, n]]], r]], x], 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Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[e1, Times[f1, Power[x, Times[n, Power[2, -1]]]]], FracPart[r]], Power[Plus[e2, Times[f2, Power[x, Times[n, Power[2, -1]]]]], FracPart[r]], Power[Power[Plus[Times[e1, e2], Times[f1, f2, Power[x, n]]], FracPart[r]], -1]], Int[Times[Power[Times[g, x], m], Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], q], Power[Plus[Times[e1, e2], Times[f1, f2, Power[x, n]]], r]], x], x], And[FreeQ[List[a, b, c, d, e1, f1, e2, f2, g, m, n, p, q, r], x], ZeroQ[Plus[n2, Times[-1, n, Power[2, -1]]]], ZeroQ[Plus[Times[e2, f1], Times[e1, f2]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[x, Times[2, n, p]], Power[Plus[c, Times[b, Power[x, 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Power[Times[d, Plus[m, 1], Plus[m, n, 1]], -1]], x], Plus[Dist[Times[2, c, p, Power[n, 2], Plus[Times[2, p], -1], Power[Times[Power[d, Times[2, n]], Plus[m, 1], Plus[m, n, 1]], -1]], Int[Times[Power[Times[d, x], Plus[m, Times[2, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, -1]]], x], x], Simp[Times[n, p, Power[Times[d, x], Plus[m, 1]], Plus[Times[2, a], Times[b, Power[x, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, -1]], Power[Times[d, Plus[m, 1], Plus[m, n, 1]], -1]], x]]], And[FreeQ[List[a, b, c, d], x], EqQ[n2, Times[2, n]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m, p], Greater[p, 1], Less[Plus[m, n], -1], Not[And[NegativeIntegerQ[Times[Plus[m, Times[2, n, Plus[p, 1]], 1], Power[n, -1]]], Greater[Plus[Times[Plus[m, Times[2, n, Plus[p, 1]], 1], Power[n, -1]], p], 0]]], IntegerQ[Times[2, p]], IntegerQ[m]]]], 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Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[d, Plus[n, -1]], Plus[m, Times[n, Plus[Times[2, p], 1]], 1], Power[Times[d, x], Plus[m, Times[-1, n], 1]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[Times[b, Power[n, 2], Plus[p, 1], Plus[Times[2, p], 1]], -1]], x], Plus[Times[-1, Dist[Times[Power[d, n], Plus[m, Times[-1, n], 1], Plus[m, Times[n, Plus[Times[2, p], 1]], 1], Power[Times[b, Power[n, 2], Plus[p, 1], Plus[Times[2, p], 1]], -1]], Int[Times[Power[Times[d, x], Plus[m, Times[-1, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]]], x], x]], Times[-1, Simp[Times[Power[Times[d, x], Plus[m, 1]], Plus[b, Times[2, c, Power[x, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p], Power[Times[b, d, n, Plus[Times[2, p], 1]], -1]], x]]]], And[FreeQ[List[a, 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Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[m, Times[n, Plus[Times[2, p], 1]], 1], Power[Times[d, x], Plus[m, 1]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[Times[2, a, d, Power[n, 2], Plus[p, 1], Plus[Times[2, p], 1]], -1]], x]], Plus[Dist[Times[Plus[m, Times[n, Plus[Times[2, p], 1]], 1], Plus[m, Times[2, n, Plus[p, 1]], 1], Power[Times[2, a, Power[n, 2], Plus[p, 1], Plus[Times[2, p], 1]], -1]], Int[Times[Power[Times[d, x], m], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]]], x], x], Times[-1, Simp[Times[Power[Times[d, x], Plus[m, 1]], Plus[Times[2, a], Times[b, Power[x, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p], Power[Times[2, a, d, n, Plus[Times[2, p], 1]], -1]], x]]]], And[FreeQ[List[a, b, c, d, m], x], EqQ[n2, Times[2, n]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m, p], Less[p, -1], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[d, Plus[n, -1]], Power[Times[d, x], Plus[m, Times[-1, n], 1]], Plus[b, Times[2, c, Power[x, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p], Power[Times[2, c, Plus[m, Times[2, n, p], 1]], -1]], x], Times[-1, Dist[Times[b, Power[d, n], Plus[m, Times[-1, n], 1], Power[Times[2, c, Plus[m, Times[2, n, p], 1]], -1]], Int[Times[Power[Times[d, x], Plus[m, Times[-1, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x]]], And[FreeQ[List[a, b, c, d, p], x], EqQ[n2, Times[2, n]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m], Greater[m, Plus[n, -1]], NonzeroQ[Plus[m, Times[2, n, p], 1]], Or[IntegerQ[Times[2, p]], PositiveIntegerQ[Times[Plus[m, 1], Power[n, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[d, x], Plus[m, 1]], Plus[b, Times[2, c, Power[x, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p], Power[Times[b, d, Plus[m, 1]], -1]], x], Times[-1, Dist[Times[2, c, Plus[m, Times[n, Plus[Times[2, p], 1]], 1], Power[Times[b, Power[d, n], Plus[m, 1]], -1]], Int[Times[Power[Times[d, x], Plus[m, n]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x]]], And[FreeQ[List[a, b, c, d, p], x], EqQ[n2, Times[2, n]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m], Less[m, -1], Or[IntegerQ[Times[2, p]], PositiveIntegerQ[Times[Plus[m, 1], Power[n, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Subst[Int[Times[Power[Plus[a, Times[b, Power[x, Times[-1, n]]], Times[c, Power[x, Times[-2, n]]]], p], Power[Power[x, Plus[m, 2]], -1]], x], x, Times[1, Power[x, -1]]]], And[FreeQ[List[a, b, c, p], x], EqQ[n2, Times[2, n]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NegativeIntegerQ[n], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[k, Denominator[m]]], Times[-1, Dist[Times[k, Power[d, -1]], Subst[Int[Times[Power[Plus[a, Times[b, Power[d, Times[-1, n]], Power[x, Times[-1, k, n]]], Times[c, Power[d, Times[-2, n]], Power[x, Times[-2, k, n]]]], p], Power[Power[x, Plus[Times[k, Plus[m, 1]], 1]], -1]], x], x, Times[1, Power[Power[Times[d, x], Times[1, Power[k, -1]]], -1]]], x]]], And[FreeQ[List[a, b, c, d, p], x], EqQ[n2, Times[2, n]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NegativeIntegerQ[n], FractionQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Times[Power[d, IntPart[m]], Power[Times[d, x], FracPart[m]], Power[Power[x, -1], FracPart[m]]], Subst[Int[Times[Power[Plus[a, Times[b, Power[x, Times[-1, n]]], Times[c, Power[x, Times[-2, n]]]], p], Power[Power[x, Plus[m, 2]], -1]], x], x, Times[1, Power[x, -1]]], x]], And[FreeQ[List[a, b, c, d, m, p], x], EqQ[n2, Times[2, n]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NegativeIntegerQ[n], Not[RationalQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], FracPart[p]], Power[Times[Power[c, IntPart[p]], Power[Plus[Times[b, Power[2, -1]], Times[c, Power[x, n]]], Times[2, FracPart[p]]]], -1]], Int[Times[Power[Times[d, x], m], Power[Plus[Times[b, Power[2, -1]], Times[c, Power[x, n]]], Times[2, p]]], x], x], And[FreeQ[List[a, b, c, d, m, n, p], x], EqQ[n2, Times[2, n]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[n, -1]], Subst[Int[Times[Power[x, Plus[Simplify[Times[Plus[m, 1], Power[n, -1]]], -1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x, Power[x, n]], x], And[FreeQ[List[a, b, c, m, n, p], x], EqQ[n2, Times[2, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegerQ[Simplify[Times[Plus[m, 1], Power[n, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[d, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[d, IntPart[m]], Power[Times[d, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, d, m, n, p], x], EqQ[n2, Times[2, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegerQ[Simplify[Times[Plus[m, 1], Power[n, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[k, GCD[Plus[m, 1], n]]], Condition[Dist[Times[1, Power[k, -1]], Subst[Int[Times[Power[x, Plus[Times[Plus[m, 1], Power[k, -1]], -1]], Power[Plus[a, Times[b, Power[x, Times[n, Power[k, -1]]]], Times[c, Power[x, Times[2, n, Power[k, -1]]]]], p]], x], x, Power[x, k]], x], Unequal[k, 1]]], And[FreeQ[List[a, b, c, p], x], EqQ[n2, Times[2, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[k, Denominator[m]]], Dist[Times[k, Power[d, -1]], Subst[Int[Times[Power[x, Plus[Times[k, Plus[m, 1]], -1]], Power[Plus[a, Times[b, Power[x, Times[k, n]], Power[Power[d, n], -1]], Times[c, Power[x, Times[2, k, n]], Power[Power[d, Times[2, n]], -1]]], p]], x], x, Power[Times[d, x], Times[1, Power[k, -1]]]], x]], And[FreeQ[List[a, b, c, d, p], x], EqQ[n2, Times[2, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], FractionQ[m], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[d, Plus[n, -1]], Power[Times[d, x], Plus[m, Times[-1, n], 1]], Plus[Times[b, n, p], Times[c, Plus[m, Times[n, Plus[Times[2, p], -1]], 1], Power[x, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p], Power[Times[c, Plus[m, Times[2, n, p], 1], Plus[m, Times[n, Plus[Times[2, p], -1]], 1]], -1]], x], Times[-1, Dist[Times[n, p, Power[d, n], Power[Times[c, Plus[m, Times[2, n, p], 1], Plus[m, Times[n, Plus[Times[2, p], -1]], 1]], -1]], Int[Times[Power[Times[d, x], Plus[m, Times[-1, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, -1]], Simp[Plus[Times[a, b, Plus[m, Times[-1, n], 1]], Times[-1, Plus[Times[2, a, c, Plus[m, Times[n, Plus[Times[2, p], -1]], 1]], Times[-1, Power[b, 2], Plus[m, Times[n, Plus[p, -1]], 1]]], Power[x, n]]], x]], x], x]]], And[FreeQ[List[a, b, c, d], x], EqQ[n2, Times[2, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m, p], Greater[p, 0], Greater[m, Plus[n, -1]], Unequal[Plus[m, Times[2, n, p], 1], 0], Unequal[Plus[m, Times[n, Plus[Times[2, p], -1]], 1], 0], Or[IntegerQ[p], And[IntegerQ[Times[2, p]], IntegerQ[m], Equal[n, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[d, x], Plus[m, 1]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p], Power[Times[d, Plus[m, 1]], -1]], x], Times[-1, Dist[Times[n, p, Power[Times[Power[d, n], Plus[m, 1]], -1]], Int[Times[Power[Times[d, x], Plus[m, n]], Plus[b, Times[2, c, Power[x, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, -1]]], x], x]]], And[FreeQ[List[a, b, c, d], x], EqQ[n2, Times[2, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m, p], Greater[p, 0], Less[m, -1], Or[IntegerQ[p], And[IntegerQ[Times[2, p]], IntegerQ[m], Equal[n, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[d, x], Plus[m, 1]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p], Power[Times[d, Plus[m, Times[2, n, p], 1]], -1]], x], Dist[Times[n, p, Power[Plus[m, Times[2, n, p], 1], -1]], Int[Times[Power[Times[d, x], m], Plus[Times[2, a], Times[b, Power[x, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, -1]]], x], x]], And[FreeQ[List[a, b, c, d, m], x], EqQ[n2, Times[2, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[p], Greater[p, 0], NonzeroQ[Plus[m, Times[2, n, p], 1]], Or[IntegerQ[p], And[IntegerQ[Times[2, p]], IntegerQ[m], Equal[n, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[d, Plus[n, -1]], Power[Times[d, x], Plus[m, Times[-1, n], 1]], Plus[b, Times[2, c, Power[x, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[Times[n, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], x], Times[-1, Dist[Times[Power[d, n], Power[Times[n, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], Int[Times[Power[Times[d, x], Plus[m, Times[-1, n]]], Plus[Times[b, Plus[m, Times[-1, n], 1]], Times[2, c, Plus[m, Times[2, n, Plus[p, 1]], 1], Power[x, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]]], x], x]]], And[FreeQ[List[a, b, c, d], x], EqQ[n2, Times[2, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m, p], Less[p, -1], Inequality[Plus[n, -1], Less, m, LessEqual, Plus[Times[2, n], -1]], Or[IntegerQ[p], And[IntegerQ[Times[2, p]], IntegerQ[m], Equal[n, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[d, Plus[Times[2, n], -1]], Power[Times[d, x], Plus[m, Times[-1, 2, n], 1]], Plus[Times[2, a], Times[b, Power[x, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[Times[n, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], x]], Dist[Times[Power[d, Times[2, n]], Power[Times[n, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], Int[Times[Power[Times[d, x], Plus[m, Times[-1, 2, n]]], Plus[Times[2, a, Plus[m, Times[-1, 2, n], 1]], Times[b, Plus[m, Times[n, Plus[Times[2, p], 1]], 1], Power[x, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]]], x], x]], And[FreeQ[List[a, b, c, d], x], EqQ[n2, Times[2, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m, p], Less[p, -1], Greater[m, Plus[Times[2, n], -1]], Or[IntegerQ[p], And[IntegerQ[Times[2, p]], IntegerQ[m], Equal[n, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Times[d, x], Plus[m, 1]], Plus[Power[b, 2], Times[-1, 2, a, c], Times[b, c, Power[x, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[Times[a, d, n, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], x]], Dist[Times[1, Power[Times[a, n, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], Int[Times[Power[Times[d, x], m], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Simp[Plus[Times[Power[b, 2], Plus[Times[n, Plus[p, 1]], m, 1]], Times[-1, 2, a, c, Plus[m, Times[2, n, Plus[p, 1]], 1]], Times[b, c, Plus[Times[2, n, p], Times[3, n], m, 1], Power[x, n]]], x]], x], x]], And[FreeQ[List[a, b, c, d, m], x], EqQ[n2, Times[2, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[p], Less[p, -1], Or[IntegerQ[p], And[IntegerQ[Times[2, p]], IntegerQ[m], Equal[n, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[d, Plus[Times[2, n], -1]], Power[Times[d, x], Plus[m, Times[-1, 2, n], 1]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[Times[c, Plus[m, Times[2, n, p], 1]], -1]], x], Times[-1, Dist[Times[Power[d, Times[2, n]], Power[Times[c, Plus[m, Times[2, n, p], 1]], -1]], Int[Times[Power[Times[d, x], Plus[m, Times[-1, 2, n]]], Simp[Plus[Times[a, Plus[m, Times[-1, 2, n], 1]], Times[b, Plus[m, Times[n, Plus[p, -1]], 1], Power[x, n]]], x], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x]]], And[FreeQ[List[a, b, c, d, p], x], EqQ[n2, Times[2, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m], Greater[m, Plus[Times[2, n], -1]], NonzeroQ[Plus[m, Times[2, n, p], 1]], Or[IntegerQ[p], And[IntegerQ[Times[2, p]], IntegerQ[m], Equal[n, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[d, x], Plus[m, 1]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[Times[a, d, Plus[m, 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[a, Power[d, n], Plus[m, 1]], -1]], Int[Times[Power[Times[d, x], Plus[m, n]], Plus[Times[b, Plus[m, Times[n, Plus[p, 1]], 1]], Times[c, Plus[m, Times[2, n, Plus[p, 1]], 1], Power[x, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x]]], And[FreeQ[List[a, b, c, d, p], x], EqQ[n2, Times[2, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m], Less[m, -1], Or[IntegerQ[p], And[IntegerQ[Times[2, p]], IntegerQ[m], Equal[n, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[d, x], Plus[m, 1]], Power[Times[a, d, Plus[m, 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[a, Power[d, n]], -1]], Int[Times[Power[Times[d, x], Plus[m, n]], Plus[b, Times[c, Power[x, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d], x], EqQ[n2, Times[2, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m], Less[m, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Int[PolynomialDivide[Power[x, m], Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], x], x], And[FreeQ[List[a, b, c], x], EqQ[n2, Times[2, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], IntegerQ[m], Greater[m, Plus[Times[3, n], -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[d, Plus[Times[2, n], -1]], Power[Times[d, x], Plus[m, Times[-1, 2, n], 1]], Power[Times[c, Plus[m, Times[-1, 2, n], 1]], -1]], x], Times[-1, Dist[Times[Power[d, Times[2, n]], Power[c, -1]], Int[Times[Power[Times[d, x], Plus[m, Times[-1, 2, n]]], Plus[a, Times[b, Power[x, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d], x], EqQ[n2, Times[2, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m], Greater[m, Plus[Times[2, n], -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Times[a, Power[c, -1]], 2]]], Plus[Dist[Times[1, Power[2, -1]], Int[Times[Plus[q, Power[x, 2]], Power[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]], -1]], x], x], Times[-1, Dist[Times[1, Power[2, -1]], Int[Times[Plus[q, Times[-1, Power[x, 2]]], Power[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]], -1]], x], x]]]], And[FreeQ[List[a, b, c], x], NegativeQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PosQ[Times[a, c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Times[a, Power[c, -1]], 2]]], With[List[Set[r, Rt[Plus[Times[2, q], Times[-1, b, Power[c, -1]]], 2]]], Plus[Times[-1, Dist[Times[1, Power[Times[2, c, r], -1]], Int[Times[Power[x, Plus[m, Times[-1, 3, Times[n, Power[2, -1]]]]], Plus[q, Times[-1, r, Power[x, Times[n, Power[2, -1]]]]], Power[Plus[q, Times[-1, r, Power[x, Times[n, Power[2, -1]]]], Power[x, n]], -1]], x], x]], Dist[Times[1, Power[Times[2, c, r], -1]], Int[Times[Power[x, Plus[m, Times[-1, 3, Times[n, Power[2, -1]]]]], Plus[q, Times[r, Power[x, Times[n, Power[2, -1]]]]], Power[Plus[q, Times[r, Power[x, Times[n, Power[2, -1]]]], Power[x, n]], -1]], x], x]]]], And[FreeQ[List[a, b, c], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[Times[n, Power[2, -1]], m], Inequality[Times[3, n, Power[2, -1]], LessEqual, m, Less, Times[2, n]], NegQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Times[a, Power[c, -1]], 2]]], With[List[Set[r, Rt[Plus[Times[2, q], Times[-1, b, Power[c, -1]]], 2]]], Plus[Dist[Times[1, Power[Times[2, c, r], -1]], Int[Times[Power[x, Plus[m, Times[-1, n, Power[2, -1]]]], Power[Plus[q, Times[-1, r, Power[x, Times[n, Power[2, -1]]]], Power[x, n]], -1]], x], x], Times[-1, Dist[Times[1, Power[Times[2, c, r], -1]], Int[Times[Power[x, Plus[m, Times[-1, n, Power[2, -1]]]], Power[Plus[q, Times[r, Power[x, Times[n, Power[2, -1]]]], Power[x, n]], -1]], x], x]]]]], And[FreeQ[List[a, b, c], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[Times[n, Power[2, -1]], m], Inequality[Times[n, Power[2, -1]], LessEqual, m, Less, Times[3, n, Power[2, -1]]], NegQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Plus[Dist[Times[Power[d, n], Power[2, -1], Plus[Times[b, Power[q, -1]], 1]], Int[Times[Power[Times[d, x], Plus[m, Times[-1, n]]], Power[Plus[Times[b, Power[2, -1]], Times[q, Power[2, -1]], Times[c, Power[x, n]]], -1]], x], x], Times[-1, Dist[Times[Power[d, n], Power[2, -1], Plus[Times[b, Power[q, -1]], -1]], Int[Times[Power[Times[d, x], Plus[m, Times[-1, n]]], Power[Plus[Times[b, Power[2, -1]], Times[-1, q, Power[2, -1]], Times[c, Power[x, n]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d], x], EqQ[n2, Times[2, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m], GreaterEqual[m, n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Plus[Dist[Times[c, Power[q, -1]], Int[Times[Power[Times[d, x], m], Power[Plus[Times[b, Power[2, -1]], Times[-1, q, Power[2, -1]], Times[c, Power[x, n]]], -1]], x], x], Times[-1, Dist[Times[c, Power[q, -1]], Int[Times[Power[Times[d, x], m], Power[Plus[Times[b, Power[2, -1]], Times[q, Power[2, -1]], Times[c, Power[x, n]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d, m], x], EqQ[n2, Times[2, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Dist[Times[2, Sqrt[Times[-1, c]]], Int[Times[Power[x, 2], Power[Times[Sqrt[Plus[b, q, Times[2, c, Power[x, 2]]]], Sqrt[Plus[Times[-1, b], q, Times[-1, 2, c, Power[x, 2]]]]], -1]], x], x]], And[FreeQ[List[a, b, c], x], PositiveQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NegativeQ[c]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Times[c, Power[a, -1]], 2]]], Plus[Dist[Times[1, Power[q, -1]], Int[Times[1, Power[Sqrt[Plus[a, Times[b, 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x], Times[-1, Simp[Times[Rt[Times[Plus[b, Times[-1, q]], Power[Times[2, a], -1]], 2], Plus[Times[2, a], Times[Plus[b, Times[-1, q]], Power[x, 2]]], Sqrt[Times[Plus[Times[2, a], Times[Plus[b, q], Power[x, 2]]], Power[Plus[Times[2, a], Times[Plus[b, Times[-1, q]], Power[x, 2]]], -1]]], Power[Times[2, c, Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]]], -1], EllipticE[ArcTan[Times[Rt[Times[Plus[b, Times[-1, q]], Power[Times[2, a], -1]], 2], x]], Times[-2, q, Power[Plus[b, Times[-1, q]], -1]]]], x]]], PosQ[Times[Plus[b, Times[-1, q]], Power[a, -1]]]]], And[FreeQ[List[a, b, c], x], PositiveQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Condition[Plus[Times[-1, Dist[Times[Plus[b, q], Power[Times[2, c], -1]], Int[Times[1, Power[Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]], -1]], x], x]], Dist[Times[1, Power[Times[2, c], -1]], Int[Times[Plus[b, q, Times[2, c, Power[x, 2]]], Power[Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]], -1]], x], x]], And[NegQ[Times[Plus[b, q], Power[a, -1]]], Not[And[NegQ[Times[Plus[b, Times[-1, q]], Power[a, -1]]], SimplerSqrtQ[Times[-1, Plus[b, Times[-1, q]], Power[Times[2, a], -1]], Times[-1, Plus[b, q], Power[Times[2, a], -1]]]]]]]], And[FreeQ[List[a, b, c], x], PositiveQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[b, 2], Times[-1, 4, 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Dist[Times[1, Power[q, -1]], Int[Times[Plus[1, Times[-1, q, Power[x, 2]]], Power[Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]], -1]], x], x]]]], And[FreeQ[List[a, b, c], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PosQ[Times[c, Power[a, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Dist[Times[Sqrt[Plus[1, Times[2, c, Power[x, 2], Power[Plus[b, Times[-1, q]], -1]]]], Sqrt[Plus[1, Times[2, c, Power[x, 2], Power[Plus[b, q], -1]]]], Power[Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]], -1]], Int[Times[Power[x, 2], Power[Times[Sqrt[Plus[1, Times[2, c, Power[x, 2], Power[Plus[b, Times[-1, q]], -1]]]], Sqrt[Plus[1, Times[2, c, Power[x, 2], Power[Plus[b, q], -1]]]]], -1]], x], x]], And[FreeQ[List[a, b, c], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NegQ[Times[c, Power[a, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Subst[Int[Times[Power[Plus[a, Times[b, Power[x, Times[-1, n]]], Times[c, Power[x, Times[-2, n]]]], p], Power[Power[x, Plus[m, 2]], -1]], x], x, Times[1, Power[x, -1]]]], And[FreeQ[List[a, b, c, p], x], EqQ[n2, Times[2, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NegativeIntegerQ[n], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, 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Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Times[Power[d, IntPart[m]], Power[Times[d, x], FracPart[m]], Power[Power[x, -1], FracPart[m]]], Subst[Int[Times[Power[Plus[a, Times[b, Power[x, Times[-1, n]]], Times[c, Power[x, Times[-2, n]]]], p], Power[Power[x, Plus[m, 2]], -1]], x], x, Times[1, Power[x, -1]]], x]], And[FreeQ[List[a, b, c, d, m, p], x], EqQ[n2, Times[2, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NegativeIntegerQ[n], Not[RationalQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[k, Denominator[n]]], Dist[k, Subst[Int[Times[Power[x, Plus[Times[k, Plus[m, 1]], -1]], Power[Plus[a, Times[b, Power[x, Times[k, n]]], Times[c, Power[x, Times[2, k, n]]]], p]], x], x, Power[x, Times[1, Power[k, -1]]]], x]], And[FreeQ[List[a, b, c, m, p], x], EqQ[n2, Times[2, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], FractionQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[d, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[d, IntPart[m]], Power[Times[d, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, d, m, p], x], EqQ[n2, Times[2, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], FractionQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Plus[m, 1], -1]], Subst[Int[Power[Plus[a, Times[b, Power[x, Simplify[Times[n, Power[Plus[m, 1], -1]]]]], Times[c, Power[x, Simplify[Times[2, n, Power[Plus[m, 1], -1]]]]]], p], x], x, Power[x, Plus[m, 1]]], x], And[FreeQ[List[a, b, c, m, n, p], x], EqQ[n2, Times[2, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegerQ[Simplify[Times[n, Power[Plus[m, 1], -1]]]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[d, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[d, IntPart[m]], Power[Times[d, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, d, m, n, p], x], EqQ[n2, Times[2, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegerQ[Simplify[Times[n, Power[Plus[m, 1], -1]]]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Plus[Dist[Times[2, c, Power[q, -1]], Int[Times[Power[Times[d, x], m], Power[Plus[b, Times[-1, q], Times[2, c, Power[x, n]]], -1]], x], x], Times[-1, Dist[Times[2, c, Power[q, -1]], Int[Times[Power[Times[d, x], m], Power[Plus[b, q, Times[2, c, Power[x, n]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d, m, n], x], EqQ[n2, Times[2, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Times[d, x], Plus[m, 1]], Plus[Power[b, 2], Times[-1, 2, a, c], Times[b, c, Power[x, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[Times[a, d, n, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], x]], Dist[Times[1, Power[Times[a, n, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], Int[Times[Power[Times[d, x], m], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Simp[Plus[Times[Power[b, 2], Plus[Times[n, Plus[p, 1]], m, 1]], Times[-1, 2, a, c, Plus[m, Times[2, n, Plus[p, 1]], 1]], Times[b, c, Plus[Times[2, n, p], Times[3, n], m, 1], Power[x, n]]], x]], x], x]], And[FreeQ[List[a, b, c, d, m, n], x], EqQ[n2, Times[2, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NegativeIntegerQ[Plus[p, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], 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Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[mn, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[x, Plus[m, Times[-1, n, p]]], Power[Plus[b, Times[a, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], And[FreeQ[List[a, b, c, m, n], x], EqQ[mn, Times[-1, n]], IntegerQ[p], PosQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[mn, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[x, Times[n, FracPart[p]]], Power[Plus[a, Times[b, Power[Power[x, n], -1]], Times[c, Power[x, n]]], FracPart[p]], Power[Power[Plus[b, Times[a, Power[x, n]], Times[c, Power[x, Times[2, n]]]], FracPart[p]], -1]], Int[Times[Power[x, Plus[m, Times[-1, n, p]]], Power[Plus[b, Times[a, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, m, n, p], x], EqQ[mn, Times[-1, n]], Not[IntegerQ[p]], PosQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[d, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[mn, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[d, IntPart[m]], Power[Times[d, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Plus[a, Times[b, Power[x, Times[-1, n]]], Times[c, Power[x, n]]], p]], x], x], And[FreeQ[List[a, b, c, d, m, n, p], x], EqQ[mn, Times[-1, n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Power[Coefficient[v, x, 1], Plus[m, 1]], -1]], Subst[Int[SimplifyIntegrand[Times[Power[Plus[x, Times[-1, Coefficient[v, x, 0]]], m], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x], x, v], x], And[FreeQ[List[a, b, c, n, p], x], EqQ[n2, Times[2, n]], LinearQ[v, x], IntegerQ[m], NonzeroQ[Plus[v, Times[-1, x]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[u, m], Power[Times[Coefficient[v, x, 1], Power[v, m]], -1]], Subst[Int[Times[Power[x, m], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x, v], x], And[FreeQ[List[a, b, c, m, n, p], x], EqQ[n2, Times[2, n]], LinearPairQ[u, v, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[x, Times[n, Plus[Times[2, p], q]]], Power[Plus[e, Times[d, Power[x, Times[-1, n]]]], q], Power[Plus[c, Times[b, Power[x, Times[-1, n]]], Times[a, Power[x, Times[-2, n]]]], p]], x], And[FreeQ[List[a, b, c, d, e, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], IntegersQ[p, q], NegQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[x, Times[n, Plus[Times[2, p], q]]], Power[Plus[e, Times[d, Power[x, Times[-1, n]]]], q], Power[Plus[c, Times[a, Power[x, Times[-2, n]]]], p]], x], And[FreeQ[List[a, c, d, e, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], IntegersQ[p, q], NegQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Subst[Int[Times[Power[Plus[d, Times[e, Power[x, Times[-1, n]]]], q], Power[Plus[a, Times[b, Power[x, Times[-1, n]]], Times[c, Power[x, Times[-2, n]]]], p], Power[Power[x, 2], -1]], x], x, Times[1, Power[x, -1]]]], And[FreeQ[List[a, b, c, d, e, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NegativeIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Subst[Int[Times[Power[Plus[d, Times[e, Power[x, Times[-1, n]]]], q], Power[Plus[a, Times[c, Power[x, Times[-2, n]]]], p], Power[Power[x, 2], -1]], x], x, Times[1, Power[x, -1]]]], And[FreeQ[List[a, c, d, e, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NegativeIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[g, Denominator[n]]], Dist[g, Subst[Int[Times[Power[x, Plus[g, -1]], Power[Plus[d, Times[e, Power[x, Times[g, n]]]], q], Power[Plus[a, Times[b, Power[x, Times[g, n]]], Times[c, Power[x, Times[2, g, n]]]], p]], x], x, Power[x, Times[1, Power[g, -1]]]], x]], And[FreeQ[List[a, b, c, d, e, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], FractionQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[g, Denominator[n]]], Dist[g, Subst[Int[Times[Power[x, Plus[g, -1]], Power[Plus[d, Times[e, Power[x, Times[g, n]]]], q], Power[Plus[a, Times[c, Power[x, Times[2, g, n]]]], p]], x], x, Power[x, Times[1, Power[g, -1]]]], x]], And[FreeQ[List[a, c, d, e, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], FractionQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[b, e], Times[-1, d, c]], Power[Plus[Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[Times[b, c, n, Plus[p, 1], Power[x, Times[2, n, Plus[p, 1]]]], -1]], x], Dist[Times[e, Power[c, -1]], Int[Times[Power[x, Times[-1, n]], Power[Plus[Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]]], x], x]], And[FreeQ[List[b, c, d, e, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], Not[IntegerQ[p]], ZeroQ[Plus[Times[n, Plus[Times[2, p], 1]], 1]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[e, Power[x, Plus[Times[-1, n], 1]], Power[Plus[Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[Times[c, Plus[Times[n, Plus[Times[2, p], 1]], 1]], -1]], x], And[FreeQ[List[b, c, d, e, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], Not[IntegerQ[p]], NonzeroQ[Plus[Times[n, Plus[Times[2, p], 1]], 1]], ZeroQ[Plus[Times[b, e, Plus[Times[n, p], 1]], Times[-1, c, d, Plus[Times[n, Plus[Times[2, p], 1]], 1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[e, Power[x, Plus[Times[-1, n], 1]], Power[Plus[Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[Times[c, Plus[Times[n, Plus[Times[2, p], 1]], 1]], -1]], x], Times[-1, Dist[Times[Plus[Times[b, e, Plus[Times[n, p], 1]], Times[-1, c, d, Plus[Times[n, Plus[Times[2, p], 1]], 1]]], Power[Times[c, Plus[Times[n, Plus[Times[2, p], 1]], 1]], -1]], Int[Power[Plus[Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p], x], x]]], And[FreeQ[List[b, c, d, e, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], Not[IntegerQ[p]], NonzeroQ[Plus[Times[n, Plus[Times[2, p], 1]], 1]], NonzeroQ[Plus[Times[b, e, Plus[Times[n, p], 1]], Times[-1, c, d, Plus[Times[n, Plus[Times[2, p], 1]], 1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], FracPart[p]], Power[Times[Power[x, Times[n, FracPart[p]]], Power[Plus[b, Times[c, Power[x, n]]], FracPart[p]]], -1]], Int[Times[Power[x, Times[n, p]], Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[b, Times[c, Power[x, n]]], p]], x], x], And[FreeQ[List[b, c, d, e, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], FracPart[p]], Power[Times[Power[Times[4, c], IntPart[p]], Power[Plus[b, Times[2, c, Power[x, n]]], Times[2, FracPart[p]]]], -1]], Int[Times[Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[b, Times[2, c, Power[x, n]]], Times[2, p]]], x], x], And[FreeQ[List[a, b, c, d, e, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[d, Times[e, Power[x, n]]], Plus[p, q]], Power[Plus[Times[a, Power[d, -1]], Times[c, Power[e, -1], Power[x, n]]], p]], x], And[FreeQ[List[a, b, c, d, e, n, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[d, Times[e, Power[x, n]]], Plus[p, q]], Power[Plus[Times[a, Power[d, -1]], Times[c, Power[e, -1], Power[x, n]]], p]], x], And[FreeQ[List[a, c, d, e, n, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], FracPart[p]], Power[Times[Power[Plus[d, Times[e, Power[x, n]]], FracPart[p]], Power[Plus[Times[a, Power[d, -1]], Times[Times[c, Power[x, n]], Power[e, -1]]], FracPart[p]]], -1]], Int[Times[Power[Plus[d, Times[e, Power[x, n]]], Plus[p, q]], Power[Plus[Times[a, Power[d, -1]], Times[c, Power[e, -1], Power[x, n]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[c, Power[x, Times[2, n]]]], FracPart[p]], Power[Times[Power[Plus[d, Times[e, Power[x, n]]], FracPart[p]], Power[Plus[Times[a, Power[d, -1]], Times[Times[c, Power[x, n]], Power[e, -1]]], FracPart[p]]], -1]], Int[Times[Power[Plus[d, Times[e, Power[x, n]]], Plus[p, q]], Power[Plus[Times[a, Power[d, -1]], Times[c, Power[e, -1], Power[x, n]]], p]], x], x], And[FreeQ[List[a, c, d, e, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[d, Times[e, Power[x, n]]], q], Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]]], x], x], And[FreeQ[List[a, b, c, d, e, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], PositiveIntegerQ[q]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[d, Times[e, Power[x, n]]], q], Plus[a, Times[c, Power[x, Times[2, n]]]]], x], x], And[FreeQ[List[a, c, d, e, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], PositiveIntegerQ[q]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]], x, Power[Plus[d, Times[e, Power[x, n]]], Plus[q, 1]], Power[Times[d, Power[e, 2], n, Plus[q, 1]], -1]], x]], Dist[Times[1, Power[Times[n, Plus[q, 1], d, Power[e, 2]], -1]], Int[Times[Power[Plus[d, Times[e, Power[x, n]]], Plus[q, 1]], Simp[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2], Plus[Times[n, Plus[q, 1]], 1]], Times[c, d, e, n, Plus[q, 1], Power[x, n]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], RationalQ[q], Less[q, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], x, Power[Plus[d, Times[e, Power[x, n]]], Plus[q, 1]], Power[Times[d, Power[e, 2], n, Plus[q, 1]], -1]], x]], Dist[Times[1, Power[Times[n, Plus[q, 1], d, Power[e, 2]], -1]], Int[Times[Power[Plus[d, Times[e, Power[x, n]]], Plus[q, 1]], Simp[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2], Plus[Times[n, Plus[q, 1]], 1]], Times[c, d, e, n, Plus[q, 1], Power[x, n]]], x]], x], x]], And[FreeQ[List[a, c, d, e, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], RationalQ[q], Less[q, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[c, Power[x, Plus[n, 1]], Power[Plus[d, Times[e, Power[x, n]]], Plus[q, 1]], Power[Times[e, Plus[Times[n, Plus[q, 2]], 1]], -1]], x], Dist[Times[1, Power[Times[e, Plus[Times[n, Plus[q, 2]], 1]], -1]], Int[Times[Power[Plus[d, Times[e, Power[x, n]]], q], Plus[Times[a, e, Plus[Times[n, Plus[q, 2]], 1]], Times[-1, Plus[Times[c, d, Plus[n, 1]], Times[-1, b, e, Plus[Times[n, Plus[q, 2]], 1]]], Power[x, n]]]], x], x]], And[FreeQ[List[a, b, c, d, e, n, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 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Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Power[e, 2], Power[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]], -1]], Int[Power[Plus[d, Times[e, Power[x, n]]], q], x], x], Dist[Times[1, Power[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]], -1]], Int[Times[Power[Plus[d, Times[e, Power[x, n]]], Plus[q, 1]], Plus[Times[c, d], Times[-1, b, e], Times[-1, c, e, Power[x, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Not[IntegerQ[q]], RationalQ[q], Less[q, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, 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Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[r, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Plus[Dist[Times[2, c, Power[r, -1]], Int[Times[Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[b, Times[-1, r], Times[2, c, Power[x, n]]], -1]], x], x], Times[-1, Dist[Times[2, c, Power[r, -1]], Int[Times[Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[b, r, Times[2, c, Power[x, n]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d, e, n, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Not[IntegerQ[q]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[r, Rt[Times[-1, a, c], 2]]], Plus[Times[-1, Dist[Times[c, Power[Times[2, r], -1]], Int[Times[Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[r, Times[-1, c, Power[x, n]]], -1]], x], x]], Times[-1, Dist[Times[c, Power[Times[2, r], -1]], Int[Times[Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[r, Times[c, Power[x, n]]], -1]], x], x]]]], And[FreeQ[List[a, c, d, e, n, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[q]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[x, Plus[Times[b, e, n, p], Times[c, d, Plus[Times[2, n, p], n, 1]], Times[c, e, Plus[Times[2, n, p], 1], Power[x, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p], Power[Times[c, Plus[Times[2, n, p], 1], Plus[Times[2, n, p], n, 1]], -1]], x], Dist[Times[n, p, Power[Times[c, Plus[Times[2, n, p], 1], Plus[Times[2, n, p], n, 1]], -1]], Int[Times[Simp[Plus[Times[2, a, c, d, Plus[Times[2, n, p], n, 1]], Times[-1, a, b, e], Times[Plus[Times[2, a, c, e, Plus[Times[2, n, p], 1]], Times[b, d, c, Plus[Times[2, n, p], n, 1]], Times[-1, Power[b, 2], e, Plus[Times[n, p], 1]]], Power[x, n]]], x], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, -1]]], x], x]], And[FreeQ[List[a, b, c, d, e, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], FractionQ[p], Greater[p, 0], 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Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Dist[Times[Sqrt[Plus[1, Times[2, c, Power[x, 2], Power[Plus[b, Times[-1, q]], -1]]]], Sqrt[Plus[1, Times[2, c, Power[x, 2], Power[Plus[b, q], -1]]]], Power[Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]], -1]], Int[Times[Plus[d, Times[e, Power[x, 2]]], Power[Times[Sqrt[Plus[1, Times[2, c, Power[x, 2], Power[Plus[b, Times[-1, q]], -1]]]], Sqrt[Plus[1, Times[2, c, Power[x, 2], Power[Plus[b, q], -1]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NegQ[Times[c, Power[a, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, 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x]], And[FreeQ[List[a, c, d, e, n, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PositiveIntegerQ[p], NonzeroQ[Plus[Times[2, n, p], Times[n, q], 1]], PositiveIntegerQ[n], Not[PositiveIntegerQ[q]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Dist[Power[Power[e, 2], -1], Int[Times[Plus[Times[c, d], Times[-1, b, e], Times[-1, c, e, Power[x, 2]]], Power[Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]], -1]], x], x]], Dist[Times[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]], Power[Power[e, 2], -1]], Int[Times[1, Power[Times[Plus[d, Times[e, Power[x, 2]]], Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Dist[Times[c, Power[Power[e, 2], -1]], Int[Times[Plus[d, Times[-1, e, Power[x, 2]]], Power[Sqrt[Plus[a, Times[c, Power[x, 4]]]], -1]], x], x]], Dist[Times[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], Power[Power[e, 2], -1]], Int[Times[1, Power[Times[Plus[d, Times[e, Power[x, 2]]], Sqrt[Plus[a, Times[c, Power[x, 4]]]]], -1]], x], x]], And[FreeQ[List[a, c, d, e], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[3, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Plus[Times[-1, Dist[Times[Power[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]], 2], Power[Times[Power[e, 3], Plus[Times[2, c, d], Times[-1, e, Plus[b, q]]]], -1]], Int[Times[Plus[b, q, Times[2, c, Power[x, 2]]], Power[Times[Plus[d, Times[e, Power[x, 2]]], Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]]], -1]], x], x]], Times[-1, Dist[Times[1, Power[Power[e, 4], -1]], Int[Times[1, Power[Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]], -1], Simp[Plus[Times[Plus[Times[c, d], Times[-1, b, e]], Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[2, a, Power[e, 2]]]], Times[-1, 2, 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2]]], Plus[Times[-1, Dist[Times[Power[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], 2], Power[Times[Power[e, 3], Plus[Times[c, d], Times[-1, e, q]]], -1]], Int[Times[Plus[q, Times[c, Power[x, 2]]], Power[Times[Plus[d, Times[e, Power[x, 2]]], Sqrt[Plus[a, Times[c, Power[x, 4]]]]], -1]], x], x]], Times[-1, Dist[Times[c, Power[Power[e, 4], -1]], Int[Times[1, Power[Sqrt[Plus[a, Times[c, Power[x, 4]]]], -1], Simp[Plus[Times[d, Plus[Times[c, Power[d, 2]], Times[2, a, Power[e, 2]]]], Times[-1, Power[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], 2], Power[Plus[Times[c, d], Times[-1, e, q]], -1]], Times[-1, e, Plus[Times[c, Power[d, 2]], Times[2, a, Power[e, 2]]], Power[x, 2]], Times[c, d, Power[e, 2], Power[x, 4]], Times[-1, c, Power[e, 3], Power[x, 6]]], x]], x], x]]]], And[FreeQ[List[a, c, d, e], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], PositiveQ[Times[-1, a, c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[a, Int[Times[Power[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]], Plus[p, -1]], Power[Plus[d, Times[e, Power[x, 2]]], -1]], x], x], Plus[Dist[b, Int[Times[Power[x, 2], Power[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]], Plus[p, -1]], Power[Plus[d, Times[e, Power[x, 2]]], -1]], x], x], Dist[c, Int[Times[Power[x, 4], Power[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]], Plus[p, -1]], Power[Plus[d, Times[e, Power[x, 2]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], PositiveIntegerQ[Plus[p, Times[-1, 1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[a, Int[Times[Power[Plus[a, Times[c, Power[x, 4]]], Plus[p, -1]], Power[Plus[d, Times[e, Power[x, 2]]], -1]], x], x], Dist[c, Int[Times[Power[x, 4], Power[Plus[a, Times[c, Power[x, 4]]], Plus[p, -1]], Power[Plus[d, Times[e, Power[x, 2]]], -1]], x], x]], And[FreeQ[List[a, c, d, e], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], PositiveIntegerQ[Plus[p, Times[-1, 1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Dist[Times[2, Sqrt[Times[-1, c]]], Int[Times[1, Power[Times[Plus[d, Times[e, Power[x, 2]]], Sqrt[Plus[b, q, Times[2, c, Power[x, 2]]]], Sqrt[Plus[Times[-1, b], q, Times[-1, 2, c, Power[x, 2]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e], x], PositiveQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NegativeQ[c]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Times[-1, a, c], 2]]], Dist[Sqrt[Times[-1, c]], Int[Times[1, Power[Times[Plus[d, Times[e, Power[x, 2]]], Sqrt[Plus[q, Times[c, Power[x, 2]]]], Sqrt[Plus[q, Times[-1, c, Power[x, 2]]]]], -1]], x], x]], And[FreeQ[List[a, c, d, e], x], PositiveQ[a], NegativeQ[c]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Plus[Dist[Times[2, c, Power[Plus[Times[2, c, d], Times[-1, e, Plus[b, Times[-1, q]]]], -1]], Int[Times[1, Power[Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]], -1]], x], x], Times[-1, Dist[Times[e, Power[Plus[Times[2, c, d], Times[-1, e, Plus[b, Times[-1, q]]]], -1]], Int[Times[Plus[b, Times[-1, q], Times[2, c, Power[x, 2]]], Power[Times[Plus[d, Times[e, Power[x, 2]]], Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d, e], x], PositiveQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], Not[NegativeQ[c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Times[-1, a, c], 2]]], Plus[Dist[Times[c, Power[Plus[Times[c, d], Times[e, q]], -1]], Int[Times[1, Power[Sqrt[Plus[a, Times[c, Power[x, 4]]]], -1]], x], x], Dist[Times[e, Power[Plus[Times[c, d], Times[e, q]], -1]], Int[Times[Plus[q, Times[-1, c, Power[x, 2]]], Power[Times[Plus[d, Times[e, Power[x, 2]]], Sqrt[Plus[a, Times[c, Power[x, 4]]]]], -1]], x], x]]], And[FreeQ[List[a, c, d, e], x], PositiveQ[Times[-1, a, c]], Not[NegativeQ[c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Times[c, Power[a, -1]], 4]]], Condition[Plus[Simp[Times[ArcTan[Times[Sqrt[Times[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]], Power[Times[d, e], -1]]], x, Power[Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]], -1]]], Power[Times[2, d, Sqrt[Times[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]], Power[Times[d, e], -1]]]], -1]], x], Plus[Times[-1, Dist[Times[Power[q, 2], Power[Plus[e, Times[-1, d, Power[q, 2]]], -1]], Int[Times[1, Power[Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]], -1]], x], x]], Simp[Times[Plus[e, Times[d, Power[q, 2]]], Plus[1, Times[Power[q, 2], Power[x, 2]]], Sqrt[Times[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]], Power[Times[a, Power[Plus[1, Times[Power[q, 2], Power[x, 2]]], 2]], -1]]], Power[Times[4, d, q, Plus[e, Times[-1, d, Power[q, 2]]], Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]]], -1], EllipticPi[Times[-1, Power[Plus[e, Times[-1, d, Power[q, 2]]], 2], Power[Times[4, d, e, Power[q, 2]], -1]], Times[2, ArcTan[Times[q, x]]], Plus[Times[1, Power[2, -1]], Times[-1, b, Power[q, 2], Power[Times[4, c], -1]]]]], x]]], NonzeroQ[Plus[e, Times[-1, d, Power[q, 2]]]]]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], PosQ[Times[c, Power[a, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Times[c, Power[a, -1]], 4]]], Condition[Plus[Simp[Times[ArcTan[Times[Sqrt[Times[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], Power[Times[d, e], -1]]], x, Power[Sqrt[Plus[a, Times[c, Power[x, 4]]]], -1]]], Power[Times[2, d, Sqrt[Times[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], Power[Times[d, e], -1]]]], -1]], x], Plus[Times[-1, Dist[Times[Power[q, 2], Power[Plus[e, Times[-1, d, Power[q, 2]]], -1]], Int[Times[1, Power[Sqrt[Plus[a, Times[c, Power[x, 4]]]], -1]], x], x]], Simp[Times[Plus[e, Times[d, Power[q, 2]]], Plus[1, Times[Power[q, 2], Power[x, 2]]], Sqrt[Times[Plus[a, Times[c, Power[x, 4]]], Power[Times[a, Power[Plus[1, Times[Power[q, 2], Power[x, 2]]], 2]], -1]]], Power[Times[4, d, q, Plus[e, Times[-1, d, Power[q, 2]]], Sqrt[Plus[a, Times[c, Power[x, 4]]]]], -1], EllipticPi[Times[-1, Power[Plus[e, Times[-1, d, Power[q, 2]]], 2], Power[Times[4, d, e, Power[q, 2]], -1]], Times[2, ArcTan[Times[q, x]]], Times[1, Power[2, -1]]]], x]]], NonzeroQ[Plus[e, Times[-1, d, Power[q, 2]]]]]], And[FreeQ[List[a, c, d, e], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], PosQ[Times[c, Power[a, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Times[-1, c, Power[a, -1]], 4]]], Simp[Times[1, Power[Times[d, Sqrt[a], q], -1], EllipticPi[Times[-1, e, Power[Times[d, Power[q, 2]], -1]], ArcSin[Times[q, x]], -1]], x]], And[FreeQ[List[a, c, d, e], x], NegQ[Times[c, Power[a, -1]]], PositiveQ[a]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Plus[1, Times[c, Power[x, 4], Power[a, -1]]]], Power[Sqrt[Plus[a, Times[c, Power[x, 4]]]], -1]], Int[Times[1, Power[Times[Plus[d, Times[e, Power[x, 2]]], Sqrt[Plus[1, Times[c, Power[x, 4], Power[a, -1]]]]], -1]], x], x], And[FreeQ[List[a, c, d, e], x], NegQ[Times[c, Power[a, -1]]], Not[PositiveQ[a]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Dist[Times[Sqrt[Plus[1, Times[2, c, Power[x, 2], Power[Plus[b, Times[-1, q]], -1]]]], Sqrt[Plus[1, Times[2, c, Power[x, 2], Power[Plus[b, q], -1]]]], Power[Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]], -1]], Int[Times[1, Power[Times[Plus[d, Times[e, Power[x, 2]]], Sqrt[Plus[1, Times[2, c, Power[x, 2], Power[Plus[b, Times[-1, q]], -1]]]], Sqrt[Plus[1, Times[2, c, Power[x, 2], Power[Plus[b, q], -1]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NegQ[Times[c, Power[a, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[x, Plus[Times[Power[b, 2], c, d], Times[-1, Power[b, 3], e], Times[-1, 2, a, Power[c, 2], d], Times[3, a, b, c, e], Times[c, Plus[Times[b, c, d], Times[-1, Power[b, 2], e], Times[2, a, c, e]], Power[x, 2]]], Power[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]], Plus[p, 1]], Power[Times[2, a, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], -1]], x]], Times[-1, Dist[Times[1, Power[Times[2, a, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]], Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], -1]], Int[Times[Times[Power[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]], Plus[p, 1]], Power[Plus[d, Times[e, Power[x, 2]]], -1]], Simp[Plus[Times[Power[b, 3], d, e, Plus[3, Times[2, p]]], Times[-1, a, b, c, d, e, Plus[11, Times[8, p]]], Times[-1, Power[b, 2], Plus[Times[2, a, Power[e, 2], Plus[p, 1]], Times[c, Power[d, 2], Plus[3, Times[2, p]]]]], Times[2, a, c, Plus[Times[4, a, Power[e, 2], Plus[p, 1]], Times[c, Power[d, 2], Plus[5, Times[4, p]]]]], Times[-1, Plus[Times[4, a, Power[c, 2], d, e], Times[-1, 2, Power[b, 2], c, d, e, Plus[2, p]], Times[-1, Power[b, 3], Power[e, 2], Plus[3, Times[2, p]]], Times[b, c, Plus[Times[c, Power[d, 2], Plus[7, Times[4, p]]], Times[a, Power[e, 2], Plus[11, Times[8, p]]]]]], Power[x, 2]], Times[-1, c, e, Plus[Times[b, c, d], Times[-1, Power[b, 2], e], Times[2, a, c, e]], Plus[7, Times[4, p]], Power[x, 4]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], NegativeIntegerQ[Plus[p, Times[1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[x, Plus[Times[-2, a, Power[c, 2], d], Times[c, Times[2, a, c, e], Power[x, 2]]], Power[Plus[a, Times[c, Power[x, 4]]], Plus[p, 1]], Power[Times[2, a, Plus[p, 1], Times[-4, a, c], Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], -1]], x]], Times[-1, Dist[Times[1, Power[Times[2, a, Plus[p, 1], Times[-4, a, c], Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], -1]], Int[Times[Times[Power[Plus[a, Times[c, Power[x, 4]]], Plus[p, 1]], Power[Plus[d, Times[e, Power[x, 2]]], -1]], Simp[Plus[Times[2, a, c, Plus[Times[4, a, Power[e, 2], Plus[p, 1]], Times[c, Power[d, 2], Plus[5, Times[4, p]]]]], Times[-1, Times[4, a, Power[c, 2], d, e], Power[x, 2]], Times[-1, c, e, Times[2, a, c, e], Plus[7, Times[4, p]], Power[x, 4]]], x]], x], x]]], And[FreeQ[List[a, c, d, e], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], NegativeIntegerQ[Plus[p, Times[1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[e, 2], x, Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]], Power[Times[2, d, Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]], Plus[d, Times[e, Power[x, 2]]]], -1]], x], Plus[Times[-1, Dist[Times[c, Power[Times[2, d, Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], -1]], Int[Times[Plus[d, Times[e, Power[x, 2]]], Power[Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]], -1]], x], x]], Dist[Times[Plus[Times[3, c, Power[d, 2]], Times[-1, 2, b, d, e], Times[a, Power[e, 2]]], Power[Times[2, d, Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], -1]], Int[Times[1, Power[Times[Plus[d, Times[e, Power[x, 2]]], Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[e, 2], x, Sqrt[Plus[a, Times[c, Power[x, 4]]]], Power[Times[2, d, Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], Plus[d, Times[e, Power[x, 2]]]], -1]], x], Plus[Times[-1, Dist[Times[c, Power[Times[2, d, Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], -1]], Int[Times[Plus[d, Times[e, Power[x, 2]]], Power[Sqrt[Plus[a, Times[c, Power[x, 4]]]], -1]], x], x]], Dist[Times[Plus[Times[3, c, Power[d, 2]], Times[a, Power[e, 2]]], Power[Times[2, d, Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], -1]], Int[Times[1, Power[Times[Plus[d, Times[e, Power[x, 2]]], Sqrt[Plus[a, Times[c, Power[x, 4]]]]], -1]], x], x]]], And[FreeQ[List[a, c, d, e], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[r, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Dist[Times[Power[a, IntPart[p]], Power[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]], FracPart[p]], Power[Times[Power[Plus[1, Times[2, c, Power[x, 2], Power[Plus[b, Times[-1, r]], -1]]], FracPart[p]], Power[Plus[1, Times[2, c, Power[x, 2], Power[Plus[b, r], -1]]], FracPart[p]]], -1]], Int[Times[Power[Plus[d, Times[e, Power[x, 2]]], q], Power[Plus[1, Times[2, c, Power[x, 2], Power[Plus[b, Times[-1, r]], -1]]], p], Power[Plus[1, Times[2, c, Power[x, 2], Power[Plus[b, r], -1]]], p]], x], x]], And[FreeQ[List[a, b, c, d, e, q], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], IntegerQ[Plus[p, Times[-1, 1, Power[2, -1]]]], NegativeIntegerQ[q]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[r, Rt[Times[-1, a, c], 2]]], Dist[Times[Power[a, IntPart[p]], Power[Plus[a, Times[c, Power[x, 4]]], FracPart[p]], Power[Times[Power[Plus[1, Times[-1, c, Power[x, 2], Power[r, -1]]], FracPart[p]], Power[Plus[1, Times[c, Power[x, 2], Power[r, -1]]], FracPart[p]]], -1]], Int[Times[Power[Plus[d, Times[e, Power[x, 2]]], q], Power[Plus[1, Times[-1, c, Power[x, 2], Power[r, -1]]], p], Power[Plus[1, Times[c, Power[x, 2], Power[r, -1]]], p]], x], x]], And[FreeQ[List[a, c, d, e, q], x], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], IntegerQ[Plus[p, Times[-1, 1, Power[2, -1]]]], NegativeIntegerQ[q]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[1, Power[Times[2, Sqrt[a], Sqrt[d], Rt[Times[-1, e, Power[d, -1]], 2]], -1], EllipticF[Times[2, ArcSin[Times[Rt[Times[-1, e, Power[d, -1]], 2], x]]], Times[b, d, Power[Times[4, a, e], -1]]]], x], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[Times[c, d], Times[-1, b, e]]], PositiveQ[a], PositiveQ[d]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Times[Plus[d, Times[e, Power[x, 2]]], Power[d, -1]]], Sqrt[Times[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]], Power[a, -1]]], Power[Times[Sqrt[Plus[d, Times[e, Power[x, 2]]]], Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]]], -1]], Int[Times[1, Power[Times[Sqrt[Plus[1, Times[e, Power[d, -1], Power[x, 2]]]], Sqrt[Plus[1, Times[b, Power[a, -1], Power[x, 2]], Times[c, Power[a, -1], Power[x, 4]]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[Times[c, d], Times[-1, b, e]]], Not[And[PositiveQ[a], PositiveQ[d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Sqrt[a], Power[Times[2, Sqrt[d], Rt[Times[-1, e, Power[d, -1]], 2]], -1], EllipticE[Times[2, ArcSin[Times[Rt[Times[-1, e, Power[d, -1]], 2], x]]], Times[b, d, Power[Times[4, a, e], -1]]]], x], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[Times[c, d], Times[-1, b, e]]], PositiveQ[a], PositiveQ[d]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]], Sqrt[Times[Plus[d, Times[e, Power[x, 2]]], Power[d, -1]]], Power[Times[Sqrt[Plus[d, Times[e, Power[x, 2]]]], Sqrt[Times[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]], Power[a, -1]]]], -1]], Int[Times[Sqrt[Plus[1, Times[b, Power[a, -1], Power[x, 2]], Times[c, Power[a, -1], Power[x, 4]]]], Power[Sqrt[Plus[1, Times[e, Power[d, -1], Power[x, 2]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[Times[c, d], Times[-1, b, e]]], Not[And[PositiveQ[a], PositiveQ[d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Or[And[IntegersQ[p, q], Not[IntegerQ[n]]], PositiveIntegerQ[p], And[PositiveIntegerQ[q], Not[IntegerQ[n]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, c, d, e, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Times[c, 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RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], And[FreeQ[List[a, b, c, d, e, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], Not[IntegersQ[n, q]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], p]], x], And[FreeQ[List[a, c, d, e, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], Not[IntegersQ[n, q]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Coefficient[u, x, 1], -1]], Subst[Int[Times[Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x, u], x], And[FreeQ[List[a, b, c, d, e, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], LinearQ[u, x], NonzeroQ[Plus[u, Times[-1, x]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Coefficient[u, x, 1], -1]], Subst[Int[Times[Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], p]], x], x, u], x], And[FreeQ[List[a, c, d, e, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], LinearQ[u, x], NonzeroQ[Plus[u, Times[-1, x]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[x, Times[-1, n, q]], Power[Plus[e, Times[d, Power[x, n]]], q], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], And[FreeQ[List[a, b, c, d, e, n, p], x], EqQ[n2, Times[2, n]], EqQ[mn, Times[-1, n]], IntegerQ[q]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[x, Times[mn, q]], Power[Plus[e, Times[d, Power[x, 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Times[-1, mn, FracPart[q]]], Power[Plus[d, Times[e, Power[x, mn]]], FracPart[q]], Power[Power[Plus[e, Times[d, Power[x, Times[-1, mn]]]], FracPart[q]], -1]], Int[Times[Power[x, Times[mn, q]], Power[Plus[e, Times[d, Power[x, Times[-1, mn]]]], q], Power[Plus[a, Times[c, Power[x, n2]]], p]], x], x], And[FreeQ[List[a, c, d, e, mn, p, q], x], EqQ[n2, Times[-2, mn]], Not[IntegerQ[q]], Not[IntegerQ[p]], PosQ[n2]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Pattern[q, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], FracPart[p]], Power[Times[Power[x, Times[2, n, FracPart[p]]], Power[Plus[c, Times[b, Power[x, Times[-1, n]]], Times[a, Power[x, Times[-2, n]]]], FracPart[p]]], -1]], Int[Times[Power[x, Times[2, n, p]], Power[Plus[d, Times[e, Power[x, Times[-1, n]]]], q], Power[Plus[c, Times[b, Power[x, Times[-1, n]]], Times[a, Power[x, Times[-2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, n, q], x], EqQ[n2, Times[2, n]], EqQ[mn, Times[-1, n]], Not[IntegerQ[q]], Not[IntegerQ[p]], NegQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[c, Power[x, n2]]], FracPart[p]], Power[Times[Power[x, Times[n2, FracPart[p]]], Power[Plus[c, Times[a, Power[x, Times[2, mn]]]], FracPart[p]]], -1]], Int[Times[Power[x, Times[n2, p]], Power[Plus[d, Times[e, Power[x, mn]]], q], Power[Plus[c, Times[a, Power[x, Times[2, mn]]]], p]], x], x], And[FreeQ[List[a, c, d, e, mn, q], x], EqQ[n2, Times[-2, mn]], Not[IntegerQ[q]], Not[IntegerQ[p]], NegQ[n2]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[mn, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[x, Times[-1, n, p]], Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[b, Times[a, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], And[FreeQ[List[a, b, c, d, e, n, q], x], EqQ[mn, Times[-1, n]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[mn, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[x, Times[n, FracPart[p]]], Power[Plus[a, Times[b, Power[Power[x, n], -1]], Times[c, Power[x, n]]], FracPart[p]], Power[Power[Plus[b, Times[a, Power[x, n]], Times[c, Power[x, Times[2, n]]]], FracPart[p]], -1]], Int[Times[Power[x, Times[-1, n, p]], Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[b, Times[a, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, n, p, q], x], EqQ[mn, Times[-1, n]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], FracPart[p]], Power[Times[Power[Times[4, c], IntPart[p]], Power[Plus[b, Times[2, c, Power[x, n]]], Times[2, FracPart[p]]]], -1]], Int[Times[Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[f, Times[g, Power[x, n]]], r], Power[Plus[b, Times[2, c, Power[x, n]]], Times[2, p]]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, n, p, q, r], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[d, Times[e, Power[x, n]]], Plus[p, q]], Power[Plus[f, Times[g, Power[x, n]]], r], Power[Plus[Times[a, Power[d, -1]], Times[c, Power[e, -1], Power[x, n]]], p]], x], And[FreeQ[List[a, b, c, d, e, f, g, n, q, r], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[d, Times[e, Power[x, n]]], Plus[p, q]], Power[Plus[f, Times[g, Power[x, n]]], r], Power[Plus[Times[a, Power[d, -1]], Times[c, Power[e, -1], Power[x, n]]], p]], x], And[FreeQ[List[a, c, d, e, f, g, n, q, r], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], FracPart[p]], Power[Times[Power[Plus[d, Times[e, Power[x, n]]], FracPart[p]], Power[Plus[Times[a, Power[d, -1]], Times[Times[c, Power[x, n]], Power[e, -1]]], FracPart[p]]], -1]], Int[Times[Power[Plus[d, Times[e, Power[x, n]]], Plus[p, q]], Power[Plus[f, Times[g, Power[x, n]]], r], Power[Plus[Times[a, Power[d, -1]], Times[c, Power[e, -1], Power[x, n]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, n, p, q, r], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[r, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[c, Power[x, Times[2, n]]]], FracPart[p]], Power[Times[Power[Plus[d, Times[e, Power[x, n]]], FracPart[p]], Power[Plus[Times[a, Power[d, -1]], Times[Times[c, Power[x, n]], Power[e, -1]]], FracPart[p]]], -1]], Int[Times[Power[Plus[d, Times[e, Power[x, n]]], Plus[p, q]], Power[Plus[f, Times[g, Power[x, n]]], r], Power[Plus[Times[a, Power[d, -1]], Times[c, Power[e, -1], Power[x, n]]], p]], x], x], And[FreeQ[List[a, c, d, e, f, g, n, p, q, r], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Condition[Plus[Dist[Times[Plus[Times[2, c, f], Times[-1, g, Plus[b, Times[-1, q]]]], Power[Plus[Times[2, c, d], Times[-1, e, Plus[b, Times[-1, q]]]], -1]], Int[Times[1, Power[Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]], -1]], x], x], Times[-1, Dist[Times[Plus[Times[e, f], Times[-1, d, g]], Power[Plus[Times[2, c, d], Times[-1, e, Plus[b, Times[-1, q]]]], -1]], Int[Times[Plus[b, Times[-1, q], Times[2, c, Power[x, 2]]], Power[Times[Plus[d, Times[e, Power[x, 2]]], Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]]], -1]], x], x]]], NonzeroQ[Plus[Times[2, c, f], Times[-1, g, Plus[b, Times[-1, q]]]]]]], And[FreeQ[List[a, b, c, d, e, f, g], x], PositiveQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Not[NegativeQ[c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Times[-1, a, c], 2]]], Condition[Plus[Dist[Times[Plus[Times[c, f], Times[g, q]], Power[Plus[Times[c, d], Times[e, q]], -1]], Int[Times[1, Power[Sqrt[Plus[a, Times[c, Power[x, 4]]]], -1]], x], x], Dist[Times[Plus[Times[e, f], Times[-1, d, g]], Power[Plus[Times[c, d], Times[e, q]], -1]], Int[Times[Plus[q, Times[-1, c, Power[x, 2]]], Power[Times[Plus[d, Times[e, Power[x, 2]]], Sqrt[Plus[a, Times[c, Power[x, 4]]]]], -1]], x], x]], NonzeroQ[Plus[Times[c, f], Times[g, q]]]]], And[FreeQ[List[a, c, d, e, f, g], x], PositiveQ[Times[-1, a, c]], NonzeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[NegativeQ[c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[Times[d1, d2], Times[e1, e2, Power[x, n]]], q], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], And[FreeQ[List[a, b, c, d1, e1, d2, e2, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[non2, Times[-1, n, Power[2, -1]]]], ZeroQ[Plus[Times[d2, e1], Times[d1, e2]]], Or[IntegerQ[q], And[PositiveQ[d1], PositiveQ[d2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[d1, Times[e1, Power[x, Times[n, Power[2, -1]]]]], FracPart[q]], Power[Plus[d2, Times[e2, Power[x, Times[n, Power[2, -1]]]]], FracPart[q]], Power[Power[Plus[Times[d1, d2], Times[e1, e2, Power[x, n]]], FracPart[q]], -1]], Int[Times[Power[Plus[Times[d1, d2], Times[e1, e2, Power[x, n]]], q], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, d1, e1, d2, e2, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[non2, Times[-1, n, Power[2, -1]]]], ZeroQ[Plus[Times[d2, e1], Times[d1, e2]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[A, Int[Times[Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x], Dist[B, Int[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x]], And[FreeQ[List[a, b, c, d, e, A, B, m, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[m, Times[-1, n], 1]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[A, Int[Times[Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], p]], x], x], Dist[B, Int[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], p]], x], x]], And[FreeQ[List[a, c, d, e, A, B, m, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[m, Times[-1, n], 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[f, m], Power[Times[n, Power[e, Plus[Times[Plus[m, 1], Power[n, -1]], -1]]], -1]], Subst[Int[Times[Power[Times[e, x], Plus[q, Times[Plus[m, 1], Power[n, -1]], -1]], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x, Power[x, n]], x], And[FreeQ[List[a, b, c, e, f, m, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], Or[IntegerQ[m], PositiveQ[f]], IntegerQ[Simplify[Times[Plus[m, 1], Power[n, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[f, m], Power[Times[n, Power[e, Plus[Times[Plus[m, 1], Power[n, -1]], -1]]], -1]], Subst[Int[Times[Power[Times[e, x], Plus[q, Times[Plus[m, 1], Power[n, -1]], -1]], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x, Power[x, n]], x], And[FreeQ[List[a, c, e, f, m, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], Or[IntegerQ[m], PositiveQ[f]], IntegerQ[Simplify[Times[Plus[m, 1], Power[n, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[f, m], Power[e, IntPart[q]], Power[Times[e, Power[x, n]], FracPart[q]], Power[Power[x, Times[n, FracPart[q]]], -1]], Int[Times[Power[x, Plus[m, Times[n, q]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, e, f, m, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], Or[IntegerQ[m], PositiveQ[f]], Not[IntegerQ[Simplify[Times[Plus[m, 1], Power[n, -1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[f, m], Power[e, IntPart[q]], Power[Times[e, Power[x, n]], FracPart[q]], Power[Power[x, Times[n, FracPart[q]]], -1]], Int[Times[Power[x, Plus[m, Times[n, q]]], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, c, e, f, m, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], Or[IntegerQ[m], PositiveQ[f]], Not[IntegerQ[Simplify[Times[Plus[m, 1], Power[n, -1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[f, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[f, IntPart[m]], Power[Times[f, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Times[e, Power[x, n]], q], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, e, f, m, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[f, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[f, IntPart[m]], Power[Times[f, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Times[e, Power[x, n]], q], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, c, e, f, m, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[n, -1]], Subst[Int[Times[Power[Plus[d, 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Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[n, -1]], Subst[Int[Times[Power[x, Plus[Times[Plus[m, 1], Power[n, -1]], -1]], Power[Plus[d, Times[e, x]], q], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x, Power[x, n]], x], And[FreeQ[List[a, b, c, d, e, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], Not[IntegerQ[p]], PositiveIntegerQ[m, n, Times[Plus[m, 1], Power[n, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], FracPart[p]], Power[Times[Power[c, IntPart[p]], Power[Plus[Times[b, Power[2, -1]], Times[c, Power[x, n]]], Times[2, FracPart[p]]]], -1]], Int[Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[Times[b, Power[2, -1]], Times[c, Power[x, n]]], Times[2, p]]], x], x], And[FreeQ[List[a, b, c, d, e, f, m, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[n, -1]], Subst[Int[Times[Power[x, Plus[Simplify[Times[Plus[m, 1], Power[n, -1]]], -1]], Power[Plus[d, Times[e, x]], q], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x, Power[x, n]], x], And[FreeQ[List[a, b, c, d, e, m, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], IntegerQ[Simplify[Times[Plus[m, 1], Power[n, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[n, -1]], Subst[Int[Times[Power[x, Plus[Simplify[Times[Plus[m, 1], Power[n, -1]]], -1]], Power[Plus[d, Times[e, x]], q], Power[Plus[a, Times[c, Power[x, 2]]], p]], x], x, Power[x, n]], x], And[FreeQ[List[a, c, d, e, m, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], IntegerQ[Simplify[Times[Plus[m, 1], Power[n, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[f, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[f, IntPart[m]], Power[Times[f, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, f, m, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], IntegerQ[Simplify[Times[Plus[m, 1], Power[n, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[f, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[f, IntPart[m]], Power[Times[f, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, c, d, e, f, m, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], IntegerQ[Simplify[Times[Plus[m, 1], Power[n, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, n]]], Plus[q, p]], Power[Plus[Times[a, Power[d, -1]], Times[c, Power[e, -1], Power[x, n]]], p]], x], And[FreeQ[List[a, b, c, d, e, f, m, n, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, n]]], Plus[q, p]], Power[Plus[Times[a, Power[d, -1]], Times[c, Power[e, -1], Power[x, n]]], p]], x], And[FreeQ[List[a, c, d, e, f, q, m, n, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], FracPart[p]], Power[Times[Power[Plus[d, Times[e, Power[x, n]]], FracPart[p]], Power[Plus[Times[a, Power[d, -1]], Times[Times[c, Power[x, n]], Power[e, -1]]], FracPart[p]]], -1]], Int[Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, n]]], Plus[q, p]], Power[Plus[Times[a, Power[d, -1]], Times[c, Power[e, -1], Power[x, n]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, f, m, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[c, Power[x, Times[2, n]]]], FracPart[p]], Power[Times[Power[Plus[d, Times[e, Power[x, n]]], FracPart[p]], Power[Plus[Times[a, Power[d, -1]], Times[Times[c, Power[x, n]], Power[e, -1]]], FracPart[p]]], -1]], Int[Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, n]]], Plus[q, p]], Power[Plus[Times[a, Power[d, -1]], Times[c, Power[e, -1], Power[x, n]]], p]], x], x], And[FreeQ[List[a, c, d, e, f, m, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[-1, d], Plus[Times[Plus[m, Times[-1, Mod[m, n]]], Power[n, -1]], -1]], Power[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]], p], Power[x, Plus[Mod[m, n], 1]], Power[Plus[d, Times[e, Power[x, n]]], Plus[q, 1]], Power[Times[n, Power[e, Plus[Times[2, p], Times[Plus[m, Times[-1, Mod[m, n]]], Power[n, -1]]]], Plus[q, 1]], -1]], x], Dist[Times[1, Power[Times[n, Power[e, Plus[Times[2, p], Times[Plus[m, Times[-1, Mod[m, n]]], Power[n, -1]]]], Plus[q, 1]], -1]], Int[Times[Power[x, Mod[m, n]], Power[Plus[d, Times[e, Power[x, n]]], Plus[q, 1]], ExpandToSum[Together[Times[1, Power[Plus[d, Times[e, Power[x, n]]], -1], Plus[Times[n, Power[e, Plus[Times[2, p], Times[Plus[m, Times[-1, Mod[m, n]]], Power[n, -1]]]], Plus[q, 1], Power[x, Plus[m, Times[-1, Mod[m, n]]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], Times[-1, Power[Times[-1, d], Plus[Times[Plus[m, Times[-1, Mod[m, n]]], Power[n, -1]], -1]], Power[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]], p], Plus[Times[d, Plus[Mod[m, n], 1]], Times[e, Plus[Mod[m, n], Times[n, Plus[q, 1]], 1], Power[x, n]]]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n, p], IntegersQ[m, q], Less[q, -1], Greater[m, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[-1, d], Plus[Times[Plus[m, Times[-1, Mod[m, n]]], Power[n, -1]], -1]], Power[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], p], Power[x, Plus[Mod[m, n], 1]], Power[Plus[d, Times[e, Power[x, n]]], Plus[q, 1]], Power[Times[n, Power[e, Plus[Times[2, p], Times[Plus[m, Times[-1, Mod[m, n]]], Power[n, -1]]]], Plus[q, 1]], -1]], x], Dist[Times[1, Power[Times[n, Power[e, Plus[Times[2, p], Times[Plus[m, Times[-1, Mod[m, n]]], Power[n, -1]]]], Plus[q, 1]], -1]], Int[Times[Power[x, Mod[m, n]], Power[Plus[d, Times[e, Power[x, n]]], Plus[q, 1]], ExpandToSum[Together[Times[1, Power[Plus[d, Times[e, Power[x, n]]], -1], Plus[Times[n, Power[e, Plus[Times[2, p], Times[Plus[m, Times[-1, Mod[m, n]]], Power[n, -1]]]], Plus[q, 1], Power[x, Plus[m, Times[-1, Mod[m, n]]]], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], p]], Times[-1, Power[Times[-1, d], Plus[Times[Plus[m, Times[-1, Mod[m, n]]], Power[n, -1]], -1]], Power[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], p], Plus[Times[d, Plus[Mod[m, n], 1]], Times[e, Plus[Mod[m, n], Times[n, Plus[q, 1]], 1], Power[x, n]]]]]]], x]], x], x]], And[FreeQ[List[a, c, d, e], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PositiveIntegerQ[n, p], IntegersQ[m, q], Less[q, -1], Greater[m, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[-1, d], Plus[Times[Plus[m, Times[-1, Mod[m, n]]], Power[n, -1]], -1]], Power[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]], p], Power[x, Plus[Mod[m, n], 1]], Power[Plus[d, Times[e, Power[x, n]]], Plus[q, 1]], Power[Times[n, Power[e, Plus[Times[2, p], Times[Plus[m, Times[-1, Mod[m, n]]], Power[n, -1]]]], Plus[q, 1]], -1]], x], Dist[Times[Power[Times[-1, d], Plus[Times[Plus[m, Times[-1, Mod[m, n]]], Power[n, -1]], -1]], Power[Times[n, Power[e, Times[2, p]], Plus[q, 1]], -1]], Int[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, n]]], Plus[q, 1]], ExpandToSum[Together[Times[1, Power[Plus[d, Times[e, Power[x, n]]], -1], Plus[Times[n, Power[Times[-1, d], Plus[Times[-1, Plus[m, Times[-1, Mod[m, n]]], Power[n, -1]], 1]], 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Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[-1, d], Plus[Times[Plus[m, Times[-1, Mod[m, n]]], Power[n, -1]], -1]], Power[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], p], Power[x, Plus[Mod[m, n], 1]], Power[Plus[d, Times[e, Power[x, n]]], Plus[q, 1]], Power[Times[n, Power[e, Plus[Times[2, p], Times[Plus[m, Times[-1, Mod[m, n]]], Power[n, -1]]]], Plus[q, 1]], -1]], x], Dist[Times[Power[Times[-1, d], Plus[Times[Plus[m, Times[-1, Mod[m, n]]], Power[n, -1]], -1]], Power[Times[n, Power[e, Times[2, p]], Plus[q, 1]], -1]], Int[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, n]]], Plus[q, 1]], ExpandToSum[Together[Times[1, Power[Plus[d, Times[e, Power[x, n]]], -1], Plus[Times[n, Power[Times[-1, d], Plus[Times[-1, Plus[m, Times[-1, Mod[m, n]]], Power[n, -1]], 1]], Power[e, Times[2, p]], Plus[q, 1], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], p]], Times[-1, Times[Power[e, Times[-1, Plus[m, Times[-1, Mod[m, n]]], Power[n, -1]]], Power[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], p], Power[x, Times[-1, Plus[m, Times[-1, Mod[m, n]]]]]], Plus[Times[d, Plus[Mod[m, n], 1]], Times[e, Plus[Mod[m, n], Times[n, Plus[q, 1]], 1], Power[x, n]]]]]]], x]], x], x]], And[FreeQ[List[a, c, d, e], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PositiveIntegerQ[n, p], IntegersQ[m, q], Less[q, -1], Less[m, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[c, p], Power[Times[f, x], Plus[m, 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RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[c, p], Power[Times[f, x], Plus[m, Times[2, n, p], Times[-1, n], 1]], Power[Plus[d, Times[e, Power[x, n]]], Plus[q, 1]], Power[Times[e, Power[f, Plus[Times[2, n, p], Times[-1, n], 1]], Plus[m, Times[2, n, p], Times[n, q], 1]], -1]], x], Dist[Times[1, Power[Times[e, Plus[m, Times[2, n, p], Times[n, q], 1]], -1]], Int[Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, n]]], q], ExpandToSum[Plus[Times[e, Plus[m, Times[2, n, p], Times[n, q], 1], Plus[Power[Plus[a, Times[c, Power[x, Times[2, n]]]], p], Times[-1, Power[c, p], Power[x, Times[2, n, p]]]]], Times[-1, d, Power[c, p], Plus[m, Times[2, n, p], Times[-1, n], 1], Power[x, Plus[Times[2, n, p], Times[-1, n]]]]], x]], x], x]], And[FreeQ[List[a, c, d, e, f, m, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PositiveIntegerQ[n, p], Greater[Times[2, n, p], Plus[n, -1]], Not[IntegerQ[q]], NonzeroQ[Plus[m, Times[2, n, p], Times[n, q], 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], 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n]]], ZeroQ[Plus[n2, Times[-1, 2, n]]], PositiveIntegerQ[n, p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[k, GCD[Plus[m, 1], n]]], Condition[Dist[Times[1, Power[k, -1]], Subst[Int[Times[Power[x, Plus[Times[Plus[m, 1], Power[k, -1]], -1]], Power[Plus[d, Times[e, Power[x, Times[n, Power[k, -1]]]]], q], Power[Plus[a, Times[b, Power[x, Times[n, Power[k, -1]]]], Times[c, Power[x, Times[2, n, Power[k, -1]]]]], p]], x], x, Power[x, k]], x], Unequal[k, 1]]], And[FreeQ[List[a, b, c, d, e, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[k, GCD[Plus[m, 1], n]]], Condition[Dist[Times[1, Power[k, -1]], Subst[Int[Times[Power[x, Plus[Times[Plus[m, 1], Power[k, -1]], -1]], Power[Plus[d, Times[e, Power[x, Times[n, Power[k, -1]]]]], q], Power[Plus[a, Times[c, Power[x, Times[2, n, Power[k, -1]]]]], p]], x], x, Power[x, k]], x], Unequal[k, 1]]], And[FreeQ[List[a, c, d, e, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PositiveIntegerQ[n], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[k, Denominator[m]]], Dist[Times[k, Power[f, -1]], Subst[Int[Times[Power[x, Plus[Times[k, Plus[m, 1]], -1]], Power[Plus[d, Times[e, Power[x, Times[k, n]], Power[Power[f, n], -1]]], q], Power[Plus[a, Times[b, Power[x, Times[k, n]], Power[Power[f, n], -1]], Times[c, Power[x, Times[2, k, n]], Power[Power[f, Times[2, n]], -1]]], p]], x], x, Power[Times[f, x], Times[1, Power[k, -1]]]], x]], And[FreeQ[List[a, b, c, d, e, f, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], FractionQ[m], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[k, Denominator[m]]], Dist[Times[k, Power[f, -1]], Subst[Int[Times[Power[x, Plus[Times[k, Plus[m, 1]], -1]], Power[Plus[d, Times[e, Power[x, Times[k, n]], Power[f, -1]]], q], Power[Plus[a, Times[c, Power[x, Times[2, k, n]], Power[f, -1]]], p]], x], x, Power[Times[f, x], Times[1, Power[k, -1]]]], x]], And[FreeQ[List[a, c, d, e, f, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], 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Int[Times[Power[Times[f, x], Plus[m, n]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, -1]], Simp[Plus[Times[2, a, e, Plus[m, 1]], Times[-1, b, d, Plus[m, Times[n, Plus[Times[2, p], 1]], 1]], Times[Plus[Times[b, e, Plus[m, 1]], Times[-1, 2, c, d, Plus[m, Times[n, Plus[Times[2, p], 1]], 1]]], Power[x, n]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m, p], Greater[p, 0], Less[m, -1], Unequal[Plus[m, Times[n, Plus[Times[2, p], 1]], 1], 0], Or[IntegerQ[p], And[IntegerQ[Times[2, p]], IntegerQ[m], Equal[n, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], p], Plus[Times[d, Plus[m, Times[n, Plus[Times[2, p], 1]], 1]], Times[e, Plus[m, 1], Power[x, n]]], Power[Times[f, Plus[m, 1], Plus[m, Times[n, Plus[Times[2, p], 1]], 1]], -1]], x], Dist[Times[2, n, p, Power[Times[Power[f, n], Plus[m, 1], Plus[m, Times[n, Plus[Times[2, p], 1]], 1]], -1]], Int[Times[Power[Times[f, x], Plus[m, n]], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], Plus[p, -1]], Plus[Times[a, e, Plus[m, 1]], Times[-1, c, d, Plus[m, Times[n, Plus[Times[2, p], 1]], 1], Power[x, n]]]], x], x]], And[FreeQ[List[a, c, d, e, f], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PositiveIntegerQ[n], RationalQ[m, p], Greater[p, 0], Less[m, -1], Unequal[Plus[m, Times[n, Plus[Times[2, p], 1]], 1], 0], Or[IntegerQ[p], And[IntegerQ[Times[2, p]], IntegerQ[m], Equal[n, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p], Plus[Times[b, e, n, p], Times[c, d, Plus[m, Times[n, Plus[Times[2, p], 1]], 1]], Times[c, e, Plus[Times[2, n, p], m, 1], Power[x, n]]], Power[Times[c, f, Plus[Times[2, n, p], m, 1], Plus[m, Times[n, Plus[Times[2, p], 1]], 1]], -1]], x], Dist[Times[n, p, Power[Times[c, Plus[Times[2, n, p], m, 1], Plus[m, Times[n, Plus[Times[2, p], 1]], 1]], -1]], 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Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], p], Plus[Times[c, d, Plus[m, Times[n, Plus[Times[2, p], 1]], 1]], Times[c, e, Plus[Times[2, n, p], m, 1], Power[x, n]]], Power[Times[c, f, Plus[Times[2, n, p], m, 1], Plus[m, Times[n, Plus[Times[2, p], 1]], 1]], -1]], x], Dist[Times[2, a, n, p, Power[Times[Plus[Times[2, n, p], m, 1], Plus[m, Times[n, Plus[Times[2, p], 1]], 1]], -1]], Int[Times[Power[Times[f, x], m], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], Plus[p, -1]], Simp[Plus[Times[d, Plus[m, Times[n, Plus[Times[2, p], 1]], 1]], Times[e, Plus[Times[2, n, p], m, 1], Power[x, n]]], x]], x], x]], And[FreeQ[List[a, c, d, e, f, m], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PositiveIntegerQ[n], RationalQ[p], Greater[p, 0], NonzeroQ[Plus[Times[2, n, p], m, 1]], NonzeroQ[Plus[m, Times[n, Plus[Times[2, p], 1]], 1]], Or[IntegerQ[p], And[IntegerQ[Times[2, p]], IntegerQ[m], Equal[n, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[f, Plus[n, -1]], Power[Times[f, x], Plus[m, Times[-1, n], 1]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Plus[Times[b, d], Times[-1, 2, a, e], Times[-1, Plus[Times[b, e], Times[-1, 2, c, d]], Power[x, n]]], Power[Times[n, Plus[p, 1], 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RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Plus[Times[d, Plus[Power[b, 2], Times[-1, 2, a, c]]], Times[-1, a, b, e], Times[Plus[Times[b, d], Times[-1, 2, a, e]], c, Power[x, n]]], Power[Times[a, f, n, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], x]], Dist[Times[1, Power[Times[a, n, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], Int[Times[Power[Times[f, x], m], 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Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Plus[d, Times[e, Power[x, n]]], Power[Times[2, a, f, n, Plus[p, 1]], -1]], x]], Dist[Times[1, Power[Times[2, a, n, Plus[p, 1]], -1]], Int[Times[Power[Times[f, x], m], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Simp[Plus[Times[d, Plus[m, Times[2, n, Plus[p, 1]], 1]], Times[e, Plus[m, Times[n, Plus[Times[2, p], 3]], 1], Power[x, n]]], x]], x], x]], And[FreeQ[List[a, c, d, e, f, m], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PositiveIntegerQ[n], RationalQ[p], Less[p, -1], Or[IntegerQ[p], And[IntegerQ[Times[2, p]], IntegerQ[m], Equal[n, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, 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Not[NegativeQ[Times[2, c, q]]]]], And[FreeQ[List[a, c, d, e, f, m], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], IntegerQ[Times[n, Power[2, -1]]], Greater[n, 2], PositiveQ[Times[a, c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Plus[Dist[Plus[Times[e, Power[2, -1]], Times[Plus[Times[2, c, d], Times[-1, b, e]], Power[Times[2, q], -1]]], Int[Times[Power[Times[f, x], m], Power[Plus[Times[b, Power[2, -1]], Times[-1, q, Power[2, -1]], Times[c, Power[x, n]]], -1]], x], x], 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n]]], -1]], x], x]], Dist[Plus[Times[e, Power[2, -1]], Times[-1, c, d, Power[Times[2, q], -1]]], Int[Times[Power[Times[f, x], m], Power[Plus[q, Times[c, Power[x, n]]], -1]], x], x]]], And[FreeQ[List[a, c, d, e, f, m], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, m], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], IntegerQ[q], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], -1]], x], x], And[FreeQ[List[a, c, d, e, f, m], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PositiveIntegerQ[n], IntegerQ[q], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Power[Times[f, x], m], Times[Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, m], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], IntegerQ[q], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Power[Times[f, x], m], Times[Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], -1]], x], x], And[FreeQ[List[a, c, d, e, f, m], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PositiveIntegerQ[n], IntegerQ[q], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Power[f, Times[2, n]], Power[Power[c, 2], -1]], Int[Times[Power[Times[f, x], Plus[m, Times[-1, 2, n]]], Plus[Times[c, d], Times[-1, b, e], Times[c, e, Power[x, n]]], Power[Plus[d, Times[e, Power[x, n]]], Plus[q, -1]]], x], x], Times[-1, Dist[Times[Power[f, Times[2, n]], Power[Power[c, 2], -1]], Int[Times[Power[Times[f, x], Plus[m, Times[-1, 2, n]]], Power[Plus[d, Times[e, Power[x, n]]], Plus[q, -1]], Simp[Plus[Times[a, Plus[Times[c, d], Times[-1, b, e]]], Times[Plus[Times[b, c, d], Times[-1, Power[b, 2], e], Times[a, c, e]], Power[x, n]]], x], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], Not[IntegerQ[q]], RationalQ[m, q], Greater[q, 0], Greater[m, Plus[Times[2, n], -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Power[f, Times[2, n]], Power[c, -1]], Int[Times[Power[Times[f, x], Plus[m, Times[-1, 2, n]]], Power[Plus[d, Times[e, Power[x, n]]], q]], x], x], Times[-1, Dist[Times[a, Power[f, Times[2, n]], Power[c, -1]], Int[Times[Power[Times[f, x], Plus[m, Times[-1, 2, n]]], Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], -1]], x], x]]], And[FreeQ[List[a, c, d, e, f, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PositiveIntegerQ[n], Not[IntegerQ[q]], RationalQ[m], Greater[m, Plus[Times[2, n], -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[e, Power[f, n], Power[c, -1]], Int[Times[Power[Times[f, x], Plus[m, Times[-1, n]]], Power[Plus[d, Times[e, Power[x, n]]], Plus[q, -1]]], x], x], Times[-1, Dist[Times[Power[f, n], Power[c, -1]], Int[Times[Power[Times[f, x], Plus[m, Times[-1, n]]], Power[Plus[d, Times[e, Power[x, n]]], Plus[q, -1]], Simp[Plus[Times[a, e], Times[-1, Plus[Times[c, d], Times[-1, b, e]], Power[x, n]]], x], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], Not[IntegerQ[q]], RationalQ[m, q], Greater[q, 0], Inequality[Plus[n, -1], Less, m, LessEqual, Plus[Times[2, n], -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[e, Power[f, n], Power[c, -1]], Int[Times[Power[Times[f, x], Plus[m, Times[-1, n]]], Power[Plus[d, Times[e, Power[x, n]]], Plus[q, -1]]], x], x], Times[-1, Dist[Times[Power[f, n], Power[c, -1]], Int[Times[Power[Times[f, x], Plus[m, Times[-1, n]]], Power[Plus[d, Times[e, Power[x, n]]], Plus[q, -1]], Simp[Plus[Times[a, e], Times[-1, c, d, Power[x, n]]], x], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], -1]], x], x]]], And[FreeQ[List[a, c, d, e, f], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PositiveIntegerQ[n], Not[IntegerQ[q]], RationalQ[m, q], Greater[q, 0], Inequality[Plus[n, -1], Less, m, LessEqual, Plus[Times[2, n], -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[d, Power[a, -1]], Int[Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, n]]], Plus[q, -1]]], x], x], Times[-1, Dist[Times[1, Power[Times[a, Power[f, n]], -1]], Int[Times[Power[Times[f, x], Plus[m, n]], Power[Plus[d, Times[e, Power[x, n]]], Plus[q, -1]], Simp[Plus[Times[b, d], Times[-1, a, e], Times[c, d, Power[x, n]]], x], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], Not[IntegerQ[q]], RationalQ[m, q], Greater[q, 0], Less[m, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[d, Power[a, -1]], Int[Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, n]]], Plus[q, -1]]], x], x], Dist[Times[1, Power[Times[a, Power[f, n]], -1]], Int[Times[Power[Times[f, x], Plus[m, n]], Power[Plus[d, Times[e, Power[x, n]]], Plus[q, -1]], Simp[Plus[Times[a, e], Times[-1, c, d, Power[x, n]]], x], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], -1]], x], x]], And[FreeQ[List[a, c, d, e, f], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PositiveIntegerQ[n], Not[IntegerQ[q]], RationalQ[m, q], Greater[q, 0], Less[m, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Power[d, 2], Power[f, Times[2, n]], Power[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]], -1]], Int[Times[Power[Times[f, x], Plus[m, Times[-1, 2, n]]], Power[Plus[d, Times[e, Power[x, n]]], q]], x], x], Times[-1, Dist[Times[Power[f, Times[2, n]], Power[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]], -1]], Int[Times[Power[Times[f, x], Plus[m, Times[-1, 2, n]]], Power[Plus[d, Times[e, Power[x, n]]], Plus[q, 1]], Simp[Plus[Times[a, d], Times[Plus[Times[b, d], Times[-1, a, e]], Power[x, n]]], x], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], Not[IntegerQ[q]], RationalQ[m, q], Less[q, -1], Greater[m, Plus[Times[2, n], -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Power[d, 2], Power[f, Times[2, n]], Power[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], -1]], Int[Times[Power[Times[f, x], Plus[m, Times[-1, 2, n]]], Power[Plus[d, Times[e, Power[x, n]]], q]], x], x], Times[-1, Dist[Times[a, Power[f, Times[2, n]], Power[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], -1]], Int[Times[Power[Times[f, x], Plus[m, Times[-1, 2, n]]], Power[Plus[d, Times[e, Power[x, n]]], Plus[q, 1]], Plus[d, Times[-1, e, Power[x, n]]], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], -1]], x], x]]], And[FreeQ[List[a, c, d, e, f], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PositiveIntegerQ[n], Not[IntegerQ[q]], RationalQ[m, q], Less[q, -1], Greater[m, Plus[Times[2, n], -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Dist[Times[d, e, Power[f, n], Power[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]], -1]], Int[Times[Power[Times[f, x], Plus[m, Times[-1, n]]], Power[Plus[d, Times[e, Power[x, n]]], q]], x], x]], Dist[Times[Power[f, n], Power[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]], -1]], Int[Times[Power[Times[f, x], Plus[m, Times[-1, n]]], Power[Plus[d, Times[e, Power[x, n]]], Plus[q, 1]], Simp[Plus[Times[a, e], Times[c, d, Power[x, n]]], x], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], Not[IntegerQ[q]], RationalQ[m, q], Less[q, -1], Inequality[Plus[n, -1], Less, m, LessEqual, Plus[Times[2, n], -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], -1], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Dist[Times[d, e, Power[f, n], Power[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], -1]], Int[Times[Power[Times[f, x], Plus[m, Times[-1, n]]], Power[Plus[d, Times[e, Power[x, n]]], q]], x], x]], Dist[Times[Power[f, n], Power[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], -1]], Int[Times[Power[Times[f, x], Plus[m, Times[-1, n]]], Power[Plus[d, Times[e, Power[x, n]]], Plus[q, 1]], Simp[Plus[Times[a, e], Times[c, d, Power[x, n]]], x], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], -1]], x], x]], And[FreeQ[List[a, c, d, e, f], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PositiveIntegerQ[n], Not[IntegerQ[q]], RationalQ[m, q], Less[q, -1], Inequality[Plus[n, -1], Less, m, LessEqual, Plus[Times[2, n], -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Power[e, 2], Power[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]], -1]], Int[Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, n]]], q]], x], x], Dist[Times[1, Power[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]], -1]], Int[Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, n]]], Plus[q, 1]], Simp[Plus[Times[c, d], Times[-1, b, e], Times[-1, c, e, Power[x, n]]], x], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f, m], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], Not[IntegerQ[q]], RationalQ[q], Less[q, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Power[e, 2], Power[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], -1]], Int[Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, n]]], q]], x], x], Dist[Times[c, Power[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], -1]], Int[Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, n]]], Plus[q, 1]], Plus[d, Times[-1, e, Power[x, n]]], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], -1]], x], x]], And[FreeQ[List[a, c, d, e, f, m], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PositiveIntegerQ[n], Not[IntegerQ[q]], RationalQ[q], Less[q, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Power[Plus[d, Times[e, Power[x, n]]], q], Times[Power[Times[f, x], m], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, q, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], Not[IntegerQ[q]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Power[Plus[d, Times[e, Power[x, n]]], q], Times[Power[Times[f, x], m], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], -1]], x], x], And[FreeQ[List[a, c, d, e, f, q, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PositiveIntegerQ[n], Not[IntegerQ[q]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, n]]], q]], Times[1, Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, m, q, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], Not[IntegerQ[q]], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, n]]], q]], Times[1, Power[Plus[a, Times[c, Power[x, Times[2, n]]]], -1]], x], x], And[FreeQ[List[a, c, d, e, f, m, q, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PositiveIntegerQ[n], Not[IntegerQ[q]], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[Power[d, 2], -1]], Int[Times[Power[Times[f, x], m], Plus[Times[a, d], Times[Plus[Times[b, d], Times[-1, a, e]], Power[x, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, -1]]], x], x], Dist[Times[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]], Power[Times[Power[d, 2], Power[f, Times[2, n]]], -1]], Int[Times[Power[Times[f, x], Plus[m, Times[2, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, -1]], Power[Plus[d, Times[e, Power[x, n]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m, p], Greater[p, 0], Less[m, Times[-1, n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[a, Power[Power[d, 2], -1]], Int[Times[Power[Times[f, x], m], Plus[d, Times[-1, e, Power[x, n]]], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], Plus[p, -1]]], x], x], Dist[Times[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], Power[Times[Power[d, 2], Power[f, Times[2, n]]], -1]], Int[Times[Power[Times[f, x], Plus[m, Times[2, n]]], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], Plus[p, -1]], Power[Plus[d, Times[e, Power[x, n]]], -1]], x], x]], And[FreeQ[List[a, c, d, e, f], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PositiveIntegerQ[n], RationalQ[m, p], Greater[p, 0], Less[m, Times[-1, n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[Times[d, e], -1]], Int[Times[Power[Times[f, x], m], Plus[Times[a, e], Times[c, d, Power[x, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, -1]]], x], x], Times[-1, Dist[Times[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]], Power[Times[d, e, Power[f, n]], -1]], Int[Times[Power[Times[f, x], Plus[m, n]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, -1]], Power[Plus[d, Times[e, Power[x, n]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m, p], Greater[p, 0], Less[m, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[Times[d, e], -1]], Int[Times[Power[Times[f, x], m], Plus[Times[a, e], Times[c, d, Power[x, n]]], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], Plus[p, -1]]], x], x], Times[-1, Dist[Times[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], Power[Times[d, e, Power[f, n]], -1]], Int[Times[Power[Times[f, x], Plus[m, n]], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], Plus[p, -1]], Power[Plus[d, Times[e, Power[x, n]]], -1]], x], x]]], And[FreeQ[List[a, c, d, e, f], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PositiveIntegerQ[n], RationalQ[m, p], Greater[p, 0], Less[m, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Dist[Times[Power[f, Times[2, n]], Power[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]], -1]], Int[Times[Power[Times[f, x], Plus[m, Times[-1, 2, n]]], Plus[Times[a, d], Times[Plus[Times[b, d], Times[-1, a, e]], Power[x, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x]], Dist[Times[Power[d, 2], Power[f, Times[2, n]], Power[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]], -1]], Int[Times[Power[Times[f, x], Plus[m, Times[-1, 2, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[Plus[d, Times[e, Power[x, n]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m, p], Less[p, -1], Greater[m, n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Dist[Times[a, Power[f, Times[2, n]], Power[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], -1]], Int[Times[Power[Times[f, x], Plus[m, Times[-1, 2, n]]], Plus[d, Times[-1, e, Power[x, n]]], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], p]], x], x]], Dist[Times[Power[d, 2], Power[f, Times[2, n]], Power[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], -1]], Int[Times[Power[Times[f, x], Plus[m, Times[-1, 2, n]]], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[Plus[d, Times[e, Power[x, n]]], -1]], x], x]], And[FreeQ[List[a, c, d, e, f], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PositiveIntegerQ[n], RationalQ[m, p], Less[p, -1], Greater[m, n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Power[f, n], Power[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]], -1]], Int[Times[Power[Times[f, x], Plus[m, Times[-1, n]]], Plus[Times[a, e], Times[c, d, Power[x, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x], Times[-1, Dist[Times[d, e, Power[f, n], Power[Plus[Times[c, Power[d, 2]], Times[-1, b, d, e], Times[a, Power[e, 2]]], -1]], Int[Times[Power[Times[f, x], Plus[m, Times[-1, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[Plus[d, Times[e, Power[x, n]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m, p], Less[p, -1], Greater[m, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Power[f, n], Power[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], -1]], Int[Times[Power[Times[f, x], Plus[m, Times[-1, n]]], Plus[Times[a, e], Times[c, d, Power[x, n]]], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], p]], x], x], Times[-1, Dist[Times[d, e, Power[f, n], Power[Plus[Times[c, Power[d, 2]], Times[a, Power[e, 2]]], -1]], Int[Times[Power[Times[f, x], Plus[m, Times[-1, n]]], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[Plus[d, Times[e, Power[x, n]]], -1]], x], x]]], And[FreeQ[List[a, c, d, e, f], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PositiveIntegerQ[n], RationalQ[m, p], Less[p, -1], Greater[m, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p], Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, n]]], q]], x], x], And[FreeQ[List[a, b, c, d, e, f, m, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], Or[PositiveIntegerQ[q], IntegersQ[m, q]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Power[Plus[a, Times[c, Power[x, Times[2, n]]]], p], Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, n]]], q]], x], x], And[FreeQ[List[a, c, d, e, f, m, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PositiveIntegerQ[n], Or[PositiveIntegerQ[q], IntegersQ[m, q]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Subst[Int[Times[Power[Plus[d, Times[e, Power[x, Times[-1, n]]]], q], Power[Plus[a, Times[b, Power[x, Times[-1, n]]], Times[c, Power[x, Times[-2, n]]]], p], Power[Power[x, Plus[m, 2]], -1]], x], x, Times[1, Power[x, -1]]]], And[FreeQ[List[a, b, c, d, e, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NegativeIntegerQ[n], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Subst[Int[Times[Power[Plus[d, Times[e, Power[x, Times[-1, n]]]], q], Power[Plus[a, Times[c, Power[x, Times[-2, n]]]], p], Power[Power[x, Plus[m, 2]], -1]], x], x, Times[1, Power[x, -1]]]], And[FreeQ[List[a, c, d, e, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NegativeIntegerQ[n], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[g, Denominator[m]]], Times[-1, Dist[Times[g, Power[f, -1]], Subst[Int[Times[Power[Plus[d, Times[e, Power[f, Times[-1, n]], Power[x, Times[-1, g, n]]]], q], Power[Plus[a, Times[b, Power[f, Times[-1, n]], Power[x, Times[-1, g, n]]], Times[c, Power[f, Times[-2, n]], Power[x, Times[-2, g, n]]]], p], Power[Power[x, Plus[Times[g, Plus[m, 1]], 1]], -1]], x], x, Times[1, Power[Power[Times[f, x], Times[1, Power[g, -1]]], -1]]], x]]], And[FreeQ[List[a, b, c, d, e, f, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NegativeIntegerQ[n], FractionQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[g, Denominator[m]]], Times[-1, Dist[Times[g, Power[f, -1]], Subst[Int[Times[Power[Plus[d, Times[e, Power[f, Times[-1, n]], Power[x, Times[-1, g, n]]]], q], Power[Plus[a, Times[c, Power[f, Times[-2, n]], Power[x, Times[-2, g, n]]]], p], Power[Power[x, Plus[Times[g, Plus[m, 1]], 1]], -1]], x], x, Times[1, Power[Power[Times[f, x], Times[1, Power[g, -1]]], -1]]], x]]], And[FreeQ[List[a, c, d, e, f, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NegativeIntegerQ[n], FractionQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Times[Power[f, IntPart[m]], Power[Times[f, x], FracPart[m]], Power[Power[x, -1], FracPart[m]]], Subst[Int[Times[Power[Plus[d, Times[e, Power[x, Times[-1, n]]]], q], Power[Plus[a, Times[b, Power[x, Times[-1, n]]], Times[c, Power[x, Times[-2, n]]]], p], Power[Power[x, Plus[m, 2]], -1]], x], x, Times[1, Power[x, -1]]], x]], And[FreeQ[List[a, b, c, d, e, f, m, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NegativeIntegerQ[n], Not[RationalQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Times[Power[f, IntPart[m]], Power[Times[f, x], FracPart[m]], Power[Power[x, -1], FracPart[m]]], Subst[Int[Times[Power[Plus[d, Times[e, Power[x, Times[-1, n]]]], q], Power[Plus[a, Times[c, Power[x, Times[-2, n]]]], p], Power[Power[x, Plus[m, 2]], -1]], x], x, Times[1, Power[x, -1]]], x]], And[FreeQ[List[a, c, d, e, f, m, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NegativeIntegerQ[n], Not[RationalQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[g, Denominator[n]]], Dist[g, Subst[Int[Times[Power[x, Plus[Times[g, Plus[m, 1]], -1]], Power[Plus[d, Times[e, Power[x, Times[g, n]]]], q], Power[Plus[a, Times[b, Power[x, Times[g, n]]], Times[c, Power[x, Times[2, g, n]]]], p]], x], x, Power[x, Times[1, Power[g, -1]]]], x]], And[FreeQ[List[a, b, c, d, e, m, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], FractionQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[g, Denominator[n]]], Dist[g, Subst[Int[Times[Power[x, Plus[Times[g, Plus[m, 1]], -1]], Power[Plus[d, Times[e, Power[x, Times[g, n]]]], q], Power[Plus[a, Times[c, Power[x, Times[2, g, n]]]], p]], x], x, Power[x, Times[1, Power[g, -1]]]], x]], And[FreeQ[List[a, c, d, e, m, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], FractionQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[f, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[f, IntPart[m]], Power[Times[f, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, f, m, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], FractionQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[f, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[f, IntPart[m]], Power[Times[f, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, c, d, e, f, m, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], FractionQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Plus[m, 1], -1]], Subst[Int[Times[Power[Plus[d, Times[e, Power[x, Simplify[Times[n, Power[Plus[m, 1], -1]]]]]], q], Power[Plus[a, Times[b, Power[x, Simplify[Times[n, Power[Plus[m, 1], -1]]]]], Times[c, Power[x, Simplify[Times[2, n, Power[Plus[m, 1], -1]]]]]], p]], x], x, Power[x, Plus[m, 1]]], x], And[FreeQ[List[a, b, c, d, e, m, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegerQ[Simplify[Times[n, Power[Plus[m, 1], -1]]]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Plus[m, 1], -1]], Subst[Int[Times[Power[Plus[d, Times[e, Power[x, Simplify[Times[n, Power[Plus[m, 1], -1]]]]]], q], Power[Plus[a, Times[c, Power[x, Simplify[Times[2, n, Power[Plus[m, 1], -1]]]]]], p]], x], x, Power[x, Plus[m, 1]]], x], And[FreeQ[List[a, c, d, e, m, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], IntegerQ[Simplify[Times[n, Power[Plus[m, 1], -1]]]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[f, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[f, IntPart[m]], Power[Times[f, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, f, m, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegerQ[Simplify[Times[n, Power[Plus[m, 1], -1]]]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[f, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[f, IntPart[m]], Power[Times[f, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, c, d, e, f, m, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], IntegerQ[Simplify[Times[n, Power[Plus[m, 1], -1]]]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[r, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Plus[Dist[Times[2, c, Power[r, -1]], Int[Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[b, Times[-1, r], Times[2, c, Power[x, n]]], -1]], x], x], Times[-1, Dist[Times[2, c, Power[r, -1]], Int[Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[b, r, Times[2, c, Power[x, n]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d, e, f, m, n, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[r, Rt[Times[-1, a, c], 2]]], Plus[Times[-1, Dist[Times[c, Power[Times[2, r], -1]], Int[Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[r, Times[-1, c, Power[x, n]]], -1]], x], x]], Times[-1, Dist[Times[c, Power[Times[2, r], -1]], Int[Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[r, Times[c, Power[x, n]]], -1]], x], x]]]], And[FreeQ[List[a, c, d, e, f, m, n, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Plus[Times[d, Plus[Power[b, 2], Times[-1, 2, a, c]]], Times[-1, a, b, e], Times[Plus[Times[b, d], Times[-1, 2, a, e]], c, Power[x, n]]], Power[Times[a, f, n, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], x]], Dist[Times[1, Power[Times[a, n, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], Int[Times[Power[Times[f, x], m], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Simp[Plus[Times[d, Plus[Times[Power[b, 2], Plus[m, Times[n, Plus[p, 1]], 1]], Times[-1, 2, a, c, Plus[m, Times[2, n, Plus[p, 1]], 1]]]], Times[-1, a, b, e, Plus[m, 1]], Times[Plus[m, Times[n, Plus[Times[2, p], 3]], 1], Plus[Times[b, d], Times[-1, 2, a, e]], c, Power[x, n]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, m, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NegativeIntegerQ[Plus[p, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Plus[d, Times[e, Power[x, n]]], Power[Times[2, a, f, n, Plus[p, 1]], -1]], x]], Dist[Times[1, Power[Times[2, a, n, Plus[p, 1]], -1]], Int[Times[Power[Times[f, x], m], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Simp[Plus[Times[d, Plus[m, Times[2, n, Plus[p, 1]], 1]], Times[e, Plus[m, Times[n, Plus[Times[2, p], 3]], 1], Power[x, n]]], x]], x], x]], And[FreeQ[List[a, c, d, e, f, m, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NegativeIntegerQ[Plus[p, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, f, m, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], Or[PositiveIntegerQ[p], PositiveIntegerQ[q]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, c, d, e, f, m, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], Or[PositiveIntegerQ[p], PositiveIntegerQ[q]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[f, m], Int[ExpandIntegrand[Times[Power[x, m], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], p]], Power[Plus[Times[d, Power[Plus[Power[d, 2], Times[-1, Power[e, 2], Power[x, Times[2, n]]]], -1]], Times[-1, e, Power[x, n], Power[Plus[Power[d, 2], Times[-1, Power[e, 2], Power[x, Times[2, n]]]], -1]]], Times[-1, q]], x], x], x], And[FreeQ[List[a, c, d, e, f, m, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NegativeIntegerQ[q], Or[IntegerQ[m], PositiveQ[f]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[f, x], m], Power[Power[x, m], -1]], Int[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, c, d, e, f, m, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NegativeIntegerQ[q], Not[Or[IntegerQ[m], PositiveQ[f]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], And[FreeQ[List[a, b, c, d, e, f, m, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], p]], x], And[FreeQ[List[a, c, d, e, f, m, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[u, m], Power[Times[Coefficient[v, x, 1], Power[v, m]], -1]], Subst[Int[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x, v], x], And[FreeQ[List[a, b, c, d, e, m, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], LinearPairQ[u, v, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[u, m], Power[Times[Coefficient[v, x, 1], Power[v, m]], -1]], Subst[Int[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], p]], x], x, v], x], And[FreeQ[List[a, c, d, e, m, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], LinearPairQ[u, v, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[x, Plus[m, Times[-1, n, q]]], Power[Plus[e, Times[d, Power[x, n]]], q], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], And[FreeQ[List[a, b, c, d, e, m, n, p], x], EqQ[n2, Times[2, n]], EqQ[mn, Times[-1, n]], IntegerQ[q]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[x, Plus[m, Times[mn, q]]], Power[Plus[e, Times[d, Power[x, Times[-1, mn]]]], q], Power[Plus[a, Times[c, Power[x, n2]]], p]], x], And[FreeQ[List[a, c, d, e, m, mn, p], x], EqQ[n2, Times[-2, mn]], IntegerQ[q]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Pattern[q, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[x, Plus[m, Times[2, n, p]]], Power[Plus[d, Times[e, Power[x, Times[-1, n]]]], q], Power[Plus[c, Times[b, Power[x, Times[-1, n]]], Times[a, Power[x, Times[-2, n]]]], p]], x], And[FreeQ[List[a, b, c, d, e, m, n, q], x], EqQ[n2, Times[2, n]], EqQ[mn, Times[-1, n]], Not[IntegerQ[q]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[x, Plus[m, Times[-1, 2, mn, p]]], Power[Plus[d, Times[e, Power[x, mn]]], q], Power[Plus[c, Times[a, Power[x, Times[2, mn]]]], p]], x], And[FreeQ[List[a, c, d, e, m, mn, q], x], EqQ[n2, Times[-2, mn]], Not[IntegerQ[q]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Pattern[q, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[x, Times[n, FracPart[q]]], Power[Plus[d, Times[e, Power[x, Times[-1, n]]]], FracPart[q]], Power[Power[Plus[e, Times[d, Power[x, n]]], FracPart[q]], -1]], Int[Times[Power[x, Plus[m, Times[-1, n, q]]], Power[Plus[e, Times[d, Power[x, n]]], q], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, m, n, p, q], x], EqQ[n2, Times[2, n]], EqQ[mn, Times[-1, n]], Not[IntegerQ[q]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Pattern[q, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[x, Times[-1, mn, FracPart[q]]], Power[Plus[d, Times[e, Power[x, mn]]], FracPart[q]], Power[Power[Plus[e, Times[d, Power[x, Times[-1, mn]]]], FracPart[q]], -1]], Int[Times[Power[x, Plus[m, Times[mn, q]]], Power[Plus[e, Times[d, Power[x, Times[-1, mn]]]], q], Power[Plus[a, Times[c, Power[x, n2]]], p]], x], x], And[FreeQ[List[a, c, d, e, m, mn, p, q], x], EqQ[n2, Times[-2, mn]], Not[IntegerQ[q]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[f, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[f, IntPart[m]], Power[Times[f, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, mn]]], q], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, f, m, n, p, q], x], EqQ[n2, Times[2, n]], EqQ[mn, Times[-1, n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[f, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[mn, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[f, IntPart[m]], Power[Times[f, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, mn]]], q], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, c, d, e, f, m, mn, p, q], x], EqQ[n2, Times[-2, mn]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[mn, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[x, Plus[m, Times[-1, n, p]]], Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[b, Times[a, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], And[FreeQ[List[a, b, c, d, e, m, n, q], x], EqQ[mn, Times[-1, n]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[mn, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[x, Times[n, FracPart[p]]], Power[Plus[a, Times[b, Power[Power[x, n], -1]], Times[c, Power[x, n]]], FracPart[p]], Power[Power[Plus[b, Times[a, Power[x, n]], Times[c, Power[x, Times[2, n]]]], FracPart[p]], -1]], Int[Times[Power[x, Plus[m, Times[-1, n, p]]], Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[b, Times[a, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, m, n, p, q], x], EqQ[mn, Times[-1, n]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[f, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[mn, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[f, IntPart[m]], Power[Times[f, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, n]]], q], Power[Plus[a, Times[b, Power[x, Times[-1, n]]], Times[c, Power[x, n]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, f, m, n, p, q], x], EqQ[mn, Times[-1, n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Times[f, x], m], Power[Plus[Times[d1, d2], Times[e1, e2, Power[x, n]]], q], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], And[FreeQ[List[a, b, c, d1, e1, d2, e2, f, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[non2, Times[-1, n, Power[2, -1]]]], ZeroQ[Plus[Times[d2, e1], Times[d1, e2]]], Or[IntegerQ[q], And[PositiveQ[d1], PositiveQ[d2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[non2, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[d1, Times[e1, Power[x, Times[n, Power[2, -1]]]]], FracPart[q]], Power[Plus[d2, Times[e2, Power[x, Times[n, Power[2, -1]]]]], FracPart[q]], Power[Power[Plus[Times[d1, d2], Times[e1, e2, Power[x, n]]], FracPart[q]], -1]], Int[Times[Power[Times[f, x], m], Power[Plus[Times[d1, d2], Times[e1, e2, Power[x, n]]], q], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, d1, e1, d2, e2, f, n, p, q], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[non2, Times[-1, n, Power[2, -1]]]], ZeroQ[Plus[Times[d2, e1], Times[d1, e2]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Power[Times[Plus[a, b, c], Power[x, n]], p], x], And[FreeQ[List[a, b, c, n, p], x], ZeroQ[Plus[n, Times[-1, q]]], ZeroQ[Plus[r, Times[-1, n]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[x, Times[p, q]], Power[Plus[a, Times[b, Power[x, Plus[n, Times[-1, q]]]], Times[c, Power[x, Times[2, Plus[n, Times[-1, q]]]]]], p]], x], And[FreeQ[List[a, b, c, n, q], x], ZeroQ[Plus[r, Times[-1, Plus[Times[2, n], Times[-1, q]]]]], PosQ[Plus[n, Times[-1, q]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Rational[1, 2]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]]], Power[Times[Power[x, Times[q, Power[2, -1]]], Sqrt[Plus[a, Times[b, Power[x, Plus[n, Times[-1, q]]]], Times[c, Power[x, Times[2, Plus[n, Times[-1, q]]]]]]]], -1]], Int[Times[Power[x, Times[q, Power[2, -1]]], Sqrt[Plus[a, Times[b, Power[x, Plus[n, Times[-1, q]]]], Times[c, Power[x, Times[2, Plus[n, Times[-1, q]]]]]]]], x], x], And[FreeQ[List[a, b, c, n, q], x], ZeroQ[Plus[r, Times[-1, Plus[Times[2, n], Times[-1, q]]]]], PosQ[Plus[n, Times[-1, q]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[x, Times[q, Power[2, -1]]], Sqrt[Plus[a, Times[b, Power[x, Plus[n, Times[-1, q]]]], Times[c, Power[x, Times[2, Plus[n, Times[-1, q]]]]]]], Power[Sqrt[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]]], -1]], Int[Times[1, Power[Times[Power[x, Times[q, Power[2, -1]]], Sqrt[Plus[a, Times[b, Power[x, Plus[n, Times[-1, q]]]], Times[c, Power[x, Times[2, Plus[n, Times[-1, q]]]]]]]], -1]], x], x], And[FreeQ[List[a, b, c, n, q], x], ZeroQ[Plus[r, Times[-1, Plus[Times[2, n], Times[-1, q]]]]], PosQ[Plus[n, Times[-1, q]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[x, Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], p], Power[Plus[Times[p, Plus[Times[2, n], Times[-1, q]]], 1], -1]], x], Dist[Times[Plus[n, Times[-1, q]], p, Power[Plus[Times[p, Plus[Times[2, n], Times[-1, q]]], 1], -1]], Int[Times[Power[x, q], Plus[Times[2, a], Times[b, Power[x, Plus[n, Times[-1, q]]]]], Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], Plus[p, -1]]], x], x]], And[FreeQ[List[a, b, c, n, q], x], ZeroQ[Plus[r, Times[-1, Plus[Times[2, n], Times[-1, q]]]]], PosQ[Plus[n, Times[-1, q]]], Not[IntegerQ[p]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], RationalQ[p], Greater[p, 0], NonzeroQ[Plus[Times[p, Plus[Times[2, n], Times[-1, q]]], 1]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[x, Plus[Times[-1, q], 1]], Plus[Power[b, 2], Times[-1, 2, a, c], Times[b, c, Power[x, Plus[n, Times[-1, q]]]]], Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], Plus[p, 1]], Power[Times[a, Plus[n, Times[-1, q]], Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], x]], Dist[Times[1, Power[Times[a, Plus[n, Times[-1, q]], Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], Int[Times[Power[x, Times[-1, q]], Plus[Times[Plus[Times[p, q], 1], Plus[Power[b, 2], Times[-1, 2, a, c]]], Times[Plus[n, Times[-1, q]], Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], Times[b, c, Plus[Times[p, q], Times[Plus[n, Times[-1, q]], Plus[Times[2, p], 3]], 1], Power[x, Plus[n, Times[-1, q]]]]], Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], Plus[p, 1]]], x], x]], And[FreeQ[List[a, b, c, n, q], x], ZeroQ[Plus[r, Times[-1, Plus[Times[2, n], Times[-1, q]]]]], PosQ[Plus[n, Times[-1, q]]], Not[IntegerQ[p]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], RationalQ[p], Less[p, -1]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], p], Power[Times[Power[x, Times[p, q]], Power[Plus[a, Times[b, Power[x, Plus[n, Times[-1, q]]]], Times[c, Power[x, Times[2, Plus[n, Times[-1, q]]]]]], p]], -1]], Int[Times[Power[x, Times[p, q]], Power[Plus[a, Times[b, Power[x, Plus[n, Times[-1, q]]]], Times[c, Power[x, Times[2, Plus[n, Times[-1, q]]]]]], p]], x], x], And[FreeQ[List[a, b, c, n, p, q], x], ZeroQ[Plus[r, Times[-1, Plus[Times[2, n], Times[-1, q]]]]], PosQ[Plus[n, Times[-1, q]]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Power[Plus[Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]], Times[a, Power[x, q]]], p], x], And[FreeQ[List[a, b, c, n, p, q], x], ZeroQ[Plus[r, Times[-1, Plus[Times[2, n], Times[-1, q]]]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[u, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[u, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Coefficient[u, x, 1], -1]], Subst[Int[Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], p], x], x, u], x], And[FreeQ[List[a, b, c, n, p, q], x], ZeroQ[Plus[r, Times[-1, Plus[Times[2, n], Times[-1, q]]]]], LinearQ[u, x], NonzeroQ[Plus[u, Times[-1, x]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[x, m], Power[Times[Plus[a, b, c], Power[x, n]], p]], x], And[FreeQ[List[a, b, c, m, n, p], x], ZeroQ[Plus[q, Times[-1, n]]], ZeroQ[Plus[r, Times[-1, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[x, Plus[m, Times[p, q]]], Power[Plus[a, Times[b, Power[x, Plus[n, Times[-1, q]]]], Times[c, Power[x, Times[2, Plus[n, Times[-1, q]]]]]], p]], x], And[FreeQ[List[a, b, c, m, n, q], x], ZeroQ[Plus[r, Times[-1, Plus[Times[2, n], Times[-1, q]]]]], IntegerQ[p], PosQ[Plus[n, Times[-1, q]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[x, Times[q, Power[2, -1]]], Sqrt[Plus[a, Times[b, Power[x, Plus[n, Times[-1, q]]]], Times[c, Power[x, Times[2, Plus[n, Times[-1, q]]]]]]], Power[Sqrt[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]]], -1]], Int[Times[Power[x, Plus[m, Times[-1, q, Power[2, -1]]]], Power[Sqrt[Plus[a, Times[b, Power[x, Plus[n, Times[-1, q]]]], Times[c, Power[x, Times[2, Plus[n, Times[-1, q]]]]]]], -1]], x], x], And[FreeQ[List[a, b, c, m, n, q], x], ZeroQ[Plus[r, Times[-1, Plus[Times[2, n], Times[-1, q]]]]], PosQ[Plus[n, Times[-1, q]]], Or[And[ZeroQ[Plus[m, -1]], ZeroQ[Plus[n, -3]], ZeroQ[Plus[q, -2]]], And[Or[ZeroQ[Plus[m, Times[1, Power[2, -1]]]], ZeroQ[Plus[m, Times[-1, 3, Power[2, -1]]]], ZeroQ[Plus[m, Times[-1, 1, Power[2, -1]]]], ZeroQ[Plus[m, Times[-1, 5, Power[2, -1]]]]], ZeroQ[Plus[n, -3]], ZeroQ[Plus[q, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[-2, Power[x, Times[Plus[n, -1], Power[2, -1]]], Plus[b, Times[2, c, x]], Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Sqrt[Plus[Times[a, Power[x, Plus[n, -1]]], Times[b, Power[x, n]], Times[c, Power[x, Plus[n, 1]]]]]], -1]], x], And[FreeQ[List[a, b, c, n], x], ZeroQ[Plus[m, Times[-1, 3, Plus[n, -1], Power[2, -1]]]], ZeroQ[Plus[q, Times[-1, n], 1]], ZeroQ[Plus[r, Times[-1, n], -1]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[x, Times[Plus[n, -1], Power[2, -1]]], Plus[Times[4, a], Times[2, b, x]], Power[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Sqrt[Plus[Times[a, Power[x, Plus[n, -1]]], Times[b, Power[x, n]], Times[c, Power[x, Plus[n, 1]]]]]], -1]], x], And[FreeQ[List[a, b, c, n], x], ZeroQ[Plus[m, Times[-1, Plus[Times[3, n], -1], Power[2, -1]]]], ZeroQ[Plus[q, Times[-1, n], 1]], ZeroQ[Plus[r, Times[-1, n], -1]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[m, Times[-1, n]]], Power[Plus[Times[a, Power[x, Plus[n, -1]]], Times[b, Power[x, n]], Times[c, Power[x, Plus[n, 1]]]], Plus[p, 1]], Power[Times[2, c, Plus[p, 1]], -1]], x], Times[-1, Dist[Times[b, Power[Times[2, c], -1]], Int[Times[Power[x, Plus[m, -1]], Power[Plus[Times[a, Power[x, Plus[n, -1]]], Times[b, Power[x, n]], Times[c, Power[x, Plus[n, 1]]]], p]], x], x]]], And[FreeQ[List[a, b, c], x], ZeroQ[Plus[r, Times[-1, Plus[Times[2, n], Times[-1, q]]]]], PosQ[Plus[n, Times[-1, q]]], Not[IntegerQ[p]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m, p, q], Equal[Plus[m, Times[p, Plus[n, -1]], -1], 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[m, Times[-1, n], q, 1]], Plus[b, Times[2, c, Power[x, Plus[n, Times[-1, q]]]]], Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], p], Power[Times[2, c, Plus[n, Times[-1, q]], Plus[Times[2, p], 1]], -1]], x], Times[-1, Dist[Times[p, Plus[Power[b, 2], Times[-1, 4, a, c]], Power[Times[2, c, Plus[Times[2, p], 1]], -1]], Int[Times[Power[x, Plus[m, q]], Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], Plus[p, -1]]], x], x]]], And[FreeQ[List[a, b, c], x], ZeroQ[Plus[r, Times[-1, Plus[Times[2, n], Times[-1, q]]]]], PosQ[Plus[n, Times[-1, q]]], Not[IntegerQ[p]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m, p, q], Greater[p, 0], Equal[Plus[m, Times[p, q], 1], Plus[n, Times[-1, q]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[m, Times[-1, n], q, 1]], Plus[Times[b, Plus[n, Times[-1, q]], p], Times[c, Plus[m, Times[p, q], Times[Plus[n, Times[-1, q]], Plus[Times[2, p], -1]], 1], Power[x, Plus[n, Times[-1, q]]]]], Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], p], Power[Times[c, Plus[m, Times[p, Plus[Times[2, n], Times[-1, q]]], 1], Plus[m, Times[p, q], Times[Plus[n, Times[-1, q]], Plus[Times[2, p], -1]], 1]], -1]], x], Dist[Times[Plus[n, Times[-1, q]], p, Power[Times[c, Plus[m, Times[p, Plus[Times[2, n], Times[-1, q]]], 1], Plus[m, Times[p, q], Times[Plus[n, Times[-1, q]], Plus[Times[2, p], -1]], 1]], -1]], Int[Times[Power[x, Plus[m, Times[-1, Plus[n, Times[-1, 2, q]]]]], Simp[Plus[Times[-1, a, b, Plus[m, Times[p, q], Times[-1, n], q, 1]], Times[Plus[Times[2, a, c, Plus[m, Times[p, q], Times[Plus[n, Times[-1, q]], Plus[Times[2, p], -1]], 1]], Times[-1, Power[b, 2], Plus[m, Times[p, q], Times[Plus[n, Times[-1, q]], Plus[p, -1]], 1]]], Power[x, Plus[n, Times[-1, q]]]]], x], Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], Plus[p, -1]]], x], x]], And[FreeQ[List[a, b, c], x], ZeroQ[Plus[r, Times[-1, Plus[Times[2, n], Times[-1, q]]]]], PosQ[Plus[n, Times[-1, q]]], Not[IntegerQ[p]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m, p, q], Greater[p, 0], Greater[Plus[m, Times[p, q], 1], Plus[n, Times[-1, q]]], Unequal[Plus[m, Times[p, Plus[Times[2, n], Times[-1, q]]], 1], 0], Unequal[Plus[m, Times[p, q], Times[Plus[n, Times[-1, q]], Plus[Times[2, p], -1]], 1], 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[m, 1]], Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], p], Power[Plus[m, Times[p, q], 1], -1]], x], Times[-1, Dist[Times[Plus[n, Times[-1, q]], p, Power[Plus[m, Times[p, q], 1], -1]], Int[Times[Power[x, Plus[m, n]], Plus[b, Times[2, c, Power[x, Plus[n, Times[-1, q]]]]], Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], Plus[p, -1]]], x], x]]], And[FreeQ[List[a, b, c], x], ZeroQ[Plus[r, Times[-1, Plus[Times[2, n], Times[-1, q]]]]], PosQ[Plus[n, Times[-1, q]]], Not[IntegerQ[p]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m, p, q], Greater[p, 0], LessEqual[Plus[m, Times[p, q], 1], Plus[Times[-1, Plus[n, Times[-1, q]]], 1]], NonzeroQ[Plus[m, Times[p, q], 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[m, 1]], Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], p], Power[Plus[m, Times[p, Plus[Times[2, n], Times[-1, q]]], 1], -1]], x], Dist[Times[Plus[n, Times[-1, q]], p, Power[Plus[m, Times[p, Plus[Times[2, n], Times[-1, q]]], 1], -1]], Int[Times[Power[x, Plus[m, q]], Plus[Times[2, a], Times[b, Power[x, Plus[n, Times[-1, q]]]]], Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], Plus[p, -1]]], x], x]], And[FreeQ[List[a, b, c], x], ZeroQ[Plus[r, Times[-1, Plus[Times[2, n], Times[-1, q]]]]], PosQ[Plus[n, Times[-1, q]]], Not[IntegerQ[p]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m, p, q], Greater[p, 0], Greater[Plus[m, Times[p, q], 1], Times[-1, Plus[n, Times[-1, q]]]], Unequal[Plus[m, Times[p, Plus[Times[2, n], Times[-1, q]]], 1], 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[x, Plus[m, Times[-1, q], 1]], Plus[Power[b, 2], Times[-1, 2, a, c], Times[b, c, Power[x, Plus[n, Times[-1, q]]]]], Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], Plus[p, 1]], Power[Times[a, Plus[n, Times[-1, q]], Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], x]], Dist[Times[Plus[Times[2, a, c], Times[-1, Power[b, 2], Plus[p, 2]]], Power[Times[a, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], Int[Times[Power[x, Plus[m, Times[-1, q]]], Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], Plus[p, 1]]], x], x]], And[FreeQ[List[a, b, c], x], ZeroQ[Plus[r, Times[-1, Plus[Times[2, n], Times[-1, q]]]]], PosQ[Plus[n, Times[-1, q]]], Not[IntegerQ[p]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m, p, q], Less[p, -1], Equal[Plus[m, Times[p, q], 1], Times[-1, Plus[n, Times[-1, q]], Plus[Times[2, p], 3]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[x, Plus[m, Times[-1, 2, n], q, 1]], Plus[Times[2, a], Times[b, Power[x, Plus[n, Times[-1, q]]]]], Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], Plus[p, 1]], Power[Times[Plus[n, Times[-1, q]], Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], x]], Dist[Times[1, Power[Times[Plus[n, Times[-1, q]], Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], Int[Times[Power[x, Plus[m, Times[-1, 2, n], q]], Plus[Times[2, a, Plus[m, Times[p, q], Times[-1, 2, Plus[n, Times[-1, q]]], 1]], Times[b, Plus[m, Times[p, q], Times[Plus[n, Times[-1, q]], Plus[Times[2, p], 1]], 1], Power[x, Plus[n, Times[-1, q]]]]], Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], Plus[p, 1]]], x], x]], And[FreeQ[List[a, b, c], x], ZeroQ[Plus[r, Times[-1, Plus[Times[2, n], Times[-1, q]]]]], PosQ[Plus[n, Times[-1, q]]], Not[IntegerQ[p]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m, p, q], Less[p, -1], Greater[Plus[m, Times[p, q], 1], Times[2, Plus[n, Times[-1, q]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[x, Plus[m, Times[-1, q], 1]], Plus[Power[b, 2], Times[-1, 2, a, c], Times[b, c, Power[x, Plus[n, Times[-1, q]]]]], Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], Plus[p, 1]], Power[Times[a, Plus[n, Times[-1, q]], Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], x]], Dist[Times[1, Power[Times[a, Plus[n, Times[-1, q]], Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], Int[Times[Power[x, Plus[m, Times[-1, q]]], Plus[Times[Power[b, 2], Plus[m, Times[p, q], Times[Plus[n, Times[-1, q]], Plus[p, 1]], 1]], Times[-1, 2, a, c, Plus[m, Times[p, q], Times[2, Plus[n, Times[-1, q]], Plus[p, 1]], 1]], Times[b, c, Plus[m, Times[p, q], Times[Plus[n, Times[-1, q]], Plus[Times[2, p], 3]], 1], Power[x, Plus[n, Times[-1, q]]]]], Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], Plus[p, 1]]], x], x]], And[FreeQ[List[a, b, c], x], ZeroQ[Plus[r, Times[-1, Plus[Times[2, n], Times[-1, q]]]]], PosQ[Plus[n, Times[-1, q]]], Not[IntegerQ[p]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m, p, q], Less[p, -1], Less[Plus[m, Times[p, q], 1], Plus[n, Times[-1, q]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[m, Times[-1, n], 1]], Plus[b, Times[2, c, Power[x, Plus[n, Times[-1, q]]]]], Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], Plus[p, 1]], Power[Times[Plus[n, Times[-1, q]], Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], x], Times[-1, Dist[Times[1, Power[Times[Plus[n, Times[-1, q]], Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], Int[Times[Power[x, Plus[m, Times[-1, n]]], Plus[Times[b, Plus[m, Times[p, q], Times[-1, n], q, 1]], Times[2, c, Plus[m, Times[p, q], Times[2, Plus[n, Times[-1, q]], Plus[p, 1]], 1], Power[x, Plus[n, Times[-1, q]]]]], Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], Plus[p, 1]]], x], x]]], And[FreeQ[List[a, b, c], x], ZeroQ[Plus[r, Times[-1, Plus[Times[2, n], Times[-1, q]]]]], PosQ[Plus[n, Times[-1, q]]], Not[IntegerQ[p]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m, p, q], Less[p, -1], Less[Plus[n, Times[-1, q]], Plus[m, Times[p, q], 1], Times[2, Plus[n, Times[-1, q]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[m, Times[-1, 2, n], q, 1]], Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], Plus[p, 1]], Power[Times[2, c, Plus[n, Times[-1, q]], Plus[p, 1]], -1]], x], Times[-1, Dist[Times[b, Power[Times[2, c], -1]], Int[Times[Power[x, Plus[m, Times[-1, n], q]], Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], p]], x], x]]], And[FreeQ[List[a, b, c], x], ZeroQ[Plus[r, Times[-1, Plus[Times[2, n], Times[-1, q]]]]], PosQ[Plus[n, Times[-1, q]]], Not[IntegerQ[p]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m, p, q], Inequality[-1, LessEqual, p, Less, 0], Equal[Plus[m, Times[p, q], 1], Times[2, Plus[n, Times[-1, q]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[x, Plus[m, Times[-1, q], 1]], Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], Plus[p, 1]], Power[Times[2, a, Plus[n, Times[-1, q]], Plus[p, 1]], -1]], x]], Times[-1, Dist[Times[b, Power[Times[2, a], -1]], Int[Times[Power[x, Plus[m, n, Times[-1, q]]], Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], p]], x], x]]], And[FreeQ[List[a, b, c], x], ZeroQ[Plus[r, Times[-1, Plus[Times[2, n], Times[-1, q]]]]], PosQ[Plus[n, Times[-1, q]]], Not[IntegerQ[p]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m, p, q], Inequality[-1, LessEqual, p, Less, 0], Equal[Plus[m, Times[p, q], 1], Times[-2, Plus[n, Times[-1, q]], Plus[p, 1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[m, Times[-1, 2, n], q, 1]], Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], Plus[p, 1]], Power[Times[c, Plus[m, Times[p, q], Times[2, Plus[n, Times[-1, q]], p], 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[c, Plus[m, Times[p, q], Times[2, Plus[n, Times[-1, q]], p], 1]], -1]], Int[Times[Power[x, Plus[m, Times[-1, 2, Plus[n, Times[-1, q]]]]], Plus[Times[a, Plus[m, Times[p, q], Times[-1, 2, Plus[n, Times[-1, q]]], 1]], Times[b, Plus[m, Times[p, q], Times[Plus[n, Times[-1, q]], Plus[p, -1]], 1], Power[x, Plus[n, Times[-1, q]]]]], Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], p]], x], x]]], And[FreeQ[List[a, b, c], x], ZeroQ[Plus[r, Times[-1, Plus[Times[2, n], Times[-1, q]]]]], PosQ[Plus[n, Times[-1, q]]], Not[IntegerQ[p]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m, p, q], Inequality[-1, LessEqual, p, Less, 0], Greater[Plus[m, Times[p, q], 1], Times[2, Plus[n, Times[-1, q]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[m, Times[-1, q], 1]], Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], Plus[p, 1]], Power[Times[a, Plus[m, Times[p, q], 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[a, Plus[m, Times[p, q], 1]], -1]], Int[Times[Power[x, Plus[m, n, Times[-1, q]]], Plus[Times[b, Plus[m, Times[p, q], Times[Plus[n, Times[-1, q]], Plus[p, 1]], 1]], Times[c, Plus[m, Times[p, q], Times[2, Plus[n, Times[-1, q]], Plus[p, 1]], 1], Power[x, Plus[n, Times[-1, q]]]]], Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], p]], x], x]]], And[FreeQ[List[a, b, c], x], ZeroQ[Plus[r, Times[-1, Plus[Times[2, n], Times[-1, q]]]]], PosQ[Plus[n, Times[-1, q]]], Not[IntegerQ[p]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[m, p, q], Inequality[-1, LessEqual, p, Less, 0], Less[Plus[m, Times[p, q], 1], 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], p], Power[Times[Power[x, Times[p, q]], Power[Plus[a, Times[b, Power[x, Plus[n, Times[-1, q]]]], Times[c, Power[x, Times[2, Plus[n, Times[-1, q]]]]]], p]], -1]], Int[Times[Power[x, Plus[m, Times[p, q]]], Power[Plus[a, Times[b, Power[x, Plus[n, Times[-1, q]]]], Times[c, Power[x, Times[2, Plus[n, Times[-1, q]]]]]], p]], x], x], And[FreeQ[List[a, b, c, m, n, p, q], x], ZeroQ[Plus[r, Times[-1, Plus[Times[2, n], Times[-1, q]]]]], Not[IntegerQ[p]], PosQ[Plus[n, Times[-1, q]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[u, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[u, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[u, Blank[]], Optional[Pattern[r, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Coefficient[u, x, 1], -1]], Subst[Int[Times[Power[x, m], Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], p]], x], x, u], x], And[FreeQ[List[a, b, c, m, n, p, q], x], ZeroQ[Plus[r, Times[-1, Plus[Times[2, n], Times[-1, q]]]]], LinearQ[u, x], NonzeroQ[Plus[u, Times[-1, x]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[x, Times[p, q]], Plus[A, Times[B, Power[x, Plus[n, Times[-1, q]]]]], Power[Plus[a, Times[b, Power[x, Plus[n, Times[-1, q]]]], Times[c, Power[x, Times[2, Plus[n, Times[-1, q]]]]]], p]], x], And[FreeQ[List[a, b, c, A, B, n, q], x], ZeroQ[Plus[r, Times[-1, Plus[n, Times[-1, q]]]]], ZeroQ[Plus[j, Times[-1, Plus[Times[2, n], Times[-1, q]]]]], IntegerQ[p], PosQ[Plus[n, Times[-1, q]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[x, Times[q, Power[2, -1]]], Sqrt[Plus[a, Times[b, Power[x, Plus[n, Times[-1, q]]]], Times[c, Power[x, Times[2, Plus[n, Times[-1, q]]]]]]], Power[Sqrt[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]]], -1]], Int[Times[Plus[A, Times[B, Power[x, Plus[n, Times[-1, q]]]]], Power[Times[Power[x, Times[q, Power[2, -1]]], Sqrt[Plus[a, Times[b, Power[x, Plus[n, Times[-1, q]]]], Times[c, Power[x, Times[2, Plus[n, Times[-1, q]]]]]]]], -1]], x], x], And[FreeQ[List[a, b, c, A, B, n, q], x], ZeroQ[Plus[j, Times[-1, Plus[n, Times[-1, q]]]]], ZeroQ[Plus[r, Times[-1, Plus[Times[2, n], Times[-1, q]]]]], PosQ[Plus[n, Times[-1, q]]], ZeroQ[Plus[n, -3]], ZeroQ[Plus[q, -2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]], Pattern[p, Blank[]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[x, Plus[Times[b, B, Plus[n, Times[-1, q]], p], Times[A, c, Plus[Times[p, q], Times[Plus[n, Times[-1, q]], Plus[Times[2, p], 1]], 1]], Times[B, c, Plus[Times[p, Plus[Times[2, n], Times[-1, q]]], 1], Power[x, Plus[n, Times[-1, q]]]]], Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], p], Power[Times[c, Plus[Times[p, Plus[Times[2, n], Times[-1, q]]], 1], Plus[Times[p, q], Times[Plus[n, Times[-1, q]], Plus[Times[2, p], 1]], 1]], -1]], x], Dist[Times[Plus[n, Times[-1, q]], p, Power[Times[c, Plus[Times[p, Plus[Times[2, n], Times[-1, q]]], 1], Plus[Times[p, q], Times[Plus[n, Times[-1, q]], Plus[Times[2, p], 1]], 1]], -1]], Int[Times[Power[x, q], Plus[Times[2, a, A, c, Plus[Times[p, q], Times[Plus[n, Times[-1, q]], Plus[Times[2, p], 1]], 1]], Times[-1, a, b, B, Plus[Times[p, q], 1]], Times[Plus[Times[2, a, B, c, Plus[Times[p, Plus[Times[2, n], Times[-1, q]]], 1]], Times[A, b, c, Plus[Times[p, q], Times[Plus[n, Times[-1, q]], Plus[Times[2, p], 1]], 1]], Times[-1, Power[b, 2], B, Plus[Times[p, q], Times[Plus[n, Times[-1, q]], p], 1]]], Power[x, Plus[n, Times[-1, q]]]]], Power[Plus[Times[a, Power[x, q]], Times[b, Power[x, n]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], Plus[p, -1]]], x], x]], And[FreeQ[List[a, b, c, A, B, n, q], x], ZeroQ[Plus[r, Times[-1, Plus[n, Times[-1, q]]]]], ZeroQ[Plus[j, Times[-1, Plus[Times[2, n], Times[-1, q]]]]], Not[IntegerQ[p]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], RationalQ[p], Greater[p, 0], NonzeroQ[Plus[Times[p, Plus[Times[2, n], Times[-1, q]]], 1]], NonzeroQ[Plus[Times[p, q], Times[Plus[n, Times[-1, q]], Plus[Times[2, p], 1]], 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]]], Pattern[p, Blank[]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[n, Plus[q, r]]], Condition[Plus[Simp[Times[x, Plus[Times[A, Plus[Times[p, q], Times[Plus[n, Times[-1, q]], Plus[Times[2, p], 1]], 1]], Times[B, Plus[Times[p, Plus[Times[2, n], Times[-1, q]]], 1], Power[x, Plus[n, Times[-1, q]]]]], Power[Plus[Times[a, Power[x, q]], Times[c, Power[x, Plus[Times[2, n], Times[-1, q]]]]], p], Power[Times[Plus[Times[p, Plus[Times[2, n], Times[-1, q]]], 1], Plus[Times[p, q], Times[Plus[n, Times[-1, q]], Plus[Times[2, p], 1]], 1]], -1]], x], Dist[Times[Plus[n, Times[-1, q]], p, Power[Times[Plus[Times[p, Plus[Times[2, n], Times[-1, q]]], 1], Plus[Times[p, q], Times[Plus[n, Times[-1, q]], Plus[Times[2, p], 1]], 1]], -1]], Int[Times[Power[x, q], Plus[Times[2, a, A, Plus[Times[p, q], Times[Plus[n, Times[-1, q]], Plus[Times[2, p], 1]], 1]], Times[Times[2, a, B, Plus[Times[p, Plus[Times[2, 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p], 1], Plus[Pq, Times[-1, Pqq, Power[x, q]]]], Times[-1, a, Pqq, Plus[m, q, Times[-1, n], 1], Power[x, Plus[q, Times[-1, n]]]]], x], Power[Plus[a, Times[b, Power[x, n]]], p]], x], x], Simp[Times[Pqq, Power[Times[c, x], Plus[m, q, Times[-1, n], 1]], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Power[Times[b, Power[c, Plus[q, Times[-1, n], 1]], Plus[m, q, Times[n, p], 1]], -1]], x]]], And[NonzeroQ[Plus[m, q, Times[n, p], 1]], GreaterEqual[Plus[q, Times[-1, n]], 0], Or[IntegerQ[Times[2, p]], IntegerQ[Plus[p, Times[Plus[q, 1], Power[Times[2, n], -1]]]]]]]], And[FreeQ[List[a, b, c, m, p], x], PolyQ[Pq, x], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Expon[Pq, x]]], Condition[With[List[Set[Pqq, Coeff[Pq, x, q]]], Plus[Dist[Times[1, Power[Times[b, Plus[q, 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Times[1, Power[Power[Times[c, x], Times[1, Power[g, -1]]], -1]]], x]]], And[FreeQ[List[a, b, c, p], x], PolyQ[Pq, x], NegativeIntegerQ[n], FractionQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Expon[Pq, x]]], Times[-1, Dist[Times[Power[Times[c, x], m], Power[Power[x, -1], m]], Subst[Int[Times[ExpandToSum[Times[Power[x, q], ReplaceAll[Pq, Rule[x, Power[x, -1]]]], x], Power[Plus[a, Times[b, Power[x, Times[-1, n]]]], p], Power[Power[x, Plus[m, q, 2]], -1]], x], x, Times[1, Power[x, -1]]], x]]], And[FreeQ[List[a, b, c, m, p], x], PolyQ[Pq, x], NegativeIntegerQ[n], Not[RationalQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, 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FractionQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[c, IntPart[m]], Power[Times[c, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Pq, Power[Plus[a, Times[b, Power[x, n]]], p]], x], x], And[FreeQ[List[a, b, c, m, p], x], PolyQ[Pq, x], FractionQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Plus[m, 1], -1]], Subst[Int[Times[ReplaceAll[SubstFor[Power[x, n], Pq, x], Rule[x, Power[x, Simplify[Times[n, Power[Plus[m, 1], -1]]]]]], Power[Plus[a, Times[b, Power[x, Simplify[Times[n, Power[Plus[m, 1], -1]]]]]], p]], x], x, Power[x, Plus[m, 1]]], x], And[FreeQ[List[a, b, m, n, p], x], PolyQ[Pq, Power[x, n]], IntegerQ[Simplify[Times[n, Power[Plus[m, 1], -1]]]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[c, IntPart[m]], Power[Times[c, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Pq, Power[Plus[a, Times[b, Power[x, n]]], p]], x], x], And[FreeQ[List[a, b, c, m, n, p], x], PolyQ[Pq, Power[x, n]], IntegerQ[Simplify[Times[n, Power[Plus[m, 1], -1]]]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[A, Int[Power[Plus[a, Times[b, Power[x, n]]], p], x], x], Dist[B, Int[Times[Power[x, m], Power[Plus[a, Times[b, Power[x, n]]], p]], x], x]], And[FreeQ[List[a, b, A, B, m, n, p], x], ZeroQ[Plus[m, Times[-1, n], 1]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Times[c, x], m], Pq, Power[Plus[a, Times[b, Power[x, n]]], p]], x], x], And[FreeQ[List[a, b, c, m, n, p], x], Or[PolyQ[Pq, x], PolyQ[Pq, Power[x, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Pq, Power[Plus[a, Times[b, Power[x, n]]], p]], x], x], And[FreeQ[List[a, b, n, p], x], Or[PolyQ[Pq, x], PolyQ[Pq, Power[x, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[u, m], Power[Times[Coeff[v, x, 1], Power[v, m]], -1]], Subst[Int[Times[Power[x, m], SubstFor[v, Pq, x], Power[Plus[a, Times[b, Power[x, n]]], p]], x], x, v], x], And[FreeQ[List[a, b, m, n, p], x], LinearPairQ[u, v, x], PolyQ[Pq, Power[v, n]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Coeff[v, x, 1], -1]], Subst[Int[Times[SubstFor[v, Pq, x], Power[Plus[a, Times[b, Power[x, n]]], p]], x], x, v], x], And[FreeQ[List[a, b, n, p], x], LinearQ[v, x], PolyQ[Pq, Power[v, n]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], 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Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a1, Times[b1, Power[x, n]]], FracPart[p]], Power[Plus[a2, Times[b2, Power[x, n]]], FracPart[p]], Power[Power[Plus[Times[a1, a2], Times[b1, b2, Power[x, Times[2, n]]]], FracPart[p]], -1]], Int[Times[Power[Times[c, x], m], Pq, Power[Plus[Times[a1, a2], Times[b1, b2, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a1, b1, a2, b2, c, m, n, p], x], PolyQ[Pq, x], ZeroQ[Plus[Times[a2, b1], Times[a1, b2]]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a1, Blank[]], Times[Optional[Pattern[b1, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a2, Blank[]], Times[Optional[Pattern[b2, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a1, Times[b1, Power[x, n]]], FracPart[p]], Power[Plus[a2, Times[b2, Power[x, n]]], FracPart[p]], Power[Power[Plus[Times[a1, a2], Times[b1, b2, Power[x, Times[2, n]]]], FracPart[p]], -1]], Int[Times[Pq, Power[Plus[Times[a1, a2], Times[b1, b2, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a1, b1, a2, b2, n, p], x], PolyQ[Pq, x], ZeroQ[Plus[Times[a2, b1], Times[a1, b2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, 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Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[e, x, Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Power[Plus[c, Times[d, Power[x, n]]], Plus[p, 1]], Power[Times[a, c], -1]], x], And[FreeQ[List[a, b, c, d, e, g, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[Times[n, Plus[p, 1]], 1]], ZeroQ[Plus[Times[a, c, g], Times[-1, b, d, e, Plus[Times[2, n, Plus[p, 1]], 1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, 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RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[A, Int[Times[Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], q]], x], x], Dist[B, Int[Times[Power[x, m], Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], q]], x], x]], And[FreeQ[List[a, b, c, d, A, B, m, n, p, q], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[m, Times[-1, n], 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[Px, Blank[]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[k, Denominator[n]]], Dist[Times[k, Power[d, -1]], Subst[Int[SimplifyIntegrand[Times[Power[x, Plus[k, -1]], Power[ReplaceAll[Px, Rule[x, Plus[Times[Power[x, k], Power[d, -1]], Times[-1, c, Power[d, -1]]]]], q], Power[Plus[a, Times[b, Power[x, Times[k, n]]]], p]], x], x], x, Power[Plus[c, Times[d, x]], Times[1, Power[k, -1]]]], x]], And[FreeQ[List[a, b, c, d, p], x], PolynomialQ[Px, x], IntegerQ[q], RationalQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[n, -1]], Subst[Int[Times[SubstFor[Power[x, n], Pq, x], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x, Power[x, n]], x], And[FreeQ[List[a, b, c, m, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, Power[x, n]], ZeroQ[Simplify[Plus[m, Times[-1, n], 1]]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Times[d, x], m], Pq, Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, d, m, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, x], PositiveIntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Pq, Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, x], PositiveIntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[d, x, Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[a, -1]], x], And[FreeQ[List[a, b, c, d, e, f, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[Times[a, e], Times[-1, b, d, Plus[Times[n, Plus[p, 1]], 1]]]], ZeroQ[Plus[Times[a, f], Times[-1, c, d, Plus[Times[2, n, Plus[p, 1]], 1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[d, x, Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[a, -1]], x], And[FreeQ[List[a, b, c, d, f, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[Times[n, Plus[p, 1]], 1]], ZeroQ[Plus[Times[c, d], Times[a, f]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[d, Power[Times[g, x], Plus[m, 1]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[Times[a, g, Plus[m, 1]], -1]], x], And[FreeQ[List[a, b, c, d, e, f, g, m, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[Times[a, e, Plus[m, 1]], Times[-1, b, d, Plus[m, Times[n, Plus[p, 1]], 1]]]], ZeroQ[Plus[Times[a, f, Plus[m, 1]], Times[-1, c, d, Plus[m, Times[2, n, Plus[p, 1]], 1]]]], NonzeroQ[Plus[m, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[d, Power[Times[g, x], Plus[m, 1]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[Times[a, g, Plus[m, 1]], -1]], x], And[FreeQ[List[a, b, c, d, f, g, m, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[m, Times[n, Plus[p, 1]], 1]], ZeroQ[Plus[Times[c, d], Times[a, f]]], NonzeroQ[Plus[m, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], FracPart[p]], Power[Times[Power[Times[4, c], IntPart[p]], Power[Plus[b, Times[2, c, Power[x, n]]], Times[2, FracPart[p]]]], -1]], Int[Times[Power[Times[d, x], m], Pq, Power[Plus[b, Times[2, c, Power[x, n]]], Times[2, p]]], x], x], And[FreeQ[List[a, b, c, d, m, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], Not[IntegerQ[Times[2, p]]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], FracPart[p]], Power[Times[Power[Times[4, c], IntPart[p]], Power[Plus[b, Times[2, c, Power[x, n]]], Times[2, FracPart[p]]]], -1]], Int[Times[Pq, Power[Plus[b, Times[2, c, Power[x, n]]], Times[2, p]]], x], x], And[FreeQ[List[a, b, c, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], Not[IntegerQ[Times[2, p]]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[n, -1]], Subst[Int[Times[Power[x, Plus[Simplify[Times[Plus[m, 1], Power[n, -1]]], -1]], SubstFor[Power[x, n], Pq, x], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], x, Power[x, n]], x], And[FreeQ[List[a, b, c, m, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, Power[x, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegerQ[Simplify[Times[Plus[m, 1], Power[n, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Times[Pattern[d, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[d, x], m], Power[Power[x, m], -1]], Int[Times[Power[x, m], Pq, Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, d, m, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, Power[x, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegerQ[Simplify[Times[Plus[m, 1], Power[n, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[d, -1]], Int[Times[Power[Times[d, x], Plus[m, 1]], ExpandToSum[Times[Pq, Power[x, -1]], x], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, d, m, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, x], ZeroQ[Coeff[Pq, x, 0]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[x, ExpandToSum[Times[Pq, Power[x, -1]], x], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], And[FreeQ[List[a, b, c, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, x], ZeroQ[Coeff[Pq, x, 0]], SumQ[Pq]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n3, Blank[]]]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[x, Plus[Times[3, a, d], Times[-1, Plus[Times[3, b, d], Times[-1, a, e], Times[2, b, d, p]], Power[x, 2]]], Power[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]], Plus[p, 1]], Power[Times[3, Power[a, 2]], -1]], x], And[FreeQ[List[a, b, c, d, e, f, g, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[n3, Times[-1, 3, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[Power[a, 2], g, Plus[n, 1]], Times[-1, c, Plus[Times[n, Plus[Times[2, p], 3]], 1], Plus[Times[a, e], Times[-1, b, d, Plus[Times[n, Plus[p, 1]], 1]]]]]], ZeroQ[Plus[Times[Power[a, 2], f, Plus[n, 1]], Times[-1, a, c, d, Plus[n, 1], Plus[Times[2, n, Plus[p, 1]], 1]], Times[-1, b, Plus[Times[n, Plus[p, 2]], 1], Plus[Times[a, e], Times[-1, b, d, Plus[Times[n, Plus[p, 1]], 1]]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n3, Blank[]]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[x, Plus[Times[3, a, d], Times[-1, Plus[Times[3, b, d], Times[2, b, d, p]], Power[x, 2]]], Power[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]], Plus[p, 1]], Power[Times[3, Power[a, 2]], -1]], x], And[FreeQ[List[a, b, c, d, f, g, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[n3, Times[-1, 3, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[Power[a, 2], g, Plus[n, 1]], Times[c, b, d, Plus[Times[n, Plus[Times[2, p], 3]], 1], Plus[Times[n, Plus[p, 1]], 1]]]], ZeroQ[Plus[Times[Power[a, 2], f, Plus[n, 1]], Times[-1, a, c, d, Plus[n, 1], Plus[Times[2, n, Plus[p, 1]], 1]], Times[Power[b, 2], d, Plus[Times[n, Plus[p, 2]], 1], Plus[Times[n, Plus[p, 1]], 1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n3, Blank[]]]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[x, Plus[Times[3, a, d], Times[-1, Plus[Times[3, b, d], Times[-1, a, e], Times[2, b, d, p]], Power[x, 2]]], Power[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]], Plus[p, 1]], Power[Times[3, Power[a, 2]], -1]], x], And[FreeQ[List[a, b, c, d, e, g, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[n3, Times[-1, 3, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[Power[a, 2], g, Plus[n, 1]], Times[-1, c, Plus[Times[n, Plus[Times[2, p], 3]], 1], Plus[Times[a, e], Times[-1, b, d, Plus[Times[n, Plus[p, 1]], 1]]]]]], ZeroQ[Plus[Times[a, c, d, Plus[n, 1], Plus[Times[2, n, Plus[p, 1]], 1]], Times[b, Plus[Times[n, Plus[p, 2]], 1], Plus[Times[a, e], Times[-1, b, d, Plus[Times[n, Plus[p, 1]], 1]]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n3, Blank[]]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[x, Plus[Times[3, a, d], Times[-1, Plus[Times[3, b, d], Times[2, b, d, p]], Power[x, 2]]], Power[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]], Plus[p, 1]], Power[Times[3, Power[a, 2]], -1]], x], And[FreeQ[List[a, b, c, d, g, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[n3, Times[-1, 3, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[Power[a, 2], g, Plus[n, 1]], Times[c, b, d, Plus[Times[n, Plus[Times[2, p], 3]], 1], Plus[Times[n, Plus[p, 1]], 1]]]], ZeroQ[Plus[Times[a, c, d, Plus[n, 1], Plus[Times[2, n, Plus[p, 1]], 1]], Times[-1, Power[b, 2], d, Plus[Times[n, Plus[p, 2]], 1], Plus[Times[n, Plus[p, 1]], 1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Rational[-3, 2]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]], Times[Optional[Pattern[h, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[s, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Simp[Times[Plus[Times[2, c, Plus[Times[b, f], Times[-1, 2, a, g]]], Times[2, h, Plus[Power[b, 2], Times[-1, 4, a, c]], Power[x, Times[n, Power[2, -1]]]], Times[2, c, Plus[Times[2, c, f], Times[-1, b, g]], Power[x, n]]], Power[Times[c, n, Plus[Power[b, 2], Times[-1, 4, a, c]], Sqrt[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]]]], -1]], x]], And[FreeQ[List[a, b, c, e, f, g, h, m, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[q, Times[-1, n, Power[2, -1]]]], ZeroQ[Plus[r, Times[-1, 3, n, Power[2, -1]]]], ZeroQ[Plus[s, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[2, m], Times[-1, n], 2]], ZeroQ[Plus[Times[c, e], Times[a, h]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[d, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Rational[-3, 2]], Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[q, Blank[]]]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[r, Blank[]]]]], Times[Optional[Pattern[h, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[s, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[d, x], m], Power[Power[x, m], -1]], Int[Times[Power[x, m], Plus[e, Times[f, Power[x, Times[n, Power[2, -1]]]], Times[g, Power[x, Times[Times[3, n], Power[2, -1]]]], Times[h, Power[x, Times[2, n]]]], Power[Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Times[3, Power[2, -1]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, h, m, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[q, Times[-1, n, Power[2, -1]]]], ZeroQ[Plus[r, Times[-1, 3, n, Power[2, -1]]]], ZeroQ[Plus[s, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[2, m], Times[-1, n], 2]], ZeroQ[Plus[Times[c, e], Times[a, h]]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[q, Expon[Pq, x]], i], Condition[Plus[Times[-1, Simp[Times[x, Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[Times[a, n, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1], Sum[Plus[Times[Plus[Times[Plus[Power[b, 2], Times[-1, 2, a, c]], Coeff[Pq, x, i]], Times[-1, a, b, Coeff[Pq, x, Plus[n, i]]]], Power[x, i]], Times[c, Plus[Times[b, Coeff[Pq, x, i]], Times[-1, 2, a, Coeff[Pq, x, Plus[n, i]]]], Power[x, Plus[n, i]]]], List[i, 0, Plus[n, -1]]]], x]], Dist[Times[1, Power[Times[a, n, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], Int[Times[Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Sum[Plus[Times[Plus[Times[Plus[Times[Power[b, 2], Plus[Times[n, Plus[p, 1]], i, 1]], Times[-1, 2, a, c, Plus[Times[2, n, Plus[p, 1]], i, 1]]], Coeff[Pq, x, i]], Times[-1, a, b, Plus[i, 1], Coeff[Pq, x, Plus[n, i]]]], Power[x, i]], Times[c, Plus[Times[n, Plus[Times[2, p], 3]], i, 1], Plus[Times[b, Coeff[Pq, x, i]], Times[-1, 2, a, Coeff[Pq, x, Plus[n, i]]]], Power[x, Plus[n, i]]]], List[i, 0, Plus[n, -1]]]], x], x]], Less[q, Times[2, n]]]], And[FreeQ[List[a, b, c], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[p], Less[p, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-3, 2]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 3]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 4]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Simp[Times[Plus[Times[c, Plus[Times[b, e], Times[-1, 2, a, f]]], Times[g, Plus[Power[b, 2], Times[-1, 4, a, c]], x], Times[c, Plus[Times[2, c, e], Times[-1, b, f]], Power[x, 2]]], Power[Times[c, Plus[Power[b, 2], Times[-1, 4, a, c]], Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]]], -1]], x]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, d], Times[a, g]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-3, 2]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 3]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 4]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Plus[Times[2, a, c, f], Times[-1, g, Plus[Power[b, 2], Times[-1, 4, a, c]], x], Times[b, c, f, Power[x, 2]]], Power[Times[c, Plus[Power[b, 2], Times[-1, 4, a, c]], Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]]], -1]], x], And[FreeQ[List[a, b, c, d, f, g], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, d], Times[a, g]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-3, 2]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 4]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Simp[Times[Plus[Times[b, c, e], Times[Plus[Power[b, 2], Times[-1, 4, a, c]], g, x], Times[2, Power[c, 2], e, Power[x, 2]]], Power[Times[c, Plus[Power[b, 2], Times[-1, 4, a, c]], Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]]], -1]], x]], And[FreeQ[List[a, b, c, d, e, g], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, d], Times[a, g]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-3, 2]], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[h, Blank[]]], Power[Pattern[x, Blank[]], 4]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Plus[Times[2, Power[a, 2], c, f], Times[a, b, c, f, Power[x, 2]], Times[a, Plus[Power[b, 2], Times[-1, 4, a, c]], h, Power[x, 3]]], Power[Times[a, c, Plus[Power[b, 2], Times[-1, 4, a, c]], Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]]], -1]], x], And[FreeQ[List[a, b, c, e, f, g, h], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, e], Times[-1, 3, a, h]]], ZeroQ[Plus[Times[c, g], Times[-1, 2, b, h]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-3, 2]], Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[h, Blank[]]], Power[Pattern[x, Blank[]], 4]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[h, Power[x, 3], Power[Times[c, Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]]], -1]], x], And[FreeQ[List[a, b, c, e, g, h], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], ZeroQ[Plus[Times[c, e], Times[-1, 3, a, h]]], ZeroQ[Plus[Times[c, g], Times[-1, 2, b, h]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-3, 2]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 3]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 4]], Times[Optional[Pattern[h, Blank[]]], Power[Pattern[x, Blank[]], 6]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Plus[Times[2, Power[a, 2], c, f], Times[c, Plus[Power[b, 2], Times[-1, 4, a, c]], d, x], Times[a, b, c, f, Power[x, 2]], Times[a, Plus[Power[b, 2], Times[-1, 4, a, c]], h, Power[x, 3]]], Power[Times[a, c, Plus[Power[b, 2], Times[-1, 4, a, c]], Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]]], -1]], x], And[FreeQ[List[a, b, c, d, e, f, g, h], x], 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Times[-1, 2, Coeff[R, x, Plus[n, i]]]], Power[x, Plus[n, i, Times[-1, m]]]]], List[i, 0, Plus[n, -1]]]], x]], x], x], Times[-1, Simp[Times[x, Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[Times[Power[a, 2], n, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]], Power[Times[b, c], Plus[Floor[Times[Plus[q, -1], Power[n, -1]]], 1]]], -1], Sum[Plus[Times[Plus[Times[Plus[Power[b, 2], Times[-1, 2, a, c]], Coeff[R, x, i]], Times[-1, a, b, Coeff[R, x, Plus[n, i]]]], Power[x, i]], Times[c, Plus[Times[b, Coeff[R, x, i]], Times[-1, 2, a, Coeff[R, x, Plus[n, i]]]], Power[x, Plus[n, i]]]], List[i, 0, Plus[n, -1]]]], x]]]], GreaterEqual[q, Times[2, n]]]], And[FreeQ[List[a, b, c], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], RationalQ[p], Less[p, -1], NegativeIntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[g, GCD[Plus[m, 1], n]]], Condition[Dist[Times[1, Power[g, -1]], Subst[Int[Times[Power[x, Plus[Times[Plus[m, 1], Power[g, -1]], -1]], ReplaceAll[Pq, Rule[x, Power[x, Times[1, Power[g, -1]]]]], Power[Plus[a, Times[b, Power[x, Times[n, Power[g, -1]]]], Times[c, Power[x, Times[2, n, Power[g, -1]]]]], p]], x], x, Power[x, g]], x], Unequal[g, 1]]], And[FreeQ[List[a, b, c, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, Power[x, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, 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Equal[Plus[q, Times[2, p], 1], 0]]], And[FreeQ[List[a, b, c], x], PolyQ[Pq, x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NegativeIntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Expon[Pq, x]]], Condition[With[List[Set[Pqq, Coeff[Pq, x, q]]], Plus[Int[Times[ExpandToSum[Plus[Pq, Times[-1, Power[c, Plus[p, Times[1, Power[2, -1]]]], Pqq, Power[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, Times[1, Power[2, -1]]]], -1]]], x], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], Simp[Times[Power[c, p], Pqq, ArcTanh[Times[Plus[b, Times[2, c, x]], Power[Times[2, Rt[c, 2], Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]], -1]]]], x]]], Equal[Plus[q, Times[2, p], 1], 0]]], And[FreeQ[List[a, b, c], x], PolyQ[Pq, x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NegativeIntegerQ[Plus[p, Times[1, Power[2, -1]]]], PosQ[c]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Expon[Pq, x]]], Condition[With[List[Set[Pqq, Coeff[Pq, x, q]]], Plus[Int[Times[ExpandToSum[Plus[Pq, Times[-1, Power[Times[-1, c], Plus[p, Times[1, Power[2, -1]]]], Pqq, Power[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], Plus[p, Times[1, Power[2, -1]]]], -1]]], x], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], p]], x], Times[-1, Simp[Times[Power[Times[-1, c], p], Pqq, ArcTan[Times[Plus[b, Times[2, c, x]], Power[Times[2, Rt[Times[-1, c], 2], Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]], -1]]]], x]]]], Equal[Plus[q, Times[2, p], 1], 0]]], And[FreeQ[List[a, b, c], x], PolyQ[Pq, x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NegativeIntegerQ[Plus[p, Times[1, Power[2, -1]]]], NegQ[c]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Expon[Pq, x]]], Condition[With[List[Set[Pqq, Coeff[Pq, x, q]]], Plus[Int[Times[Power[Times[d, x], m], ExpandToSum[Plus[Pq, Times[-1, Pqq, Power[x, q]], Times[-1, Pqq, Plus[Times[a, Plus[m, q, Times[-1, 2, n], 1], Power[x, Plus[q, Times[-1, 2, n]]]], Times[b, Plus[m, q, Times[n, Plus[p, -1]], 1], Power[x, Plus[q, Times[-1, n]]]]], Power[Times[c, Plus[m, q, Times[2, n, p], 1]], -1]]], x], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], Simp[Times[Pqq, Power[Times[d, x], Plus[m, q, Times[-1, 2, n], 1]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[Times[c, Power[d, Plus[q, Times[-1, 2, n], 1]], Plus[m, q, Times[2, n, p], 1]], -1]], x]]], And[GreaterEqual[q, Times[2, n]], Unequal[Plus[m, q, Times[2, n, p], 1], 0], Or[IntegerQ[Times[2, p]], And[Equal[n, 1], IntegerQ[Times[4, p]]], IntegerQ[Plus[p, Times[Plus[q, 1], Power[Times[2, n], -1]]]]]]]], And[FreeQ[List[a, b, c, d, m, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, Power[x, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Expon[Pq, x]]], Condition[With[List[Set[Pqq, Coeff[Pq, x, q]]], Plus[Int[Times[ExpandToSum[Plus[Pq, Times[-1, Pqq, Power[x, q]], Times[-1, Pqq, Plus[Times[a, Plus[q, Times[-1, 2, n], 1], Power[x, Plus[q, Times[-1, 2, n]]]], Times[b, Plus[q, Times[n, Plus[p, -1]], 1], Power[x, Plus[q, Times[-1, n]]]]], Power[Times[c, Plus[q, Times[2, n, p], 1]], -1]]], x], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], Simp[Times[Pqq, Power[x, Plus[q, Times[-1, 2, n], 1]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[Times[c, Plus[q, Times[2, n, p], 1]], -1]], x]]], And[GreaterEqual[q, Times[2, n]], Unequal[Plus[q, Times[2, n, p], 1], 0], Or[IntegerQ[Times[2, p]], And[Equal[n, 1], IntegerQ[Times[4, p]]], IntegerQ[Plus[p, Times[Plus[q, 1], Power[Times[2, n], -1]]]]]]]], And[FreeQ[List[a, b, c, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, Power[x, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[q, Expon[Pq, x]], j, k], Int[Sum[Times[1, Power[Power[d, j], -1], Power[Times[d, x], Plus[m, j]], Sum[Times[Coeff[Pq, x, Plus[j, Times[k, n]]], Power[x, Times[k, n]]], List[k, 0, Plus[Times[Plus[q, Times[-1, j]], Power[n, -1]], 1]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], List[j, 0, Plus[n, -1]]], x]], And[FreeQ[List[a, b, c, d, m, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], Not[PolyQ[Pq, Power[x, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[q, Expon[Pq, x]], j, k], Int[Sum[Times[Power[x, j], Sum[Times[Coeff[Pq, x, Plus[j, Times[k, n]]], Power[x, Times[k, n]]], List[k, 0, Plus[Times[Plus[q, Times[-1, j]], Power[n, -1]], 1]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], List[j, 0, Plus[n, -1]]], x]], And[FreeQ[List[a, b, c, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n], Not[PolyQ[Pq, Power[x, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Int[RationalFunctionExpand[Times[Power[Times[d, x], m], Pq, Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, m], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Int[RationalFunctionExpand[Times[Pq, Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], -1]], x], x], And[FreeQ[List[a, b, c], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Expon[Pq, x]]], Times[-1, Subst[Int[Times[ExpandToSum[Times[Power[x, q], ReplaceAll[Pq, Rule[x, Power[x, -1]]]], x], Power[Plus[a, Times[b, Power[x, Times[-1, n]]], Times[c, Power[x, Times[-2, n]]]], p], Power[Power[x, Plus[m, q, 2]], -1]], x], x, Times[1, Power[x, -1]]]]], And[FreeQ[List[a, b, c, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NegativeIntegerQ[n], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[g, Denominator[m]], Set[q, Expon[Pq, x]]], Times[-1, Dist[Times[g, Power[d, -1]], Subst[Int[Times[ExpandToSum[Times[Power[x, Times[g, q]], ReplaceAll[Pq, Rule[x, Times[Power[d, -1], Power[x, Times[-1, g]]]]]], x], Power[Plus[a, Times[b, Power[d, Times[-1, n]], Power[x, Times[-1, g, n]]], Times[c, Power[d, Times[-2, n]], Power[x, Times[-2, g, n]]]], p], Power[Power[x, Plus[Times[g, Plus[m, q, 1]], 1]], -1]], x], x, Times[1, Power[Power[Times[d, x], Times[1, Power[g, -1]]], -1]]], x]]], And[FreeQ[List[a, b, c, d, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NegativeIntegerQ[n], FractionQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Expon[Pq, x]]], Times[-1, Dist[Times[Power[Times[d, x], m], Power[Power[x, -1], m]], Subst[Int[Times[ExpandToSum[Times[Power[x, q], ReplaceAll[Pq, Rule[x, Power[x, -1]]]], x], Power[Plus[a, Times[b, Power[x, Times[-1, n]]], Times[c, Power[x, Times[-2, n]]]], p], Power[Power[x, Plus[m, q, 2]], -1]], x], x, Times[1, Power[x, -1]]], x]]], And[FreeQ[List[a, b, c, d, m, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NegativeIntegerQ[n], Not[RationalQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[g, Denominator[n]]], Dist[g, Subst[Int[Times[Power[x, Plus[Times[g, Plus[m, 1]], -1]], ReplaceAll[Pq, Rule[x, Power[x, g]]], Power[Plus[a, Times[b, Power[x, Times[g, n]]], Times[c, Power[x, Times[2, g, n]]]], p]], x], x, Power[x, Times[1, Power[g, -1]]]], x]], And[FreeQ[List[a, b, c, m, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], FractionQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[g, Denominator[n]]], Dist[g, Subst[Int[Times[Power[x, Plus[g, -1]], ReplaceAll[Pq, Rule[x, Power[x, g]]], Power[Plus[a, Times[b, Power[x, Times[g, n]]], Times[c, Power[x, Times[2, g, n]]]], p]], x], x, Power[x, Times[1, Power[g, -1]]]], x]], And[FreeQ[List[a, b, c, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], FractionQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Times[Pattern[d, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[d, Plus[m, Times[-1, 1, Power[2, -1]]]], Sqrt[Times[d, x]], Power[Sqrt[x], -1]], Int[Times[Power[x, m], Pq, Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, d, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], FractionQ[n], PositiveIntegerQ[Plus[m, Times[1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Times[Pattern[d, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[d, Plus[m, Times[1, Power[2, -1]]]], Sqrt[x], Power[Sqrt[Times[d, x]], -1]], Int[Times[Power[x, m], Pq, Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, d, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], FractionQ[n], NegativeIntegerQ[Plus[m, Times[-1, 1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Times[Pattern[d, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[d, x], m], Power[Power[x, m], -1]], Int[Times[Power[x, m], Pq, Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, d, m, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], FractionQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Plus[m, 1], -1]], Subst[Int[Times[ReplaceAll[SubstFor[Power[x, n], Pq, x], Rule[x, Power[x, Simplify[Times[n, Power[Plus[m, 1], -1]]]]]], Power[Plus[a, Times[b, Power[x, Simplify[Times[n, Power[Plus[m, 1], -1]]]]], Times[c, Power[x, Simplify[Times[2, n, Power[Plus[m, 1], -1]]]]]], p]], x], x, Power[x, Plus[m, 1]]], x], And[FreeQ[List[a, b, c, m, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, Power[x, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegerQ[Simplify[Times[n, Power[Plus[m, 1], -1]]]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Times[Pattern[d, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[d, x], m], Power[Power[x, m], -1]], Int[Times[Power[x, m], Pq, Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, d, m, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, Power[x, n]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegerQ[Simplify[Times[n, Power[Plus[m, 1], -1]]]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Plus[Dist[Times[2, c, Power[q, -1]], Int[Times[Power[Times[d, x], m], Pq, Power[Plus[b, Times[-1, q], Times[2, c, Power[x, n]]], -1]], x], x], Times[-1, Dist[Times[2, c, Power[q, -1]], Int[Times[Power[Times[d, x], m], Pq, Power[Plus[b, q, Times[2, c, Power[x, n]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d, m, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Plus[Dist[Times[2, c, Power[q, -1]], Int[Times[Pq, Power[Plus[b, Times[-1, q], Times[2, c, Power[x, n]]], -1]], x], x], Times[-1, Dist[Times[2, c, Power[q, -1]], Int[Times[Pq, Power[Plus[b, q, Times[2, c, Power[x, n]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Times[d, x], m], Pq, Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, d, m, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, x], NegativeIntegerQ[Plus[p, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Pq, Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], PolyQ[Pq, x], NegativeIntegerQ[Plus[p, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Pq, Power[Times[d, x], m], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], And[FreeQ[List[a, b, c, d, m, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], Or[PolyQ[Pq, x], PolyQ[Pq, Power[x, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Pq, Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], And[FreeQ[List[a, b, c, n, p], x], ZeroQ[Plus[n2, 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Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Coefficient[v, x, 1], -1]], Subst[Int[Times[SubstFor[v, Pq, x], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x, v], x], And[FreeQ[List[a, b, c, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], LinearQ[v, x], PolyQ[Pq, Power[v, n]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], Plus[p, 1]], Power[Times[b, Plus[n, Times[-1, j]], Plus[p, 1], Power[x, Plus[n, -1]]], -1]], x], And[FreeQ[List[a, b, j, n, p], x], Not[IntegerQ[p]], NonzeroQ[Plus[n, Times[-1, j]]], ZeroQ[Plus[Times[j, p], Times[-1, n], j, 1]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], Plus[p, 1]], Power[Times[a, Plus[n, Times[-1, j]], Plus[p, 1], Power[x, Plus[j, -1]]], -1]], x]], Dist[Times[Plus[Times[n, p], n, Times[-1, j], 1], Power[Times[a, Plus[n, Times[-1, j]], Plus[p, 1]], -1]], Int[Times[Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], Plus[p, 1]], Power[Power[x, j], -1]], x], x]], And[FreeQ[List[a, b, j, n], x], Not[IntegerQ[p]], NonzeroQ[Plus[n, Times[-1, j]]], NegativeIntegerQ[Simplify[Times[Plus[Times[n, p], n, Times[-1, j], 1], Power[Plus[n, Times[-1, j]], -1]]]], RationalQ[p], Less[p, -1]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], Plus[p, 1]], Power[Times[a, Plus[Times[j, p], 1], Power[x, Plus[j, -1]]], -1]], x], Times[-1, Dist[Times[b, Plus[Times[n, p], n, Times[-1, j], 1], Power[Times[a, Plus[Times[j, p], 1]], -1]], Int[Times[Power[x, Plus[n, Times[-1, j]]], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], p]], x], x]]], And[FreeQ[List[a, b, j, n, p], x], Not[IntegerQ[p]], NonzeroQ[Plus[n, Times[-1, j]]], NegativeIntegerQ[Simplify[Times[Plus[Times[n, p], n, Times[-1, j], 1], Power[Plus[n, Times[-1, j]], -1]]]], NonzeroQ[Plus[Times[j, p], 1]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[x, Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], p], Power[Plus[Times[j, p], 1], -1]], x], Times[-1, Dist[Times[b, Plus[n, Times[-1, j]], p, Power[Plus[Times[j, p], 1], -1]], Int[Times[Power[x, n], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], Plus[p, -1]]], x], x]]], And[FreeQ[List[a, b], x], Not[IntegerQ[p]], RationalQ[j, n, p], Less[0, j, n], Greater[p, 0], Less[Plus[Times[j, p], 1], 0]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[x, Power[Plus[Times[a, Power[x, j]], Times[b, 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Times[-1, n]], -1]], Subst[Int[Times[1, Power[Plus[1, Times[-1, a, Power[x, 2]]], -1]], x], x, Times[x, Power[Sqrt[Plus[Times[a, Power[x, 2]], Times[b, Power[x, n]]]], -1]]], x], And[FreeQ[List[a, b, n], x], NonzeroQ[Plus[n, -2]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], Plus[p, 1]], Power[Times[a, Plus[n, Times[-1, j]], Plus[p, 1], Power[x, Plus[j, -1]]], -1]], x]], Dist[Times[Plus[Times[n, p], n, Times[-1, j], 1], Power[Times[a, Plus[n, Times[-1, j]], Plus[p, 1]], -1]], Int[Times[Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], Plus[p, 1]], Power[Power[x, j], -1]], x], x]], And[FreeQ[List[a, b, j, n], x], NegativeIntegerQ[Plus[p, Times[1, 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Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], FracPart[p]], Power[Times[Power[x, Times[j, FracPart[p]]], Power[Plus[a, Times[b, Power[x, Plus[n, Times[-1, j]]]]], FracPart[p]]], -1]], Int[Times[Power[x, Times[j, p]], Power[Plus[a, Times[b, Power[x, Plus[n, Times[-1, j]]]]], p]], x], x], And[FreeQ[List[a, b, j, n, p], x], Not[IntegerQ[p]], NonzeroQ[Plus[n, Times[-1, j]]], PosQ[Plus[n, Times[-1, j]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[u, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[u, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Coefficient[u, x, 1], -1]], Subst[Int[Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], p], x], x, u], x], And[FreeQ[List[a, b, j, n, p], x], LinearQ[u, x], NonzeroQ[Plus[u, Times[-1, x]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[n, -1]], Subst[Int[Power[Plus[Times[a, Power[x, Simplify[Times[j, Power[n, -1]]]]], Times[b, x]], p], x], x, Power[x, n]], x], And[FreeQ[List[a, b, j, m, n, p], x], Not[IntegerQ[p]], NonzeroQ[Plus[n, Times[-1, j]]], IntegerQ[Simplify[Times[j, Power[n, -1]]]], ZeroQ[Simplify[Plus[m, Times[-1, n], 1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Simp[Times[Power[c, Plus[j, -1]], Power[Times[c, x], Plus[m, Times[-1, j], 1]], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], Plus[p, 1]], Power[Times[a, Plus[n, Times[-1, j]], Plus[p, 1]], -1]], x]], And[FreeQ[List[a, b, c, j, m, n, p], x], Not[IntegerQ[p]], NonzeroQ[Plus[n, Times[-1, j]]], ZeroQ[Plus[m, Times[n, p], n, Times[-1, j], 1]], Or[IntegerQ[j], PositiveQ[c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[c, Plus[j, -1]], Power[Times[c, x], Plus[m, Times[-1, j], 1]], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], Plus[p, 1]], Power[Times[a, Plus[n, Times[-1, j]], Plus[p, 1]], -1]], x]], Dist[Times[Power[c, j], Plus[m, Times[n, p], n, Times[-1, j], 1], Power[Times[a, Plus[n, Times[-1, j]], Plus[p, 1]], -1]], Int[Times[Power[Times[c, x], Plus[m, Times[-1, j]]], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], Plus[p, 1]]], x], x]], And[FreeQ[List[a, b, c, j, m, n], x], Not[IntegerQ[p]], NonzeroQ[Plus[n, Times[-1, j]]], NegativeIntegerQ[Simplify[Times[Plus[m, Times[n, p], n, Times[-1, j], 1], Power[Plus[n, Times[-1, j]], -1]]]], RationalQ[p], Less[p, -1], Or[IntegerQ[j], PositiveQ[c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[c, Plus[j, -1]], Power[Times[c, x], Plus[m, Times[-1, j], 1]], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], Plus[p, 1]], Power[Times[a, Plus[m, Times[j, p], 1]], -1]], x], Times[-1, Dist[Times[b, Plus[m, Times[n, p], n, Times[-1, j], 1], Power[Times[a, Power[c, Plus[n, Times[-1, j]]], Plus[m, Times[j, p], 1]], -1]], Int[Times[Power[Times[c, x], Plus[m, n, Times[-1, j]]], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], p]], x], x]]], And[FreeQ[List[a, b, c, j, m, n, p], x], Not[IntegerQ[p]], NonzeroQ[Plus[n, Times[-1, j]]], NegativeIntegerQ[Simplify[Times[Plus[m, Times[n, p], n, Times[-1, j], 1], Power[Plus[n, Times[-1, j]], -1]]]], NonzeroQ[Plus[m, Times[j, p], 1]], Or[IntegersQ[j, n], PositiveQ[c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[c, IntPart[m]], Power[Times[c, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], p]], x], x], And[FreeQ[List[a, b, c, j, m, n, p], x], Not[IntegerQ[p]], NonzeroQ[Plus[n, Times[-1, j]]], NegativeIntegerQ[Simplify[Times[Plus[m, Times[n, p], n, Times[-1, j], 1], Power[Plus[n, Times[-1, j]], -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[n, -1]], Subst[Int[Times[Power[x, Plus[Simplify[Times[Plus[m, 1], Power[n, -1]]], -1]], Power[Plus[Times[a, Power[x, Simplify[Times[j, Power[n, -1]]]]], Times[b, x]], p]], x], x, Power[x, n]], x], And[FreeQ[List[a, b, j, m, n, p], x], Not[IntegerQ[p]], NonzeroQ[Plus[n, Times[-1, j]]], IntegerQ[Simplify[Times[j, Power[n, -1]]]], IntegerQ[Simplify[Times[Plus[m, 1], Power[n, -1]]]], NonzeroQ[Plus[Power[n, 2], -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[c, IntPart[m]], Power[Times[c, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Plus[Times[a, 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Power[x, j]], Times[b, Power[x, n]]], Plus[p, -1]]], x], x]]], And[FreeQ[List[a, b, c], x], Not[IntegerQ[p]], RationalQ[j, m, n, p], Less[0, j, n], Or[IntegersQ[j, n], PositiveQ[c]], Greater[p, 0], Less[Plus[m, Times[j, p], 1], 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[c, x], Plus[m, 1]], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], p], Power[Times[c, Plus[m, Times[n, p], 1]], -1]], x], Dist[Times[a, Plus[n, Times[-1, j]], p, Power[Times[Power[c, j], Plus[m, Times[n, p], 1]], -1]], Int[Times[Power[Times[c, x], Plus[m, j]], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], Plus[p, -1]]], x], x]], And[FreeQ[List[a, b, c, m], x], Not[IntegerQ[p]], RationalQ[j, n, p], Less[0, j, n], Or[IntegersQ[j, n], PositiveQ[c]], Greater[p, 0], NonzeroQ[Plus[m, Times[n, p], 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[c, Plus[n, -1]], Power[Times[c, x], Plus[m, Times[-1, n], 1]], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], Plus[p, 1]], Power[Times[b, Plus[n, Times[-1, j]], Plus[p, 1]], -1]], x], Times[-1, Dist[Times[Power[c, n], Plus[m, Times[j, p], Times[-1, n], j, 1], Power[Times[b, Plus[n, Times[-1, j]], Plus[p, 1]], -1]], Int[Times[Power[Times[c, x], Plus[m, Times[-1, n]]], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], Plus[p, 1]]], x], x]]], And[FreeQ[List[a, b, c], x], Not[IntegerQ[p]], RationalQ[j, m, n, p], Less[0, j, n], Or[IntegersQ[j, n], PositiveQ[c]], Less[p, -1], Greater[Plus[m, Times[j, p], 1], Plus[n, Times[-1, j]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[c, Plus[j, -1]], Power[Times[c, x], Plus[m, Times[-1, j], 1]], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], Plus[p, 1]], Power[Times[a, Plus[n, Times[-1, j]], Plus[p, 1]], -1]], x]], Dist[Times[Power[c, j], Plus[m, Times[n, p], n, Times[-1, j], 1], Power[Times[a, Plus[n, Times[-1, j]], Plus[p, 1]], -1]], Int[Times[Power[Times[c, x], Plus[m, Times[-1, j]]], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], Plus[p, 1]]], x], x]], And[FreeQ[List[a, b, c, m], x], Not[IntegerQ[p]], RationalQ[j, n, p], Less[0, j, n], Or[IntegersQ[j, n], PositiveQ[c]], Less[p, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[c, Plus[n, -1]], Power[Times[c, x], Plus[m, Times[-1, n], 1]], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], Plus[p, 1]], Power[Times[b, Plus[m, Times[n, p], 1]], -1]], x], Times[-1, Dist[Times[a, Power[c, Plus[n, Times[-1, j]]], Plus[m, Times[j, p], Times[-1, n], j, 1], Power[Times[b, Plus[m, Times[n, p], 1]], -1]], Int[Times[Power[Times[c, x], Plus[m, Times[-1, Plus[n, Times[-1, j]]]]], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], p]], x], x]]], And[FreeQ[List[a, b, c, m, p], x], Not[IntegerQ[p]], RationalQ[j, n], Less[0, j, n], Or[IntegersQ[j, n], PositiveQ[c]], PositiveQ[Plus[m, Times[j, p], 1, Times[-1, n], j]], NonzeroQ[Plus[m, Times[n, p], 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[c, Plus[j, -1]], Power[Times[c, x], Plus[m, Times[-1, j], 1]], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], Plus[p, 1]], Power[Times[a, Plus[m, Times[j, p], 1]], -1]], x], Times[-1, Dist[Times[b, Plus[m, Times[n, p], n, Times[-1, j], 1], Power[Times[a, Power[c, Plus[n, Times[-1, j]]], Plus[m, Times[j, p], 1]], -1]], Int[Times[Power[Times[c, x], Plus[m, n, Times[-1, j]]], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], p]], x], x]]], And[FreeQ[List[a, b, c, m, p], x], Not[IntegerQ[p]], RationalQ[j, n], Less[0, j, n], Or[IntegersQ[j, n], PositiveQ[c]], NegativeQ[Plus[m, Times[j, p], 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Plus[m, 1], -1]], Subst[Int[Power[Plus[Times[a, Power[x, Simplify[Times[j, Power[Plus[m, 1], -1]]]]], Times[b, Power[x, Simplify[Times[n, Power[Plus[m, 1], -1]]]]]], p], x], x, Power[x, Plus[m, 1]]], x], And[FreeQ[List[a, b, j, m, n, p], x], Not[IntegerQ[p]], NonzeroQ[Plus[n, Times[-1, j]]], IntegerQ[Simplify[Times[j, Power[n, -1]]]], NonzeroQ[Plus[m, 1]], IntegerQ[Simplify[Times[n, Power[Plus[m, 1], -1]]]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[c, IntPart[m]], Power[Times[c, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], p]], x], x], And[FreeQ[List[a, b, c, j, m, n, p], x], Not[IntegerQ[p]], NonzeroQ[Plus[n, Times[-1, j]]], IntegerQ[Simplify[Times[j, Power[n, -1]]]], NonzeroQ[Plus[m, 1]], IntegerQ[Simplify[Times[n, Power[Plus[m, 1], -1]]]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[c, x], Plus[m, 1]], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], p], Power[Times[c, p, Plus[n, Times[-1, j]]], -1]], x], Dist[Times[a, Power[Power[c, j], -1]], Int[Times[Power[Times[c, x], Plus[m, j]], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], Plus[p, -1]]], x], x]], And[FreeQ[List[a, b, c, j, m, n], x], PositiveIntegerQ[Plus[p, Times[1, Power[2, -1]]]], NonzeroQ[Plus[n, Times[-1, j]]], ZeroQ[Simplify[Plus[m, Times[j, p], 1]]], Or[IntegerQ[j], PositiveQ[c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[-2, Power[Plus[n, Times[-1, j]], -1]], Subst[Int[Times[1, Power[Plus[1, Times[-1, a, Power[x, 2]]], -1]], x], x, Times[Power[x, Times[j, Power[2, -1]]], Power[Sqrt[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]]], -1]]], x], And[FreeQ[List[a, b, j, n], x], ZeroQ[Plus[m, Times[-1, j, Power[2, -1]], 1]], NonzeroQ[Plus[n, Times[-1, j]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[c, Plus[j, -1]], Power[Times[c, x], Plus[m, Times[-1, j], 1]], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], Plus[p, 1]], Power[Times[a, Plus[n, Times[-1, j]], Plus[p, 1]], -1]], x]], Dist[Times[Power[c, j], Plus[m, Times[n, p], n, Times[-1, j], 1], Power[Times[a, Plus[n, Times[-1, j]], Plus[p, 1]], -1]], Int[Times[Power[Times[c, x], Plus[m, Times[-1, j]]], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], Plus[p, 1]]], x], x]], And[FreeQ[List[a, b, c, j, m, n], x], NegativeIntegerQ[Plus[p, Times[1, Power[2, -1]]]], NonzeroQ[Plus[n, Times[-1, j]]], ZeroQ[Simplify[Plus[m, Times[j, p], 1]]], Or[IntegerQ[j], PositiveQ[c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[c, IntPart[m]], Power[Times[c, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], p]], x], x], And[FreeQ[List[a, b, c, j, m, n, p], x], IntegerQ[Plus[p, Times[1, Power[2, -1]]]], NonzeroQ[Plus[n, Times[-1, j]]], ZeroQ[Simplify[Plus[m, Times[j, p], 1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[c, IntPart[m]], Power[Times[c, x], FracPart[m]], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], FracPart[p]], Power[Times[Power[x, Plus[FracPart[m], Times[j, FracPart[p]]]], Power[Plus[a, Times[b, Power[x, Plus[n, Times[-1, j]]]]], FracPart[p]]], -1]], Int[Times[Power[x, Plus[m, Times[j, p]]], Power[Plus[a, Times[b, Power[x, Plus[n, Times[-1, j]]]]], p]], x], x], And[FreeQ[List[a, b, c, j, m, n, p], x], Not[IntegerQ[p]], NonzeroQ[Plus[n, Times[-1, j]]], PosQ[Plus[n, Times[-1, j]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[u, m], Power[Times[Coefficient[v, x, 1], Power[v, m]], -1]], Subst[Int[Times[Power[x, m], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], p]], x], x, v], x], And[FreeQ[List[a, b, j, m, n, p], x], LinearPairQ[u, v, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[k, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Pattern[j, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[n, -1]], Subst[Int[Times[Power[x, Plus[Simplify[Times[Plus[m, 1], Power[n, -1]]], -1]], Power[Plus[Times[a, Power[x, Simplify[Times[j, Power[n, -1]]]]], Times[b, Power[x, Simplify[Times[k, Power[n, -1]]]]]], p], Power[Plus[c, Times[d, x]], q]], x], x, Power[x, n]], x], And[FreeQ[List[a, b, c, d, j, k, m, n, p, q], x], Not[IntegerQ[p]], NonzeroQ[Plus[k, Times[-1, j]]], IntegerQ[Simplify[Times[j, Power[n, -1]]]], IntegerQ[Simplify[Times[k, Power[n, -1]]]], IntegerQ[Simplify[Times[Plus[m, 1], Power[n, -1]]]], NonzeroQ[Plus[Power[n, 2], -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[k, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Pattern[j, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[e, IntPart[m]], Power[Times[e, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, k]]], p], Power[Plus[c, Times[d, Power[x, n]]], q]], x], x], And[FreeQ[List[a, b, c, d, e, j, k, m, n, p, q], x], Not[IntegerQ[p]], NonzeroQ[Plus[k, Times[-1, j]]], IntegerQ[Simplify[Times[j, Power[n, -1]]]], IntegerQ[Simplify[Times[k, Power[n, -1]]]], IntegerQ[Simplify[Times[Plus[m, 1], Power[n, -1]]]], NonzeroQ[Plus[Power[n, 2], -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[jn, Blank[]]]]]], Pattern[p, Blank[]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[c, Power[e, Plus[j, -1]], Power[Times[e, x], Plus[m, Times[-1, j], 1]], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, Plus[j, n]]]], Plus[p, 1]], Power[Times[a, Plus[m, Times[j, p], 1]], -1]], x], And[FreeQ[List[a, b, c, d, e, j, m, n, p], x], ZeroQ[Plus[jn, Times[-1, j], Times[-1, n]]], Not[IntegerQ[p]], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Times[a, d, Plus[m, Times[j, p], 1]], Times[-1, b, c, Plus[m, n, Times[p, Plus[j, n]], 1]]]], Or[PositiveQ[e], IntegersQ[j]], NonzeroQ[Plus[m, Times[j, p], 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[jn, Blank[]]]]]], Pattern[p, Blank[]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[e, Plus[j, -1]], Plus[Times[b, c], Times[-1, a, d]], Power[Times[e, x], Plus[m, Times[-1, j], 1]], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, Plus[j, n]]]], Plus[p, 1]], Power[Times[a, b, n, Plus[p, 1]], -1]], x]], Times[-1, Dist[Times[Power[e, j], Plus[Times[a, d, Plus[m, Times[j, p], 1]], Times[-1, b, c, Plus[m, n, Times[p, Plus[j, n]], 1]]], Power[Times[a, b, n, Plus[p, 1]], -1]], Int[Times[Power[Times[e, x], Plus[m, Times[-1, j]]], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, Plus[j, n]]]], Plus[p, 1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, j, m, n], x], ZeroQ[Plus[jn, Times[-1, j], Times[-1, n]]], Not[IntegerQ[p]], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[j, m, p], Less[p, -1], Inequality[0, Less, j, LessEqual, m], Or[PositiveQ[e], IntegerQ[j]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[jn, Blank[]]]]]], Pattern[p, Blank[]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[c, Power[e, Plus[j, -1]], Power[Times[e, x], Plus[m, Times[-1, j], 1]], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, Plus[j, n]]]], Plus[p, 1]], Power[Times[a, Plus[m, Times[j, p], 1]], -1]], x], Dist[Times[Plus[Times[a, d, Plus[m, Times[j, p], 1]], Times[-1, b, c, Plus[m, n, Times[p, Plus[j, n]], 1]]], Power[Times[a, Power[e, n], Plus[m, Times[j, p], 1]], -1]], Int[Times[Power[Times[e, x], Plus[m, n]], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, Plus[j, n]]]], p]], x], x]], And[FreeQ[List[a, b, c, d, e, j, p], x], ZeroQ[Plus[jn, Times[-1, j], Times[-1, n]]], Not[IntegerQ[p]], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[m, n], Greater[n, 0], Or[Less[Plus[m, Times[j, p]], -1], And[IntegersQ[Plus[m, Times[-1, 1, Power[2, -1]]], Plus[p, Times[-1, 1, 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p], 1]], Times[-1, b, c, Plus[m, n, Times[p, Plus[j, n]], 1]]], Power[Times[b, Plus[m, n, Times[p, Plus[j, n]], 1]], -1]], Int[Times[Power[Times[e, x], m], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, Plus[j, n]]]], p]], x], x]]], And[FreeQ[List[a, b, c, d, e, j, m, n, p], x], ZeroQ[Plus[jn, Times[-1, j], Times[-1, n]]], Not[IntegerQ[p]], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[m, n, Times[p, Plus[j, n]], 1]], Or[PositiveQ[e], IntegerQ[j]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[k, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Pattern[j, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Plus[m, 1], -1]], Subst[Int[Times[Power[Plus[Times[a, Power[x, Simplify[Times[j, Power[Plus[m, 1], -1]]]]], Times[b, Power[x, Simplify[Times[k, Power[Plus[m, 1], -1]]]]]], p], Power[Plus[c, Times[d, Power[x, Simplify[Times[n, Power[Plus[m, 1], -1]]]]]], q]], x], x, Power[x, Plus[m, 1]]], x], And[FreeQ[List[a, b, c, d, j, k, m, n, p, q], x], Not[IntegerQ[p]], NonzeroQ[Plus[k, Times[-1, j]]], IntegerQ[Simplify[Times[j, Power[n, -1]]]], IntegerQ[Simplify[Times[k, Power[n, -1]]]], NonzeroQ[Plus[m, 1]], IntegerQ[Simplify[Times[n, Power[Plus[m, 1], -1]]]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[k, Blank[]]]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Pattern[j, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[e, IntPart[m]], Power[Times[e, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, k]]], p], Power[Plus[c, Times[d, Power[x, n]]], q]], x], x], And[FreeQ[List[a, b, c, d, e, j, k, m, n, p, q], x], Not[IntegerQ[p]], NonzeroQ[Plus[k, Times[-1, j]]], IntegerQ[Simplify[Times[j, Power[n, -1]]]], IntegerQ[Simplify[Times[k, Power[n, -1]]]], NonzeroQ[Plus[m, 1]], IntegerQ[Simplify[Times[n, Power[Plus[m, 1], -1]]]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[jn, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[e, IntPart[m]], Power[Times[e, x], FracPart[m]], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, Plus[j, n]]]], FracPart[p]], Power[Times[Power[x, Plus[FracPart[m], Times[j, FracPart[p]]]], Power[Plus[a, Times[b, Power[x, n]]], FracPart[p]]], -1]], Int[Times[Power[x, Plus[m, Times[j, p]]], Power[Plus[a, Times[b, Power[x, n]]], p], Power[Plus[c, Times[d, Power[x, n]]], q]], x], x], And[FreeQ[List[a, b, c, d, e, j, m, n, p, q], x], ZeroQ[Plus[jn, Times[-1, j], Times[-1, n]]], Not[IntegerQ[p]], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], Not[And[ZeroQ[Plus[n, -1]], ZeroQ[Plus[j, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[d, Denominator[n]]], Dist[d, Subst[Int[Times[Power[x, Plus[d, -1]], ReplaceAll[SubstFor[Power[x, n], Pq, x], Rule[x, Power[x, Times[d, n]]]], Power[Plus[Times[a, Power[x, Times[d, j]]], Times[b, Power[x, Times[d, n]]]], p]], x], x, Power[x, Times[1, Power[d, -1]]]], x]], And[FreeQ[List[a, b, j, n, p], x], PolyQ[Pq, Power[x, n]], Not[IntegerQ[p]], NonzeroQ[Plus[n, Times[-1, j]]], RationalQ[j, n], IntegerQ[Times[j, Power[n, -1]]], Less[-1, n, 1]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[n, -1]], Subst[Int[Times[Power[x, Plus[Simplify[Times[Plus[m, 1], Power[n, -1]]], -1]], SubstFor[Power[x, n], Pq, x], Power[Plus[Times[a, Power[x, Simplify[Times[j, Power[n, -1]]]]], Times[b, x]], p]], x], x, Power[x, n]], x], And[FreeQ[List[a, b, j, m, n, p], x], PolyQ[Pq, Power[x, n]], Not[IntegerQ[p]], NonzeroQ[Plus[n, Times[-1, j]]], IntegerQ[Simplify[Times[j, Power[n, -1]]]], IntegerQ[Simplify[Times[Plus[m, 1], Power[n, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[c, Times[Sign[m], Quotient[m, Sign[m]]]], Power[Times[c, x], Mod[m, Sign[m]]], Power[Power[x, Mod[m, Sign[m]]], -1]], Int[Times[Power[x, m], Pq, Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], p]], x], x], And[FreeQ[List[a, b, c, j, n, p], x], PolyQ[Pq, Power[x, n]], Not[IntegerQ[p]], NonzeroQ[Plus[n, Times[-1, j]]], IntegerQ[Simplify[Times[j, Power[n, -1]]]], IntegerQ[Simplify[Times[Plus[m, 1], Power[n, -1]]]], RationalQ[m], Greater[Power[m, 2], 1]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[c, x], m], Power[Power[x, m], -1]], Int[Times[Power[x, m], Pq, Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], p]], x], x], And[FreeQ[List[a, b, c, j, m, n, p], x], PolyQ[Pq, Power[x, n]], Not[IntegerQ[p]], NonzeroQ[Plus[n, Times[-1, j]]], IntegerQ[Simplify[Times[j, Power[n, -1]]]], IntegerQ[Simplify[Times[Plus[m, 1], Power[n, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[g, GCD[Plus[m, 1], n]]], Condition[Dist[Times[1, Power[g, -1]], Subst[Int[Times[Power[x, Plus[Times[Plus[m, 1], Power[g, -1]], -1]], ReplaceAll[Pq, Rule[x, Power[x, Times[1, Power[g, -1]]]]], Power[Plus[Times[a, Power[x, Times[j, Power[g, -1]]]], Times[b, Power[x, Times[n, Power[g, -1]]]]], p]], x], x, Power[x, g]], x], Unequal[g, 1]]], And[FreeQ[List[a, b, p], x], PolyQ[Pq, Power[x, n]], Not[IntegerQ[p]], PositiveIntegerQ[j, n, Times[j, Power[n, -1]]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Expon[Pq, x]]], Condition[With[List[Set[Pqq, Coeff[Pq, x, q]]], Plus[Int[Times[Power[Times[c, x], m], ExpandToSum[Plus[Pq, Times[-1, Pqq, Power[x, q]], Times[-1, a, Pqq, Plus[m, q, Times[-1, n], 1], Power[x, Plus[q, Times[-1, n]]], Power[Times[b, Plus[m, q, Times[n, p], 1]], -1]]], x], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], p]], x], Simp[Times[Pqq, Power[Times[c, x], Plus[m, q, Times[-1, n], 1]], Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], Plus[p, 1]], Power[Times[b, Power[c, Plus[q, Times[-1, n], 1]], Plus[m, q, Times[n, p], 1]], -1]], x]]], And[Greater[q, Plus[n, -1]], Unequal[Plus[m, q, Times[n, p], 1], 0], Or[IntegerQ[Times[2, p]], IntegerQ[Plus[p, Times[Plus[q, 1], Power[Times[2, n], -1]]]]]]]], And[FreeQ[List[a, b, c, m, p], x], PolyQ[Pq, x], Not[IntegerQ[p]], PositiveIntegerQ[j, n], Less[j, n]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Plus[m, 1], -1]], Subst[Int[Times[ReplaceAll[SubstFor[Power[x, n], Pq, x], Rule[x, Power[x, Simplify[Times[n, Power[Plus[m, 1], -1]]]]]], Power[Plus[Times[a, Power[x, Simplify[Times[j, Power[Plus[m, 1], -1]]]]], Times[b, Power[x, Simplify[Times[n, Power[Plus[m, 1], -1]]]]]], p]], x], x, Power[x, Plus[m, 1]]], x], And[FreeQ[List[a, b, j, m, n, p], x], PolyQ[Pq, Power[x, n]], Not[IntegerQ[p]], NonzeroQ[Plus[n, Times[-1, j]]], IntegerQ[Simplify[Times[j, Power[n, -1]]]], IntegerQ[Simplify[Times[n, Power[Plus[m, 1], -1]]]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Times[Pattern[c, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[c, Times[Sign[m], Quotient[m, Sign[m]]]], Power[Times[c, x], Mod[m, Sign[m]]], Power[Power[x, Mod[m, Sign[m]]], -1]], Int[Times[Power[x, m], Pq, Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], p]], x], x], And[FreeQ[List[a, b, c, j, n, p], x], PolyQ[Pq, Power[x, n]], Not[IntegerQ[p]], NonzeroQ[Plus[n, Times[-1, j]]], IntegerQ[Simplify[Times[j, 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Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Times[c, x], m], Pq, Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], p]], x], x], And[FreeQ[List[a, b, c, j, m, n, p], x], Or[PolyQ[Pq, x], PolyQ[Pq, Power[x, n]]], Not[IntegerQ[p]], NonzeroQ[Plus[n, Times[-1, j]]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[Pq, Blank[]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[j, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Pq, Power[Plus[Times[a, Power[x, j]], Times[b, Power[x, n]]], p]], x], x], And[FreeQ[List[a, b, j, n, p], x], Or[PolyQ[Pq, x], PolyQ[Pq, Power[x, n]]], Not[IntegerQ[p]], NonzeroQ[Plus[n, Times[-1, j]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Times[Power[3, Times[3, p]], Power[a, Times[2, p]]], -1]], Int[Times[Power[Plus[Times[3, a], Times[-1, b, x]], p], Power[Plus[Times[3, a], Times[2, b, x]], Times[2, p]]], x], x], And[FreeQ[List[a, b, d], x], IntegerQ[p], ZeroQ[Plus[Times[4, Power[b, 3]], Times[27, Power[a, 2], d]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandToSum[Power[Plus[a, Times[b, x], 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Condition[With[List[Set[r, Rt[Plus[Times[-27, a, Power[d, 2]], Times[3, Sqrt[3], d, Sqrt[Plus[Times[4, Power[b, 3], d], Times[27, Power[a, 2], Power[d, 2]]]]]], 3]]], Dist[Times[1, Power[Times[Power[3, Times[3, p]], Power[d, Times[2, p]]], -1]], Int[Times[Power[Plus[Times[Plus[Times[6, b, d], Times[-1, Power[2, Times[1, Power[3, -1]]], Power[r, 2]]], Power[Times[Power[2, Times[2, Power[3, -1]]], r], -1]], Times[3, d, x]], p], Power[Plus[Times[Plus[Times[6, Plus[1, Times[I, Sqrt[3]]], b, d], Times[-1, Power[2, Times[1, Power[3, -1]]], Plus[1, Times[-1, I, Sqrt[3]]], Power[r, 2]]], Power[Times[2, Power[2, Times[2, Power[3, -1]]], r], -1]], Times[-1, 3, d, x]], p], Power[Plus[Times[Plus[Times[6, Plus[1, Times[-1, I, Sqrt[3]]], b, d], Times[-1, Power[2, Times[1, Power[3, -1]]], Plus[1, Times[I, Sqrt[3]]], Power[r, 2]]], Power[Times[2, Power[2, Times[2, Power[3, -1]]], r], -1]], Times[-1, 3, d, x]], p]], x], x]], And[FreeQ[List[a, b, d], x], NegativeIntegerQ[p], NonzeroQ[Plus[Times[4, Power[b, 3]], Times[27, Power[a, 2], d]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, x], Times[d, Power[x, 3]]], p], Power[Times[Power[Plus[Times[3, a], Times[-1, b, x]], p], Power[Plus[Times[3, a], Times[2, b, x]], Times[2, p]]], -1]], Int[Times[Power[Plus[Times[3, a], Times[-1, b, x]], p], Power[Plus[Times[3, a], Times[2, b, x]], Times[2, p]]], x], x], And[FreeQ[List[a, b, d, p], x], Not[IntegerQ[p]], ZeroQ[Plus[Times[4, Power[b, 3]], Times[27, Power[a, 2], d]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, NonfreeFactors[Factor[Plus[a, Times[b, x], Times[d, Power[x, 3]]]], x]]], Condition[Dist[Times[Power[Plus[a, Times[b, x], Times[d, Power[x, 3]]], p], Power[DistributeDegree[u, p], -1]], Int[DistributeDegree[u, p], x], x], ProductQ[u]]], And[FreeQ[List[a, b, d, p], x], Not[IntegerQ[p]], NonzeroQ[Plus[Times[4, Power[b, 3]], Times[27, Power[a, 2], d]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[r, Rt[Plus[Times[-27, a, Power[d, 2]], Times[3, Sqrt[3], d, Sqrt[Plus[Times[4, Power[b, 3], d], Times[27, Power[a, 2], Power[d, 2]]]]]], 3]]], Dist[Times[Power[Plus[a, Times[b, x], Times[d, Power[x, 3]]], p], Power[Times[Power[Plus[Times[Plus[Times[6, b, d], Times[-1, Power[2, Times[1, Power[3, -1]]], Power[r, 2]]], Power[Times[Power[2, Times[2, Power[3, -1]]], r], -1]], Times[3, d, x]], p], Power[Plus[Times[Plus[Times[6, Plus[1, Times[I, Sqrt[3]]], b, d], Times[-1, Power[2, Times[1, Power[3, -1]]], Plus[1, Times[-1, I, Sqrt[3]]], Power[r, 2]]], Power[Times[2, Power[2, Times[2, Power[3, -1]]], r], -1]], Times[-1, 3, d, x]], p], Power[Plus[Times[Plus[Times[6, Plus[1, Times[-1, I, Sqrt[3]]], b, d], Times[-1, Power[2, Times[1, Power[3, -1]]], Plus[1, Times[I, Sqrt[3]]], Power[r, 2]]], Power[Times[2, Power[2, Times[2, Power[3, -1]]], r], -1]], Times[-1, 3, d, x]], p]], -1]], Int[Times[Power[Plus[Times[Plus[Times[6, b, d], Times[-1, Power[2, Times[1, Power[3, -1]]], Power[r, 2]]], Power[Times[Power[2, Times[2, Power[3, -1]]], r], -1]], Times[3, d, x]], p], Power[Plus[Times[Plus[Times[6, Plus[1, Times[I, Sqrt[3]]], b, d], Times[-1, Power[2, Times[1, Power[3, -1]]], Plus[1, Times[-1, I, Sqrt[3]]], Power[r, 2]]], Power[Times[2, Power[2, Times[2, Power[3, -1]]], r], -1]], Times[-1, 3, d, x]], p], Power[Plus[Times[Plus[Times[6, Plus[1, Times[-1, I, Sqrt[3]]], b, d], Times[-1, Power[2, Times[1, Power[3, -1]]], Plus[1, Times[I, Sqrt[3]]], Power[r, 2]]], Power[Times[2, Power[2, Times[2, Power[3, -1]]], r], -1]], Times[-1, 3, d, x]], p]], x], x]], And[FreeQ[List[a, b, d, p], x], Not[IntegerQ[p]], NonzeroQ[Plus[Times[4, Power[b, 3]], Times[27, Power[a, 2], d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Times[Power[3, Times[3, p]], Power[a, Times[2, p]]], -1]], Int[Times[Power[Plus[e, Times[f, x]], m], Power[Plus[Times[3, a], Times[-1, b, x]], p], Power[Plus[Times[3, a], Times[2, b, x]], Times[2, p]]], x], x], And[FreeQ[List[a, b, d, e, f, m], x], IntegerQ[p], ZeroQ[Plus[Times[4, Power[b, 3]], Times[27, Power[a, 2], d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[e, Times[f, x]], m], Power[Plus[a, Times[b, x], Times[d, Power[x, 3]]], p]], x], x], And[FreeQ[List[a, b, d, e, f, m], x], PositiveIntegerQ[p], NonzeroQ[Plus[Times[4, Power[b, 3]], Times[27, Power[a, 2], d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, Factor[Plus[a, Times[b, x], Times[d, Power[x, 3]]]]]], Condition[Dist[Power[FreeFactors[u, x], p], Int[Times[Power[Plus[e, Times[f, x]], m], DistributeDegree[NonfreeFactors[u, x], p]], x], x], ProductQ[NonfreeFactors[u, x]]]], And[FreeQ[List[a, b, d, e, f, m], x], NegativeIntegerQ[p], NonzeroQ[Plus[Times[4, Power[b, 3]], Times[27, Power[a, 2], d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[r, Rt[Plus[Times[-27, a, Power[d, 2]], Times[3, Sqrt[3], d, Sqrt[Plus[Times[4, Power[b, 3], d], Times[27, Power[a, 2], Power[d, 2]]]]]], 3]]], Dist[Times[1, Power[Times[Power[3, Times[3, p]], Power[d, Times[2, p]]], -1]], Int[Times[Power[Plus[e, Times[f, x]], m], Power[Plus[Times[Plus[Times[6, b, d], Times[-1, Power[2, Times[1, Power[3, -1]]], Power[r, 2]]], Power[Times[Power[2, Times[2, Power[3, -1]]], r], -1]], Times[3, d, x]], p], Power[Plus[Times[Plus[Times[6, Plus[1, Times[I, Sqrt[3]]], b, d], Times[-1, Power[2, Times[1, Power[3, -1]]], Plus[1, Times[-1, I, Sqrt[3]]], Power[r, 2]]], Power[Times[2, Power[2, Times[2, Power[3, -1]]], r], -1]], Times[-1, 3, d, x]], p], Power[Plus[Times[Plus[Times[6, Plus[1, Times[-1, I, Sqrt[3]]], b, d], Times[-1, Power[2, Times[1, Power[3, -1]]], Plus[1, Times[I, Sqrt[3]]], Power[r, 2]]], Power[Times[2, Power[2, Times[2, Power[3, -1]]], r], -1]], Times[-1, 3, d, x]], p]], x], x]], And[FreeQ[List[a, b, d, e, f, m], x], NegativeIntegerQ[p], NonzeroQ[Plus[Times[4, Power[b, 3]], Times[27, Power[a, 2], d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, x], Times[d, Power[x, 3]]], p], Power[Times[Power[Plus[Times[3, a], Times[-1, b, x]], p], Power[Plus[Times[3, a], Times[2, b, x]], Times[2, p]]], -1]], Int[Times[Power[Plus[e, Times[f, x]], m], Power[Plus[Times[3, a], Times[-1, b, x]], p], Power[Plus[Times[3, a], Times[2, b, x]], Times[2, p]]], x], x], And[FreeQ[List[a, b, d, e, f, m, p], x], Not[IntegerQ[p]], ZeroQ[Plus[Times[4, Power[b, 3]], Times[27, Power[a, 2], d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, NonfreeFactors[Factor[Plus[a, Times[b, x], Times[d, Power[x, 3]]]], x]]], Condition[Dist[Times[Power[Plus[a, Times[b, x], Times[d, Power[x, 3]]], p], Power[DistributeDegree[u, p], -1]], Int[Times[Power[Plus[e, Times[f, x]], m], DistributeDegree[u, p]], x], x], ProductQ[u]]], And[FreeQ[List[a, b, d, e, f, m, p], x], Not[IntegerQ[p]], NonzeroQ[Plus[Times[4, Power[b, 3]], Times[27, Power[a, 2], d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[r, Rt[Plus[Times[-27, a, Power[d, 2]], Times[3, Sqrt[3], d, Sqrt[Plus[Times[4, Power[b, 3], d], Times[27, Power[a, 2], Power[d, 2]]]]]], 3]]], Dist[Times[Power[Plus[a, Times[b, x], Times[d, Power[x, 3]]], p], Power[Times[Power[Plus[Times[Plus[Times[6, b, d], Times[-1, Power[2, Times[1, Power[3, -1]]], Power[r, 2]]], Power[Times[Power[2, Times[2, Power[3, -1]]], r], -1]], Times[3, d, x]], p], Power[Plus[Times[Plus[Times[6, Plus[1, Times[I, Sqrt[3]]], b, d], Times[-1, Power[2, Times[1, Power[3, -1]]], Plus[1, Times[-1, I, Sqrt[3]]], Power[r, 2]]], Power[Times[2, Power[2, Times[2, Power[3, -1]]], r], -1]], Times[-1, 3, d, x]], p], Power[Plus[Times[Plus[Times[6, Plus[1, Times[-1, I, Sqrt[3]]], b, d], Times[-1, Power[2, Times[1, Power[3, -1]]], Plus[1, Times[I, Sqrt[3]]], Power[r, 2]]], Power[Times[2, Power[2, Times[2, Power[3, -1]]], r], -1]], Times[-1, 3, d, x]], p]], -1]], Int[Times[Power[Plus[e, Times[f, x]], m], Power[Plus[Times[Plus[Times[6, b, d], Times[-1, Power[2, Times[1, Power[3, -1]]], Power[r, 2]]], Power[Times[Power[2, Times[2, Power[3, -1]]], r], -1]], Times[3, d, x]], p], Power[Plus[Times[Plus[Times[6, Plus[1, Times[I, Sqrt[3]]], b, d], Times[-1, Power[2, Times[1, Power[3, -1]]], Plus[1, Times[-1, I, Sqrt[3]]], Power[r, 2]]], Power[Times[2, Power[2, Times[2, Power[3, -1]]], r], -1]], Times[-1, 3, d, x]], p], Power[Plus[Times[Plus[Times[6, Plus[1, Times[-1, I, Sqrt[3]]], b, d], Times[-1, Power[2, Times[1, Power[3, -1]]], Plus[1, Times[I, Sqrt[3]]], Power[r, 2]]], Power[Times[2, Power[2, Times[2, Power[3, -1]]], r], -1]], Times[-1, 3, d, x]], p]], x], x]], And[FreeQ[List[a, b, d, e, f, m, p], x], Not[IntegerQ[p]], NonzeroQ[Plus[Times[4, Power[b, 3]], Times[27, Power[a, 2], d]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Power[Times[Power[3, Times[3, p]], Power[d, Times[2, p]]], -1], Int[Times[Power[Plus[c, Times[-1, 3, d, x]], p], Power[Plus[Times[2, c], Times[3, d, x]], Times[2, p]]], x], x]], And[FreeQ[List[a, c, d], x], IntegerQ[p], ZeroQ[Plus[Times[4, Power[c, 3]], Times[27, a, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandToSum[Power[Plus[a, Times[c, Power[x, 2]], Times[d, Power[x, 3]]], p], x], x], And[FreeQ[List[a, c, d], x], PositiveIntegerQ[p], NonzeroQ[Plus[Times[4, Power[c, 3]], Times[27, a, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, Factor[Plus[a, Times[c, Power[x, 2]], Times[d, Power[x, 3]]]]]], Condition[Dist[Power[FreeFactors[u, x], p], Int[DistributeDegree[NonfreeFactors[u, x], p], x], x], ProductQ[NonfreeFactors[u, x]]]], And[FreeQ[List[a, c, d], x], NegativeIntegerQ[p], NonzeroQ[Plus[Times[4, Power[c, 3]], Times[27, a, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[r, Rt[Plus[Times[-2, Power[c, 3]], Times[-1, 27, a, Power[d, 2]], Times[3, Sqrt[3], d, Sqrt[Plus[Times[4, a, Power[c, 3]], Times[27, Power[a, 2], Power[d, 2]]]]]], 3]]], Dist[Times[1, Power[Times[Power[3, Times[3, p]], Power[d, Times[2, p]]], -1]], Int[Times[Power[Plus[c, Times[-1, Plus[Times[2, Power[c, 2]], Times[Power[2, Times[1, Power[3, -1]]], Power[r, 2]]], Power[Times[Power[2, Times[2, Power[3, -1]]], r], -1]], Times[3, d, x]], p], Power[Plus[c, Times[Plus[Times[2, Plus[1, Times[I, Sqrt[3]]], Power[c, 2]], Times[Power[2, Times[1, Power[3, -1]]], Plus[1, Times[-1, I, Sqrt[3]]], Power[r, 2]]], Power[Times[2, Power[2, Times[2, Power[3, -1]]], r], -1]], Times[3, d, x]], p], Power[Plus[c, Times[Plus[Times[2, Plus[1, Times[-1, I, Sqrt[3]]], Power[c, 2]], Times[Power[2, Times[1, Power[3, -1]]], Plus[1, Times[I, Sqrt[3]]], Power[r, 2]]], Power[Times[2, Power[2, Times[2, Power[3, -1]]], r], -1]], Times[3, d, x]], p]], x], x]], And[FreeQ[List[a, c, d], x], NegativeIntegerQ[p], NonzeroQ[Plus[Times[4, Power[c, 3]], Times[27, a, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[c, Power[x, 2]], Times[d, Power[x, 3]]], p], Power[Times[Power[Plus[c, Times[-1, 3, d, x]], p], Power[Plus[Times[2, c], Times[3, d, x]], Times[2, p]]], -1]], Int[Times[Power[Plus[c, Times[-1, 3, d, x]], p], Power[Plus[Times[2, c], Times[3, d, x]], Times[2, p]]], x], x], And[FreeQ[List[a, c, d, p], x], Not[IntegerQ[p]], ZeroQ[Plus[Times[4, Power[c, 3]], Times[27, a, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, NonfreeFactors[Factor[Plus[a, Times[c, Power[x, 2]], Times[d, Power[x, 3]]]], x]]], Condition[Dist[Times[Power[Plus[a, Times[c, Power[x, 2]], Times[d, Power[x, 3]]], p], Power[DistributeDegree[u, p], -1]], Int[DistributeDegree[u, p], x], x], ProductQ[u]]], And[FreeQ[List[a, c, d, p], x], Not[IntegerQ[p]], NonzeroQ[Plus[Times[4, Power[c, 3]], Times[27, a, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[r, Rt[Plus[Times[-2, Power[c, 3]], Times[-1, 27, a, Power[d, 2]], Times[3, Sqrt[3], d, Sqrt[Plus[Times[4, a, Power[c, 3]], Times[27, Power[a, 2], Power[d, 2]]]]]], 3]]], Dist[Times[Power[Plus[a, Times[c, Power[x, 2]], Times[d, Power[x, 3]]], p], Power[Times[Power[Plus[c, Times[-1, Plus[Times[2, Power[c, 2]], Times[Power[2, Times[1, Power[3, -1]]], Power[r, 2]]], Power[Times[Power[2, Times[2, Power[3, -1]]], r], -1]], Times[3, d, x]], p], Power[Plus[c, Times[Plus[Times[2, Plus[1, Times[I, Sqrt[3]]], Power[c, 2]], Times[Power[2, Times[1, Power[3, -1]]], Plus[1, Times[-1, I, Sqrt[3]]], Power[r, 2]]], Power[Times[2, Power[2, Times[2, Power[3, -1]]], r], -1]], Times[3, d, x]], p], Power[Plus[c, Times[Plus[Times[2, Plus[1, Times[-1, I, Sqrt[3]]], Power[c, 2]], Times[Power[2, Times[1, Power[3, -1]]], Plus[1, Times[I, Sqrt[3]]], Power[r, 2]]], Power[Times[2, Power[2, Times[2, Power[3, -1]]], r], -1]], Times[3, d, x]], p]], -1]], Int[Times[Power[Plus[c, Times[-1, Plus[Times[2, Power[c, 2]], Times[Power[2, Times[1, Power[3, -1]]], Power[r, 2]]], Power[Times[Power[2, Times[2, Power[3, -1]]], r], -1]], Times[3, d, x]], p], Power[Plus[c, Times[Plus[Times[2, Plus[1, Times[I, Sqrt[3]]], Power[c, 2]], Times[Power[2, Times[1, Power[3, -1]]], Plus[1, Times[-1, I, Sqrt[3]]], Power[r, 2]]], Power[Times[2, Power[2, Times[2, Power[3, -1]]], r], -1]], Times[3, d, x]], p], Power[Plus[c, Times[Plus[Times[2, Plus[1, Times[-1, I, Sqrt[3]]], Power[c, 2]], Times[Power[2, Times[1, Power[3, -1]]], Plus[1, Times[I, Sqrt[3]]], Power[r, 2]]], Power[Times[2, Power[2, Times[2, Power[3, -1]]], r], -1]], Times[3, d, x]], p]], x], x]], And[FreeQ[List[a, c, d, p], x], Not[IntegerQ[p]], NonzeroQ[Plus[Times[4, Power[c, 3]], Times[27, a, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Power[Times[Power[3, Times[3, p]], Power[d, Times[2, p]]], -1], Int[Times[Power[Plus[e, Times[f, x]], m], Power[Plus[c, Times[-1, 3, d, x]], p], Power[Plus[Times[2, c], Times[3, d, x]], Times[2, p]]], x], x]], And[FreeQ[List[a, c, d, e, f, m], x], IntegerQ[p], ZeroQ[Plus[Times[4, Power[c, 3]], Times[27, a, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[e, Times[f, x]], m], Power[Plus[a, Times[c, Power[x, 2]], Times[d, Power[x, 3]]], p]], x], x], And[FreeQ[List[a, c, d, e, f, m], x], PositiveIntegerQ[p], NonzeroQ[Plus[Times[4, Power[c, 3]], Times[27, a, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], 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Power[Times[2, Power[2, Times[2, Power[3, -1]]], r], -1]], Times[3, d, x]], p], Power[Plus[c, Times[Plus[Times[2, Plus[1, Times[-1, I, Sqrt[3]]], Power[c, 2]], Times[-1, 6, Plus[1, Times[-1, I, Sqrt[3]]], b, d], Times[I, Power[2, Times[1, Power[3, -1]]], Plus[Times[-1, I], Sqrt[3]], Power[r, 2]]], Power[Times[2, Power[2, Times[2, Power[3, -1]]], r], -1]], Times[3, d, x]], p]], x], x]], And[FreeQ[List[a, b, c, d, p], x], Not[IntegerQ[p]], NonzeroQ[Plus[Power[c, 2], Times[-1, 3, b, d]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 3, a, c]]]]]], RuleDelayed[HoldPattern[Int[Power[Pattern[u, Blank[]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Power[ExpandToSum[u, x], p], x], And[FreeQ[p, x], PolyQ[u, x, 3], Not[CubicMatchQ[u, x]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, 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Blank[Symbol]]]], Condition[With[List[Set[r, Rt[Plus[Power[b, 3], Times[-1, 3, a, b, c]], 3]]], Dist[Times[1, Power[Times[Power[3, p], Power[b, p], Power[c, p]], -1]], Int[Times[Power[Plus[e, Times[f, x]], m], Power[Plus[b, Times[-1, r], Times[c, x]], p], Power[Plus[b, Times[Plus[1, Times[-1, I, Sqrt[3]]], r, Power[2, -1]], Times[c, x]], p], Power[Plus[b, Times[Plus[1, Times[I, Sqrt[3]]], r, Power[2, -1]], Times[c, x]], p]], x], x]], And[FreeQ[List[a, b, c, d, e, f, m], x], IntegerQ[p], ZeroQ[Plus[Power[c, 2], Times[-1, 3, b, d]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 3, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[r, Rt[Plus[Power[c, 3], Times[-1, 3, b, c, d]], 3]]], Dist[Times[1, Power[Times[Power[3, p], Power[b, p], Power[c, p]], -1]], Int[Times[Power[Plus[e, Times[f, x]], m], Power[Plus[b, Times[Plus[c, Times[-1, r]], x]], p], Power[Plus[b, Times[Plus[c, Times[Plus[1, Times[-1, I, Sqrt[3]]], r, Power[2, -1]]], x]], p], Power[Plus[b, Times[Plus[c, Times[Plus[1, Times[I, Sqrt[3]]], r, Power[2, -1]]], x]], p]], x], x]], And[FreeQ[List[a, b, c, d, e, f, m], x], IntegerQ[p], NonzeroQ[Plus[Power[c, 2], Times[-1, 3, b, d]]], ZeroQ[Plus[Power[b, 2], Times[-1, 3, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[e, Times[f, x]], m], Power[Plus[a, Times[b, x], Times[c, Power[x, 2]], Times[d, Power[x, 3]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, f, m], x], PositiveIntegerQ[p], NonzeroQ[Plus[Power[c, 2], Times[-1, 3, b, d]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 3, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, Factor[Plus[a, Times[b, x], Times[c, Power[x, 2]], Times[d, Power[x, 3]]]]]], Condition[Dist[Power[FreeFactors[u, x], p], Int[Times[Power[Plus[e, Times[f, x]], m], DistributeDegree[NonfreeFactors[u, x], p]], x], x], ProductQ[NonfreeFactors[u, x]]]], And[FreeQ[List[a, b, c, d, e, f, m], x], NegativeIntegerQ[p], NonzeroQ[Plus[Power[c, 2], Times[-1, 3, b, d]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 3, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Times[Power[3, Times[3, p]], Power[d, Times[2, p]]], -1]], Subst[Int[Power[Plus[Times[2, Power[c, 3]], Times[-1, 9, b, c, d], Times[27, a, Power[d, 2]], Times[-1, 9, d, Plus[Power[c, 2], Times[-1, 3, b, d]], x], Times[27, Power[d, 3], Power[x, 3]]], p], x], x, Plus[x, Times[c, Power[Times[3, d], -1]]]], x], And[FreeQ[List[a, b, c, d, e, f, m], x], NegativeIntegerQ[p], NonzeroQ[Plus[Power[c, 2], Times[-1, 3, b, d]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 3, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]], Times[d, Power[x, 3]]], p], Power[Power[Plus[b, Times[c, x]], Times[3, p]], -1]], Int[Times[Power[Plus[e, Times[f, x]], m], Power[Plus[b, Times[c, x]], Times[3, p]]], x], x], And[FreeQ[List[a, b, c, d, e, f, m, p], x], Not[IntegerQ[p]], ZeroQ[Plus[Power[c, 2], Times[-1, 3, b, d]]], ZeroQ[Plus[Power[b, 2], Times[-1, 3, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[r, Rt[Plus[Power[b, 3], Times[-1, 3, a, b, c]], 3]]], Dist[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]], Times[d, Power[x, 3]]], p], Power[Times[Power[Plus[b, Times[-1, r], Times[c, x]], p], Power[Plus[b, Times[Plus[1, Times[-1, I, Sqrt[3]]], r, Power[2, -1]], Times[c, x]], p], Power[Plus[b, Times[Plus[1, Times[I, Sqrt[3]]], r, Power[2, -1]], Times[c, x]], p]], -1]], Int[Times[Power[Plus[e, Times[f, x]], m], Power[Plus[b, Times[-1, r], Times[c, x]], p], Power[Plus[b, Times[Plus[1, Times[-1, I, Sqrt[3]]], r, Power[2, -1]], Times[c, x]], p], Power[Plus[b, Times[Plus[1, Times[I, Sqrt[3]]], r, Power[2, -1]], Times[c, x]], p]], x], x]], And[FreeQ[List[a, b, c, d, e, f, m, p], x], Not[IntegerQ[p]], ZeroQ[Plus[Power[c, 2], Times[-1, 3, b, d]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 3, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[r, Rt[Plus[Power[c, 3], Times[-1, 3, b, c, d]], 3]]], Dist[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]], Times[d, Power[x, 3]]], p], Power[Times[Power[Plus[b, Times[Plus[c, Times[-1, r]], x]], p], Power[Plus[b, Times[Plus[c, Times[Plus[1, Times[-1, I, Sqrt[3]]], r, Power[2, -1]]], x]], p], Power[Plus[b, Times[Plus[c, Times[Plus[1, Times[I, Sqrt[3]]], r, Power[2, -1]]], x]], p]], -1]], Int[Times[Power[Plus[e, Times[f, x]], m], Power[Plus[b, Times[Plus[c, Times[-1, r]], x]], p], Power[Plus[b, Times[Plus[c, Times[Plus[1, Times[-1, I, Sqrt[3]]], r, Power[2, -1]]], x]], p], Power[Plus[b, Times[Plus[c, Times[Plus[1, Times[I, Sqrt[3]]], r, Power[2, -1]]], x]], p]], x], x]], And[FreeQ[List[a, b, c, d, e, f, m, p], x], Not[IntegerQ[p]], NonzeroQ[Plus[Power[c, 2], Times[-1, 3, b, d]]], ZeroQ[Plus[Power[b, 2], Times[-1, 3, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, NonfreeFactors[Factor[Plus[a, Times[b, x], Times[c, Power[x, 2]], Times[d, Power[x, 3]]]], x]]], Condition[Dist[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]], Times[d, Power[x, 3]]], p], Power[DistributeDegree[u, p], -1]], Int[Times[Power[Plus[e, Times[f, x]], m], DistributeDegree[u, p]], x], x], ProductQ[u]]], And[FreeQ[List[a, b, c, d, e, f, m, p], x], Not[IntegerQ[p]], NonzeroQ[Plus[Power[c, 2], Times[-1, 3, b, d]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 3, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[r, Rt[Plus[Times[-2, Power[c, 3]], Times[9, b, c, d], Times[-1, 27, a, Power[d, 2]], Times[3, Sqrt[3], d, Sqrt[Plus[Times[-1, Power[b, 2], Power[c, 2]], Times[4, a, Power[c, 3]], Times[4, Power[b, 3], d], Times[-1, 18, a, b, c, d], Times[27, Power[a, 2], Power[d, 2]]]]]], 3]]], Dist[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]], Times[d, Power[x, 3]]], p], Power[Times[Power[Plus[c, Times[-1, Plus[Times[2, Power[c, 2]], Times[-1, 6, b, d], Times[Power[2, Times[1, Power[3, -1]]], Power[r, 2]]], Power[Times[Power[2, Times[2, Power[3, -1]]], r], -1]], Times[3, d, x]], p], Power[Plus[c, Times[Plus[Times[2, Plus[1, Times[I, Sqrt[3]]], Power[c, 2]], Times[-1, 6, Plus[1, Times[I, Sqrt[3]]], b, d], Times[-1, I, Power[2, Times[1, Power[3, -1]]], Plus[I, Sqrt[3]], Power[r, 2]]], Power[Times[2, Power[2, Times[2, Power[3, -1]]], r], -1]], Times[3, d, x]], p], Power[Plus[c, Times[Plus[Times[2, Plus[1, Times[-1, I, Sqrt[3]]], Power[c, 2]], Times[-1, 6, Plus[1, Times[-1, I, Sqrt[3]]], b, d], Times[I, Power[2, Times[1, Power[3, -1]]], Plus[Times[-1, I], Sqrt[3]], Power[r, 2]]], Power[Times[2, Power[2, Times[2, Power[3, -1]]], r], -1]], Times[3, d, x]], p]], -1]], Int[Times[Power[Plus[e, Times[f, x]], m], Power[Plus[c, Times[-1, Plus[Times[2, Power[c, 2]], Times[-1, 6, b, d], Times[Power[2, Times[1, Power[3, -1]]], Power[r, 2]]], Power[Times[Power[2, Times[2, Power[3, -1]]], r], -1]], Times[3, d, x]], p], Power[Plus[c, Times[Plus[Times[2, Plus[1, Times[I, Sqrt[3]]], Power[c, 2]], Times[-1, 6, Plus[1, Times[I, Sqrt[3]]], b, d], Times[-1, I, Power[2, Times[1, Power[3, -1]]], Plus[I, Sqrt[3]], Power[r, 2]]], Power[Times[2, Power[2, Times[2, Power[3, -1]]], r], -1]], Times[3, d, x]], p], Power[Plus[c, Times[Plus[Times[2, Plus[1, Times[-1, I, Sqrt[3]]], Power[c, 2]], Times[-1, 6, Plus[1, Times[-1, I, Sqrt[3]]], b, d], Times[I, Power[2, Times[1, Power[3, -1]]], Plus[Times[-1, I], Sqrt[3]], Power[r, 2]]], Power[Times[2, Power[2, Times[2, Power[3, -1]]], r], -1]], Times[3, d, x]], p]], x], x]], And[FreeQ[List[a, b, c, d, e, f, m, p], x], Not[IntegerQ[p]], NonzeroQ[Plus[Power[c, 2], Times[-1, 3, b, d]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 3, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Pattern[v, Blank[]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[ExpandToSum[u, x], m], Power[ExpandToSum[v, x], p]], x], And[FreeQ[List[m, p], x], LinearQ[u, x], PolyQ[v, x, 3], Not[And[LinearMatchQ[u, x], CubicMatchQ[v, x]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[a, f, Power[Times[d, Rt[Times[Power[a, 2], Plus[Times[2, a], Times[-1, c]]], 2]], -1], ArcTan[Times[Plus[Times[a, b], Times[Plus[Times[4, Power[a, 2]], Power[b, 2], Times[-1, 2, a, c]], x], Times[a, b, Power[x, 2]]], Power[Times[2, Rt[Times[Power[a, 2], Plus[Times[2, a], Times[-1, c]]], 2], Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]], Times[b, Power[x, 3]], Times[a, Power[x, 4]]]]], -1]]]], x], And[FreeQ[List[a, b, c, d, e, f, g], x], ZeroQ[Plus[Times[b, d], Times[-1, a, e]]], ZeroQ[Plus[f, g]], PosQ[Times[Power[a, 2], Plus[Times[2, a], Times[-1, c]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 3]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Simp[Times[a, f, Power[Times[d, Rt[Times[-1, Power[a, 2], Plus[Times[2, a], Times[-1, c]]], 2]], -1], ArcTanh[Times[Plus[Times[a, b], Times[Plus[Times[4, Power[a, 2]], Power[b, 2], Times[-1, 2, a, c]], x], Times[a, b, Power[x, 2]]], Power[Times[2, Rt[Times[-1, Power[a, 2], Plus[Times[2, a], Times[-1, c]]], 2], Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]], Times[b, Power[x, 3]], Times[a, Power[x, 4]]]]], -1]]]], x]], And[FreeQ[List[a, b, c, d, e, f, g], x], ZeroQ[Plus[Times[b, d], Times[-1, a, e]]], ZeroQ[Plus[f, g]], NegQ[Times[Power[a, 2], Plus[Times[2, a], Times[-1, c]]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Subst[Int[SimplifyIntegrand[Power[Plus[a, Times[Power[d, 4], Power[Times[256, Power[e, 3]], -1]], Times[-1, b, d, Power[Times[8, e], -1]], Times[Plus[c, Times[-1, 3, Power[d, 2], Power[Times[8, e], -1]]], Power[x, 2]], Times[e, Power[x, 4]]], p], x], x], x, Plus[Times[d, Power[Times[4, e], -1]], x]], And[FreeQ[List[a, b, c, d, e, p], x], ZeroQ[Plus[Power[d, 3], Times[-1, 4, c, d, e], Times[8, b, Power[e, 2]]]], UnsameQ[p, 2], UnsameQ[p, 3]]]], RuleDelayed[HoldPattern[Int[Power[Pattern[v, Blank[]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[a, Coefficient[v, x, 0]], Set[b, Coefficient[v, x, 1]], Set[c, Coefficient[v, x, 2]], Set[d, Coefficient[v, x, 3]], Set[e, Coefficient[v, x, 4]]], Condition[Subst[Int[SimplifyIntegrand[Power[Plus[a, Times[Power[d, 4], Power[Times[256, Power[e, 3]], -1]], Times[-1, b, d, Power[Times[8, e], -1]], Times[Plus[c, Times[-1, 3, Power[d, 2], Power[Times[8, e], -1]]], Power[x, 2]], Times[e, Power[x, 4]]], p], x], x], x, Plus[Times[d, Power[Times[4, e], -1]], x]], And[ZeroQ[Plus[Power[d, 3], Times[-1, 4, c, d, e], Times[8, b, Power[e, 2]]]], NonzeroQ[d]]]], And[FreeQ[p, x], PolynomialQ[v, x], Equal[Exponent[v, x], 4], UnsameQ[p, 2], UnsameQ[p, 3]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 3]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Subst[Int[SimplifyIntegrand[Times[ReplaceAll[u, Rule[x, Plus[Times[-1, d, Power[Times[4, e], -1]], x]]], Power[Plus[a, Times[Power[d, 4], Power[Times[256, Power[e, 3]], -1]], Times[-1, b, d, Power[Times[8, e], -1]], Times[Plus[c, Times[-1, 3, Power[d, 2], Power[Times[8, e], -1]]], Power[x, 2]], Times[e, Power[x, 4]]], p]], x], x], x, Plus[Times[d, Power[Times[4, e], -1]], x]], And[FreeQ[List[a, b, c, d, e, p], x], PolynomialQ[u, x], ZeroQ[Plus[Power[d, 3], Times[-1, 4, c, d, e], Times[8, b, Power[e, 2]]]], Not[PositiveIntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Power[Pattern[v, Blank[]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[a, Coefficient[v, x, 0]], Set[b, Coefficient[v, x, 1]], Set[c, Coefficient[v, x, 2]], Set[d, Coefficient[v, x, 3]], Set[e, Coefficient[v, x, 4]]], Condition[Subst[Int[SimplifyIntegrand[Times[ReplaceAll[u, Rule[x, Plus[Times[-1, d, Power[Times[4, e], -1]], x]]], Power[Plus[a, Times[Power[d, 4], Power[Times[256, Power[e, 3]], -1]], Times[-1, b, d, Power[Times[8, e], -1]], Times[Plus[c, Times[-1, 3, Power[d, 2], Power[Times[8, e], -1]]], Power[x, 2]], Times[e, Power[x, 4]]], p]], x], x], x, Plus[Times[d, Power[Times[4, e], -1]], x]], And[ZeroQ[Plus[Power[d, 3], Times[-1, 4, c, d, e], Times[8, b, Power[e, 2]]]], NonzeroQ[d]]]], And[FreeQ[p, x], PolynomialQ[u, x], PolynomialQ[v, x], Equal[Exponent[v, x], 4], Not[PositiveIntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], 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And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[Times[c, d], Times[a, e]]], PosQ[Times[a, c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 4]]], Rational[1, 2]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 4]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Sqrt[Plus[Power[b, 2], Times[-1, 4, a, c]]]]], Plus[Times[-1, Simp[Times[a, Sqrt[Plus[b, q]], Power[Times[2, Sqrt[2], Rt[Times[-1, a, c], 2], d], -1], ArcTan[Times[Sqrt[Plus[b, q]], x, Plus[b, Times[-1, q], Times[2, c, Power[x, 2]]], Power[Times[2, Sqrt[2], Rt[Times[-1, a, c], 2], Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]]], -1]]]], x]], Simp[Times[a, Sqrt[Plus[Times[-1, b], q]], Power[Times[2, Sqrt[2], Rt[Times[-1, a, c], 2], d], -1], ArcTanh[Times[Sqrt[Plus[Times[-1, b], q]], x, Plus[b, q, Times[2, c, Power[x, 2]]], Power[Times[2, Sqrt[2], Rt[Times[-1, a, c], 2], Sqrt[Plus[a, Times[b, Power[x, 2]], Times[c, Power[x, 4]]]]], -1]]]], x]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[Times[c, d], Times[a, e]]], NegQ[Times[a, c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]], Power[Plus[Pattern[e, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[a, Int[Times[1, Power[Times[Plus[Power[a, 2], Times[-1, Power[b, 2], Power[x, 2]]], Sqrt[Plus[c, Times[d, Power[x, 2]]]], Sqrt[Plus[e, Times[f, Power[x, 2]]]]], -1]], x], x], Times[-1, Dist[b, Int[Times[x, Power[Times[Plus[Power[a, 2], Times[-1, Power[b, 2], Power[x, 2]]], Sqrt[Plus[c, Times[d, Power[x, 2]]]], Sqrt[Plus[e, Times[f, Power[x, 2]]]]], -1]], x], x]]], FreeQ[List[a, b, c, d, e, f], x]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[2, Plus[Times[f, Plus[Times[5, b, c, Power[g, 2]], Times[-1, 2, Power[b, 2], g, h], Times[-1, 3, a, c, g, h], Times[2, a, b, Power[h, 2]]]], Times[c, f, Plus[Times[10, c, Power[g, 2]], Times[-1, b, g, h], Times[a, Power[h, 2]]], x], Times[9, Power[c, 2], f, g, h, Power[x, 2]], Times[3, Power[c, 2], f, Power[h, 2], Power[x, 3]], Times[-1, Plus[Times[e, g], Times[-1, d, h]], Plus[Times[5, c, g], Times[-1, 2, b, h], Times[c, h, x]], Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]]], Power[Times[15, Power[c, 2], f, Plus[g, Times[h, x]]], -1], Sqrt[Plus[d, Times[e, x], Times[f, Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]]]]], x], And[FreeQ[List[a, b, c, d, e, f, g, h], x], ZeroQ[Plus[Power[Plus[Times[e, g], Times[-1, d, h]], 2], Times[-1, Power[f, 2], Plus[Times[c, Power[g, 2]], Times[-1, b, g, h], Times[a, Power[h, 2]]]]]], ZeroQ[Plus[Times[2, Power[e, 2], g], Times[-1, 2, d, e, h], Times[-1, Power[f, 2], Plus[Times[2, c, g], Times[-1, b, h]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[u, Blank[]], Times[Optional[Pattern[f, Blank[]]], Plus[Optional[Pattern[j, Blank[]]], Times[Optional[Pattern[k, Blank[]]], Power[Pattern[v, Blank[]], Rational[1, 2]]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[g, Times[h, x]], m], Power[Plus[ExpandToSum[Plus[u, Times[f, j]], x], Times[f, k, Sqrt[ExpandToSum[v, x]]]], n]], x], And[FreeQ[List[f, g, h, j, k, m, n], x], LinearQ[u, x], QuadraticQ[v, x], Not[And[LinearMatchQ[u, x], QuadraticMatchQ[v, x], Or[ZeroQ[j], ZeroQ[Plus[f, -1]]]]], ZeroQ[Plus[Power[Plus[Times[Coefficient[u, x, 1], g], Times[-1, h, Plus[Coefficient[u, x, 0], Times[f, j]]]], 2], Times[-1, Power[f, 2], Power[k, 2], Plus[Times[Coefficient[v, x, 2], Power[g, 2]], Times[-1, Coefficient[v, x, 1], g, h], Times[Coefficient[v, x, 0], Power[h, 2]]]]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[2, Subst[Int[Times[Power[Plus[g, Times[h, Power[x, n]]], p], Plus[Times[Power[d, 2], e], Times[-1, Plus[Times[b, d], Times[-1, a, e]], Power[f, 2]], Times[-1, Plus[Times[2, d, e], Times[-1, b, Power[f, 2]]], x], Times[e, Power[x, 2]]], Power[Power[Plus[Times[-2, d, e], Times[b, Power[f, 2]], Times[2, e, x]], 2], -1]], x], x, Plus[d, Times[e, x], Times[f, Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]]]], x], And[FreeQ[List[a, b, c, d, e, f, g, h, n], x], ZeroQ[Plus[Power[e, 2], Times[-1, c, Power[f, 2]]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Times[2, e], -1]], Subst[Int[Times[Power[Plus[g, Times[h, Power[x, n]]], p], Plus[Power[d, 2], Times[a, Power[f, 2]], Times[-1, 2, d, x], Power[x, 2]], Power[Power[Plus[d, Times[-1, x]], 2], -1]], x], x, Plus[d, Times[e, x], Times[f, Sqrt[Plus[a, Times[c, Power[x, 2]]]]]]], x], And[FreeQ[List[a, c, d, e, f, g, h, n], x], ZeroQ[Plus[Power[e, 2], Times[-1, c, Power[f, 2]]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Power[Plus[Pattern[u, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[v, Blank[]], Rational[1, 2]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Power[Plus[g, Times[h, Power[Plus[ExpandToSum[u, x], Times[f, Sqrt[ExpandToSum[v, x]]]], n]]], p], x], And[FreeQ[List[f, g, h, n], x], LinearQ[u, x], QuadraticQ[v, x], Not[And[LinearMatchQ[u, x], QuadraticMatchQ[v, x]]], ZeroQ[Plus[Power[Coefficient[u, x, 1], 2], Times[-1, Coefficient[v, x, 2], Power[f, 2]]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Times[Power[2, Plus[m, 1]], Power[e, Plus[m, 1]]], -1]], Subst[Int[Times[Power[x, Plus[n, Times[-1, m], -2]], Plus[Times[a, Power[f, 2]], Power[x, 2]], Power[Plus[Times[-1, a, Power[f, 2], h], Times[2, e, g, x], Times[h, Power[x, 2]]], m]], x], x, Plus[Times[e, x], Times[f, Sqrt[Plus[a, Times[c, Power[x, 2]]]]]]], x], And[FreeQ[List[a, c, e, f, g, h, n], x], ZeroQ[Plus[Power[e, 2], Times[-1, c, Power[f, 2]]]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[i, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Times[Power[2, Plus[Times[2, m], p, 1]], Power[e, Plus[p, 1]], Power[f, Times[2, m]]], -1], Power[Times[i, Power[c, -1]], m]], Subst[Int[Times[Power[x, Plus[n, Times[-1, 2, m], Times[-1, p], -2]], Power[Plus[Times[-1, a, Power[f, 2]], Power[x, 2]], p], Power[Plus[Times[a, Power[f, 2]], Power[x, 2]], Plus[Times[2, m], 1]]], x], x, Plus[Times[e, x], Times[f, Sqrt[Plus[a, Times[c, Power[x, 2]]]]]]], x], And[FreeQ[List[a, c, e, f, g, i, n], x], ZeroQ[Plus[Power[e, 2], Times[-1, c, Power[f, 2]]]], ZeroQ[Plus[Times[c, g], Times[-1, a, i]]], IntegersQ[p, Times[2, m]], Or[IntegerQ[m], PositiveQ[Times[i, Power[c, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[i, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[2, Power[Power[f, Times[2, m]], -1], Power[Times[i, 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Times[Optional[Pattern[f, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Times[Power[2, Plus[Times[2, m], 1]], e, Power[f, Times[2, m]]], -1], Power[Times[i, Power[c, -1]], m]], Subst[Int[Times[Power[x, n], Power[Plus[Power[d, 2], Times[a, Power[f, 2]], Times[-1, 2, d, x], Power[x, 2]], Plus[Times[2, m], 1]], Power[Power[Plus[Times[-1, d], x], Times[2, Plus[m, 1]]], -1]], x], x, Plus[d, Times[e, x], Times[f, Sqrt[Plus[a, Times[c, Power[x, 2]]]]]]], x], And[FreeQ[List[a, c, d, e, f, g, i, n], x], ZeroQ[Plus[Power[e, 2], Times[-1, c, Power[f, 2]]]], ZeroQ[Plus[Times[c, g], Times[-1, a, i]]], IntegerQ[Times[2, m]], Or[IntegerQ[m], PositiveQ[Times[i, Power[c, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[i, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[i, Power[c, -1]], Plus[m, Times[-1, 1, Power[2, -1]]]], Sqrt[Plus[g, Times[h, x], Times[i, Power[x, 2]]]], Power[Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], -1]], Int[Times[Power[Plus[a, Times[b, x], Times[c, Power[x, 2]]], m], Power[Plus[d, Times[e, x], Times[f, Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, h, i, n], x], ZeroQ[Plus[Power[e, 2], Times[-1, c, Power[f, 2]]]], ZeroQ[Plus[Times[c, g], Times[-1, a, i]]], ZeroQ[Plus[Times[c, h], Times[-1, b, i]]], PositiveIntegerQ[Plus[m, Times[1, Power[2, -1]]]], Not[PositiveQ[Times[i, Power[c, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[g, Blank[]], Times[Optional[Pattern[i, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[i, Power[c, -1]], Plus[m, Times[-1, 1, Power[2, -1]]]], Sqrt[Plus[g, Times[i, Power[x, 2]]]], Power[Sqrt[Plus[a, Times[c, Power[x, 2]]]], -1]], Int[Times[Power[Plus[a, Times[c, Power[x, 2]]], m], Power[Plus[d, Times[e, x], Times[f, Sqrt[Plus[a, Times[c, Power[x, 2]]]]]], n]], x], x], And[FreeQ[List[a, c, d, e, f, g, i, n], x], ZeroQ[Plus[Power[e, 2], Times[-1, c, Power[f, 2]]]], ZeroQ[Plus[Times[c, g], Times[-1, a, i]]], PositiveIntegerQ[Plus[m, Times[1, Power[2, -1]]]], Not[PositiveQ[Times[i, Power[c, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[i, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[i, Power[c, -1]], Plus[m, Times[1, Power[2, -1]]]], Sqrt[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], Power[Sqrt[Plus[g, Times[h, x], Times[i, 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Times[b, Power[x, n]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, n], x], ZeroQ[Plus[Times[b, c], Times[-1, a, d]], 0]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[A, 2], Plus[n, -1]], Subst[Int[Times[1, Power[Plus[a, Times[Power[A, 2], b, Power[Plus[n, -1], 2], Power[x, 2]]], -1]], x], x, Times[x, Power[Plus[Times[A, Plus[n, -1]], Times[-1, B, Power[x, n]]], -1]]], x], And[FreeQ[List[a, b, c, d, A, B, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[n, -2]], ZeroQ[Plus[Times[a, Power[B, 2]], Times[-1, Power[A, 2], d, Power[Plus[n, -1], 2]]]], ZeroQ[Plus[Times[B, c], Times[2, A, d, Plus[n, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[k, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[A, 2], Plus[m, Times[-1, n], 1], Power[Plus[m, 1], -1]], Subst[Int[Times[1, Power[Plus[a, Times[Power[A, 2], b, Power[Plus[m, Times[-1, n], 1], 2], Power[x, 2]]], -1]], x], x, Times[Power[x, Plus[m, 1]], Power[Plus[Times[A, Plus[m, Times[-1, n], 1]], Times[B, Plus[m, 1], Power[x, n]]], -1]]], x], And[FreeQ[List[a, b, c, d, A, B, m, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[k, Times[-1, 2, Plus[m, 1]]]], ZeroQ[Plus[Times[a, Power[B, 2], Power[Plus[m, 1], 2]], Times[-1, Power[A, 2], d, Power[Plus[m, Times[-1, n], 1], 2]]]], ZeroQ[Plus[Times[B, c, Plus[m, 1]], Times[-1, 2, A, d, Plus[m, Times[-1, n], 1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]], Times[Pattern[g, Blank[]], Power[Pattern[x, Blank[]], Pattern[n3, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[x, Plus[Times[Power[b, 2], c, d], Times[-1, 2, a, c, Plus[Times[c, d], Times[-1, a, f]]], Times[-1, a, b, Plus[Times[c, e], Times[a, g]]], Times[Plus[Times[b, c, Plus[Times[c, d], Times[a, f]]], Times[-1, a, Power[b, 2], g], Times[-1, 2, a, c, Plus[Times[c, e], Times[-1, a, g]]]], Power[x, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[Times[a, c, n, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], x]], Times[-1, Dist[Times[1, Power[Times[a, c, n, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], Int[Times[Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Simp[Plus[Times[a, b, Plus[Times[c, e], Times[a, g]]], Times[-1, Power[b, 2], c, d, Plus[n, Times[n, p], 1]], Times[-1, 2, a, c, Plus[Times[a, f], Times[-1, c, d, Plus[Times[2, n, Plus[p, 1]], 1]]]], Times[Plus[Times[a, Power[b, 2], g, Plus[Times[n, Plus[p, 2]], 1]], Times[-1, b, c, Plus[Times[c, d], Times[a, f]], Plus[Times[n, Plus[Times[2, p], 3]], 1]], Times[-1, 2, a, c, Plus[Times[a, g, Plus[n, 1]], Times[-1, c, e, Plus[Times[n, Plus[Times[2, p], 3]], 1]]]]], Power[x, n]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[n3, Times[-1, 3, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NegativeIntegerQ[Plus[p, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[x, Plus[Times[Power[b, 2], d], Times[-1, 2, a, Plus[Times[c, d], Times[-1, a, f]]], Times[-1, a, b, e], Times[Plus[Times[b, Plus[Times[c, d], Times[a, f]]], Times[-1, 2, a, c, e]], Power[x, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[Times[a, n, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], x]], Times[-1, Dist[Times[1, Power[Times[a, n, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], Int[Times[Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Simp[Plus[Times[a, b, e], Times[-1, Power[b, 2], d, Plus[n, Times[n, p], 1]], Times[-1, 2, a, Plus[Times[a, f], Times[-1, c, d, Plus[Times[2, n, Plus[p, 1]], 1]]]], Times[-1, Plus[Times[b, Plus[Times[c, d], Times[a, f]], Plus[Times[n, Plus[Times[2, p], 3]], 1]], Times[-1, 2, a, c, e, Plus[Times[n, Plus[Times[2, p], 3]], 1]]], Power[x, n]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NegativeIntegerQ[Plus[p, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Pattern[g, Blank[]], Power[Pattern[x, Blank[]], Pattern[n3, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[x, Plus[Times[Power[b, 2], c, d], Times[-1, 2, a, Power[c, 2], d], Times[-1, a, b, Plus[Times[c, e], Times[a, g]]], Times[Plus[Times[b, Power[c, 2], d], Times[-1, a, Power[b, 2], g], Times[-1, 2, a, c, Plus[Times[c, e], Times[-1, a, g]]]], Power[x, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[Times[a, c, n, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], x]], Times[-1, Dist[Times[1, Power[Times[a, c, n, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], Int[Times[Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Simp[Plus[Times[a, b, Plus[Times[c, e], Times[a, g]]], Times[-1, Power[b, 2], c, d, Plus[n, Times[n, p], 1]], Times[2, a, Power[c, 2], d, Plus[Times[2, n, Plus[p, 1]], 1]], Times[Plus[Times[a, Power[b, 2], g, Plus[Times[n, Plus[p, 2]], 1]], Times[-1, b, Power[c, 2], d, Plus[Times[n, Plus[Times[2, p], 3]], 1]], Times[-1, 2, a, c, Plus[Times[a, g, Plus[n, 1]], Times[-1, c, e, Plus[Times[n, Plus[Times[2, p], 3]], 1]]]]], Power[x, n]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, g, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[n3, Times[-1, 3, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NegativeIntegerQ[Plus[p, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]], Times[Pattern[g, Blank[]], Power[Pattern[x, Blank[]], Pattern[n3, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[x, Plus[Times[Power[b, 2], c, d], Times[-1, 2, a, c, Plus[Times[c, d], Times[-1, a, f]]], Times[-1, Power[a, 2], b, g], Times[Plus[Times[b, c, Plus[Times[c, d], Times[a, f]]], Times[-1, a, Power[b, 2], g], Times[2, Power[a, 2], c, g]], Power[x, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[Times[a, c, n, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], x]], Times[-1, Dist[Times[1, Power[Times[a, c, n, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], Int[Times[Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Simp[Plus[Times[Power[a, 2], b, g], Times[-1, Power[b, 2], c, d, Plus[n, Times[n, p], 1]], Times[-1, 2, a, c, Plus[Times[a, f], Times[-1, c, d, Plus[Times[2, n, Plus[p, 1]], 1]]]], Times[Plus[Times[a, Power[b, 2], g, Plus[Times[n, Plus[p, 2]], 1]], Times[-1, b, c, Plus[Times[c, d], Times[a, f]], Plus[Times[n, Plus[Times[2, p], 3]], 1]], Times[-1, 2, Power[a, 2], c, g, Plus[n, 1]]], Power[x, n]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, f, g, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[n3, Times[-1, 3, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NegativeIntegerQ[Plus[p, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[x, Plus[Times[Power[b, 2], d], Times[-1, 2, a, Plus[Times[c, d], Times[-1, a, f]]], Times[b, Plus[Times[c, d], Times[a, f]], Power[x, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[Times[a, n, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], x]], Dist[Times[1, Power[Times[a, n, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], Int[Times[Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Simp[Plus[Times[Power[b, 2], d, Plus[n, Times[n, p], 1]], Times[2, a, Plus[Times[a, f], Times[-1, c, d, Plus[Times[2, n, Plus[p, 1]], 1]]]], Times[b, Plus[Times[c, d], Times[a, f]], Plus[Times[n, Plus[Times[2, p], 3]], 1], Power[x, n]]], x]], x], x]], And[FreeQ[List[a, b, c, d, f, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NegativeIntegerQ[Plus[p, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]], Plus[Pattern[d, Blank[]], Times[Pattern[g, Blank[]], Power[Pattern[x, Blank[]], Pattern[n3, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[x, Plus[Times[Power[b, 2], c, d], Times[-1, 2, a, Power[c, 2], d], Times[-1, Power[a, 2], b, g], Times[Plus[Times[b, Power[c, 2], d], Times[-1, a, Power[b, 2], g], Times[2, Power[a, 2], c, g]], Power[x, n]]], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[Times[a, c, n, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], x]], Times[-1, Dist[Times[1, Power[Times[a, c, n, Plus[p, 1], Plus[Power[b, 2], Times[-1, 4, a, c]]], -1]], Int[Times[Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Simp[Plus[Times[Power[a, 2], b, g], Times[-1, Power[b, 2], c, d, Plus[n, Times[n, p], 1]], Times[2, a, Power[c, 2], d, Plus[Times[2, n, Plus[p, 1]], 1]], Times[Plus[Times[a, Power[b, 2], g, Plus[Times[n, Plus[p, 2]], 1]], Times[-1, b, Power[c, 2], d, Plus[Times[n, Plus[Times[2, p], 3]], 1]], Times[-1, 2, a, c, Times[a, g, Plus[n, 1]]]], Power[x, n]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, g, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[n3, Times[-1, 3, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], NegativeIntegerQ[Plus[p, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]], Times[Pattern[g, Blank[]], Power[Pattern[x, Blank[]], Pattern[n3, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[x, Plus[Times[-2, a, c, Plus[Times[c, d], Times[-1, a, f]]], Times[Times[-2, a, c, Plus[Times[c, e], Times[-1, a, g]]], Power[x, n]]], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[Times[a, c, n, Plus[p, 1], Times[-4, a, c]], -1]], x]], Times[-1, Dist[Times[1, Power[Times[a, c, n, Plus[p, 1], Times[-4, a, c]], -1]], Int[Times[Power[Plus[a, Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Simp[Plus[Times[-2, a, c, Plus[Times[a, f], Times[-1, c, d, Plus[Times[2, n, Plus[p, 1]], 1]]]], Times[Times[-2, a, c, Plus[Times[a, g, Plus[n, 1]], Times[-1, c, e, Plus[Times[n, Plus[Times[2, p], 3]], 1]]]], Power[x, n]]], x]], x], x]]], And[FreeQ[List[a, c, d, e, f, g, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[n3, Times[-1, 3, n]]], NegativeIntegerQ[Plus[p, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[x, Plus[Times[-2, a, Plus[Times[c, d], Times[-1, a, f]]], Times[Times[-2, a, c, e], Power[x, n]]], Power[Plus[a, Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Power[Times[a, n, Plus[p, 1], Times[-4, a, c]], -1]], x]], Times[-1, Dist[Times[1, Power[Times[a, n, Plus[p, 1], Times[-4, a, c]], -1]], Int[Times[Power[Plus[a, Times[c, Power[x, Times[2, n]]]], Plus[p, 1]], Simp[Plus[Times[-2, a, Plus[Times[a, f], Times[-1, c, d, Plus[Times[2, n, Plus[p, 1]], 1]]]], Times[-1, Times[-2, a, c, e, Plus[Times[n, Plus[Times[2, p], 3]], 1]], Power[x, n]]], x]], x], x]]], And[FreeQ[List[a, c, d, e, f, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NegativeIntegerQ[Plus[p, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n2, Blank[]]]]], Pattern[p, Blank[]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, 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Power[Plus[Optional[Pattern[g, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Power[Pattern[x, Blank[]], Pattern[r, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[r, -1]], Subst[Int[Times[Power[Plus[g, Times[h, x]], m], Power[Plus[a, Times[b, Log[Times[c, Power[Times[d, Power[Plus[e, Times[f, x]], p]], q]]]]], n]], x], x, Power[x, r]], x], And[FreeQ[List[a, b, c, d, e, f, g, h, m, n, p, q, r], x], ZeroQ[Plus[r1, Times[-1, Plus[r, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, IntHide[Times[1, Power[Plus[d, Times[e, Power[x, 2]]], -1]], x]]], Plus[Dist[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]], u, x], Times[-1, Dist[Times[b, n], Int[Times[u, Power[x, -1]], x], x]]]], FreeQ[List[a, b, c, d, e, n], x]]], RuleDelayed[HoldPattern[Int[Times[Log[Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[mn, Blank[]]]]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[1, Power[Times[d, n], -1], PolyLog[2, Times[-1, Together[Times[b, c, Plus[d, Times[e, Power[x, n]]], Power[Times[d, Power[x, n]], -1]]]]]], x], And[FreeQ[List[a, b, c, d, e, n], x], ZeroQ[Plus[n, mn]], ZeroQ[Plus[d, Times[-1, a, c, d], Times[b, c, e]]]]]], RuleDelayed[HoldPattern[Int[Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Pattern[mn, Blank[]]], Plus[Optional[Pattern[b, Blank[]]], Times[Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[1, Power[Times[d, n], -1], PolyLog[2, Times[-1, Together[Times[b, c, Plus[d, Times[e, Power[x, n]]], Power[Times[d, Power[x, n]], -1]]]]]], x], And[FreeQ[List[a, b, c, d, e, n], x], ZeroQ[Plus[n, mn]], ZeroQ[Plus[d, Times[-1, a, c, d], Times[b, c, e]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[Px, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Px, Power[Plus[a, Times[b, Log[Times[c, Power[Times[d, Power[Plus[e, Times[f, x]], p]], q]]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, f, n, p, q], x], PolynomialQ[Px, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[RFx, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, ExpandIntegrand[Power[Plus[a, Times[b, Log[Times[c, Power[Times[d, Power[Plus[e, Times[f, x]], p]], q]]]]], n], RFx, x]]], Condition[Int[u, x], SumQ[u]]], And[FreeQ[List[a, b, c, d, e, f, p, q], x], RationalFunctionQ[RFx, x], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[RFx, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, ExpandIntegrand[Times[RFx, Power[Plus[a, Times[b, Log[Times[c, Power[Times[d, Power[Plus[e, Times[f, x]], p]], q]]]]], n]], x]]], Condition[Int[u, x], SumQ[u]]], And[FreeQ[List[a, b, c, d, e, f, p, q], x], RationalFunctionQ[RFx, x], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[u, Power[Plus[a, Times[b, Log[Times[c, Power[Times[d, Power[Plus[f, Times[2, g, x]], Times[2, p]], Power[Times[Power[4, p], Power[g, p]], -1]], q]]]]], n]], x], And[FreeQ[List[a, b, c, d, e, f, g, q, n], x], ZeroQ[Plus[Power[f, 2], Times[-1, 4, e, g]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[p, Blank[]]]]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[u, Power[Plus[a, Times[b, Log[Times[c, Power[Times[d, Power[ExpandToSum[v, x], p]], q]]]]], n]], x], And[FreeQ[List[a, b, c, d, n, p, q], x], LinearQ[v, x], Not[MatchQ[Times[c, Power[Times[d, Power[v, p]], q]], Condition[Times[Optional[Pattern[cc, Blank[]]], Power[Times[Optional[Pattern[dd, Blank[]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], x]], Optional[Pattern[pp, Blank[]]]]], Optional[Pattern[qq, Blank[]]]]], FreeQ[List[cc, dd, e, f, pp, qq], x]]]]]]], RuleDelayed[HoldPattern[Int[Power[Log[Times[Optional[Pattern[a, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Subst[Int[Power[Log[Power[x, Times[n, p, q]]], r], x], Power[x, Times[n, p, q]], Times[a, Power[Times[b, Power[Times[c, Power[x, n]], p]], q]]], FreeQ[List[a, b, c, n, p, q, r], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Times[Optional[Pattern[a, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[q, Blank[]]]]], Optional[Pattern[r, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Subst[Int[Times[Power[x, m], Power[Log[Power[x, Times[n, p, q]]], r]], x], Power[x, Times[n, p, q]], Times[a, Power[Times[b, Power[Times[c, Power[x, n]], p]], q]]], And[FreeQ[List[a, b, c, m, n, p, q, r], x], NonzeroQ[Plus[m, 1]], Not[SameQ[Power[x, Times[n, p, q]], Times[a, Power[Times[b, Power[Times[c, Power[x, n]], p]], q]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[e1, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[Log[Times[e, Power[Times[e1, b, Power[d, -1]], n]]], p], Int[u, x], x], And[FreeQ[List[a, b, c, d, e, n, p, e1], x], ZeroQ[Plus[Times[b, c], Times[-1, a, d]]]]]], RuleDelayed[HoldPattern[Int[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[e1, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n1, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n2, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[a, Times[b, x]], Power[b, -1], Power[Log[Times[e, Power[Times[e1, Power[Plus[a, Times[b, x]], n1], Power[Power[Plus[c, Times[d, x]], n1], -1]], n]]], p]], x], Times[-1, Dist[Times[n, n1, p, Plus[Times[b, c], Times[-1, a, d]], Power[b, -1]], Int[Times[Power[Log[Times[e, Power[Times[e1, Power[Plus[a, Times[b, x]], n1], Power[Power[Plus[c, Times[d, x]], n1], -1]], n]]], Plus[p, -1]], Power[Plus[c, Times[d, x]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, n, e1, n1], x], ZeroQ[Plus[n1, n2]], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], PositiveIntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Log[Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[1, Power[g, -1], PolyLog[2, Times[Together[Plus[c, Times[-1, a, e]]], Power[Plus[c, Times[d, x]], -1]]]], x], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Times[d, f], Times[-1, c, g]]], ZeroQ[Plus[d, Times[-1, b, e]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[e1, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n1, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n2, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[g, -1], Log[Times[Plus[Times[b, c], Times[-1, a, d]], Power[Times[b, Plus[c, Times[d, x]]], -1]]], Power[Log[Times[e, Power[Times[e1, Power[Plus[a, Times[b, x]], n1], Power[Power[Plus[c, Times[d, x]], n1], -1]], n]]], p]], x]], Dist[Times[n, n1, p, Plus[Times[b, c], Times[-1, a, d]], Power[g, -1]], Int[Times[Log[Times[Plus[Times[b, c], Times[-1, a, d]], Power[Times[b, Plus[c, Times[d, x]]], -1]]], Power[Times[Plus[a, Times[b, x]], Plus[c, Times[d, x]]], -1], Power[Log[Times[e, Power[Times[e1, Power[Plus[a, Times[b, x]], n1], Power[Power[Plus[c, Times[d, x]], n1], -1]], n]]], Plus[p, -1]]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, n, e1, n1], x], ZeroQ[Plus[n1, n2]], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Times[d, f], Times[-1, c, g]]], PositiveIntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[e1, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n1, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n2, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[g, -1], Log[Times[-1, Plus[Times[b, c], Times[-1, a, d]], Power[Times[d, Plus[a, Times[b, x]]], -1]]], Power[Log[Times[e, Power[Times[e1, Power[Plus[a, Times[b, x]], n1], Power[Power[Plus[c, Times[d, x]], n1], -1]], n]]], p]], x]], Dist[Times[n, n1, p, Plus[Times[b, c], Times[-1, a, d]], Power[g, -1]], Int[Times[Log[Times[-1, Plus[Times[b, c], Times[-1, a, d]], Power[Times[d, Plus[a, Times[b, x]]], -1]]], Power[Times[Plus[a, Times[b, x]], Plus[c, Times[d, x]]], -1], Power[Log[Times[e, Power[Times[e1, Power[Plus[a, Times[b, x]], n1], Power[Power[Plus[c, Times[d, x]], n1], -1]], n]]], Plus[p, -1]]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, n, e1, n1], x], ZeroQ[Plus[n1, n2]], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Times[b, f], Times[-1, a, g]]], PositiveIntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[e1, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n1, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n2, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Log[Plus[f, Times[g, x]]], Power[g, -1], Log[Times[e, Power[Times[e1, Power[Plus[a, Times[b, x]], n1], Power[Power[Plus[c, Times[d, x]], n1], -1]], n]]]], x], Times[-1, Dist[Times[n, n1, Plus[Times[b, c], Times[-1, a, d]], Power[g, -1]], Int[Times[Log[Plus[f, Times[g, x]]], Power[Times[Plus[a, Times[b, x]], Plus[c, Times[d, x]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g, n, e1, n1], x], ZeroQ[Plus[n1, n2]], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Times[d, f], Times[-1, c, g]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[e1, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n1, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n2, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[d, Power[g, -1]], Int[Times[1, Power[Plus[c, Times[d, x]], -1], Power[Log[Times[e, Power[Times[e1, Power[Plus[a, Times[b, x]], n1], Power[Power[Plus[c, Times[d, x]], n1], -1]], n]]], p]], x], x], Times[-1, Dist[Times[Plus[Times[d, f], Times[-1, c, g]], Power[g, -1]], Int[Times[Times[1, Power[Times[Plus[c, Times[d, x]], Plus[f, Times[g, x]]], -1]], Power[Log[Times[e, Power[Times[e1, Power[Plus[a, Times[b, x]], n1], Power[Power[Plus[c, Times[d, x]], n1], -1]], n]]], p]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g, n, e1, n1], x], ZeroQ[Plus[n1, n2]], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Times[d, f], Times[-1, c, g]]], IntegerQ[p], Greater[p, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], -2]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[d, 2], Power[Times[Plus[Times[b, c], Times[-1, a, d]], e, Power[g, 2]], -1], LogIntegral[Times[e, Plus[a, Times[b, x]], Power[Plus[c, Times[d, x]], -1]]]], x], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Times[d, f], Times[-1, c, g]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[e1, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n1, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n2, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], -2]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[d, 2], Plus[a, Times[b, x]], Power[Times[Power[g, 2], n, n1, Plus[Times[b, c], Times[-1, a, d]], Plus[c, Times[d, x]]], -1], ExpIntegralEi[Times[1, Power[Times[n, n1], -1], Log[Times[e, Power[Times[e1, Power[Plus[a, Times[b, x]], n1], Power[Power[Plus[c, Times[d, x]], n1], -1]], n]]]]], Power[Power[Times[e, Power[Times[e1, Power[Plus[a, Times[b, x]], n1], Power[Plus[c, Times[d, x]], n2]], n]], Times[1, Power[Times[n, n1], -1]]], -1]], x], And[FreeQ[List[a, b, c, d, e, f, g, n, e1, n1], x], ZeroQ[Plus[n1, n2]], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Times[d, f], Times[-1, c, g]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[e1, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n1, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n2, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], -2]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[b, 2], Plus[c, Times[d, x]], Power[Times[Power[g, 2], n, n1, Plus[Times[b, c], Times[-1, a, d]], Plus[a, Times[b, x]]], -1], Power[Times[e, Power[Times[e1, Power[Plus[a, Times[b, x]], n1], Power[Plus[c, Times[d, x]], n2]], n]], Times[1, Power[Times[n, n1], -1]]], ExpIntegralEi[Times[-1, Power[Times[n, n1], -1], Log[Times[e, Power[Times[e1, Power[Plus[a, Times[b, x]], n1], Power[Power[Plus[c, Times[d, x]], n1], -1]], n]]]]]], x], And[FreeQ[List[a, b, c, d, e, f, g, n, e1, n1], x], ZeroQ[Plus[n1, n2]], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Times[b, f], Times[-1, a, g]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[e1, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n1, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n2, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], -2]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[a, Times[b, x]], Power[Times[Plus[Times[b, f], Times[-1, a, g]], Plus[f, Times[g, x]]], -1], Power[Log[Times[e, Power[Times[e1, Power[Plus[a, Times[b, x]], n1], Power[Power[Plus[c, Times[d, x]], n1], -1]], n]]], p]], x], Times[-1, Dist[Times[n, n1, p, Plus[Times[b, c], Times[-1, a, d]], Power[Plus[Times[b, f], Times[-1, a, g]], -1]], Int[Times[1, Power[Times[Plus[c, Times[d, x]], Plus[f, Times[g, x]]], -1], Power[Log[Times[e, Power[Times[e1, Power[Plus[a, Times[b, x]], n1], Power[Power[Plus[c, Times[d, x]], n1], -1]], n]]], Plus[p, -1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g, n, e1, n1], x], ZeroQ[Plus[n1, n2]], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Times[b, f], Times[-1, a, g]]], PositiveIntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[e1, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n1, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n2, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], -2]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[c, Times[d, x]], Power[Times[Plus[Times[d, f], Times[-1, c, g]], Plus[f, Times[g, x]]], -1], Power[Log[Times[e, Power[Times[e1, Power[Plus[a, Times[b, x]], n1], Power[Power[Plus[c, Times[d, x]], n1], -1]], n]]], p]], x], Times[-1, Dist[Times[n, n1, p, Plus[Times[b, c], Times[-1, a, d]], Power[Plus[Times[d, f], Times[-1, c, g]], -1]], Int[Times[1, Power[Times[Plus[a, Times[b, x]], Plus[f, Times[g, x]]], -1], Power[Log[Times[e, Power[Times[e1, Power[Plus[a, Times[b, x]], n1], Power[Power[Plus[c, Times[d, x]], n1], -1]], n]]], Plus[p, -1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g, n, e1, n1], x], ZeroQ[Plus[n1, n2]], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Times[d, f], Times[-1, c, g]]], PositiveIntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[e1, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n1, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n2, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], -3]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[b, Power[Plus[Times[b, f], Times[-1, a, g]], -1]], Int[Times[1, Power[Power[Plus[f, Times[g, x]], 2], -1], Power[Log[Times[e, Power[Times[e1, Power[Plus[a, Times[b, x]], n1], Power[Power[Plus[c, Times[d, x]], n1], -1]], n]]], p]], x], x], Times[-1, Dist[Times[g, Power[Plus[Times[b, f], Times[-1, a, g]], -1]], Int[Times[Plus[a, Times[b, x]], Power[Power[Plus[f, Times[g, x]], 3], -1], Power[Log[Times[e, Power[Times[e1, Power[Plus[a, Times[b, x]], n1], Power[Power[Plus[c, Times[d, x]], n1], -1]], n]]], p]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g, n, p, e1, n1], x], ZeroQ[Plus[n1, n2]], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Times[b, f], Times[-1, a, g]]], ZeroQ[Plus[Times[d, f], Times[-1, c, g]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[e1, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n1, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n2, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], -3]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[d, Power[Plus[Times[d, f], Times[-1, c, g]], -1]], Int[Times[1, Power[Power[Plus[f, Times[g, x]], 2], -1], Power[Log[Times[e, Power[Times[e1, Power[Plus[a, Times[b, x]], n1], Power[Power[Plus[c, Times[d, x]], n1], -1]], n]]], p]], x], x], Times[-1, Dist[Times[g, Power[Plus[Times[d, f], Times[-1, c, g]], -1]], Int[Times[Plus[c, Times[d, x]], Power[Power[Plus[f, Times[g, x]], 3], -1], Power[Log[Times[e, Power[Times[e1, Power[Plus[a, Times[b, x]], n1], Power[Power[Plus[c, Times[d, x]], n1], -1]], n]]], p]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g, n, p, e1, n1], x], ZeroQ[Plus[n1, n2]], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Times[d, f], Times[-1, c, g]]], ZeroQ[Plus[Times[b, f], Times[-1, a, g]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[e1, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n1, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n2, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[f, Times[g, x]], Plus[m, 1]], Power[Times[g, Plus[m, 1]], -1], Power[Log[Times[e, Power[Times[e1, Power[Plus[a, Times[b, x]], n1], Power[Power[Plus[c, Times[d, x]], n1], -1]], n]]], p]], x], Times[-1, Dist[Times[n, n1, p, Plus[Times[b, c], Times[-1, a, d]], Power[Times[g, Plus[m, 1]], -1]], Int[Times[Power[Plus[f, Times[g, x]], Plus[m, 1]], Power[Times[Plus[a, Times[b, x]], Plus[c, Times[d, x]]], -1], Power[Log[Times[e, Power[Times[e1, Power[Plus[a, Times[b, x]], n1], Power[Power[Plus[c, Times[d, x]], n1], -1]], n]]], Plus[p, -1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g, n, e1, n1], x], ZeroQ[Plus[n1, n2]], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], PositiveIntegerQ[p], IntegerQ[m], NonzeroQ[Plus[m, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m2, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Log[Times[e, Power[u, n]]], p], Power[Times[Plus[m, 1], Plus[Times[b, c], Times[-1, a, d]], Power[Plus[c, Times[d, x]], Plus[m, 1]]], -1]], x], Times[-1, Dist[Times[n, p, Power[Plus[m, 1], -1]], Int[Times[Power[Plus[a, Times[b, x]], m], Power[Power[Plus[c, Times[d, x]], Plus[m, 2]], -1], Power[Log[Times[e, Power[u, n]]], Plus[p, -1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, n, Simplify[Times[u, Plus[c, Times[d, x]], Power[Plus[a, Times[b, x]], -1]]]], x], ZeroQ[Plus[m2, m, 2]], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[m, 1]], RationalQ[p], Greater[p, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Pattern[u, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m2, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Log[u], p], Power[Times[Plus[m, 1], Plus[Times[b, c], Times[-1, a, d]], Power[Plus[c, Times[d, x]], Plus[m, 1]]], -1]], x], Times[-1, Dist[Times[p, Power[Plus[m, 1], -1]], Int[Times[Power[Plus[a, Times[b, x]], m], Power[Power[Plus[c, Times[d, x]], Plus[m, 2]], -1], Power[Log[u], Plus[p, -1]]], x], x]]], And[FreeQ[List[a, b, c, d, Simplify[Times[u, Plus[c, Times[d, x]], Power[Plus[a, Times[b, x]], -1]]]], x], ZeroQ[Plus[m2, m, 2]], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[m, 1]], RationalQ[p], Greater[p, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m2, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Times[n, Plus[Times[b, c], Times[-1, a, d]], Power[Plus[c, Times[d, x]], Plus[m, 1]]], -1], ExpIntegralEi[Times[Plus[m, 1], Power[n, -1], Log[Times[e, Power[u, n]]]]], Power[Power[Times[e, Power[u, n]], Times[Plus[m, 1], Power[n, -1]]], -1]], x], And[FreeQ[List[a, b, c, d, e, n, Simplify[Times[u, Plus[c, Times[d, x]], Power[Plus[a, Times[b, x]], -1]]]], x], ZeroQ[Plus[m2, m, 2]], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[m, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Pattern[u, Blank[]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m2, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Times[Plus[Times[b, c], Times[-1, a, d]], Power[Plus[c, Times[d, x]], Plus[m, 1]]], -1], ExpIntegralEi[Times[Plus[m, 1], Log[u]]], Power[Power[u, Plus[m, 1]], -1]], x], And[FreeQ[List[a, b, c, d, Simplify[Times[u, Plus[c, Times[d, x]], Power[Plus[a, Times[b, x]], -1]]]], x], ZeroQ[Plus[m2, m, 2]], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[m, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m2, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Log[Times[e, Power[u, n]]], Plus[p, 1]], Power[Times[n, Plus[p, 1], Plus[Times[b, c], Times[-1, a, d]], Power[Plus[c, Times[d, x]], Plus[m, 1]]], -1]], x], Times[-1, Dist[Times[Plus[m, 1], Power[Times[n, Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[b, x]], m], Power[Power[Plus[c, Times[d, x]], Plus[m, 2]], -1], Power[Log[Times[e, Power[u, n]]], Plus[p, 1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, n, Simplify[Times[u, Plus[c, Times[d, x]], Power[Plus[a, Times[b, x]], -1]]]], x], ZeroQ[Plus[m2, m, 2]], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[m, 1]], RationalQ[p], Less[p, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Pattern[u, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m2, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[a, Times[b, x]], Plus[m, 1]], Power[Log[u], Plus[p, 1]], Power[Times[Plus[p, 1], Plus[Times[b, c], Times[-1, a, d]], Power[Plus[c, Times[d, x]], Plus[m, 1]]], -1]], x], Times[-1, Dist[Times[Plus[m, 1], Power[Plus[p, 1], -1]], Int[Times[Power[Plus[a, Times[b, x]], m], Power[Power[Plus[c, Times[d, x]], Plus[m, 2]], -1], Power[Log[u], Plus[p, 1]]], x], x]]], And[FreeQ[List[a, b, c, d, Simplify[Times[u, Plus[c, Times[d, x]], Power[Plus[a, Times[b, x]], -1]]]], x], ZeroQ[Plus[m2, m, 2]], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[m, 1]], RationalQ[p], Less[p, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[e1, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n1, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n2, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[d, Power[g, -1]], Int[Times[1, Power[Power[Plus[c, Times[d, x]], 2], -1], Power[Log[Times[e, Power[Times[e1, Power[Plus[a, Times[b, x]], n1], Power[Power[Plus[c, Times[d, x]], n1], -1]], n]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, n, e1, n1], x], ZeroQ[Plus[n1, n2]], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Times[b, f], Times[-1, a, g]]], ZeroQ[Plus[Times[d, f], Times[-1, c, g]]]]]], RuleDelayed[HoldPattern[Int[Times[Log[Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[1, Power[Plus[Times[d, f], Times[-1, c, g]], -1], PolyLog[2, Simplify[Times[Plus[c, Times[-1, a, e]], Plus[f, Times[g, x]], Power[Times[f, Plus[c, Times[d, x]]], -1]]]]], x], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Times[b, f], Times[-1, a, g]]], NonzeroQ[Plus[Times[d, f], Times[-1, c, g]]], ZeroQ[Plus[Times[d, f], Times[-1, c, g], Times[-1, e, Plus[Times[b, f], Times[-1, a, g]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[e1, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n1, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n2, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Plus[Times[d, f], Times[-1, c, g]], -1], Log[Times[Plus[Times[b, c], Times[-1, a, d]], Plus[f, Times[g, x]], Power[Times[Plus[Times[b, f], Times[-1, a, g]], Plus[c, Times[d, x]]], -1]]], Power[Log[Times[e, Power[Times[e1, Power[Plus[a, Times[b, x]], n1], Power[Power[Plus[c, Times[d, x]], n1], -1]], n]]], p]], x]], Dist[Times[n, n1, p, Plus[Times[b, c], Times[-1, a, d]], Power[Plus[Times[d, f], Times[-1, c, g]], -1]], Int[Times[Times[1, Power[Times[Plus[a, Times[b, x]], Plus[c, Times[d, x]]], -1]], Log[Times[Plus[Times[b, c], Times[-1, a, d]], Plus[f, Times[g, x]], Power[Times[Plus[Times[b, f], Times[-1, a, g]], Plus[c, Times[d, x]]], -1]]], Power[Log[Times[e, Power[Times[e1, Power[Plus[a, Times[b, x]], n1], Power[Power[Plus[c, Times[d, x]], n1], -1]], n]]], Plus[p, -1]]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, n, e1, n1], x], ZeroQ[Plus[n1, n2]], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Times[b, f], Times[-1, a, g]]], NonzeroQ[Plus[Times[d, f], Times[-1, c, g]]], PositiveIntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Log[Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[c, Power[Times[2, d, f], -1], PolyLog[2, Simplify[Times[Plus[c, Times[-1, a, e]], Plus[c, Times[-1, d, x]], Power[Times[c, Plus[c, Times[d, x]]], -1]]]]], x], And[FreeQ[List[a, b, c, d, e, f, g], x], ZeroQ[Plus[Times[Power[d, 2], f], Times[Power[c, 2], g]]], ZeroQ[Plus[Times[2, c, d], Times[-1, b, c, e], Times[-1, a, d, e]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Times[Optional[Pattern[e1, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n1, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n2, Blank[]]]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[d, 2], Int[Times[1, Power[Times[Plus[c, Times[d, x]], Plus[Times[d, g], 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Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[b, Power[Times[g, n, n1, Plus[Times[b, c], Times[-1, a, d]]], -1]], Subst[Int[Power[x, p], x], x, Log[Times[e, Power[Times[e1, Power[Plus[a, Times[b, x]], n1], Power[Power[Plus[c, Times[d, x]], n1], -1]], n]]]], x], And[FreeQ[List[a, b, c, d, e, f, g, n, p, e1, n1], x], ZeroQ[Plus[n1, n2]], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Times[d, f], Times[-1, c, g]]], ZeroQ[Plus[Times[b, f], Times[-1, a, g]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Log[Pattern[v, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[PolyLog[2, Together[Plus[1, Times[-1, v]]]], Power[Plus[Times[b, c], Times[-1, a, d]], -1], Power[Log[Times[e, Power[u, n]]], p]], x]], Dist[Times[n, p], Int[Times[PolyLog[2, Together[Plus[1, Times[-1, v]]]], Power[Times[Plus[a, Times[b, x]], Plus[c, Times[d, x]]], -1], Power[Log[Times[e, Power[u, n]]], Plus[p, -1]]], x], x]], And[FreeQ[List[a, b, c, d, e, n, Simplify[Times[u, Power[Plus[1, Times[-1, v]], -1]]], Simplify[Times[u, Plus[c, Times[d, x]], Power[Plus[a, Times[b, x]], -1]]]], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[p], Greater[p, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Pattern[u, Blank[]]], Optional[Pattern[p, Blank[]]]], Log[Pattern[v, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[PolyLog[2, Together[Plus[1, Times[-1, v]]]], Power[Plus[Times[b, c], Times[-1, a, d]], -1], Power[Log[u], p]], x]], Dist[p, Int[Times[PolyLog[2, Together[Plus[1, Times[-1, v]]]], Power[Times[Plus[a, Times[b, x]], Plus[c, Times[d, x]]], -1], Power[Log[u], Plus[p, -1]]], x], x]], And[FreeQ[List[a, b, c, d, Simplify[Times[u, Power[Plus[1, Times[-1, v]], -1]]], Simplify[Times[u, Plus[c, Times[d, x]], Power[Plus[a, Times[b, x]], -1]]]], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[p], Greater[p, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Log[Pattern[v, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[f, Simplify[Times[Plus[1, Times[-1, v]], Power[u, -1]]]]], Plus[Simp[Times[Log[v], Power[Times[n, Plus[p, 1], Plus[Times[b, c], Times[-1, a, d]]], -1], Power[Log[Times[e, Power[u, n]]], Plus[p, 1]]], x], Dist[Times[f, Power[Times[n, Plus[p, 1]], -1]], Int[Times[1, Power[Times[Plus[c, Times[d, x]], Plus[c, Times[-1, a, f], Plus[d, Times[-1, b, f]]]], -1], Power[Log[Times[e, Power[u, n]]], Plus[p, 1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, n, Simplify[Times[u, Power[Plus[1, Times[-1, v]], -1]]], Simplify[Times[u, Plus[c, Times[d, x]], Power[Plus[a, Times[b, x]], -1]]]], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[p], Less[p, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Pattern[u, Blank[]]], Pattern[p, Blank[]]], Log[Pattern[v, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[f, Simplify[Times[Plus[1, Times[-1, v]], Power[u, -1]]]]], Plus[Simp[Times[Log[v], Power[Times[Plus[p, 1], Plus[Times[b, c], Times[-1, a, d]]], -1], Power[Log[u], Plus[p, 1]]], x], Dist[Times[f, Power[Plus[p, 1], -1]], Int[Times[1, Power[Times[Plus[c, Times[d, x]], Plus[c, Times[-1, a, f], Plus[d, Times[-1, b, f]]]], -1], Power[Log[u], Plus[p, 1]]], x], x]]], And[FreeQ[List[a, b, c, d, Simplify[Times[u, Power[Plus[1, Times[-1, v]], -1]]], Simplify[Times[u, Plus[c, Times[d, x]], Power[Plus[a, Times[b, x]], -1]]]], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[p], Less[p, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Log[Pattern[v, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, 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Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[PolyLog[2, Together[Plus[1, Times[-1, v]]]], Power[Plus[Times[b, c], Times[-1, a, d]], -1], Power[Log[u], p]], x], Times[-1, Dist[p, Int[Times[PolyLog[2, Together[Plus[1, Times[-1, v]]]], Power[Times[Plus[a, Times[b, x]], Plus[c, Times[d, x]]], -1], Power[Log[u], Plus[p, -1]]], x], x]]], And[FreeQ[List[a, b, c, d, Simplify[Times[u, Plus[1, Times[-1, v]]]], Simplify[Times[u, Plus[c, Times[d, x]], Power[Plus[a, Times[b, x]], -1]]]], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[p], Greater[p, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Log[Pattern[v, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, 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Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], PolyLog[Pattern[q, Blank[]], Pattern[v, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[PolyLog[Plus[q, 1], v], Power[Plus[Times[b, c], Times[-1, a, d]], -1], Power[Log[Times[e, Power[u, n]]], p]], x], Times[-1, Dist[Times[n, p], Int[Times[PolyLog[Plus[q, 1], v], Power[Times[Plus[a, Times[b, x]], Plus[c, Times[d, x]]], -1], Power[Log[Times[e, Power[u, n]]], Plus[p, -1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, n, q, Simplify[Times[u, Power[v, -1]]], Simplify[Times[u, Plus[c, Times[d, x]], Power[Plus[a, Times[b, x]], -1]]]], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[p], Greater[p, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Pattern[u, Blank[]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], PolyLog[Pattern[q, Blank[]], Pattern[v, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[PolyLog[Plus[q, 1], v], Power[Plus[Times[b, c], Times[-1, a, d]], -1], Power[Log[u], p]], x], Times[-1, Dist[p, Int[Times[PolyLog[Plus[q, 1], v], Power[Times[Plus[a, Times[b, x]], Plus[c, Times[d, x]]], -1], Power[Log[u], Plus[p, -1]]], x], x]]], And[FreeQ[List[a, b, c, d, q, Simplify[Times[u, Power[v, -1]]], Simplify[Times[u, Plus[c, Times[d, x]], Power[Plus[a, Times[b, x]], -1]]]], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[p], Greater[p, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, 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PolyLog[Pattern[q, Blank[]], Pattern[v, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[PolyLog[q, v], Power[Times[Plus[p, 1], Plus[Times[b, c], Times[-1, a, d]]], -1], Power[Log[u], Plus[p, 1]]], x], Times[-1, Dist[Times[1, Power[Plus[p, 1], -1]], Int[Times[PolyLog[Plus[q, -1], v], Power[Times[Plus[a, Times[b, x]], Plus[c, Times[d, x]]], -1], Power[Log[u], Plus[p, 1]]], x], x]]], And[FreeQ[List[a, b, c, d, q, Simplify[Times[u, Power[v, -1]]], Simplify[Times[u, Plus[c, Times[d, x]], Power[Plus[a, Times[b, x]], -1]]]], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[p], Less[p, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Times[Optional[Pattern[e, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], PolyLog[Pattern[q, Blank[]], Pattern[v, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[PolyLog[Plus[q, 1], v], Power[Plus[Times[b, c], Times[-1, a, d]], -1], Power[Log[Times[e, Power[u, n]]], p]], x]], Dist[Times[n, p], Int[Times[PolyLog[Plus[q, 1], v], Power[Times[Plus[a, Times[b, x]], Plus[c, Times[d, x]]], -1], Power[Log[Times[e, Power[u, n]]], Plus[p, -1]]], x], x]], And[FreeQ[List[a, b, c, d, e, n, q, Simplify[Times[u, v]], Simplify[Times[u, Plus[c, Times[d, x]], Power[Plus[a, Times[b, x]], -1]]]], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[p], Greater[p, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Pattern[u, Blank[]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], PolyLog[Pattern[q, Blank[]], Pattern[v, Blank[]]]], Pattern[x, 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Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Times[Optional[Pattern[g, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[a, -1]], Int[Times[Power[Sec[Plus[e, Times[f, x]]], 2], Power[Times[g, Tan[Plus[e, Times[f, x]]]], p]], x], x], Times[-1, Dist[Times[1, Power[Times[b, g], -1]], Int[Times[Sec[Plus[e, Times[f, x]]], Power[Times[g, Tan[Plus[e, Times[f, x]]]], Plus[p, 1]]], x], x]]], And[FreeQ[List[a, b, e, f, g, p], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[p, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[f, -1]], Subst[Int[Times[Power[x, p], Power[Plus[a, x], Plus[m, Times[-1, Plus[p, 1], Power[2, -1]]]], Power[Power[Plus[a, Times[-1, x]], Times[Plus[p, 1], Power[2, -1]]], -1]], x], x, Times[b, Sin[Plus[e, Times[f, x]]]]], x], And[FreeQ[List[a, b, e, f, m], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegerQ[Times[Plus[p, 1], Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[a, p], 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2]]]], IntegersQ[m, Times[p, Power[2, -1]]], Or[Less[p, 0], Greater[Plus[m, Times[-1, p, Power[2, -1]]], 0]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Power[Times[g, Tan[Plus[e, Times[f, x]]]], p], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], x], x], And[FreeQ[List[a, b, e, f, g, p], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], PositiveIntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], 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Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Power[Times[g, Tan[Plus[e, Times[f, x]]]], p], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], x], x], And[FreeQ[List[a, b, e, f, g, p], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], PositiveIntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -2]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Plus[1, Times[-1, Power[Sin[Plus[e, Times[f, x]]], 2]]], Power[Power[Sin[Plus[e, Times[f, x]]], 2], -1]], x], And[FreeQ[List[a, b, e, f, m], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -4]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[3, a, f, Power[Sin[Plus[e, Times[f, x]]], 3]], -1]], x]], Plus[Times[-1, Dist[Times[1, Power[Times[3, Power[a, 2], b, Plus[m, 1]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Power[Sin[Plus[e, Times[f, x]]], 3], -1], Simp[Plus[Times[6, Power[a, 2]], Times[-1, Power[b, 2], Plus[m, -1], Plus[m, -2]], Times[a, b, Plus[m, 1], Sin[Plus[e, Times[f, x]]]], Times[-1, Plus[Times[3, Power[a, 2]], Times[-1, Power[b, 2], m, Plus[m, -2]]], 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2], Plus[m, -1], Plus[Times[2, m], 1]]], Power[Sin[Plus[e, Times[f, x]]], 2]]], x]], x], x], Simp[Times[Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[b, f, m, Power[Sin[Plus[e, Times[f, x]]], 2]], -1]], x], Simp[Times[a, Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[Power[b, 2], f, m, Plus[m, -1], Power[Sin[Plus[e, Times[f, x]]], 3]], -1]], x], Times[-1, Simp[Times[b, Plus[m, -4], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[20, Power[a, 2], f, Power[Sin[Plus[e, Times[f, x]]], 4]], -1]], x]]]], And[FreeQ[List[a, b, e, f, m], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[m, -1]], IntegerQ[Times[2, m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], 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RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Times[Optional[Pattern[g, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[a, -1]], Int[Times[Power[Times[g, Tan[Plus[e, Times[f, x]]]], p], Power[Power[Cos[Plus[e, Times[f, x]]], 2], -1]], x], x], Plus[Times[-1, Dist[Times[b, Power[Times[Power[a, 2], g], -1]], Int[Times[Power[Times[g, Tan[Plus[e, Times[f, x]]]], Plus[p, 1]], Power[Cos[Plus[e, Times[f, x]]], -1]], x], x]], Times[-1, Dist[Times[Plus[Power[a, 2], Times[-1, Power[b, 2]]], Power[Times[Power[a, 2], Power[g, 2]], -1]], Int[Times[Power[Times[g, Tan[Plus[e, Times[f, x]]]], Plus[p, 2]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], -1]], x], x]]]], And[FreeQ[List[a, b, e, f, g], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegersQ[Times[2, p]], Less[p, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Times[Optional[Pattern[g, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Cos[Plus[e, Times[f, x]]]], Sqrt[Times[g, Tan[Plus[e, Times[f, x]]]]], Power[Sqrt[Sin[Plus[e, Times[f, x]]]], -1]], Int[Times[Sqrt[Sin[Plus[e, Times[f, x]]]], Power[Times[Sqrt[Cos[Plus[e, Times[f, x]]]], Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x], And[FreeQ[List[a, b, e, f, g], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], 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Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Sin[Plus[e, Times[f, x]]], p], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Power[Plus[1, Times[-1, Power[Sin[Plus[e, Times[f, x]]], 2]]], Times[p, Power[2, -1]]], -1]], x], x], And[FreeQ[List[a, b, e, f], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegersQ[m, Times[p, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], 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sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[2, a, c], Times[b, d]], x, Power[2, -1]], x], Plus[Times[-1, Simp[Times[Plus[Times[b, c], Times[a, d]], Cos[Plus[e, Times[f, x]]], Power[f, -1]], x]], Times[-1, Simp[Times[b, d, Cos[Plus[e, Times[f, x]]], Sin[Plus[e, Times[f, x]]], Power[Times[2, f], -1]], x]]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, x, Power[d, -1]], x], Times[-1, Dist[Times[Plus[Times[b, c], Times[-1, a, d]], Power[d, -1]], Int[Times[1, Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[a, m], Power[c, m]], Int[Times[Power[Cos[Plus[e, Times[f, x]]], Times[2, m]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, Times[-1, m]]]], x], x], And[FreeQ[List[a, b, c, d, e, f, n], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegerQ[m], Not[And[IntegerQ[n], Or[And[Less[m, 0], Greater[n, 0]], Less[0, n, m], Less[m, n, 0]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[a, c, Cos[Plus[e, Times[f, x]]], Power[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]]], -1]], Int[Times[Cos[Plus[e, Times[f, x]]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[-2, b, Cos[Plus[e, Times[f, x]]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Power[Times[f, Plus[Times[2, n], 1], Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], And[FreeQ[List[a, b, c, d, e, f, n], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[n, Times[1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[-2, b, Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Power[Times[f, Plus[Times[2, n], 1]], -1]], x], Times[-1, Dist[Times[b, Plus[Times[2, m], -1], Power[Times[d, Plus[Times[2, n], 1]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], PositiveIntegerQ[Plus[m, Times[-1, 1, Power[2, -1]]]], RationalQ[n], Less[n, -1], Not[And[NegativeIntegerQ[Plus[m, n]], Greater[Plus[Times[2, m], n, 1], 0]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[b, Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Power[Times[f, Plus[m, n]], -1]], x]], Dist[Times[a, Plus[Times[2, m], -1], Power[Plus[m, n], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n]], x], x]], And[FreeQ[List[a, b, c, d, e, f, n], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], PositiveIntegerQ[Plus[m, Times[-1, 1, Power[2, -1]]]], Not[And[RationalQ[n], Less[n, -1]]], Not[And[PositiveIntegerQ[Plus[n, Times[-1, 1, Power[2, -1]]]], Less[n, m]]], Not[And[NegativeIntegerQ[Plus[m, n]], Greater[Plus[Times[2, m], n, 1], 0]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Cos[Plus[e, Times[f, x]]], Power[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]]], -1]], Int[Times[1, Power[Cos[Plus[e, Times[f, x]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[b, Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Power[Times[a, f, Plus[Times[2, m], 1]], -1]], x], And[FreeQ[List[a, b, c, d, e, f, m, n], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[m, n, 1]], NonzeroQ[Plus[m, Times[1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Power[Times[a, f, Plus[Times[2, m], 1]], -1]], x], Dist[Times[Plus[m, n, 1], Power[Times[a, Plus[Times[2, m], 1]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n]], x], x]], And[FreeQ[List[a, b, c, d, e, f, m, n], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NegativeIntegerQ[Simplify[Plus[m, n, 1]]], NonzeroQ[Plus[m, Times[1, Power[2, -1]]]], Or[SumSimplerQ[m, 1], Not[SumSimplerQ[n, 1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Power[Times[a, f, Plus[Times[2, m], 1]], -1]], x], Dist[Times[Plus[m, n, 1], Power[Times[a, Plus[Times[2, m], 1]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n]], x], x]], And[FreeQ[List[a, b, c, d, e, f, n], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m], Less[m, -1], Not[And[RationalQ[n], Less[m, n, -1]]], IntegersQ[Times[2, m], Times[2, n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[a, IntPart[m]], Power[c, IntPart[m]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], FracPart[m]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], FracPart[m]], Power[Power[Cos[Plus[e, Times[f, x]]], Times[2, FracPart[m]]], -1]], Int[Times[Power[Cos[Plus[e, Times[f, x]]], Times[2, m]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, Times[-1, m]]]], x], x], And[FreeQ[List[a, b, c, d, e, f, m, n], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], Or[FractionQ[m], Not[FractionQ[n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], 2], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[b, 2], Cos[Plus[e, Times[f, x]]], Power[Times[d, f], -1]], x]], Dist[Times[1, Power[d, -1]], Int[Times[Simp[Plus[Times[Power[a, 2], d], Times[-1, b, Plus[Times[b, c], Times[-1, 2, a, d]], Sin[Plus[e, Times[f, x]]]]], x], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[b, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Int[Times[1, Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], -1]], x], x], Times[-1, Dist[Times[d, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Int[Times[1, Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[c, Int[Power[Times[b, Sin[Plus[e, Times[f, x]]]], m], x], x], Dist[Times[d, Power[b, -1]], Int[Power[Times[b, Sin[Plus[e, Times[f, x]]]], Plus[m, 1]], x], x]], FreeQ[List[b, c, d, e, f, m], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Simp[Times[d, Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Times[f, Plus[m, 1]], -1]], x]], And[FreeQ[List[a, b, c, d, e, f, m], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[Times[a, d, m], Times[b, c, Plus[m, 1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[b, c], Times[-1, a, d]], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Times[a, f, Plus[Times[2, m], 1]], -1]], x], Dist[Times[Plus[Times[a, d, m], Times[b, c, Plus[m, 1]]], Power[Times[a, b, Plus[Times[2, m], 1]], -1]], Int[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m], Less[m, Times[-1, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[d, Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Times[f, Plus[m, 1]], -1]], x]], Dist[Times[Plus[Times[a, d, m], Times[b, c, Plus[m, 1]]], Power[Times[b, Plus[m, 1]], -1]], Int[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], x], x]], And[FreeQ[List[a, b, c, d, e, f, m], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], Not[And[RationalQ[m], Less[m, Times[-1, Power[2, -1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Plus[Times[b, c], Times[-1, a, d]], Power[b, -1]], Int[Times[1, Power[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x], Dist[Times[d, Power[b, -1]], Int[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[d, Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Times[f, Plus[m, 1]], -1]], x]], Dist[Times[1, Power[Plus[m, 1], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, -1]], Simp[Plus[Times[b, d, m], Times[a, c, Plus[m, 1]], Times[Plus[Times[a, d, m], Times[b, c, Plus[m, 1]]], Sin[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m], Greater[m, 0], IntegerQ[Times[2, m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[b, c], Times[-1, a, d]], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[f, Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], x]], Dist[Times[1, Power[Times[Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Simp[Plus[Times[Plus[Times[a, c], Times[-1, b, d]], Plus[m, 1]], Times[-1, Plus[Times[b, c], Times[-1, a, d]], Plus[m, 2], Sin[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m], Less[m, -1], IntegerQ[Times[2, m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[c, Cos[Plus[e, Times[f, x]]], Power[Times[f, Sqrt[Plus[1, Sin[Plus[e, Times[f, x]]]]], Sqrt[Plus[1, Times[-1, Sin[Plus[e, Times[f, x]]]]]]], -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], m], Sqrt[Plus[1, Times[d, Power[c, -1], x]]], Power[Sqrt[Plus[1, Times[-1, d, Power[c, -1], x]]], -1]], x], x, Sin[Plus[e, Times[f, x]]]], x], And[FreeQ[List[a, b, c, d, e, f, m], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], Not[IntegerQ[Times[2, m]]], ZeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Plus[Times[b, c], Times[-1, a, d]], Power[b, -1]], Int[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], x], x], Dist[Times[d, Power[b, -1]], Int[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], x], x]], And[FreeQ[List[a, b, c, d, e, f, m], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrig[Times[Power[Plus[a, Times[b, sin[Plus[e, Times[f, x]]]]], m], Power[Times[d, sin[Plus[e, Times[f, x]]]], n]], x], x], And[FreeQ[List[a, b, d, e, f, n], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], PositiveIntegerQ[m], RationalQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Times[a, f, Plus[Times[2, m], 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[Power[a, 2], Plus[Times[2, m], 1]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Plus[Times[a, m], Times[-1, b, Plus[Times[2, m], 1], Sin[Plus[e, Times[f, x]]]]]], x], x]]], And[FreeQ[List[a, b, e, f], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m], Less[m, Times[-1, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[b, f, Plus[m, 2]], -1]], x]], Dist[Times[1, Power[Times[b, Plus[m, 2]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Plus[Times[b, Plus[m, 1]], Times[-1, a, Sin[Plus[e, Times[f, x]]]]]], x], x]], And[FreeQ[List[a, b, e, f, m], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], Not[And[RationalQ[m], Less[m, Times[-1, Power[2, -1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], 2]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[b, c], Times[-1, a, d]], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Power[Times[a, f, Plus[Times[2, m], 1]], -1]], x], Dist[Times[1, Power[Times[a, b, Plus[Times[2, m], 1]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Simp[Plus[Times[a, c, d, Plus[m, -1]], Times[b, Plus[Power[d, 2], Times[Power[c, 2], Plus[m, 1]]]], Times[d, Plus[Times[a, d, Plus[m, -1]], Times[b, c, Plus[m, 2]]], Sin[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m], Less[m, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], 2]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[d, 2], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[b, f, Plus[m, 2]], -1]], x]], Dist[Times[1, Power[Times[b, Plus[m, 2]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Simp[Plus[Times[b, Plus[Times[Power[d, 2], Plus[m, 1]], 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Times[b, Sin[Plus[e, Times[f, x]]]]]], -1]]], x], And[FreeQ[List[a, b, d, e, f], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[d, Times[-1, a, Power[b, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[-2, b, Power[f, -1]], Subst[Int[Times[1, Power[Plus[b, Times[d, Power[x, 2]]], -1]], x], x, Times[b, Cos[Plus[e, Times[f, x]]], Power[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]]], -1]]], x], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[a, 2], Cos[Plus[e, Times[f, x]]], Power[Times[f, Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Plus[a, Times[-1, b, Sin[Plus[e, Times[f, x]]]]]]], -1]], Subst[Int[Times[Power[Plus[c, Times[d, x]], n], Power[Sqrt[Plus[a, Times[-1, b, x]]], -1]], x], x, Sin[Plus[e, Times[f, x]]]], x], And[FreeQ[List[a, b, c, d, e, f, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], Not[IntegerQ[Times[2, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[d, Power[b, -1]], Int[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x], Dist[Times[Plus[Times[b, c], Times[-1, a, d]], Power[b, -1]], Int[Times[1, Power[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[-2, d, Cos[Plus[e, Times[f, x]]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, -1]], Power[Times[f, Plus[Times[2, n], -1], Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], Times[-1, Dist[Times[1, Power[Times[b, Plus[Times[2, n], -1]], -1]], Int[Times[Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, -2]], Power[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], -1], Simp[Plus[Times[a, c, d], Times[-1, b, Plus[Times[2, Power[d, 2], Plus[n, -1]], Times[Power[c, 2], Plus[Times[2, n], -1]]]], Times[d, Plus[Times[a, d], Times[-1, b, c, Plus[Times[4, n], -3]]], Sin[Plus[e, Times[f, x]]]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], RationalQ[n], Greater[n, 1], IntegerQ[Times[2, n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[d, Cos[Plus[e, Times[f, x]]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[f, Plus[n, 1], Plus[Power[c, 2], Times[-1, Power[d, 2]]], Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]]], -1]], x]], Times[-1, Dist[Times[1, Power[Times[2, b, Plus[n, 1], Plus[Power[c, 2], Times[-1, Power[d, 2]]]], -1]], Int[Times[Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Simp[Plus[Times[a, d], Times[-1, 2, b, c, Plus[n, 1]], Times[b, d, Plus[Times[2, n], 3], Sin[Plus[e, Times[f, x]]]]], x], Power[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], RationalQ[n], Less[n, -1], IntegerQ[Times[2, n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[b, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Int[Times[1, Power[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x], Times[-1, Dist[Times[d, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Int[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Times[Sqrt[2], Power[Times[Sqrt[a], f], -1]], Subst[Int[Times[1, Power[Sqrt[Plus[1, Times[-1, Power[x, 2]]]], -1]], x], x, Times[b, Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], -1]]], x]], And[FreeQ[List[a, b, d, e, f], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[d, Times[-1, a, Power[b, -1]]]], PositiveQ[a]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[-2, a, Power[f, -1]], Subst[Int[Times[1, Power[Plus[Times[2, Power[b, 2]], Times[-1, Plus[Times[a, c], Times[-1, b, d]], Power[x, 2]]], -1]], x], x, Times[b, Cos[Plus[e, Times[f, x]]], Power[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]]], -1]]], x], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[d, Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, -1]], Power[Times[f, Plus[m, n]], -1]], x]], Dist[Times[1, Power[Times[b, Plus[m, n]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, -2]], Simp[Plus[Times[d, Plus[Times[a, c, m], Times[b, d, Plus[n, -1]]]], Times[b, Power[c, 2], Plus[m, n]], Times[d, Plus[Times[a, d, m], Times[b, c, Plus[m, Times[2, n], -1]]], Sin[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, m], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], RationalQ[n], Greater[n, 1], IntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[a, m], Cos[Plus[e, Times[f, x]]], Power[Times[f, Sqrt[Plus[1, Sin[Plus[e, Times[f, x]]]]], Sqrt[Plus[1, Times[-1, Sin[Plus[e, Times[f, x]]]]]]], -1]], Subst[Int[Times[Power[Plus[1, Times[b, Power[a, -1], x]], Plus[m, Times[-1, 1, Power[2, -1]]]], Power[Plus[c, Times[d, x]], n], Power[Sqrt[Plus[1, Times[-1, b, Power[a, -1], x]]], -1]], x], x, Sin[Plus[e, Times[f, x]]]], x], And[FreeQ[List[a, b, c, d, e, f, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Times[b, Power[Times[d, Power[b, -1]], n], Cos[Plus[e, Times[f, x]]], Power[Times[f, Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Plus[a, Times[-1, b, Sin[Plus[e, Times[f, x]]]]]]], -1]], Subst[Int[Times[Power[Plus[a, Times[-1, x]], n], Power[Plus[Times[2, a], Times[-1, x]], Plus[m, Times[-1, 1, Power[2, -1]]]], Power[Sqrt[x], -1]], x], x, Plus[a, Times[-1, b, Sin[Plus[e, Times[f, x]]]]]], x]], And[FreeQ[List[a, b, d, e, f, m, n], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], Not[IntegerQ[m]], PositiveQ[a], PositiveQ[Times[d, Power[b, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[d, Power[b, -1]], IntPart[n]], Power[Times[d, Sin[Plus[e, Times[f, x]]]], FracPart[n]], Power[Power[Times[b, Sin[Plus[e, Times[f, x]]]], FracPart[n]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Times[b, Sin[Plus[e, Times[f, x]]]], n]], x], x], And[FreeQ[List[a, b, d, e, f, m, n], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], Not[IntegerQ[m]], PositiveQ[a], Not[PositiveQ[Times[d, Power[b, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[a, IntPart[m]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], FracPart[m]], Power[Power[Plus[1, Times[b, Power[a, -1], Sin[Plus[e, Times[f, x]]]]], FracPart[m]], -1]], Int[Times[Power[Plus[1, Times[b, Power[a, -1], Sin[Plus[e, Times[f, x]]]]], m], Power[Times[d, Sin[Plus[e, Times[f, x]]]], n]], x], x], And[FreeQ[List[a, b, d, e, f, m, n], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], Not[IntegerQ[m]], Not[PositiveQ[a]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[a, 2], Cos[Plus[e, Times[f, x]]], Power[Times[f, Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Plus[a, Times[-1, b, Sin[Plus[e, Times[f, x]]]]]]], -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], Plus[m, Times[-1, 1, Power[2, -1]]]], Power[Plus[c, Times[d, x]], n], Power[Sqrt[Plus[a, Times[-1, b, x]]], -1]], x], x, Sin[Plus[e, Times[f, x]]]], x], And[FreeQ[List[a, b, c, d, e, f, m, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], 2]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[2, c, d, Power[b, -1]], Int[Power[Times[b, Sin[Plus[e, Times[f, x]]]], Plus[m, 1]], x], x], Int[Times[Power[Times[b, Sin[Plus[e, Times[f, x]]]], m], Plus[Power[c, 2], Times[Power[d, 2], Power[Sin[Plus[e, Times[f, x]]], 2]]]], x]], FreeQ[List[b, c, d, e, f, m], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], 2]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[Power[b, 2], Power[c, 2]], Times[-1, 2, a, b, c, d], Times[Power[a, 2], Power[d, 2]]], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[b, f, Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], x]], Times[-1, Dist[Times[1, Power[Times[b, Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Simp[Plus[Times[b, Plus[m, 1], Plus[Times[2, b, c, d], Times[-1, a, Plus[Power[c, 2], Power[d, 2]]]]], Times[Plus[Times[Power[a, 2], Power[d, 2]], Times[-1, 2, a, b, c, d, Plus[m, 2]], Times[Power[b, 2], Plus[Times[Power[d, 2], Plus[m, 1]], Times[Power[c, 2], Plus[m, 2]]]]], Sin[Plus[e, Times[f, x]]]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m], Less[m, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], 2]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[d, 2], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[b, f, Plus[m, 2]], -1]], x]], Dist[Times[1, Power[Times[b, Plus[m, 2]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Simp[Plus[Times[b, Plus[Times[Power[d, 2], Plus[m, 1]], Times[Power[c, 2], Plus[m, 2]]]], Times[-1, d, Plus[Times[a, d], Times[-1, 2, b, c, Plus[m, 2]]], Sin[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, m], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], Not[And[RationalQ[m], Less[m, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[Power[b, 2], Power[c, 2]], Times[-1, 2, a, b, c, d], Times[Power[a, 2], Power[d, 2]]], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, 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Plus[Times[2, a, b, c], Times[-1, d, Plus[Power[a, 2], Power[b, 2]]]]]], Power[Sin[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], RationalQ[m, n], Greater[m, 2], Less[n, -1], Or[IntegerQ[m], IntegersQ[Times[2, m], Times[2, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[b, 2], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, -2]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[d, f, Plus[m, n]], -1]], x]], Dist[Times[1, Power[Times[d, Plus[m, n]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, -3]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Simp[Plus[Times[Power[a, 3], d, Plus[m, n]], Times[Power[b, 2], Plus[Times[b, c, Plus[m, -2]], Times[a, d, Plus[n, 1]]]], Times[-1, b, Plus[Times[a, b, c], Times[-1, Power[b, 2], d, Plus[m, n, -1]], Times[-1, 3, Power[a, 2], d, Plus[m, n]]], Sin[Plus[e, Times[f, x]]]], Times[-1, Power[b, 2], Plus[Times[b, c, Plus[m, -1]], Times[-1, a, d, Plus[Times[3, m], Times[2, n], -2]]], Power[Sin[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], RationalQ[m], Greater[m, 2], Or[IntegerQ[m], IntegersQ[Times[2, m], Times[2, n]]], Not[And[IntegerQ[n], Greater[n, 2], Or[Not[IntegerQ[m]], And[ZeroQ[a], NonzeroQ[c]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[-2, a, d, Cos[Plus[e, Times[f, x]]], Power[Times[f, Plus[Power[a, 2], Times[-1, Power[b, 2]]], Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]], x], Times[-1, Dist[Times[Power[d, 2], Power[Plus[Power[a, 2], Times[-1, Power[b, 2]]], -1]], Int[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Power[Times[d, Sin[Plus[e, Times[f, x]]]], Times[3, Power[2, -1]]], -1]], x], x]]], And[FreeQ[List[a, b, d, e, f], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Plus[c, Times[-1, d]], Power[Plus[a, Times[-1, b]], -1]], Int[Times[1, Power[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], x], Times[-1, Dist[Times[Plus[Times[b, c], Times[-1, a, d]], Power[Plus[a, Times[-1, b]], -1]], Int[Times[Plus[1, Sin[Plus[e, Times[f, x]]]], Power[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Times[3, Power[2, -1]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[b, Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Power[Times[f, Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], x]], Dist[Times[1, Power[Times[Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, -1]], Simp[Plus[Times[a, c, Plus[m, 1]], Times[b, d, n], Times[Plus[Times[a, d, Plus[m, 1]], Times[-1, b, c, Plus[m, 2]]], Sin[Plus[e, Times[f, x]]]], Times[-1, b, d, Plus[m, n, 2], Power[Sin[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], RationalQ[m, n], Less[m, -1], Less[0, n, 1], IntegersQ[Times[2, m], Times[2, n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[3, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[d, Power[b, -1]], Int[Times[Sqrt[Times[d, Sin[Plus[e, Times[f, x]]]]], Power[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x], Times[-1, Dist[Times[a, d, Power[b, -1]], Int[Times[Sqrt[Times[d, Sin[Plus[e, Times[f, x]]]]], Power[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Times[3, Power[2, -1]]], -1]], x], x]]], And[FreeQ[List[a, b, d, e, f], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[3, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Power[d, 2], Power[Power[b, 2], -1]], Int[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x], Dist[Times[Plus[Times[b, c], Times[-1, a, d]], Power[Power[b, 2], -1]], Int[Times[Simp[Plus[Times[b, c], Times[a, d], Times[2, b, d, Sin[Plus[e, Times[f, x]]]]], x], Power[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Times[3, Power[2, -1]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[b, c], Times[-1, a, d]], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, -1]], Power[Times[f, Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], x]], Dist[Times[1, Power[Times[Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, -2]], Simp[Plus[Times[c, Plus[Times[a, c], Times[-1, b, d]], Plus[m, 1]], Times[d, Plus[Times[b, c], Times[-1, a, d]], Plus[n, -1]], Times[Plus[Times[d, Plus[Times[a, c], Times[-1, b, d]], Plus[m, 1]], Times[-1, c, Plus[Times[b, c], Times[-1, a, d]], Plus[m, 2]]], Sin[Plus[e, Times[f, x]]]], Times[-1, d, Plus[Times[b, c], Times[-1, a, d]], Plus[m, n, 1], Power[Sin[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], RationalQ[m, n], Less[m, -1], Less[1, n, 2], IntegersQ[Times[2, m], Times[2, n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[2, b, Cos[Plus[e, Times[f, x]]], Power[Times[f, Plus[Power[a, 2], Times[-1, Power[b, 2]]], Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]], x], Dist[Times[d, Power[Plus[Power[a, 2], Times[-1, Power[b, 2]]], -1]], Int[Times[Plus[b, Times[a, Sin[Plus[e, Times[f, x]]]]], Power[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Times[d, Sin[Plus[e, Times[f, x]]]], Times[3, Power[2, -1]]]], -1]], x], x]], And[FreeQ[List[a, b, d, e, f], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[Plus[a, Times[-1, b]], -1]], Int[Times[1, Power[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], x], Times[-1, Dist[Times[b, Power[Plus[a, Times[-1, b]], -1]], Int[Times[Plus[1, Sin[Plus[e, Times[f, x]]]], Power[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Times[3, Power[2, -1]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[b, 2], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[f, Plus[m, 1], Plus[Times[b, c], Times[-1, a, d]], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], x]], Dist[Times[1, Power[Times[Plus[m, 1], Plus[Times[b, c], Times[-1, a, d]], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Simp[Plus[Times[a, Plus[Times[b, c], Times[-1, a, d]], Plus[m, 1]], Times[Power[b, 2], d, Plus[m, n, 2]], Times[-1, Plus[Times[Power[b, 2], c], Times[b, Plus[Times[b, c], Times[-1, a, d]], Plus[m, 1]]], Sin[Plus[e, Times[f, x]]]], Times[-1, Power[b, 2], d, Plus[m, n, 3], Power[Sin[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], RationalQ[m], Less[m, -1], IntegersQ[Times[2, m], Times[2, n]], Or[And[ZeroQ[a], IntegerQ[m], Not[IntegerQ[n]]], Not[And[IntegerQ[Times[2, n]], Less[n, -1], Or[And[IntegerQ[n], Not[IntegerQ[m]]], ZeroQ[a]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[d, Power[b, -1]], Int[Times[1, Power[Sqrt[Plus[c, Times[d, 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Times[-1, Dist[Times[Plus[Times[b, c], Times[-1, a, d]], Power[d, -1]], Int[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[2, Power[Times[f, Plus[a, b], Sqrt[Plus[c, d]]], -1], EllipticPi[Times[2, b, Power[Plus[a, b], -1]], Times[1, Power[2, -1], Plus[e, Times[-1, Pi, Power[2, -1]], Times[f, x]]], Times[2, d, Power[Plus[c, d], -1]]]], x], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], PositiveQ[Plus[c, d]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[2, Power[Times[f, Plus[a, Times[-1, b]], Sqrt[Plus[c, Times[-1, d]]]], -1], EllipticPi[Times[-2, b, Power[Plus[a, Times[-1, b]], -1]], Times[1, Power[2, -1], Plus[e, Times[Pi, Power[2, -1]], Times[f, x]]], Times[-2, d, Power[Plus[c, Times[-1, d]], -1]]]], x], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], PositiveQ[Plus[c, Times[-1, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Times[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Power[Plus[c, d], -1]]], Power[Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]], Int[Times[1, Power[Times[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Sqrt[Plus[Times[c, Power[Plus[c, d], -1]], Times[d, Power[Plus[c, d], -1], Sin[Plus[e, Times[f, x]]]]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], Not[PositiveQ[Plus[c, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[2, c, Rt[Times[b, Plus[c, d]], 2], Tan[Plus[e, Times[f, x]]], Sqrt[Plus[1, Csc[Plus[e, Times[f, x]]]]], Sqrt[Plus[1, Times[-1, Csc[Plus[e, Times[f, x]]]]]], Power[Times[d, f, Sqrt[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]], -1], EllipticPi[Times[Plus[c, d], Power[d, -1]], ArcSin[Times[Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], Power[Sqrt[Times[b, Sin[Plus[e, Times[f, x]]]]], -1], Power[Rt[Times[Plus[c, d], Power[b, -1]], 2], -1]]], Times[-1, Plus[c, d], Power[Plus[c, Times[-1, d]], -1]]]], x], And[FreeQ[List[b, c, d, e, f], x], PositiveQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], PosQ[Times[Plus[c, d], Power[b, -1]]], PositiveQ[Power[c, 2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[2, b, Tan[Plus[e, Times[f, x]]], Power[Times[d, f], -1], Rt[Times[Plus[c, d], Power[b, -1]], 2], Sqrt[Times[c, Plus[1, Csc[Plus[e, Times[f, x]]]], Power[Plus[c, Times[-1, d]], -1]]], Sqrt[Times[c, Plus[1, Times[-1, Csc[Plus[e, Times[f, x]]]]], Power[Plus[c, d], -1]]], EllipticPi[Times[Plus[c, d], Power[d, -1]], ArcSin[Times[Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], Power[Sqrt[Times[b, Sin[Plus[e, Times[f, x]]]]], -1], Power[Rt[Times[Plus[c, d], Power[b, -1]], 2], -1]]], Times[-1, Plus[c, d], Power[Plus[c, Times[-1, d]], -1]]]], x], And[FreeQ[List[b, c, d, e, f], x], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], PosQ[Times[Plus[c, d], Power[b, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Times[b, Sin[Plus[e, Times[f, x]]]]], Power[Sqrt[Times[-1, b, Sin[Plus[e, Times[f, x]]]]], -1]], Int[Times[Sqrt[Times[-1, b, Sin[Plus[e, Times[f, x]]]]], Power[Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x], And[FreeQ[List[b, c, d, e, f], x], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], NegQ[Times[Plus[c, d], Power[b, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[2, Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Power[Times[d, f, Rt[Times[Plus[a, b], Power[Plus[c, d], -1]], 2], Cos[Plus[e, Times[f, x]]]], -1], Sqrt[Times[Plus[Times[b, c], Times[-1, a, d]], Plus[1, Sin[Plus[e, Times[f, x]]]], Power[Times[Plus[c, Times[-1, d]], Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], -1]]], Sqrt[Times[-1, Plus[Times[b, c], Times[-1, a, d]], Plus[1, Times[-1, Sin[Plus[e, Times[f, x]]]]], Power[Times[Plus[c, d], Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], -1]]], EllipticPi[Times[b, Plus[c, d], Power[Times[d, Plus[a, b]], -1]], ArcSin[Times[Rt[Times[Plus[a, b], Power[Plus[c, d], -1]], 2], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], Power[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], -1]]], Times[Plus[a, Times[-1, b]], Plus[c, d], Power[Times[Plus[a, b], Plus[c, Times[-1, d]]], -1]]]], x], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], PosQ[Times[Plus[a, b], Power[Plus[c, d], -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Plus[Times[-1, c], Times[-1, d, Sin[Plus[e, Times[f, x]]]]]], Power[Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]], Int[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Sqrt[Plus[Times[-1, c], Times[-1, d, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], NegQ[Times[Plus[a, b], Power[Plus[c, d], -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[-2, d, Power[Times[f, Sqrt[Plus[a, Times[b, d]]]], -1], EllipticF[ArcSin[Times[Cos[Plus[e, Times[f, x]]], Power[Plus[1, Times[d, Sin[Plus[e, Times[f, x]]]]], -1]]], Times[-1, Plus[a, Times[-1, b, d]], Power[Plus[a, Times[b, d]], -1]]]], x], And[FreeQ[List[a, b, d, e, f], x], NegativeQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[Power[d, 2], -1]], PositiveQ[Times[b, d]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Times[Sign[b], Sin[Plus[e, Times[f, x]]]]], Power[Sqrt[Times[d, Sin[Plus[e, Times[f, x]]]]], -1]], Int[Times[1, Power[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Times[Sign[b], Sin[Plus[e, Times[f, x]]]]]], -1]], x], x], And[FreeQ[List[a, b, d, e, f], x], NegativeQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], PositiveQ[Power[b, 2]], Not[And[ZeroQ[Plus[Power[d, 2], -1]], PositiveQ[Times[b, d]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[-2, Sqrt[Power[a, 2]], Sqrt[Times[-1, Power[Cot[Plus[e, Times[f, x]]], 2]]], Power[Times[a, f, Sqrt[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], Cot[Plus[e, Times[f, x]]]], -1], Rt[Times[Plus[a, b], Power[d, -1]], 2], EllipticF[ArcSin[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Sqrt[Times[d, Sin[Plus[e, Times[f, x]]]]], -1], Power[Rt[Times[Plus[a, b], Power[d, -1]], 2], -1]]], Times[-1, Plus[a, b], Power[Plus[a, Times[-1, b]], -1]]]], x], And[FreeQ[List[a, b, d, e, f], x], PositiveQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], PosQ[Times[Plus[a, b], Power[d, -1]]], PositiveQ[Power[a, 2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[-2, Tan[Plus[e, Times[f, x]]], Power[Times[a, f], -1], Rt[Times[Plus[a, b], Power[d, -1]], 2], Sqrt[Times[a, Plus[1, Times[-1, Csc[Plus[e, Times[f, x]]]]], Power[Plus[a, b], -1]]], Sqrt[Times[a, Plus[1, Csc[Plus[e, Times[f, x]]]], Power[Plus[a, Times[-1, b]], -1]]], EllipticF[ArcSin[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Sqrt[Times[d, Sin[Plus[e, Times[f, x]]]]], -1], Power[Rt[Times[Plus[a, b], Power[d, -1]], 2], -1]]], Times[-1, Plus[a, b], Power[Plus[a, Times[-1, b]], -1]]]], x], And[FreeQ[List[a, b, d, e, f], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], PosQ[Times[Plus[a, b], Power[d, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Times[-1, d, Sin[Plus[e, Times[f, x]]]]], Power[Sqrt[Times[d, Sin[Plus[e, Times[f, x]]]]], -1]], Int[Times[1, Power[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Times[-1, d, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x], And[FreeQ[List[a, b, d, e, f], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NegQ[Times[Plus[a, b], Power[d, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[2, Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Power[Times[f, Plus[Times[b, c], Times[-1, a, d]], Rt[Times[Plus[c, d], Power[Plus[a, b], -1]], 2], Cos[Plus[e, Times[f, x]]]], -1], Sqrt[Times[Plus[Times[b, c], Times[-1, a, d]], Plus[1, Times[-1, Sin[Plus[e, Times[f, x]]]]], Power[Times[Plus[a, b], Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]]], Sqrt[Times[-1, Plus[Times[b, c], Times[-1, a, d]], Plus[1, Sin[Plus[e, Times[f, x]]]], Power[Times[Plus[a, Times[-1, b]], Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]]], EllipticF[ArcSin[Times[Rt[Times[Plus[c, d], Power[Plus[a, b], -1]], 2], Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]]]], Times[Plus[a, b], Plus[c, Times[-1, d]], Power[Times[Plus[a, Times[-1, b]], Plus[c, d]], -1]]]], x], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], PosQ[Times[Plus[c, d], Power[Plus[a, b], -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Plus[Times[-1, a], Times[-1, b, Sin[Plus[e, Times[f, x]]]]]], Power[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], -1]], Int[Times[1, Power[Times[Sqrt[Plus[Times[-1, a], Times[-1, b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], NegQ[Times[Plus[c, d], Power[Plus[a, b], -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[3, 2]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Dist[Times[a, d, Power[Times[2, b], -1]], Int[Times[Sqrt[Times[d, Sin[Plus[e, Times[f, x]]]]], Power[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x]], Dist[Times[d, Power[Times[2, b], -1]], Int[Times[Sqrt[Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[a, Times[2, b, Sin[Plus[e, Times[f, x]]]]], Power[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x]], And[FreeQ[List[a, b, d, e, f], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[b, Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Power[Times[f, Plus[m, n]], -1]], x]], Dist[Times[1, Power[Times[d, Plus[m, n]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, -2]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, -1]], Simp[Plus[Times[Power[a, 2], c, d, Plus[m, n]], Times[b, d, Plus[Times[b, c, Plus[m, -1]], Times[a, d, n]]], Times[Plus[Times[a, d, Plus[Times[2, b, c], Times[a, d]], Plus[m, n]], Times[-1, b, d, Plus[Times[a, c], Times[-1, b, d, Plus[m, n, -1]]]]], Sin[Plus[e, Times[f, x]]]], Times[b, d, Plus[Times[b, c, n], Times[a, d, Plus[Times[2, m], n, -1]]], Power[Sin[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], RationalQ[m, n], Less[0, m, 2], Less[-1, n, 2], NonzeroQ[Plus[m, n]], Or[IntegerQ[m], IntegersQ[Times[2, m], Times[2, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[b, Power[d, -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]]], x], x], Times[-1, Dist[Times[Plus[Times[b, c], Times[-1, a, d]], Power[d, -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], PositiveIntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n]], x], And[FreeQ[List[a, b, c, d, e, f, m, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[c, IntPart[n]], Power[Times[c, Power[Times[d, Sin[Plus[e, Times[f, x]]]], p]], FracPart[n]], Power[Power[Times[d, Sin[Plus[e, Times[f, x]]]], Times[p, FracPart[n]]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Times[d, Sin[Plus[e, Times[f, x]]]], Times[n, p]]], x], x], And[FreeQ[List[a, b, c, d, e, f, m, n, p], x], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[c, Blank[]]], Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[p, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[c, IntPart[n]], Power[Times[c, Power[Times[d, Cos[Plus[e, Times[f, x]]]], p]], FracPart[n]], Power[Power[Times[d, Cos[Plus[e, Times[f, x]]]], Times[p, FracPart[n]]], -1]], Int[Times[Power[Plus[a, Times[b, Cos[Plus[e, Times[f, x]]]]], m], Power[Times[d, Cos[Plus[e, Times[f, x]]]], Times[n, p]]], x], x], And[FreeQ[List[a, b, c, d, e, f, m, n, p], x], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[d, Times[c, Sin[Plus[e, Times[f, x]]]]], n], Power[Power[Sin[Plus[e, Times[f, x]]], n], -1]], x], And[FreeQ[List[a, b, c, d, e, f, m], x], IntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[b, Times[a, Csc[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], n], Power[Power[Csc[Plus[e, Times[f, x]]], m], -1]], x], And[FreeQ[List[a, b, c, d, e, f, n], x], Not[IntegerQ[n]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[b, Times[a, Sec[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sec[Plus[e, Times[f, x]]]]], n], Power[Power[Sec[Plus[e, Times[f, x]]], m], -1]], x], And[FreeQ[List[a, b, c, d, e, f, n], x], Not[IntegerQ[n]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Sin[Plus[e, Times[f, x]]], n], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], n], Power[Power[Plus[d, Times[c, Sin[Plus[e, Times[f, x]]]]], n], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[d, Times[c, Sin[Plus[e, Times[f, x]]]]], n], Power[Power[Sin[Plus[e, Times[f, x]]], n], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, m, n], x], Not[IntegerQ[n]], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Cos[Plus[e, Times[f, x]]], n], Power[Plus[c, Times[d, Sec[Plus[e, Times[f, x]]]]], n], Power[Power[Plus[d, Times[c, Cos[Plus[e, Times[f, x]]]]], n], -1]], Int[Times[Power[Plus[a, Times[b, Cos[Plus[e, Times[f, x]]]]], m], Power[Plus[d, Times[c, Cos[Plus[e, Times[f, x]]]]], n], Power[Power[Cos[Plus[e, Times[f, x]]], n], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, m, n], x], Not[IntegerQ[n]], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Times[b, f], -1]], Subst[Int[Times[Power[Plus[a, x], m], Power[Plus[c, Times[d, Power[b, -1], x]], n]], x], x, Times[b, Sin[Plus[e, Times[f, x]]]]], x], FreeQ[List[a, b, c, d, e, f, m, n], x]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[a, Int[Times[Power[Cos[Plus[e, Times[f, x]]], p], Power[Times[d, Sin[Plus[e, Times[f, x]]]], n]], x], x], Dist[Times[b, Power[d, -1]], Int[Times[Power[Cos[Plus[e, Times[f, x]]], p], Power[Times[d, Sin[Plus[e, Times[f, x]]]], Plus[n, 1]]], x], x]], And[FreeQ[List[a, b, d, e, f, n, p], x], IntegerQ[Times[Plus[p, -1], Power[2, -1]]], IntegerQ[n], Or[And[Less[p, 0], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]], Less[0, n, Plus[p, -1]], Less[Plus[p, 1], Times[-1, n], Plus[Times[2, p], 1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[a, -1]], Int[Times[Power[Cos[Plus[e, Times[f, x]]], Plus[p, -2]], Power[Times[d, Sin[Plus[e, Times[f, x]]]], n]], x], x], Times[-1, Dist[Times[1, Power[Times[b, d], -1]], Int[Times[Power[Cos[Plus[e, Times[f, x]]], Plus[p, -2]], Power[Times[d, Sin[Plus[e, Times[f, x]]]], Plus[n, 1]]], x], x]]], And[FreeQ[List[a, b, d, e, f, n, p], x], IntegerQ[Times[Plus[p, -1], Power[2, -1]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegerQ[n], Or[Less[0, n, Times[Plus[p, 1], Power[2, -1]]], Inequality[p, LessEqual, Times[-1, n], Less, Plus[Times[2, p], -3]], Inequality[0, Less, n, LessEqual, Times[-1, p]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Times[Power[b, p], f], -1]], Subst[Int[Times[Power[Plus[a, x], Plus[m, Times[Plus[p, -1], Power[2, -1]]]], Power[Plus[a, Times[-1, x]], Times[Plus[p, -1], Power[2, -1]]], Power[Plus[c, Times[d, Power[b, -1], x]], n]], x], x, Times[b, Sin[Plus[e, Times[f, x]]]]], x], And[FreeQ[List[a, b, e, f, c, d, m, n], x], IntegerQ[Times[Plus[p, -1], Power[2, -1]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Times[Power[b, p], f], -1]], Subst[Int[Times[Power[Plus[a, x], m], Power[Plus[c, Times[d, Power[b, -1], x]], n], Power[Plus[Power[b, 2], Times[-1, Power[x, 2]]], Times[Plus[p, -1], Power[2, -1]]]], x], x, Times[b, Sin[Plus[e, Times[f, x]]]]], x], And[FreeQ[List[a, b, c, d, e, f, m, n], x], IntegerQ[Times[Plus[p, -1], Power[2, -1]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[a, Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], p], Power[Times[d, Sin[Plus[e, Times[f, x]]]], n]], x], x], Dist[Times[b, Power[d, -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], p], Power[Times[d, Sin[Plus[e, Times[f, x]]]], Plus[n, 1]]], x], x]], FreeQ[List[a, b, d, e, f, g, n, p], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Power[g, 2], Power[a, -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, -2]], Power[Times[d, Sin[Plus[e, Times[f, x]]]], n]], x], x], Times[-1, Dist[Times[Power[g, 2], Power[Times[b, d], -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, -2]], Power[Times[d, Sin[Plus[e, Times[f, x]]]], Plus[n, 1]]], x], x]]], And[FreeQ[List[a, b, d, e, f, g, n, p], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[a, m], Power[c, m], Power[Power[g, Times[2, m]], -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[Times[2, m], p]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, Times[-1, m]]]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, n, p], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegerQ[m], Not[And[IntegerQ[n], Less[Power[n, 2], Power[m, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Times[Power[a, Times[p, Power[2, -1]]], Power[c, Times[p, Power[2, -1]]]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, Times[p, Power[2, -1]]]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, Times[p, Power[2, -1]]]]], x], x], And[FreeQ[List[a, b, c, d, e, f, n, p], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegerQ[Times[p, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[g, Cos[Plus[e, Times[f, x]]], Power[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]]], -1]], Int[Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, p], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[a, IntPart[m]], Power[c, IntPart[m]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], FracPart[m]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], FracPart[m]], Power[Times[Power[g, Times[2, IntPart[m]]], Power[Times[g, Cos[Plus[e, Times[f, x]]]], Times[2, FracPart[m]]]], -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[Times[2, m], p]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, m, n, p], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[Times[2, m], p, -1]], ZeroQ[Plus[m, Times[-1, n], -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[b, Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, 1]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Power[Times[f, g, Plus[m, Times[-1, n], -1]], -1]], x], And[FreeQ[List[a, b, c, d, e, f, g, m, n, p], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[Times[2, m], p, -1]], NonzeroQ[Plus[m, Times[-1, n], -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[-2, b, Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, 1]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Power[Times[f, g, Plus[Times[2, n], p, 1]], -1]], x], Times[-1, Dist[Times[b, Plus[Times[2, m], p, -1], Power[Times[d, Plus[Times[2, n], p, 1]], -1]], 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RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[-2, b, Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, 1]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Power[Times[f, g, Plus[Times[2, n], p, 1]], -1]], x], Times[-1, Dist[Times[b, Plus[Times[2, m], p, -1], Power[Times[d, Plus[Times[2, n], p, 1]], -1]], 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Plus[Times[2, m], 3], Sin[Plus[e, Times[f, x]]]]]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m], Less[m, Times[-3, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[d, Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 2]], Power[Times[Power[b, 2], f, Plus[m, 3]], -1]], x], Times[-1, Dist[Times[1, Power[Times[Power[b, 2], Plus[m, 3]], -1]], 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Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[b, c], Times[-1, a, d]], Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, 1]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Times[a, f, g, Plus[Times[2, m], p, 1]], -1]], x], Dist[Times[Plus[Times[a, d, m], Times[b, c, Plus[m, p, 1]]], Power[Times[a, b, Plus[Times[2, m], p, 1]], -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], p], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, m, p], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], Or[And[RationalQ[m], Less[m, -1]], NegativeIntegerQ[Simplify[Plus[m, p]]]], NonzeroQ[Plus[Times[2, m], p, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[d, Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, 1]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Times[f, g, Plus[m, p, 1]], -1]], x]], Dist[Times[Plus[Times[a, d, m], Times[b, c, Plus[m, p, 1]]], Power[Times[b, Plus[m, p, 1]], -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], p], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, m, p], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[m, p, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, 1]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Plus[d, Times[c, Sin[Plus[e, Times[f, x]]]]], Power[Times[f, g, Plus[p, 1]], -1]], x]], Dist[Times[1, Power[Times[Power[g, 2], Plus[p, 1]], -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, 2]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, -1]], Simp[Plus[Times[a, c, Plus[p, 2]], Times[b, d, m], Times[b, c, Plus[m, p, 2], Sin[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m, p], Greater[m, 0], Less[p, -1], IntegerQ[Times[2, m]], Not[And[Equal[m, 1], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], SimplerQ[Plus[c, Times[d, x]], Plus[a, Times[b, x]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[d, Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, 1]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Times[f, g, Plus[m, p, 1]], -1]], x]], Dist[Times[1, Power[Plus[m, p, 1], -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], p], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, -1]], Simp[Plus[Times[a, c, Plus[m, p, 1]], Times[b, d, m], Times[Plus[Times[a, d, m], Times[b, c, Plus[m, p, 1]]], Sin[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, p], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m], Greater[m, 0], Not[And[RationalQ[p], Less[p, -1]]], IntegerQ[Times[2, m]], Not[And[Equal[m, 1], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], SimplerQ[Plus[c, Times[d, x]], Plus[a, Times[b, x]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[g, Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, -1]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Plus[Times[b, c, Plus[m, p, 1]], Times[-1, a, d, p], Times[b, d, Plus[m, 1], Sin[Plus[e, Times[f, x]]]]], Power[Times[Power[b, 2], f, Plus[m, 1], Plus[m, p, 1]], -1]], x], Dist[Times[Power[g, 2], Plus[p, -1], Power[Times[Power[b, 2], Plus[m, 1], Plus[m, p, 1]], -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, -2]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Simp[Plus[Times[b, d, Plus[m, 1]], Times[Plus[Times[b, c, Plus[m, p, 1]], Times[-1, a, d, p]], Sin[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m, p], Less[m, -1], Greater[p, 1], NonzeroQ[Plus[m, p, 1]], IntegerQ[Times[2, m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[b, c], Times[-1, a, d]], Power[Times[g, Cos[Plus[e, 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sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[g, Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, -1]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Plus[Times[b, c, Plus[m, p, 1]], Times[-1, a, d, p], Times[b, d, Plus[m, p], Sin[Plus[e, Times[f, x]]]]], Power[Times[Power[b, 2], f, Plus[m, p], Plus[m, p, 1]], -1]], x], Dist[Times[Power[g, 2], Plus[p, -1], Power[Times[Power[b, 2], Plus[m, p], Plus[m, p, 1]], -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, -2]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Simp[Plus[Times[b, Plus[Times[a, d, m], Times[b, c, Plus[m, p, 1]]]], Times[Plus[Times[a, b, c, Plus[m, p, 1]], Times[-1, d, Plus[Times[Power[a, 2], p], Times[-1, Power[b, 2], Plus[m, p]]]]], Sin[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, m], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[p], Greater[p, 1], NonzeroQ[Plus[m, p]], NonzeroQ[Plus[m, p, 1]], IntegerQ[Times[2, m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, 1]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Plus[Times[b, c], Times[-1, a, d], Times[-1, Plus[Times[a, c], Times[-1, b, d]], Sin[Plus[e, Times[f, x]]]]], Power[Times[f, g, Plus[Power[a, 2], Times[-1, Power[b, 2]]], Plus[p, 1]], -1]], x], Dist[Times[1, Power[Times[Power[g, 2], Plus[Power[a, 2], Times[-1, Power[b, 2]]], Plus[p, 1]], -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, 2]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Simp[Plus[Times[c, Plus[Times[Power[a, 2], Plus[p, 2]], Times[-1, Power[b, 2], Plus[m, p, 2]]]], Times[a, b, d, m], Times[b, Plus[Times[a, c], Times[-1, b, d]], Plus[m, p, 3], Sin[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, m], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[p], Less[p, -1], IntegerQ[Times[2, m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], 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Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[c, g, Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, -1]], Power[Times[f, Power[Plus[1, Sin[Plus[e, Times[f, x]]]], Times[Plus[p, -1], Power[2, -1]]], Power[Plus[1, Times[-1, Sin[Plus[e, Times[f, x]]]]], Times[Plus[p, -1], Power[2, -1]]]], -1]], Subst[Int[Times[Power[Plus[1, Times[d, Power[c, -1], x]], Times[Plus[p, 1], Power[2, -1]]], Power[Plus[1, Times[-1, d, Power[c, -1], x]], Times[Plus[p, -1], Power[2, -1]]], Power[Plus[a, Times[b, x]], m]], x], x, Sin[Plus[e, Times[f, x]]]], x], And[FreeQ[List[a, b, c, d, e, f, m, p], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[a, Times[2, m]], Int[Times[Power[Times[d, Sin[Plus[e, Times[f, x]]]], n], Power[Power[Plus[a, Times[-1, b, Sin[Plus[e, Times[f, x]]]]], m], -1]], x], x], And[FreeQ[List[a, b, d, e, f, n], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegersQ[m, p], Equal[Plus[Times[2, m], p], 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, 1]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[2, b, f, g, Plus[m, 1]], -1]], x]], Dist[Times[a, Power[Times[2, Power[g, 2]], -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, 2]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, -1]]], x], x]], And[FreeQ[List[a, b, e, f, g, m, p], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[m, Times[-1, p]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, 1]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Times[a, f, g, m], -1]], x], Times[-1, Dist[Times[1, Power[Power[g, 2], -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, 2]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m]], x], x]]], And[FreeQ[List[a, b, e, f, g, m, p], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[m, p, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Power[a, p], -1]], Int[ExpandTrig[Times[Power[Times[d, sin[Plus[e, Times[f, x]]]], n], Power[Plus[a, Times[-1, b, sin[Plus[e, Times[f, x]]]]], Times[p, Power[2, -1]]], Power[Plus[a, Times[b, sin[Plus[e, Times[f, x]]]]], Plus[m, Times[p, Power[2, -1]]]]], x], x], x], And[FreeQ[List[a, b, d, e, f], x], ZeroQ[Plus[Power[a, 2], Times[-1, 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x], And[FreeQ[List[a, b, d, e, f, g, n, p], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], PositiveIntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Power[b, 2], -1]], Int[Times[Power[Times[d, Sin[Plus[e, Times[f, x]]]], n], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Plus[a, Times[-1, b, Sin[Plus[e, Times[f, x]]]]]], x], x], And[FreeQ[List[a, b, d, e, f, m, n], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], Or[NegativeIntegerQ[m], Not[PositiveIntegerQ[n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[Times[a, Power[g, -1]], Times[2, m]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[Times[2, m], p]], Power[Times[d, Sin[Plus[e, Times[f, x]]]], n], Power[Power[Plus[a, Times[-1, b, Sin[Plus[e, Times[f, x]]]]], m], -1]], x], x], And[FreeQ[List[a, b, d, e, f, g, n, p], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NegativeIntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[Times[a, Power[g, -1]], Times[2, m]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[Times[2, m], p]], Power[Times[d, Sin[Plus[e, Times[f, x]]]], n], Power[Power[Plus[a, Times[-1, b, Sin[Plus[e, Times[f, x]]]]], m], -1]], x], x], And[FreeQ[List[a, b, d, e, f, g, n], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegerQ[m], RationalQ[p], Or[Equal[Plus[Times[2, m], p], 0], And[Greater[Plus[Times[2, m], p], 0], Less[p, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, 1]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Times[a, f, g, Plus[Times[2, m], p, 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[Power[a, 2], Plus[Times[2, m], p, 1]], -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], p], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Plus[Times[a, m], Times[-1, b, Plus[Times[2, m], p, 1], Sin[Plus[e, Times[f, x]]]]]], x], x]]], And[FreeQ[List[a, b, e, f, g, p], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m], LessEqual[m, Times[-1, Power[2, -1]]], NonzeroQ[Plus[Times[2, m], p, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, 1]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[b, f, g, Plus[m, p, 2]], -1]], x]], Dist[Times[1, Power[Times[b, Plus[m, p, 2]], -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], p], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Plus[Times[b, Plus[m, 1]], Times[-1, a, Plus[p, 1], Sin[Plus[e, Times[f, x]]]]]], x], x]], And[FreeQ[List[a, b, e, f, g, m, p], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[m, p, 2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Power[b, 2], -1]], Int[Times[Power[Times[d, Sin[Plus[e, Times[f, x]]]], n], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Plus[a, Times[-1, b, Sin[Plus[e, Times[f, x]]]]]], x], x], And[FreeQ[List[a, b, d, e, f, m, n], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegersQ[Times[2, m], Times[2, n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 4], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[-2, Power[Times[a, b, d], -1]], Int[Times[Power[Times[d, Sin[Plus[e, Times[f, x]]]], Plus[n, 1]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 2]]], x], x], Dist[Times[1, Power[Power[a, 2], -1]], Int[Times[Power[Times[d, Sin[Plus[e, Times[f, x]]]], n], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 2]], Plus[1, Power[Sin[Plus[e, Times[f, x]]], 2]]], x], x]], And[FreeQ[List[a, b, d, e, f, n], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m], Less[m, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 4], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[Power[d, 4], -1]], Int[Times[Power[Times[d, Sin[Plus[e, Times[f, x]]]], Plus[n, 4]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m]], x], x], Int[Times[Power[Times[d, Sin[Plus[e, Times[f, x]]]], n], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Plus[1, Times[-1, 2, Power[Sin[Plus[e, Times[f, x]]], 2]]]], x]], And[FreeQ[List[a, b, d, e, f, m, n], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], Not[PositiveIntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[a, m], Cos[Plus[e, Times[f, x]]], Power[Times[f, Sqrt[Plus[1, Sin[Plus[e, Times[f, x]]]]], Sqrt[Plus[1, Times[-1, Sin[Plus[e, Times[f, x]]]]]]], -1]], Subst[Int[Times[Power[Times[d, x], n], Power[Plus[1, Times[b, Power[a, -1], x]], Plus[m, Times[Plus[p, -1], Power[2, -1]]]], Power[Plus[1, Times[-1, b, Power[a, -1], x]], Times[Plus[p, -1], Power[2, -1]]]], x], x, Sin[Plus[e, Times[f, x]]]], x], And[FreeQ[List[a, b, d, e, f, n], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegerQ[Times[p, Power[2, -1]]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Cos[Plus[e, Times[f, x]]], Power[Times[Power[a, Plus[p, -2]], f, Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Plus[a, Times[-1, b, Sin[Plus[e, Times[f, x]]]]]]], -1]], Subst[Int[Times[Power[Times[d, x], n], Power[Plus[a, Times[b, x]], Plus[m, Times[p, Power[2, -1]], Times[-1, 1, Power[2, -1]]]], Power[Plus[a, Times[-1, b, x]], Plus[Times[p, Power[2, -1]], Times[-1, 1, Power[2, -1]]]]], x], x, Sin[Plus[e, Times[f, x]]]], x], And[FreeQ[List[a, b, d, e, f, m, n], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegerQ[Times[p, Power[2, -1]]], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrig[Power[Times[g, cos[Plus[e, Times[f, x]]]], p], Times[Power[Times[d, sin[Plus[e, Times[f, x]]]], n], Power[Plus[a, Times[b, sin[Plus[e, Times[f, x]]]]], m]], x], x], And[FreeQ[List[a, b, d, e, f, g, n, p], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], PositiveIntegerQ[m], Or[IntegerQ[p], PositiveIntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[a, m], g, Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, -1]], Power[Times[f, Power[Plus[1, Sin[Plus[e, Times[f, x]]]], Times[Plus[p, -1], Power[2, -1]]], Power[Plus[1, Times[-1, Sin[Plus[e, Times[f, x]]]]], Times[Plus[p, -1], Power[2, -1]]]], -1]], Subst[Int[Times[Power[Times[d, x], n], Power[Plus[1, Times[b, Power[a, -1], x]], Plus[m, Times[Plus[p, -1], Power[2, -1]]]], Power[Plus[1, Times[-1, b, Power[a, -1], x]], Times[Plus[p, -1], Power[2, -1]]]], x], x, Sin[Plus[e, Times[f, x]]]], x], And[FreeQ[List[a, b, d, e, f, n, p], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[g, Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, -1]], Power[Times[f, Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Times[Plus[p, -1], Power[2, -1]]], Power[Plus[a, Times[-1, b, Sin[Plus[e, Times[f, x]]]]], Times[Plus[p, -1], Power[2, -1]]]], -1]], Subst[Int[Times[Power[Times[d, x], n], Power[Plus[a, Times[b, x]], Plus[m, Times[Plus[p, -1], Power[2, -1]]]], Power[Plus[a, Times[-1, b, x]], Times[Plus[p, -1], Power[2, -1]]]], x], x, Sin[Plus[e, Times[f, x]]]], x], And[FreeQ[List[a, b, d, e, f, m, n, p], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[g, Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, -1]], Sqrt[Times[d, Sin[Plus[e, Times[f, x]]]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[a, d, f, Plus[m, 1]], -1]], x]], Dist[Times[Power[g, 2], Plus[Times[2, m], 3], Power[Times[2, a, Plus[m, 1]], -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, -2]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Sqrt[Times[d, Sin[Plus[e, Times[f, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, d, e, f, g], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m], Less[m, -1], ZeroQ[Plus[m, p, Times[1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[2, Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, 1]], Sqrt[Times[d, Sin[Plus[e, Times[f, x]]]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Times[d, f, g, Plus[Times[2, m], 1]], -1]], x], Dist[Times[2, a, m, Power[Times[Power[g, 2], Plus[Times[2, m], 1]], -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, 2]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Sqrt[Times[d, Sin[Plus[e, Times[f, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, e, f, g], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m], Greater[m, 0], ZeroQ[Plus[m, p, Times[3, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Times[d, Sin[Plus[e, Times[f, x]]]], n], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Plus[1, Times[-1, Power[Sin[Plus[e, Times[f, x]]], 2]]]], x], And[FreeQ[List[a, b, d, e, f, m, n], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], Or[PositiveIntegerQ[m], IntegersQ[Times[2, m], Times[2, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 4], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], 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Power[Times[d, Sin[Plus[e, Times[f, x]]]], Plus[n, 2]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[Power[a, 2], b, Power[d, 2], f, Plus[n, 1], Plus[m, 1]], -1]], x]]]], And[FreeQ[List[a, b, d, e, f], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegersQ[Times[2, m], Times[2, n]], Less[m, -1], Less[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 4], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Power[a, 2], Times[-1, Power[b, 2]]], Cos[Plus[e, Times[f, x]]], 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2]], Simp[Plus[Times[Power[a, 2], n, Plus[n, 2]], Times[-1, Power[b, 2], Plus[m, n, 2], Plus[m, n, 3]], Times[a, b, m, Sin[Plus[e, Times[f, x]]]], Times[-1, Plus[Times[Power[a, 2], Plus[n, 1], Plus[n, 2]], Times[-1, Power[b, 2], Plus[m, n, 2], Plus[m, n, 4]]], Power[Sin[Plus[e, Times[f, x]]], 2]]], x]], x], x]], Times[-1, Simp[Times[b, Plus[m, n, 2], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[d, Sin[Plus[e, Times[f, x]]]], Plus[n, 2]], Power[Times[Power[a, 2], Power[d, 2], f, Plus[n, 1], Plus[n, 2]], -1]], x]]]], And[FreeQ[List[a, b, d, e, f, m], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], Or[PositiveIntegerQ[m], IntegersQ[Times[2, m], Times[2, n]]], Not[Less[m, -1]], RationalQ[n], Less[n, -1], Or[Less[n, -2], ZeroQ[Plus[m, n, 4]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 4], Power[Times[Optional[Pattern[d, 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IntegersQ[Times[2, m], Times[2, n]], NonzeroQ[Plus[n, 1]], NonzeroQ[Plus[n, 2]], NonzeroQ[Plus[m, n, 5]], NonzeroQ[Plus[m, n, 6]], Not[PositiveIntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrig[Times[Power[Times[d, sin[Plus[e, Times[f, x]]]], n], Power[Plus[a, Times[b, sin[Plus[e, Times[f, x]]]]], m], Power[Plus[1, Times[-1, Power[sin[Plus[e, Times[f, x]]], 2]]], Times[p, Power[2, -1]]]], x], x], And[FreeQ[List[a, b, d, e, f], x], 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Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Power[g, 2], Power[Times[a, b], -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, -2]], Power[Times[d, Sin[Plus[e, Times[f, x]]]], n], Plus[b, Times[-1, a, Sin[Plus[e, Times[f, x]]]]]], x], x], Dist[Times[Power[g, 2], Plus[Power[a, 2], Times[-1, Power[b, 2]]], Power[Times[a, b, d], -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, -2]], Power[Times[d, Sin[Plus[e, Times[f, x]]]], Plus[n, 1]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, d, e, f, g], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegersQ[Times[2, n], Times[2, p]], Greater[p, 1], Or[Less[n, -1], And[Equal[p, Times[3, Power[2, -1]]], Equal[n, Times[-1, Power[2, -1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Power[g, 2], Power[Power[b, 2], -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, -2]], Power[Times[d, Sin[Plus[e, Times[f, x]]]], n], Plus[a, Times[-1, b, Sin[Plus[e, Times[f, x]]]]]], x], x], Times[-1, Dist[Times[Power[g, 2], Plus[Power[a, 2], Times[-1, Power[b, 2]]], Power[Power[b, 2], -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, -2]], Power[Times[d, Sin[Plus[e, Times[f, x]]]], n], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], -1]], x], x]]], And[FreeQ[List[a, b, d, e, f, g], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegersQ[Times[2, n], Times[2, p]], Greater[p, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[a, Power[d, 2], Power[Plus[Power[a, 2], Times[-1, Power[b, 2]]], -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], p], Power[Times[d, Sin[Plus[e, Times[f, x]]]], Plus[n, -2]]], x], x], Plus[Times[-1, Dist[Times[b, d, Power[Plus[Power[a, 2], Times[-1, Power[b, 2]]], -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], p], Power[Times[d, Sin[Plus[e, Times[f, x]]]], Plus[n, -1]]], x], x]], Times[-1, Dist[Times[Power[a, 2], Power[d, 2], Power[Times[Power[g, 2], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, 2]], Power[Times[d, Sin[Plus[e, Times[f, x]]]], Plus[n, -2]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], -1]], x], x]]]], And[FreeQ[List[a, b, d, e, f, g], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegersQ[Times[2, n], Times[2, p]], Less[p, -1], Greater[n, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Dist[Times[d, Power[Plus[Power[a, 2], Times[-1, Power[b, 2]]], -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], p], Power[Times[d, Sin[Plus[e, Times[f, x]]]], Plus[n, -1]], Plus[b, Times[-1, a, Sin[Plus[e, Times[f, x]]]]]], x], x]], Dist[Times[a, b, d, Power[Times[Power[g, 2], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, 2]], Power[Times[d, Sin[Plus[e, Times[f, x]]]], Plus[n, -1]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, d, e, f, g], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegersQ[Times[2, n], Times[2, p]], Less[p, -1], Greater[n, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[Plus[Power[a, 2], Times[-1, Power[b, 2]]], -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], p], Power[Times[d, Sin[Plus[e, Times[f, x]]]], n], Plus[a, Times[-1, b, Sin[Plus[e, Times[f, x]]]]]], x], x], Times[-1, Dist[Times[Power[b, 2], Power[Times[Power[g, 2], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, 2]], Power[Times[d, Sin[Plus[e, Times[f, x]]]], n], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], -1]], x], x]]], And[FreeQ[List[a, b, d, e, f, g], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegersQ[Times[2, n], Times[2, p]], Less[p, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Rational[1, 2]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[-4, Sqrt[2], g, Power[f, -1]], Subst[Int[Times[Power[x, 2], Power[Times[Plus[Times[Plus[a, b], Power[g, 2]], Times[Plus[a, Times[-1, b]], Power[x, 4]]], Sqrt[Plus[1, Times[-1, Power[x, 4], Power[Power[g, 2], -1]]]]], -1]], x], x, Times[Sqrt[Times[g, Cos[Plus[e, Times[f, x]]]]], Power[Sqrt[Plus[1, Sin[Plus[e, Times[f, x]]]]], -1]]], x], And[FreeQ[List[a, b, e, f, g], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Rational[1, 2]], Power[Times[Pattern[d, Blank[]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Sin[Plus[e, Times[f, x]]]], Power[Sqrt[Times[d, Sin[Plus[e, Times[f, x]]]]], -1]], Int[Times[Sqrt[Times[g, Cos[Plus[e, Times[f, x]]]]], Power[Times[Sqrt[Sin[Plus[e, Times[f, x]]]], Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x], And[FreeQ[List[a, b, d, e, f, g], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Rational[-1, 2]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Times[-1, Power[a, 2]], Power[b, 2]], 2]]], Plus[Dist[Times[2, Sqrt[2], d, Plus[b, q], Power[Times[f, q], -1]], Subst[Int[Times[1, Power[Times[Plus[Times[d, Plus[b, q]], Times[a, Power[x, 2]]], Sqrt[Plus[1, Times[-1, Power[x, 4], Power[Power[d, 2], -1]]]]], -1]], x], x, Times[Sqrt[Times[d, Sin[Plus[e, Times[f, x]]]]], Power[Sqrt[Plus[1, Cos[Plus[e, Times[f, x]]]]], -1]]], x], Times[-1, Dist[Times[2, Sqrt[2], d, Plus[b, Times[-1, q]], Power[Times[f, q], -1]], Subst[Int[Times[1, Power[Times[Plus[Times[d, Plus[b, Times[-1, q]]], Times[a, Power[x, 2]]], Sqrt[Plus[1, Times[-1, Power[x, 4], Power[Power[d, 2], -1]]]]], -1]], x], x, Times[Sqrt[Times[d, Sin[Plus[e, Times[f, x]]]]], Power[Sqrt[Plus[1, Cos[Plus[e, Times[f, x]]]]], -1]]], x]]]], And[FreeQ[List[a, b, d, e, f], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Rational[-1, 2]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Cos[Plus[e, Times[f, x]]]], Power[Sqrt[Times[g, Cos[Plus[e, Times[f, x]]]]], -1]], Int[Times[Sqrt[Times[d, Sin[Plus[e, Times[f, x]]]]], Power[Times[Sqrt[Cos[Plus[e, Times[f, x]]]], Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x], And[FreeQ[List[a, b, d, e, f, g], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[d, Power[b, -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], p], Power[Times[d, Sin[Plus[e, Times[f, x]]]], Plus[n, -1]]], x], x], Times[-1, Dist[Times[a, d, Power[b, -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], p], Power[Times[d, Sin[Plus[e, Times[f, x]]]], Plus[n, -1]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], -1]], x], x]]], And[FreeQ[List[a, b, d, e, f, g], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegersQ[Times[2, n], Times[2, p]], Less[-1, p, 1], Greater[n, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[a, -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], p], Power[Times[d, Sin[Plus[e, Times[f, x]]]], n]], x], x], Times[-1, Dist[Times[b, Power[Times[a, d], -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], p], Power[Times[d, Sin[Plus[e, Times[f, x]]]], Plus[n, 1]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], -1]], x], x]]], And[FreeQ[List[a, b, d, e, f, g], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegersQ[Times[2, n], Times[2, p]], Less[-1, p, 1], Less[n, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrig[Power[Times[g, cos[Plus[e, Times[f, x]]]], p], Times[Power[Times[d, sin[Plus[e, Times[f, x]]]], n], Power[Plus[a, Times[b, sin[Plus[e, Times[f, x]]]]], m]], x], x], And[FreeQ[List[a, b, d, e, f, g, n, p], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegerQ[m], Or[Greater[m, 0], IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Power[g, 2], Power[a, -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, -2]], Power[Times[d, Sin[Plus[e, Times[f, x]]]], n], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]]], x], x], Plus[Times[-1, Dist[Times[b, Power[g, 2], Power[Times[Power[a, 2], d], -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, -2]], Power[Times[d, Sin[Plus[e, Times[f, x]]]], Plus[n, 1]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]]], x], x]], Times[-1, Dist[Times[Power[g, 2], Plus[Power[a, 2], Times[-1, Power[b, 2]]], Power[Times[Power[a, 2], Power[d, 2]], -1]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, -2]], Power[Times[d, Sin[Plus[e, Times[f, x]]]], Plus[n, 2]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m]], x], x]]]], And[FreeQ[List[a, b, d, e, f, g], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegersQ[m, Times[2, n], Times[2, p]], Less[m, 0], Greater[p, 1], Or[LessEqual[n, -2], And[Equal[m, -1], Equal[n, Times[-3, Power[2, -1]]], Equal[p, Times[3, Power[2, -1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[a, Times[2, m]], Int[Times[Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Power[Power[Plus[a, Times[-1, b, Sin[Plus[e, Times[f, x]]]]], m], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, n], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegersQ[m, p], Equal[Plus[Times[2, m], p], 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[Times[a, Power[g, -1]], Times[2, m]], Int[Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[Times[2, m], p]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Power[Power[Plus[a, Times[-1, b, Sin[Plus[e, Times[f, x]]]]], m], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, n], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegerQ[m], RationalQ[p], Or[Equal[Plus[Times[2, m], p], 0], And[Greater[Plus[Times[2, m], p], 0], Less[p, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Power[b, 2], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Plus[a, Times[-1, b, Sin[Plus[e, Times[f, x]]]]]], x], x], And[FreeQ[List[a, b, c, d, e, f, m, n], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegersQ[Times[2, m], Times[2, n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[a, m], Cos[Plus[e, Times[f, x]]], Power[Times[f, Sqrt[Plus[1, Sin[Plus[e, Times[f, x]]]]], Sqrt[Plus[1, Times[-1, Sin[Plus[e, Times[f, x]]]]]]], -1]], Subst[Int[Times[Power[Plus[1, Times[b, Power[a, -1], x]], Plus[m, Times[Plus[p, -1], Power[2, -1]]]], Power[Plus[1, Times[-1, b, Power[a, -1], x]], Times[Plus[p, -1], Power[2, -1]]], Power[Plus[c, Times[d, x]], n]], x], x, Sin[Plus[e, Times[f, x]]]], x], And[FreeQ[List[a, b, c, d, e, f, n], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegerQ[Times[p, Power[2, -1]]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Cos[Plus[e, Times[f, x]]], Power[Times[Power[a, Plus[p, -2]], f, Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Plus[a, Times[-1, b, Sin[Plus[e, Times[f, x]]]]]]], -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], Plus[m, Times[p, Power[2, -1]], Times[-1, 1, Power[2, -1]]]], Power[Plus[a, Times[-1, b, x]], Plus[Times[p, Power[2, -1]], Times[-1, 1, Power[2, -1]]]], Power[Plus[c, Times[d, x]], n]], x], x, Sin[Plus[e, Times[f, x]]]], x], And[FreeQ[List[a, b, c, d, e, f, m, n], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegerQ[Times[p, Power[2, -1]]], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrig[Power[Times[g, cos[Plus[e, Times[f, x]]]], p], Times[Power[Plus[a, Times[b, sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, sin[Plus[e, Times[f, x]]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, n, p], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], PositiveIntegerQ[m], Or[IntegerQ[p], PositiveIntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[a, m], g, Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, -1]], Power[Times[f, Power[Plus[1, Sin[Plus[e, Times[f, x]]]], Times[Plus[p, -1], Power[2, -1]]], Power[Plus[1, Times[-1, Sin[Plus[e, Times[f, x]]]]], Times[Plus[p, -1], Power[2, -1]]]], -1]], Subst[Int[Times[Power[Plus[1, Times[b, Power[a, -1], x]], Plus[m, Times[Plus[p, -1], Power[2, -1]]]], Power[Plus[1, Times[-1, b, Power[a, -1], x]], Times[Plus[p, -1], Power[2, -1]]], Power[Plus[c, Times[d, x]], n]], x], x, Sin[Plus[e, Times[f, x]]]], x], And[FreeQ[List[a, b, c, d, e, f, n, p], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[g, Power[Times[g, Cos[Plus[e, Times[f, x]]]], Plus[p, -1]], Power[Times[f, Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Times[Plus[p, -1], Power[2, -1]]], Power[Plus[a, Times[-1, b, Sin[Plus[e, Times[f, x]]]]], Times[Plus[p, -1], Power[2, -1]]]], -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], Plus[m, Times[Plus[p, -1], Power[2, -1]]]], Power[Plus[a, Times[-1, b, x]], Times[Plus[p, -1], Power[2, -1]]], Power[Plus[c, Times[d, x]], n]], x], x, Sin[Plus[e, Times[f, x]]]], x], And[FreeQ[List[a, b, c, d, e, f, m, n, p], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Plus[1, Times[-1, Power[Sin[Plus[e, Times[f, x]]], 2]]]], x], And[FreeQ[List[a, b, c, d, e, f, m, n], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], Or[PositiveIntegerQ[m], IntegersQ[Times[2, m], Times[2, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrig[Times[Power[Plus[a, Times[b, sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, sin[Plus[e, Times[f, x]]]]], n], Power[Plus[1, Times[-1, Power[sin[Plus[e, Times[f, x]]], 2]]], Times[p, Power[2, -1]]]], x], x], And[FreeQ[List[a, b, c, d, e, f, m, n], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], PositiveIntegerQ[Times[p, Power[2, -1]]], Or[PositiveIntegerQ[m], IntegersQ[Times[2, m], Times[2, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrig[Times[Power[Times[g, cos[Plus[e, Times[f, x]]]], p], Power[Plus[a, Times[b, sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, sin[Plus[e, Times[f, x]]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, p], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegersQ[Times[2, m], Times[2, n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Times[g, Cos[Plus[e, Times[f, x]]]], p], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n]], x], And[FreeQ[List[a, b, c, d, e, f, g, m, n, p], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[g, Times[2, IntPart[p]]], Power[Times[g, Cos[Plus[e, Times[f, x]]]], FracPart[p]], Power[Times[g, Sec[Plus[e, Times[f, x]]]], FracPart[p]]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Power[Power[Times[g, Cos[Plus[e, Times[f, x]]]], p], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, m, n, p], x], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[g, Times[2, IntPart[p]]], Power[Times[g, Sin[Plus[e, Times[f, x]]]], FracPart[p]], Power[Times[g, Csc[Plus[e, Times[f, x]]]], FracPart[p]]], Int[Times[Power[Plus[a, Times[b, Cos[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Cos[Plus[e, Times[f, x]]]]], n], Power[Power[Times[g, Sin[Plus[e, Times[f, x]]]], p], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, m, n, p], x], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[g, Power[d, -1]], Int[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Sqrt[Times[g, Sin[Plus[e, Times[f, x]]]]], -1]], x], x], Times[-1, Dist[Times[c, g, Power[d, -1]], Int[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Times[Sqrt[Times[g, Sin[Plus[e, Times[f, x]]]]], Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], Or[ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[b, Power[d, -1]], Int[Times[Sqrt[Times[g, Sin[Plus[e, Times[f, x]]]]], Power[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x], Times[-1, Dist[Times[Plus[Times[b, c], Times[-1, a, d]], Power[d, -1]], Int[Times[Sqrt[Times[g, Sin[Plus[e, Times[f, x]]]]], Power[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[-2, b, Power[f, -1]], Subst[Int[Times[1, Power[Plus[Times[b, c], Times[a, d], Times[c, g, Power[x, 2]]], -1]], x], x, Times[b, Cos[Plus[e, Times[f, x]]], Power[Times[Sqrt[Times[g, Sin[Plus[e, Times[f, x]]]]], Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]]], -1]]], x], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Simp[Times[Sqrt[Plus[a, b]], Power[Times[c, f], -1], EllipticE[ArcSin[Times[Cos[Plus[e, Times[f, x]]], Power[Plus[1, Sin[Plus[e, Times[f, x]]]], -1]]], Times[-1, Plus[a, Times[-1, b]], Power[Plus[a, b], -1]]]], x]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[d, Times[-1, c]]], PositiveQ[Plus[Power[b, 2], Times[-1, Power[a, 2]]]], PositiveQ[b]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Simp[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Times[d, Sin[Plus[e, Times[f, x]]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], -1]]], Power[Times[d, f, Sqrt[Times[g, Sin[Plus[e, Times[f, x]]]]], Sqrt[Times[Power[c, 2], Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Power[Times[Plus[Times[a, c], Times[b, d]], Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]]]], -1], EllipticE[ArcSin[Times[c, Cos[Plus[e, Times[f, x]]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], -1]]], Times[Plus[Times[b, c], Times[-1, a, d]], Power[Plus[Times[b, c], Times[a, d]], -1]]]], x]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[a, Power[c, -1]], Int[Times[1, Power[Times[Sqrt[Times[g, Sin[Plus[e, Times[f, x]]]]], Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], x], Dist[Times[Plus[Times[b, c], Times[-1, a, d]], Power[Times[c, g], -1]], Int[Times[Sqrt[Times[g, Sin[Plus[e, Times[f, x]]]]], Power[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[c, -1]], Int[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Sin[Plus[e, Times[f, x]]], -1]], x], x], Times[-1, Dist[Times[d, Power[c, -1]], Int[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[a, Power[c, -1]], Int[Times[1, Power[Times[Sin[Plus[e, Times[f, x]]], Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], x], Dist[Times[Plus[Times[b, c], Times[-1, a, d]], Power[c, -1]], Int[Times[1, Power[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Dist[Times[a, g, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Int[Times[1, Power[Times[Sqrt[Times[g, Sin[Plus[e, Times[f, x]]]]], Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], x]], Dist[Times[c, g, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Int[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Times[Sqrt[Times[g, Sin[Plus[e, Times[f, x]]]]], Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], Or[ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[2, Sqrt[Times[-1, Power[Cot[Plus[e, Times[f, x]]], 2]]], Sqrt[Times[g, Sin[Plus[e, Times[f, x]]]]], Power[Times[f, Plus[c, d], Cot[Plus[e, Times[f, x]]], Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]]], -1], Sqrt[Times[Plus[b, Times[a, Csc[Plus[e, Times[f, x]]]]], Power[Plus[a, b], -1]]], EllipticPi[Times[2, c, Power[Plus[c, d], -1]], ArcSin[Times[Sqrt[Plus[1, Times[-1, Csc[Plus[e, Times[f, x]]]]]], Power[Sqrt[2], -1]]], Times[2, a, Power[Plus[a, b], -1]]]], x], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[b, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Int[Times[1, Power[Times[Sqrt[Times[g, Sin[Plus[e, Times[f, x]]]]], Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], x], Times[-1, Dist[Times[d, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Int[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Times[Sqrt[Times[g, Sin[Plus[e, Times[f, x]]]]], Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], Or[ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[c, -1]], Int[Times[1, Power[Times[Sqrt[Times[g, Sin[Plus[e, Times[f, x]]]]], Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], x], Times[-1, Dist[Times[d, Power[Times[c, g], -1]], Int[Times[Sqrt[Times[g, Sin[Plus[e, Times[f, x]]]]], Power[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Power[d, 2], Power[Times[c, Plus[Times[b, c], Times[-1, a, d]]], -1]], Int[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], -1]], x], x], Dist[Times[1, Power[Times[c, Plus[Times[b, c], Times[-1, a, d]]], -1]], Int[Times[Plus[Times[b, c], Times[-1, a, d], Times[-1, b, d, Sin[Plus[e, Times[f, x]]]]], Power[Times[Sin[Plus[e, Times[f, x]]], Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[c, -1]], Int[Times[1, Power[Times[Sin[Plus[e, Times[f, x]]], Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], x], Times[-1, Dist[Times[d, Power[c, -1]], Int[Times[1, Power[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Dist[Times[d, Power[c, -1]], Int[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x]], Dist[Times[1, Power[c, -1]], Int[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], Power[Sin[Plus[e, Times[f, x]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[Times[b, c], Times[a, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[-2, a, Power[f, -1]], Subst[Int[Times[1, Power[Plus[1, Times[-1, a, c, Power[x, 2]]], -1]], x], x, Times[Cos[Plus[e, Times[f, x]]], Power[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]]], -1]]], x], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Times[b, c], Times[a, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Plus[Times[b, c], Times[-1, a, d]], Power[c, -1]], Int[Times[1, Power[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], x], Dist[Times[a, Power[c, -1]], Int[Times[Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], Power[Times[Sin[Plus[e, Times[f, x]]], Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[-2, Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Power[Times[c, f, Rt[Times[Plus[a, b], Power[Plus[c, d], -1]], 2], Cos[Plus[e, Times[f, x]]]], -1], Sqrt[Times[-1, Plus[Times[b, c], Times[-1, a, d]], Plus[1, Times[-1, Sin[Plus[e, Times[f, x]]]]], Power[Times[Plus[c, d], Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], -1]]], Sqrt[Times[Plus[Times[b, c], Times[-1, a, d]], Plus[1, Sin[Plus[e, Times[f, x]]]], Power[Times[Plus[c, Times[-1, d]], Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], -1]]], EllipticPi[Times[a, Plus[c, d], Power[Times[c, Plus[a, b]], -1]], ArcSin[Times[Rt[Times[Plus[a, b], Power[Plus[c, d], -1]], 2], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], Power[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], -1]]], Times[Plus[a, Times[-1, b]], Plus[c, d], Power[Times[Plus[a, b], Plus[c, Times[-1, d]]], -1]]]], x], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Cos[Plus[e, Times[f, x]]], Power[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]]], -1]], Int[Times[1, Power[Times[Cos[Plus[e, Times[f, x]]], Sin[Plus[e, Times[f, x]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Dist[Times[b, Power[a, -1]], Int[Times[1, Power[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], x]], Dist[Times[1, Power[a, -1]], Int[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Times[Sin[Plus[e, Times[f, x]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], Or[NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], Power[Cos[Plus[e, Times[f, x]]], -1]], Int[Cot[Plus[e, Times[f, x]]], x], x], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[d, Int[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x], Dist[c, Int[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Times[Sin[Plus[e, Times[f, x]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], Or[NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[a, n], Power[c, n]], Int[Times[Power[Tan[Plus[e, Times[f, x]]], p], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, Times[-1, n]]]], x], x], And[FreeQ[List[a, b, c, d, e, f, m], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[p, Times[2, n]]], IntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Plus[a, Times[-1, b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Times[f, Cos[Plus[e, Times[f, x]]]], -1]], Subst[Int[Times[Power[Times[g, x], p], Power[Plus[a, Times[b, x]], Plus[m, Times[-1, 1, Power[2, -1]]]], Power[Plus[c, Times[d, x]], n], Power[Sqrt[Plus[a, Times[-1, b, x]]], -1]], x], x, Sin[Plus[e, Times[f, x]]]], x], And[FreeQ[List[a, b, c, d, e, f, g, m, n, p], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], IntegerQ[Plus[m, Times[-1, 1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrig[Times[Power[Times[g, sin[Plus[e, Times[f, x]]]], p], Power[Plus[a, Times[b, sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, sin[Plus[e, Times[f, x]]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, n, p], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], Or[IntegersQ[m, n], IntegersQ[m, p], IntegersQ[n, p]], NonzeroQ[Plus[p, -2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Times[g, Sin[Plus[e, Times[f, x]]]], p], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n]], x], And[FreeQ[List[a, b, c, d, e, f, g, m, n, p], x], NonzeroQ[Plus[p, -2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[g, Plus[m, n]], Int[Times[Power[Times[g, Sin[Plus[e, Times[f, x]]]], Plus[p, Times[-1, m], Times[-1, n]]], Power[Plus[b, Times[a, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[d, Times[c, Sin[Plus[e, Times[f, x]]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, p], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], Not[IntegerQ[p]], IntegerQ[m], IntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[g, Csc[Plus[e, Times[f, x]]]], p], Power[Times[g, Sin[Plus[e, Times[f, x]]]], p]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], n], Power[Power[Times[g, Csc[Plus[e, Times[f, x]]]], p], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, m, n, p], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], Not[IntegerQ[p]], Not[And[IntegerQ[m], IntegerQ[n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[g, n], Int[Times[Power[Times[g, Sin[Plus[e, Times[f, x]]]], Plus[p, Times[-1, n]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[d, Times[c, Sin[Plus[e, Times[f, x]]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, m, p], x], IntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[b, Times[a, Csc[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], n], Power[Power[Csc[Plus[e, Times[f, x]]], Plus[m, p]], -1]], x], And[FreeQ[List[a, b, c, d, e, f, n], x], Not[IntegerQ[n]], IntegerQ[m], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Csc[Plus[e, Times[f, x]]], p], Power[Times[g, Sin[Plus[e, Times[f, x]]]], p]], Int[Times[Power[Plus[b, Times[a, Csc[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], n], Power[Power[Csc[Plus[e, Times[f, x]]], Plus[m, p]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, n, p], x], Not[IntegerQ[n]], IntegerQ[m], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[g, Sin[Plus[e, Times[f, x]]]], n], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], n], Power[Power[Plus[d, Times[c, Sin[Plus[e, Times[f, x]]]]], n], -1]], Int[Times[Power[Times[g, Sin[Plus[e, Times[f, x]]]], Plus[p, Times[-1, n]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[d, Times[c, Sin[Plus[e, Times[f, x]]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, m, n, p], x], Not[IntegerQ[n]], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[g, Plus[m, n]], Int[Times[Power[Times[g, Csc[Plus[e, Times[f, x]]]], Plus[p, Times[-1, m], Times[-1, n]]], Power[Plus[b, Times[a, Csc[Plus[e, Times[f, x]]]]], m], Power[Plus[d, Times[c, Csc[Plus[e, Times[f, x]]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, p], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], Not[IntegerQ[p]], IntegerQ[m], IntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[g, Csc[Plus[e, Times[f, x]]]], p], Power[Times[g, Sin[Plus[e, Times[f, x]]]], p]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Power[Power[Times[g, Sin[Plus[e, Times[f, x]]]], p], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, m, n, p], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], Not[IntegerQ[p]], Not[And[IntegerQ[m], IntegerQ[n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[g, m], Int[Times[Power[Times[g, Csc[Plus[e, Times[f, x]]]], Plus[p, Times[-1, m]]], Power[Plus[b, Times[a, Csc[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, n, p], x], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[d, Times[c, Sin[Plus[e, Times[f, x]]]]], n], Power[Power[Sin[Plus[e, Times[f, x]]], Plus[n, p]], -1]], x], And[FreeQ[List[a, b, c, d, e, f, m], x], Not[IntegerQ[m]], IntegerQ[n], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Sin[Plus[e, Times[f, x]]], p], Power[Times[g, Csc[Plus[e, Times[f, x]]]], p]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[d, Times[c, Sin[Plus[e, Times[f, x]]]]], n], Power[Power[Sin[Plus[e, Times[f, x]]], Plus[n, p]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, m, p], x], Not[IntegerQ[m]], IntegerQ[n], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Times[g, Csc[Plus[e, Times[f, x]]]], m], Power[Power[Plus[b, Times[a, Csc[Plus[e, Times[f, x]]]]], m], -1]], Int[Times[Power[Times[g, Csc[Plus[e, Times[f, x]]]], Plus[p, Times[-1, m]]], Power[Plus[b, Times[a, Csc[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, m, n, p], x], Not[IntegerQ[m]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrig[Times[Power[sin[Plus[e, Times[f, x]]], n], Power[Plus[a, Times[b, sin[Plus[e, Times[f, x]]]]], m], Plus[A, Times[B, sin[Plus[e, Times[f, x]]]]]], x], x], And[FreeQ[List[a, b, e, f, A, B], x], ZeroQ[Plus[Times[A, b], Times[a, B]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegerQ[m], IntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[a, m], Power[c, m]], Int[Times[Power[Cos[Plus[e, Times[f, x]]], Times[2, m]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, Times[-1, m]]], Plus[A, Times[B, Sin[Plus[e, Times[f, x]]]]]], x], x], And[FreeQ[List[a, b, c, d, e, f, A, B, n], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegerQ[m], Not[And[IntegerQ[n], Or[And[Less[m, 0], Greater[n, 0]], Less[0, n, m], Less[m, n, 0]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Plus[Times[A, c], Times[Plus[Times[B, c], Times[A, d]], Sin[Plus[e, Times[f, x]]]], Times[B, d, Power[Sin[Plus[e, Times[f, x]]], 2]]]], x], And[FreeQ[List[a, b, c, d, e, f, A, B, m], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Plus[Times[A, b], Times[a, B]], Power[Times[2, a, b], -1]], Int[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x], Dist[Times[Plus[Times[B, c], Times[A, d]], Power[Times[2, c, d], -1]], Int[Times[Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], Power[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Simp[Times[B, Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Power[Times[f, Plus[m, n, 1]], -1]], x]], And[FreeQ[List[a, b, c, d, e, f, A, B, m, n], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[Times[A, b, Plus[m, n, 1]], Times[a, B, Plus[m, Times[-1, n]]]]], NonzeroQ[Plus[m, Times[1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[B, Power[d, -1]], Int[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]]], x], x], Times[-1, Dist[Times[Plus[Times[B, c], Times[-1, A, d]], Power[d, -1]], Int[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, A, B, n], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[A, b], Times[-1, a, B]], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Power[Times[a, f, Plus[Times[2, m], 1]], -1]], x], Dist[Times[Plus[Times[a, B, Plus[m, Times[-1, n]]], Times[A, b, Plus[m, n, 1]]], Power[Times[a, b, Plus[Times[2, m], 1]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B, m, n], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], Or[And[RationalQ[m], Less[m, Times[-1, Power[2, -1]]]], And[NegativeIntegerQ[Plus[m, n]], Not[SumSimplerQ[n, 1]]]], NonzeroQ[Plus[Times[2, m], 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[B, Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Power[Times[f, Plus[m, n, 1]], -1]], x]], Times[-1, Dist[Times[Plus[Times[B, c, Plus[m, Times[-1, n]]], Times[-1, A, d, Plus[m, n, 1]]], Power[Times[d, Plus[m, n, 1]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, A, B, m, n], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], Not[And[RationalQ[m], Less[m, Times[-1, Power[2, -1]]]]], NonzeroQ[Plus[m, n, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Plus[Times[B, c], Times[-1, A, d]], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[f, Plus[n, 1], Plus[Power[c, 2], Times[-1, Power[d, 2]]]], -1]], x], And[FreeQ[List[a, b, c, d, e, f, A, B, m, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], ZeroQ[Plus[m, n, 2]], ZeroQ[Plus[Times[A, Plus[Times[a, d, m], Times[b, c, Plus[n, 1]]]], Times[-1, B, Plus[Times[a, c, m], Times[b, d, Plus[n, 1]]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[b, 2], Plus[Times[B, c], Times[-1, A, d]], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[d, f, Plus[n, 1], Plus[Times[b, c], Times[a, d]]], -1]], x]], Times[-1, Dist[Times[b, Power[Times[d, Plus[n, 1], Plus[Times[b, c], Times[a, d]]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Simp[Plus[Times[a, A, d, Plus[m, Times[-1, n], -2]], Times[-1, B, Plus[Times[a, c, Plus[m, -1]], Times[b, d, Plus[n, 1]]]], Times[-1, Plus[Times[A, b, d, Plus[m, n, 1]], Times[-1, B, Plus[Times[b, c, m], Times[-1, a, d, Plus[n, 1]]]]], Sin[Plus[e, Times[f, x]]]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, A, B], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], RationalQ[m, n], Greater[m, Times[1, Power[2, -1]]], Less[n, -1], IntegerQ[Times[2, m]], Or[IntegerQ[Times[2, n]], ZeroQ[c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[b, B, Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[d, f, Plus[m, n, 1]], -1]], x]], Dist[Times[1, Power[Times[d, Plus[m, n, 1]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Simp[Plus[Times[a, A, d, Plus[m, n, 1]], Times[B, Plus[Times[a, c, Plus[m, -1]], Times[b, d, Plus[n, 1]]]], Times[Plus[Times[A, b, d, Plus[m, n, 1]], Times[-1, B, Plus[Times[b, c, m], Times[-1, a, d, Plus[Times[2, m], n]]]]], Sin[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], RationalQ[m], Greater[m, Times[1, Power[2, -1]]], Not[And[RationalQ[n], Less[n, -1]]], IntegerQ[Times[2, m]], Or[IntegerQ[Times[2, n]], ZeroQ[c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[A, b], Times[-1, a, B]], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Power[Times[a, f, Plus[Times[2, m], 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[a, b, Plus[Times[2, m], 1]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, -1]], Simp[Plus[Times[A, Plus[Times[a, d, n], Times[-1, b, c, Plus[m, 1]]]], Times[-1, B, Plus[Times[a, c, m], Times[b, d, n]]], Times[-1, d, Plus[Times[a, B, Plus[m, Times[-1, n]]], Times[A, b, Plus[m, n, 1]]], Sin[Plus[e, Times[f, x]]]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, A, B], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], RationalQ[m, n], Less[m, Times[-1, Power[2, -1]]], Greater[n, 0], IntegerQ[Times[2, m]], Or[IntegerQ[Times[2, n]], ZeroQ[c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, Plus[Times[A, b], Times[-1, a, B]], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[a, f, Plus[Times[2, m], 1], Plus[Times[b, c], Times[-1, a, d]]], -1]], x], Dist[Times[1, Power[Times[a, Plus[Times[2, m], 1], Plus[Times[b, c], Times[-1, a, d]]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Simp[Plus[Times[B, Plus[Times[a, c, m], Times[b, d, Plus[n, 1]]]], Times[A, Plus[Times[b, c, Plus[m, 1]], Times[-1, a, d, Plus[Times[2, m], n, 2]]]], Times[d, Plus[Times[A, b], Times[-1, a, B]], Plus[m, n, 2], Sin[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], RationalQ[m], Less[m, Times[-1, Power[2, -1]]], Not[And[RationalQ[n], Greater[n, 0]]], IntegerQ[Times[2, m]], Or[IntegerQ[Times[2, n]], ZeroQ[c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[-2, b, B, Cos[Plus[e, Times[f, x]]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[d, f, Plus[Times[2, n], 3], Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], And[FreeQ[List[a, b, c, d, e, f, A, B, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], ZeroQ[Plus[Times[A, b, d, Plus[Times[2, n], 3]], Times[-1, B, Plus[Times[b, c], Times[-1, 2, a, d, Plus[n, 1]]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[b, 2], Plus[Times[B, c], Times[-1, A, d]], Cos[Plus[e, Times[f, x]]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[d, f, Plus[n, 1], Plus[Times[b, c], Times[a, d]], Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]]], -1]], x]], Dist[Times[Plus[Times[A, b, d, Plus[Times[2, n], 3]], Times[-1, B, Plus[Times[b, c], Times[-1, 2, a, d, Plus[n, 1]]]]], Power[Times[2, d, Plus[n, 1], Plus[Times[b, c], Times[a, d]]], -1]], Int[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], RationalQ[n], Less[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[-2, b, B, Cos[Plus[e, Times[f, x]]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[d, f, Plus[Times[2, n], 3], Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], Dist[Times[Plus[Times[A, b, d, Plus[Times[2, n], 3]], Times[-1, B, Plus[Times[b, c], Times[-1, 2, a, d, Plus[n, 1]]]]], Power[Times[b, d, Plus[Times[2, n], 3]], -1]], Int[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], Not[And[RationalQ[n], Less[n, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Plus[Times[A, b], Times[-1, a, B]], Power[b, -1]], Int[Times[1, Power[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], x], Dist[Times[B, Power[b, -1]], Int[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[B, Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Power[Times[f, Plus[m, n, 1]], -1]], x]], Dist[Times[1, Power[Times[b, Plus[m, n, 1]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, -1]], Simp[Plus[Times[A, b, c, Plus[m, n, 1]], Times[B, Plus[Times[a, c, m], Times[b, d, n]]], Times[Plus[Times[A, b, d, Plus[m, n, 1]], Times[B, Plus[Times[a, d, m], Times[b, c, n]]]], Sin[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B, m], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], RationalQ[n], Greater[n, 0], Or[IntegerQ[n], ZeroQ[Plus[m, Times[1, Power[2, -1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[B, c], Times[-1, A, d]], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[f, Plus[n, 1], Plus[Power[c, 2], Times[-1, Power[d, 2]]]], -1]], x], Dist[Times[1, Power[Times[b, Plus[n, 1], Plus[Power[c, 2], Times[-1, Power[d, 2]]]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Simp[Plus[Times[A, Plus[Times[a, d, m], Times[b, c, Plus[n, 1]]]], Times[-1, B, Plus[Times[a, c, m], Times[b, d, Plus[n, 1]]]], Times[b, Plus[Times[B, c], Times[-1, A, d]], Plus[m, n, 2], Sin[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B, m], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], RationalQ[n], Less[n, -1], Or[IntegerQ[n], ZeroQ[Plus[m, Times[1, Power[2, -1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Plus[Times[A, b], Times[-1, a, B]], Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Int[Times[1, Power[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x], Dist[Times[Plus[Times[B, c], Times[-1, A, d]], Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Int[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[B, Power[d, -1]], Int[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], x], x], Times[-1, Dist[Times[Plus[Times[B, c], Times[-1, A, d]], Power[d, -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, A, B, m], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], NonzeroQ[Plus[m, Times[1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Plus[Times[A, b], Times[-1, a, B]], Power[b, -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n]], x], x], Dist[Times[B, Power[b, -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B, m, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], NonzeroQ[Plus[Times[A, b], Times[a, B]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[b, c], Times[-1, a, d]], Plus[Times[B, c], Times[-1, A, d]], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[d, f, Plus[n, 1], Plus[Power[c, 2], Times[-1, Power[d, 2]]]], -1]], x]], Dist[Times[1, Power[Times[d, Plus[n, 1], Plus[Power[c, 2], Times[-1, Power[d, 2]]]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, -2]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Simp[Plus[Times[b, Plus[Times[b, c], Times[-1, a, d]], Plus[Times[B, c], Times[-1, A, d]], Plus[m, -1]], Times[a, d, Plus[Times[a, A, c], Times[b, B, c], Times[-1, Plus[Times[A, b], Times[a, B]], d]], Plus[n, 1]], Times[Plus[Times[b, Plus[Times[b, d, Plus[Times[B, c], Times[-1, A, d]]], Times[a, Plus[Times[A, c, d], Times[B, Plus[Power[c, 2], Times[-1, 2, Power[d, 2]]]]]]], Plus[n, 1]], Times[-1, a, Plus[Times[b, c], Times[-1, a, d]], Plus[Times[B, c], Times[-1, A, d]], Plus[n, 2]]], Sin[Plus[e, Times[f, x]]]], Times[b, Plus[Times[d, Plus[Times[A, b, c], Times[a, B, c], Times[-1, a, A, d]], Plus[m, n, 1]], Times[-1, b, B, Plus[Times[Power[c, 2], m], Times[Power[d, 2], Plus[n, 1]]]]], Power[Sin[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], RationalQ[m, n], Greater[m, 1], Less[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[b, B, Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[d, f, Plus[m, n, 1]], -1]], x]], Dist[Times[1, Power[Times[d, Plus[m, n, 1]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, -2]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Simp[Plus[Times[Power[a, 2], A, d, Plus[m, n, 1]], Times[b, B, Plus[Times[b, c, Plus[m, -1]], Times[a, d, Plus[n, 1]]]], Times[Plus[Times[a, d, Plus[Times[2, A, b], Times[a, B]], Plus[m, n, 1]], Times[-1, b, B, Plus[Times[a, c], Times[-1, b, d, Plus[m, n]]]]], Sin[Plus[e, Times[f, x]]]], Times[b, Plus[Times[A, b, d, Plus[m, n, 1]], Times[-1, B, Plus[Times[b, c, m], Times[-1, a, d, Plus[Times[2, m], n]]]]], Power[Sin[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], RationalQ[m], Greater[m, 1], Not[And[IntegerQ[n], Greater[n, 1], Or[Not[IntegerQ[m]], And[ZeroQ[a], NonzeroQ[c]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-3, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[B, d, Power[Power[b, 2], -1]], Int[Times[Sqrt[Times[b, Sin[Plus[e, Times[f, x]]]]], Power[Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x], Int[Times[Plus[Times[A, c], Times[Plus[Times[B, c], Times[A, d]], Sin[Plus[e, Times[f, x]]]]], Power[Times[Power[Times[b, Sin[Plus[e, Times[f, x]]]], Times[3, Power[2, -1]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]]], -1]], x]], And[FreeQ[List[b, c, d, e, f, A, B], x], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[B, Power[b, -1]], Int[Times[Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], Power[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x], Dist[Times[Plus[Times[A, b], Times[-1, a, B]], Power[b, -1]], Int[Times[Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], Power[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Times[3, Power[2, -1]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[2, Plus[Times[A, b], Times[-1, a, B]], Cos[Plus[e, Times[f, x]]], Power[Times[f, Plus[Power[a, 2], Times[-1, Power[b, 2]]], Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]], x], Dist[Times[d, Power[Plus[Power[a, 2], Times[-1, Power[b, 2]]], -1]], Int[Times[Plus[Times[A, b], Times[-1, a, B], Times[Plus[Times[a, A], Times[-1, b, B]], Sin[Plus[e, Times[f, x]]]]], Power[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Times[d, Sin[Plus[e, Times[f, x]]]], Times[3, Power[2, -1]]]], -1]], x], x]], And[FreeQ[List[a, b, d, e, f, A, B], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-3, 2]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[-2, A, Plus[c, Times[-1, d]], Tan[Plus[e, Times[f, x]]], Power[Times[f, b, Power[c, 2]], -1], Rt[Times[Plus[c, d], Power[b, -1]], 2], Sqrt[Times[c, Plus[1, Csc[Plus[e, Times[f, x]]]], Power[Plus[c, Times[-1, d]], -1]]], Sqrt[Times[c, Plus[1, Times[-1, Csc[Plus[e, Times[f, x]]]]], Power[Plus[c, d], -1]]], EllipticE[ArcSin[Times[Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], Power[Sqrt[Times[b, Sin[Plus[e, Times[f, x]]]]], -1], Power[Rt[Times[Plus[c, d], Power[b, -1]], 2], -1]]], Times[-1, Plus[c, d], Power[Plus[c, Times[-1, d]], -1]]]], x], And[FreeQ[List[b, c, d, e, f, A, B], x], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], ZeroQ[Plus[A, Times[-1, B]]], PosQ[Times[Plus[c, d], Power[b, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-3, 2]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Times[Sqrt[Times[-1, b, Sin[Plus[e, Times[f, x]]]]], Power[Sqrt[Times[b, Sin[Plus[e, Times[f, x]]]]], -1]], Int[Times[Plus[A, Times[B, Sin[Plus[e, Times[f, x]]]]], Power[Times[Power[Times[-1, b, Sin[Plus[e, Times[f, x]]]], Times[3, Power[2, -1]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], x]], And[FreeQ[List[b, c, d, e, f, A, B], x], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], ZeroQ[Plus[A, Times[-1, B]]], NegQ[Times[Plus[c, d], Power[b, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[-2, A, Plus[c, Times[-1, d]], Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Power[Times[f, 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Power[Plus[a, Times[-1, b]], -1]], Int[Times[1, Power[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], x], Times[-1, Dist[Times[Plus[Times[A, b], Times[-1, a, B]], Power[Plus[a, Times[-1, b]], -1]], Int[Times[Plus[1, Sin[Plus[e, Times[f, x]]]], Power[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Times[3, Power[2, -1]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, A, B], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], NonzeroQ[Plus[A, Times[-1, B]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[B, a], Times[-1, A, b]], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Power[Times[f, Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], x], Dist[Times[1, Power[Times[Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, -1]], Simp[Plus[Times[c, Plus[Times[a, A], Times[-1, b, B]], Plus[m, 1]], Times[d, n, Plus[Times[A, b], 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Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[A, Power[b, 2]], Times[-1, a, b, B]], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[1, n]], Power[Times[f, Plus[m, 1], Plus[Times[b, c], Times[-1, a, d]], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], x]], Dist[Times[1, Power[Times[Plus[m, 1], Plus[Times[b, c], Times[-1, a, d]], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Simp[Plus[Times[Plus[Times[a, A], Times[-1, b, B]], Plus[Times[b, c], Times[-1, a, d]], Plus[m, 1]], Times[b, d, Plus[Times[A, b], Times[-1, a, B]], Plus[m, n, 2]], Times[Plus[Times[A, b], Times[-1, a, B]], Plus[Times[a, d, Plus[m, 1]], Times[-1, b, c, Plus[m, 2]]], Sin[Plus[e, Times[f, x]]]], Times[-1, b, d, Plus[Times[A, b], Times[-1, a, B]], Plus[m, n, 3], Power[Sin[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], RationalQ[m], Less[m, -1], Or[And[ZeroQ[a], IntegerQ[m], Not[IntegerQ[n]]], Not[And[IntegerQ[Times[2, n]], Less[n, -1], Or[And[IntegerQ[n], Not[IntegerQ[m]]], ZeroQ[a]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], 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Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[B, Power[d, -1]], Int[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], x], x], Times[-1, Dist[Times[Plus[Times[B, c], Times[-1, A, d]], Power[d, -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, A, B, m], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[-2, B, Cos[Plus[e, Times[f, x]]], Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Power[Times[f, Plus[Times[2, n], 3]], -1]], x], Dist[Times[1, Power[Plus[Times[2, n], 3], -1]], Int[Times[Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, -1]], Power[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], -1], Simp[Plus[Times[a, A, c, Plus[Times[2, n], 3]], Times[B, Plus[Times[b, c], Times[2, a, d, n]]], Times[Plus[Times[B, Plus[Times[a, c], Times[b, d]], Plus[Times[2, n], 1]], Times[A, Plus[Times[b, c], Times[a, d]], Plus[Times[2, n], 3]]], Sin[Plus[e, Times[f, x]]]], Times[Plus[Times[A, b, d, Plus[Times[2, n], 3]], Times[B, Plus[Times[a, d], Times[2, b, c, n]]]], Power[Sin[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], RationalQ[n], Equal[Power[n, 2], Times[1, Power[4, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[4, A, Power[Times[f, Sqrt[Plus[a, b]]], -1], EllipticPi[-1, Times[-1, ArcSin[Times[Cos[Plus[e, Times[f, x]]], Power[Plus[1, Sin[Plus[e, Times[f, x]]]], -1]]]], Times[-1, Plus[a, Times[-1, b]], Power[Plus[a, b], -1]]]], x], And[FreeQ[List[a, b, e, f, A, B], x], PositiveQ[b], PositiveQ[Plus[Power[b, 2], Times[-1, Power[a, 2]]]], ZeroQ[Plus[A, Times[-1, B]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[d, Blank[]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Sin[Plus[e, Times[f, x]]]], Power[Sqrt[Times[d, Sin[Plus[e, Times[f, x]]]]], -1]], Int[Times[Plus[A, Times[B, Sin[Plus[e, Times[f, x]]]]], Power[Times[Sqrt[Sin[Plus[e, Times[f, x]]]], Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], x], And[FreeQ[List[a, b, e, f, d, A, B], x], PositiveQ[b], PositiveQ[Plus[Power[b, 2], Times[-1, Power[a, 2]]]], ZeroQ[Plus[A, Times[-1, B]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[B, Power[d, -1]], Int[Times[Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], Power[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x], Times[-1, Dist[Times[Plus[Times[B, c], Times[-1, A, d]], Power[d, -1]], Int[Times[1, Power[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, A, B], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], 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Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], Power[Times[f, Cos[Plus[e, Times[f, x]]]], -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], Plus[m, Times[-1, 1, Power[2, -1]]]], Power[Plus[c, Times[d, x]], Plus[n, Times[-1, 1, Power[2, -1]]]], Power[Plus[A, Times[B, x]], p]], x], x, Sin[Plus[e, Times[f, x]]]], x], And[FreeQ[List[a, b, c, d, e, f, A, B, m, n, p], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]]], Pattern[p, Blank[]]], Power[Plus[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Times[Sqrt[Plus[a, Times[b, Cos[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Cos[Plus[e, Times[f, x]]]]]], Power[Times[f, Sin[Plus[e, Times[f, x]]]], -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], Plus[m, Times[-1, 1, Power[2, -1]]]], Power[Plus[c, Times[d, x]], Plus[n, Times[-1, 1, Power[2, -1]]]], Power[Plus[A, Times[B, x]], p]], x], x, Cos[Plus[e, Times[f, x]]]], x]], And[FreeQ[List[a, b, c, d, e, f, A, B, m, n, p], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[b, -1]], Int[Times[Power[Times[b, Sin[Plus[e, Times[f, x]]]], Plus[m, 1]], Plus[B, Times[C, Sin[Plus[e, Times[f, x]]]]]], x], x], FreeQ[List[b, e, f, B, C, m], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[A, Cos[Plus[e, Times[f, x]]], Power[Times[b, Sin[Plus[e, Times[f, x]]]], Plus[m, 1]], Power[Times[b, f, Plus[m, 1]], -1]], x], And[FreeQ[List[b, e, f, A, C, m], x], ZeroQ[Plus[Times[A, Plus[m, 2]], Times[C, Plus[m, 1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[A, Cos[Plus[e, Times[f, x]]], Power[Times[b, Sin[Plus[e, Times[f, x]]]], Plus[m, 1]], Power[Times[b, f, Plus[m, 1]], -1]], x], Dist[Times[Plus[Times[A, Plus[m, 2]], Times[C, Plus[m, 1]]], Power[Times[Power[b, 2], Plus[m, 1]], -1]], Int[Power[Times[b, Sin[Plus[e, Times[f, x]]]], Plus[m, 2]], x], x]], And[FreeQ[List[b, e, f, A, C], x], RationalQ[m], Less[m, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Power[f, -1], Subst[Int[Times[Power[Plus[1, Times[-1, Power[x, 2]]], Times[Plus[m, -1], Power[2, -1]]], Plus[A, C, Times[-1, C, Power[x, 2]]]], x], x, Cos[Plus[e, Times[f, x]]]], x]], And[FreeQ[List[e, f, A, C], x], PositiveIntegerQ[Times[Plus[m, 1], Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[C, Cos[Plus[e, Times[f, x]]], Power[Times[b, Sin[Plus[e, Times[f, x]]]], Plus[m, 1]], Power[Times[b, f, Plus[m, 2]], -1]], x]], Dist[Times[Plus[Times[A, Plus[m, 2]], Times[C, Plus[m, 1]]], Power[Plus[m, 2], -1]], Int[Power[Times[b, Sin[Plus[e, Times[f, x]]]], m], x], x]], And[FreeQ[List[b, e, f, A, C, m], x], Not[And[RationalQ[m], Less[m, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Power[b, 2], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Simp[Plus[Times[b, B], Times[-1, a, C], Times[b, C, Sin[Plus[e, Times[f, x]]]]], x]], x], x], And[FreeQ[List[a, b, e, f, A, B, C, m], x], ZeroQ[Plus[Times[A, Power[b, 2]], Times[-1, a, b, B], Times[Power[a, 2], C]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[C, Power[Power[b, 2], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Simp[Plus[Times[-1, a], Times[b, Sin[Plus[e, Times[f, x]]]]], x]], x], x], And[FreeQ[List[a, b, e, f, A, C, m], x], ZeroQ[Plus[Times[A, Power[b, 2]], Times[Power[a, 2], C]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Plus[A, Times[-1, C]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Plus[1, Sin[Plus[e, Times[f, x]]]]], x], x], Dist[C, Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[1, Sin[Plus[e, Times[f, x]]]], 2]], x], x]], And[FreeQ[List[a, b, e, f, A, B, C, m], x], ZeroQ[Plus[A, Times[-1, B], C]], Not[IntegerQ[Times[2, m]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Plus[A, Times[-1, C]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Plus[1, Sin[Plus[e, Times[f, x]]]]], x], x], Dist[C, Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[1, Sin[Plus[e, Times[f, x]]]], 2]], x], x]], And[FreeQ[List[a, b, e, f, A, C, m], x], ZeroQ[Plus[A, C]], Not[IntegerQ[Times[2, m]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[A, b], Times[-1, a, B], Times[b, C]], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Times[a, f, Plus[Times[2, m], 1]], -1]], x], Dist[Times[1, Power[Times[Power[a, 2], Plus[Times[2, m], 1]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Simp[Plus[Times[a, A, Plus[m, 1]], Times[m, Plus[Times[b, B], Times[-1, a, C]]], Times[b, C, Plus[Times[2, m], 1], Sin[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, e, f, A, B, C], x], RationalQ[m], Less[m, -1], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, Plus[A, C], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Times[a, f, Plus[Times[2, m], 1]], -1]], x], Dist[Times[1, Power[Times[Power[a, 2], Plus[Times[2, m], 1]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Simp[Plus[Times[a, A, Plus[m, 1]], Times[-1, a, C, m], Times[b, C, Plus[Times[2, m], 1], Sin[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, e, f, A, C], x], RationalQ[m], Less[m, -1], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], 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x]], Dist[Times[1, Power[Times[b, Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Simp[Plus[Times[a, b, Plus[A, C], Plus[m, 1]], Times[-1, Plus[Times[A, Power[b, 2]], Times[Power[a, 2], C], Times[Power[b, 2], Plus[A, C], Plus[m, 1]]], Sin[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, e, f, A, C], x], RationalQ[m], Less[m, -1], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[C, Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[b, f, Plus[m, 2]], -1]], x]], Dist[Times[1, Power[Times[b, Plus[m, 2]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Simp[Plus[Times[A, b, Plus[m, 2]], Times[b, C, Plus[m, 1]], Times[Plus[Times[b, B, Plus[m, 2]], Times[-1, a, C]], Sin[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, e, f, A, B, C, m], x], Not[And[RationalQ[m], Less[m, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[C, Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[b, f, Plus[m, 2]], -1]], x]], Dist[Times[1, Power[Times[b, Plus[m, 2]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Simp[Plus[Times[A, b, Plus[m, 2]], Times[b, C, Plus[m, 1]], Times[-1, a, C, Sin[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, e, f, A, C, m], x], Not[And[RationalQ[m], Less[m, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[b, Power[Sin[Plus[e, Times[f, x]]], p]], m], Power[Power[Times[b, Sin[Plus[e, Times[f, x]]]], Times[m, p]], -1]], Int[Times[Power[Times[b, Sin[Plus[e, Times[f, x]]]], Times[m, p]], Plus[A, Times[B, Sin[Plus[e, Times[f, x]]]], Times[C, Power[Sin[Plus[e, Times[f, x]]], 2]]]], x], x], And[FreeQ[List[b, e, f, A, B, C, m, p], x], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Optional[Pattern[b, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[b, Power[Cos[Plus[e, Times[f, x]]], p]], m], Power[Power[Times[b, Cos[Plus[e, Times[f, x]]]], Times[m, p]], -1]], Int[Times[Power[Times[b, Cos[Plus[e, Times[f, x]]]], Times[m, p]], Plus[A, Times[B, Cos[Plus[e, Times[f, x]]]], Times[C, Power[Cos[Plus[e, Times[f, x]]], 2]]]], x], x], And[FreeQ[List[b, e, f, A, B, C, m, p], x], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[b, Power[Sin[Plus[e, Times[f, x]]], p]], m], Power[Power[Times[b, Sin[Plus[e, Times[f, x]]]], Times[m, p]], -1]], Int[Times[Power[Times[b, Sin[Plus[e, Times[f, x]]]], Times[m, p]], Plus[A, Times[C, Power[Sin[Plus[e, Times[f, x]]], 2]]]], x], x], And[FreeQ[List[b, e, f, A, C, m, p], x], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Optional[Pattern[b, Blank[]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[b, Power[Cos[Plus[e, Times[f, x]]], p]], m], Power[Power[Times[b, Cos[Plus[e, Times[f, x]]]], Times[m, p]], -1]], Int[Times[Power[Times[b, Cos[Plus[e, Times[f, 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Blank[]]]]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[C, d, Cos[Plus[e, Times[f, x]]], Sin[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[b, f, Plus[m, 3]], -1]], x]], Dist[Times[1, Power[Times[b, Plus[m, 3]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Simp[Plus[Times[a, C, d], Times[A, b, c, Plus[m, 3]], Times[b, d, Plus[Times[C, Plus[m, 2]], Times[A, Plus[m, 3]]], Sin[Plus[e, Times[f, x]]]], Times[-1, Plus[Times[2, a, C, d], Times[-1, b, c, C, Plus[m, 3]]], Power[Sin[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, C, m], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], Not[And[RationalQ[m], Less[m, -1]]]]]], 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x], Dist[Plus[A, C], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, C, m], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], Not[And[RationalQ[m], Less[m, Times[-1, Power[2, -1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], 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Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[a, A], Times[-1, b, B], Times[a, C]], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[f, Plus[Times[b, c], Times[-1, a, d]], Plus[Times[2, m], 1]], -1]], x], Dist[Times[1, Power[Times[b, Plus[Times[b, c], Times[-1, a, d]], Plus[Times[2, m], 1]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Simp[Plus[Times[A, Plus[Times[a, c, Plus[m, 1]], Times[-1, b, d, Plus[Times[2, m], n, 2]]]], Times[B, Plus[Times[b, c, m], Times[a, d, 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Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[a, Plus[A, C], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[f, Plus[Times[b, c], Times[-1, a, d]], Plus[Times[2, m], 1]], -1]], x], Dist[Times[1, Power[Times[b, Plus[Times[b, c], Times[-1, a, d]], Plus[Times[2, m], 1]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Simp[Plus[Times[A, Plus[Times[a, c, Plus[m, 1]], Times[-1, b, d, Plus[Times[2, m], n, 2]]]], Times[-1, C, Plus[Times[a, c, m], Times[b, d, Plus[n, 1]]]], Times[Plus[Times[a, A, d, Plus[m, n, 2]], Times[C, Plus[Times[b, c, Plus[Times[2, m], 1]], Times[-1, a, d, Plus[m, Times[-1, n], -1]]]]], Sin[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, C, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], RationalQ[m], Less[m, Times[-1, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[Power[c, 2], C], Times[-1, B, c, d], Times[A, Power[d, 2]]], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[d, f, Plus[n, 1], Plus[Power[c, 2], Times[-1, Power[d, 2]]]], -1]], x]], Dist[Times[1, Power[Times[b, d, Plus[n, 1], Plus[Power[c, 2], Times[-1, Power[d, 2]]]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Simp[Plus[Times[A, d, Plus[Times[a, d, m], Times[b, c, Plus[n, 1]]]], Times[Plus[Times[c, C], Times[-1, B, d]], Plus[Times[a, c, m], Times[b, d, Plus[n, 1]]]], Times[b, Plus[Times[d, Plus[Times[B, c], Times[-1, A, d]], Plus[m, n, 2]], Times[-1, C, Plus[Times[Power[c, 2], Plus[m, 1]], Times[Power[d, 2], Plus[n, 1]]]]], Sin[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B, C, m], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], Not[And[RationalQ[m], Less[m, Times[-1, Power[2, -1]]]]], Or[And[RationalQ[n], Less[n, -1]], ZeroQ[Plus[m, n, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[Power[c, 2], C], Times[A, Power[d, 2]]], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[d, f, Plus[n, 1], Plus[Power[c, 2], Times[-1, Power[d, 2]]]], -1]], x]], Dist[Times[1, Power[Times[b, d, Plus[n, 1], Plus[Power[c, 2], Times[-1, Power[d, 2]]]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Simp[Plus[Times[A, d, Plus[Times[a, d, m], Times[b, c, Plus[n, 1]]]], Times[c, C, Plus[Times[a, c, m], Times[b, d, Plus[n, 1]]]], Times[-1, b, Plus[Times[A, Power[d, 2], Plus[m, n, 2]], Times[C, Plus[Times[Power[c, 2], Plus[m, 1]], Times[Power[d, 2], Plus[n, 1]]]]], Sin[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, C, m], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], Not[And[RationalQ[m], Less[m, Times[-1, Power[2, -1]]]]], Or[And[RationalQ[n], Less[n, -1]], ZeroQ[Plus[m, n, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[C, Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[d, f, Plus[m, n, 2]], -1]], x]], Dist[Times[1, Power[Times[b, d, Plus[m, n, 2]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Simp[Plus[Times[A, b, d, Plus[m, n, 2]], Times[C, Plus[Times[a, c, m], Times[b, d, Plus[n, 1]]]], Times[Plus[Times[C, Plus[Times[a, d, m], Times[-1, b, c, Plus[m, 1]]]], Times[b, B, d, Plus[m, n, 2]]], Sin[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B, C, m, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], Not[And[RationalQ[m], Less[m, Times[-1, Power[2, -1]]]]], NonzeroQ[Plus[m, n, 2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[C, Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[d, f, Plus[m, n, 2]], -1]], x]], Dist[Times[1, Power[Times[b, d, Plus[m, n, 2]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Simp[Plus[Times[A, b, d, Plus[m, n, 2]], Times[C, Plus[Times[a, c, m], Times[b, d, Plus[n, 1]]]], Times[C, Plus[Times[a, d, m], Times[-1, b, c, Plus[m, 1]]], Sin[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, C, m, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], Not[And[RationalQ[m], Less[m, Times[-1, Power[2, -1]]]]], NonzeroQ[Plus[m, n, 2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[Power[c, 2], C], Times[-1, B, c, d], Times[A, Power[d, 2]]], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[d, f, Plus[n, 1], Plus[Power[c, 2], Times[-1, Power[d, 2]]]], -1]], x]], Dist[Times[1, Power[Times[d, Plus[n, 1], Plus[Power[c, 2], Times[-1, Power[d, 2]]]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Simp[Plus[Times[A, d, Plus[Times[b, d, m], Times[a, c, Plus[n, 1]]]], Times[Plus[Times[c, C], Times[-1, B, d]], Plus[Times[b, c, m], Times[a, d, Plus[n, 1]]]], Times[-1, Plus[Times[d, Plus[Times[A, Plus[Times[a, d, Plus[n, 2]], Times[-1, b, c, Plus[n, 1]]]], Times[B, Plus[Times[b, d, Plus[n, 1]], Times[-1, a, c, Plus[n, 2]]]]]], Times[-1, C, Plus[Times[b, c, d, Plus[n, 1]], Times[-1, a, Plus[Power[c, 2], Times[Power[d, 2], Plus[n, 1]]]]]]], Sin[Plus[e, Times[f, x]]]], Times[b, Plus[Times[d, Plus[Times[B, c], Times[-1, A, d]], Plus[m, n, 2]], Times[-1, C, Plus[Times[Power[c, 2], Plus[m, 1]], Times[Power[d, 2], Plus[n, 1]]]]], Power[Sin[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B, C], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], RationalQ[m, n], Greater[m, 0], Less[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[Power[c, 2], C], Times[A, Power[d, 2]]], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[d, f, Plus[n, 1], Plus[Power[c, 2], Times[-1, Power[d, 2]]]], -1]], x]], Dist[Times[1, Power[Times[d, Plus[n, 1], Plus[Power[c, 2], Times[-1, Power[d, 2]]]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Simp[Plus[Times[A, d, Plus[Times[b, d, m], Times[a, c, Plus[n, 1]]]], Times[c, C, Plus[Times[b, c, m], Times[a, d, Plus[n, 1]]]], Times[-1, Plus[Times[A, d, Plus[Times[a, d, Plus[n, 2]], Times[-1, b, c, Plus[n, 1]]]], Times[-1, C, Plus[Times[b, c, d, Plus[n, 1]], Times[-1, a, Plus[Power[c, 2], Times[Power[d, 2], Plus[n, 1]]]]]]], Sin[Plus[e, Times[f, x]]]], Times[-1, b, Plus[Times[A, Power[d, 2], Plus[m, n, 2]], Times[C, Plus[Times[Power[c, 2], Plus[m, 1]], Times[Power[d, 2], Plus[n, 1]]]]], Power[Sin[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, C], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], RationalQ[m, n], Greater[m, 0], Less[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[C, Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[d, f, Plus[m, n, 2]], -1]], x]], Dist[Times[1, Power[Times[d, Plus[m, n, 2]], -1]], Int[Times[Power[Plus[a, Times[b, 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Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[C, Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[d, f, Plus[m, n, 2]], -1]], x]], Dist[Times[1, Power[Times[d, Plus[m, n, 2]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Simp[Plus[Times[a, A, d, Plus[m, n, 2]], Times[C, Plus[Times[b, c, m], Times[a, d, Plus[n, 1]]]], Times[Plus[Times[A, b, d, Plus[m, n, 2]], Times[-1, C, Plus[Times[a, c], Times[-1, b, d, Plus[m, n, 1]]]]], Sin[Plus[e, Times[f, x]]]], Times[C, Plus[Times[a, d, m], Times[-1, b, c, Plus[m, 1]]], Power[Sin[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, C, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], RationalQ[m], Greater[m, 0], Not[And[IntegerQ[n], Greater[n, 0], Or[Not[IntegerQ[m]], And[ZeroQ[a], NonzeroQ[c]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[C, Power[Times[b, d], -1]], Int[Times[Sqrt[Times[d, Sin[Plus[e, Times[f, x]]]]], Power[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x], Dist[Times[1, Power[b, -1]], Int[Times[Plus[Times[A, b], Times[Plus[Times[b, B], Times[-1, a, C]], Sin[Plus[e, Times[f, x]]]]], Power[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Times[3, Power[2, -1]]], Sqrt[Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x]], And[FreeQ[List[a, b, d, e, f, A, B, C], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[C, Power[Times[b, d], -1]], Int[Times[Sqrt[Times[d, Sin[Plus[e, Times[f, x]]]]], Power[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x], Dist[Times[1, Power[b, -1]], Int[Times[Plus[Times[A, b], Times[-1, a, C, Sin[Plus[e, Times[f, x]]]]], Power[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Times[3, Power[2, -1]]], Sqrt[Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x]], And[FreeQ[List[a, b, d, e, f, A, C], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[C, Power[Power[b, 2], -1]], Int[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x], Dist[Times[1, Power[Power[b, 2], -1]], Int[Times[Plus[Times[A, Power[b, 2]], Times[-1, Power[a, 2], C], Times[b, Plus[Times[b, B], Times[-1, 2, a, C]], Sin[Plus[e, Times[f, x]]]]], Power[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Times[3, Power[2, -1]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B, C], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[C, Power[Power[b, 2], -1]], Int[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Power[Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x], Dist[Times[1, Power[Power[b, 2], -1]], Int[Times[Plus[Times[A, Power[b, 2]], Times[-1, Power[a, 2], C], Times[-1, 2, a, b, C, Sin[Plus[e, Times[f, x]]]]], Power[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Times[3, Power[2, -1]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, C], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[A, Power[b, 2]], Times[-1, a, b, B], Times[Power[a, 2], C]], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[f, Plus[m, 1], Plus[Times[b, c], Times[-1, a, d]], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], x]], Dist[Times[1, Power[Times[Plus[m, 1], Plus[Times[b, c], Times[-1, a, d]], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Simp[Plus[Times[Plus[m, 1], Plus[Times[b, c], Times[-1, a, d]], Plus[Times[a, A], Times[-1, b, B], Times[a, C]]], Times[d, Plus[Times[A, Power[b, 2]], Times[-1, a, b, B], Times[Power[a, 2], C]], Plus[m, n, 2]], Times[-1, Plus[Times[c, Plus[Times[A, Power[b, 2]], Times[-1, a, b, B], Times[Power[a, 2], C]]], Times[Plus[m, 1], Plus[Times[b, c], Times[-1, a, d]], Plus[Times[A, b], Times[-1, a, B], Times[b, C]]]], Sin[Plus[e, Times[f, x]]]], Times[-1, d, Plus[Times[A, Power[b, 2]], Times[-1, a, b, B], Times[Power[a, 2], C]], Plus[m, n, 3], Power[Sin[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B, C, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], RationalQ[m], Less[m, -1], Or[And[ZeroQ[a], IntegerQ[m], Not[IntegerQ[n]]], Not[And[IntegerQ[Times[2, n]], Less[n, -1], Or[And[IntegerQ[n], Not[IntegerQ[m]]], ZeroQ[a]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[A, Power[b, 2]], Times[Power[a, 2], C]], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[f, Plus[m, 1], Plus[Times[b, c], Times[-1, a, d]], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], x]], Dist[Times[1, Power[Times[Plus[m, 1], Plus[Times[b, c], Times[-1, a, d]], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], Int[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Simp[Plus[Times[a, Plus[m, 1], Plus[Times[b, c], Times[-1, a, d]], Plus[A, C]], Times[d, Plus[Times[A, Power[b, 2]], Times[Power[a, 2], C]], Plus[m, n, 2]], Times[-1, Plus[Times[c, Plus[Times[A, Power[b, 2]], Times[Power[a, 2], C]]], Times[b, Plus[m, 1], Plus[Times[b, c], Times[-1, a, d]], Plus[A, C]]], Sin[Plus[e, Times[f, x]]]], Times[-1, d, Plus[Times[A, Power[b, 2]], Times[Power[a, 2], C]], Plus[m, n, 3], Power[Sin[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, C, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], RationalQ[m], Less[m, -1], Or[And[ZeroQ[a], IntegerQ[m], Not[IntegerQ[n]]], Not[And[IntegerQ[Times[2, n]], Less[n, -1], Or[And[IntegerQ[n], Not[IntegerQ[m]]], ZeroQ[a]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[C, x, Power[Times[b, d], -1]], x], Plus[Dist[Times[Plus[Times[A, Power[b, 2]], Times[-1, a, b, B], Times[Power[a, 2], C]], Power[Times[b, Plus[Times[b, c], Times[-1, a, d]]], -1]], Int[Times[1, Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], -1]], x], x], Times[-1, Dist[Times[Plus[Times[Power[c, 2], C], Times[-1, B, c, d], Times[A, Power[d, 2]]], Power[Times[d, Plus[Times[b, c], Times[-1, a, d]]], -1]], Int[Times[1, Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d, e, f, A, B, C], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[C, x, Power[Times[b, d], -1]], x], Plus[Dist[Times[Plus[Times[A, Power[b, 2]], Times[Power[a, 2], C]], Power[Times[b, Plus[Times[b, c], Times[-1, a, d]]], -1]], Int[Times[1, Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], -1]], x], x], Times[-1, Dist[Times[Plus[Times[Power[c, 2], C], Times[A, Power[d, 2]]], Power[Times[d, Plus[Times[b, c], Times[-1, a, d]]], -1]], Int[Times[1, Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d, e, f, A, C], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[C, Power[Times[b, d], -1]], Int[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], x], x], Times[-1, Dist[Times[1, Power[Times[b, d], -1]], Int[Times[Simp[Plus[Times[a, c, C], Times[-1, A, b, d], Times[Plus[Times[b, c, C], Times[-1, b, B, d], Times[a, C, d]], Sin[Plus[e, Times[f, x]]]]], x], Power[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, A, B, C], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[C, Power[Times[b, d], -1]], Int[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], x], x], Times[-1, Dist[Times[1, Power[Times[b, d], -1]], Int[Times[Simp[Plus[Times[a, c, C], Times[-1, A, b, d], Times[Plus[Times[b, c, C], Times[a, C, d]], Sin[Plus[e, Times[f, x]]]]], x], Power[Times[Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, A, C], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[C, Cos[Plus[e, Times[f, x]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], Power[Times[d, f, Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]]], -1]], x]], Dist[Times[1, Power[Times[2, d], -1]], Int[Times[1, Power[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Times[3, Power[2, -1]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]]], -1], Simp[Plus[Times[2, a, A, d], Times[-1, C, Plus[Times[b, c], Times[-1, a, d]]], Times[-1, 2, Plus[Times[a, c, C], Times[-1, d, Plus[Times[A, b], Times[a, B]]]], Sin[Plus[e, Times[f, x]]]], Times[Plus[Times[2, b, B, d], Times[-1, C, Plus[Times[b, c], Times[a, d]]]], Power[Sin[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B, C], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[C, Cos[Plus[e, Times[f, x]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]], Power[Times[d, f, Sqrt[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]]], -1]], x]], Dist[Times[1, Power[Times[2, d], -1]], Int[Times[1, Power[Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Times[3, Power[2, -1]]], Sqrt[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]]]], -1], Simp[Plus[Times[2, a, A, d], Times[-1, C, Plus[Times[b, c], Times[-1, a, d]]], Times[-1, 2, Plus[Times[a, c, C], Times[-1, A, b, d]], Sin[Plus[e, Times[f, x]]]], Times[-1, C, Plus[Times[b, c], Times[a, d]], Power[Sin[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, C], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Plus[A, Times[B, Sin[Plus[e, Times[f, x]]]], Times[C, Power[Sin[Plus[e, Times[f, x]]], 2]]]], x], And[FreeQ[List[a, b, c, d, e, f, A, B, C, m, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Plus[A, Times[C, Power[Sin[Plus[e, Times[f, x]]], 2]]]], x], And[FreeQ[List[a, b, c, d, e, f, A, C, m, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[b, Power[Sin[Plus[e, Times[f, x]]], p]], m], Power[Power[Times[b, Sin[Plus[e, Times[f, x]]]], Times[m, p]], -1]], Int[Times[Power[Times[b, Sin[Plus[e, Times[f, x]]]], Times[m, p]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Plus[A, Times[B, Sin[Plus[e, Times[f, x]]]], Times[C, Power[Sin[Plus[e, Times[f, x]]], 2]]]], x], x], And[FreeQ[List[b, c, d, e, f, A, B, C, m, n, p], x], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Optional[Pattern[b, 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e, f, A, B, C, m, n, p], x], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[b, Power[Sin[Plus[e, Times[f, x]]], p]], m], Power[Power[Times[b, Sin[Plus[e, Times[f, x]]]], Times[m, p]], -1]], Int[Times[Power[Times[b, Sin[Plus[e, Times[f, x]]]], Times[m, p]], Power[Plus[c, Times[d, Sin[Plus[e, Times[f, x]]]]], n], Plus[A, Times[C, 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Blank[]]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[b, Cos[Plus[c, Times[d, x]]]], Times[-1, a, Sin[Plus[c, Times[d, x]]]]], Power[Plus[Times[a, Cos[Plus[c, Times[d, x]]]], Times[b, Sin[Plus[c, Times[d, x]]]]], Plus[n, 1]], Power[Times[d, Plus[n, 1], Plus[Power[a, 2], Power[b, 2]]], -1]], x], Dist[Times[Plus[n, 2], Power[Times[Plus[n, 1], Plus[Power[a, 2], Power[b, 2]]], -1]], Int[Power[Plus[Times[a, Cos[Plus[c, Times[d, x]]]], Times[b, Sin[Plus[c, Times[d, x]]]]], Plus[n, 2]], x], x]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], RationalQ[n], Less[n, -1], Unequal[n, -2]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, 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Times[d, x], Times[-1, ArcTan[a, b]]]], n], x], x], And[FreeQ[List[a, b, c, d, n], x], Not[And[RationalQ[n], Or[GreaterEqual[n, 1], LessEqual[n, -1]]]], Not[Or[PositiveQ[Plus[Power[a, 2], Power[b, 2]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[a, Power[Plus[Times[a, Cos[Plus[c, Times[d, x]]]], Times[b, Sin[Plus[c, Times[d, x]]]]], Plus[n, -1]], Power[Times[d, Plus[n, -1], Power[Sin[Plus[c, Times[d, x]]], Plus[n, -1]]], -1]], x]], Dist[Times[2, b], 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Cos[Plus[c, Times[d, x]]]], Times[b, Sin[Plus[c, Times[d, x]]]]], n], Power[Times[2, b, d, n, Power[Sin[Plus[c, Times[d, x]]], n]], -1], Hypergeometric2F1[1, n, Plus[n, 1], Times[Plus[b, Times[a, Cot[Plus[c, Times[d, x]]]]], Power[Times[2, b], -1]]]], x], And[FreeQ[List[a, b, c, d, n], x], ZeroQ[Plus[m, n]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Simp[Times[b, Power[Plus[Times[a, Cos[Plus[c, Times[d, x]]]], Times[b, Sin[Plus[c, Times[d, x]]]]], n], Power[Times[2, a, d, n, Power[Cos[Plus[c, Times[d, x]]], n]], -1], Hypergeometric2F1[1, n, Plus[n, 1], Times[Plus[a, Times[b, Tan[Plus[c, Times[d, x]]]]], Power[Times[2, a], -1]]]], x]], And[FreeQ[List[a, b, c, d, n], x], ZeroQ[Plus[m, n]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Power[Plus[b, Times[a, Cot[Plus[c, Times[d, x]]]]], n], x], And[FreeQ[List[a, b, c, d], x], ZeroQ[Plus[m, n]], IntegerQ[n], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Power[Plus[a, Times[b, Tan[Plus[c, Times[d, x]]]]], n], x], And[FreeQ[List[a, b, c, d], x], ZeroQ[Plus[m, n]], IntegerQ[n], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[d, -1]], Subst[Int[Times[Power[x, m], Power[Plus[a, Times[b, x]], n], Power[Power[Plus[1, Power[x, 2]], Times[Plus[m, n, 2], Power[2, -1]]], -1]], x], x, Tan[Plus[c, Times[d, x]]]], x], And[FreeQ[List[a, b, c, d], x], IntegerQ[n], IntegerQ[Times[Plus[m, n], Power[2, -1]]], Unequal[n, -1], Not[And[Greater[n, 0], Greater[m, 1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Power[d, -1], Subst[Int[Times[Power[x, m], Power[Plus[b, Times[a, x]], n], Power[Power[Plus[1, Power[x, 2]], Times[Plus[m, n, 2], Power[2, -1]]], -1]], x], x, Cot[Plus[c, Times[d, x]]]], x]], And[FreeQ[List[a, b, c, d], x], IntegerQ[n], IntegerQ[Times[Plus[m, n], Power[2, -1]]], Unequal[n, -1], Not[And[Greater[n, 0], Greater[m, 1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrig[Times[Power[sin[Plus[c, Times[d, x]]], m], Power[Plus[Times[a, cos[Plus[c, Times[d, x]]]], Times[b, sin[Plus[c, Times[d, x]]]]], n]], x], x], And[FreeQ[List[a, b, c, d], x], IntegerQ[m], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrig[Times[Power[cos[Plus[c, Times[d, x]]], m], Power[Plus[Times[a, cos[Plus[c, Times[d, x]]]], Times[b, sin[Plus[c, Times[d, x]]]]], n]], x], x], And[FreeQ[List[a, b, c, d], x], IntegerQ[m], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[a, n], Power[b, n]], Int[Times[Power[Sin[Plus[c, Times[d, x]]], m], Power[Plus[Times[b, Cos[Plus[c, Times[d, x]]]], Times[a, Sin[Plus[c, Times[d, x]]]]], Times[-1, n]]], x], x], And[FreeQ[List[a, b, c, d, m], x], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], NegativeIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[a, n], Power[b, n]], Int[Times[Power[Cos[Plus[c, Times[d, x]]], m], Power[Plus[Times[b, Cos[Plus[c, Times[d, x]]]], Times[a, Sin[Plus[c, Times[d, x]]]]], Times[-1, n]]], x], x], And[FreeQ[List[a, b, c, d, m], x], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], NegativeIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Plus[Times[a, Cos[Plus[c, Times[d, x]]]], Times[b, Sin[Plus[c, Times[d, x]]]]], Plus[n, 1]], Power[Times[a, d, Plus[n, 1]], -1]], x]], Plus[Dist[Times[1, Power[Power[a, 2], -1]], Int[Times[Power[Plus[Times[a, Cos[Plus[c, Times[d, x]]]], Times[b, Sin[Plus[c, Times[d, x]]]]], Plus[n, 2]], Power[Sin[Plus[c, Times[d, x]]], -1]], x], x], Times[-1, Dist[Times[b, Power[Power[a, 2], -1]], Int[Power[Plus[Times[a, Cos[Plus[c, Times[d, x]]]], Times[b, Sin[Plus[c, Times[d, x]]]]], Plus[n, 1]], x], x]]]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], RationalQ[n], Less[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[Times[a, Cos[Plus[c, Times[d, x]]]], Times[b, Sin[Plus[c, Times[d, x]]]]], Plus[n, 1]], Power[Times[b, d, Plus[n, 1]], -1]], x], Plus[Dist[Times[1, Power[Power[b, 2], -1]], Int[Times[Power[Plus[Times[a, Cos[Plus[c, Times[d, x]]]], Times[b, Sin[Plus[c, Times[d, x]]]]], Plus[n, 2]], Power[Cos[Plus[c, Times[d, x]]], -1]], x], x], Times[-1, Dist[Times[a, Power[Power[b, 2], -1]], Int[Power[Plus[Times[a, Cos[Plus[c, Times[d, x]]]], Times[b, Sin[Plus[c, Times[d, x]]]]], Plus[n, 1]], x], x]]]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], RationalQ[n], Less[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Dist[Plus[Power[a, 2], Power[b, 2]], Int[Times[Power[Sin[Plus[c, Times[d, x]]], Plus[m, 2]], Power[Plus[Times[a, Cos[Plus[c, Times[d, x]]]], Times[b, Sin[Plus[c, Times[d, x]]]]], Plus[n, -2]]], x], x]], Plus[Dist[Power[a, 2], Int[Times[Power[Sin[Plus[c, Times[d, x]]], m], Power[Plus[Times[a, Cos[Plus[c, Times[d, x]]]], Times[b, Sin[Plus[c, Times[d, x]]]]], Plus[n, -2]]], x], x], Dist[Times[2, b], Int[Times[Power[Sin[Plus[c, Times[d, x]]], Plus[m, 1]], Power[Plus[Times[a, Cos[Plus[c, Times[d, x]]]], Times[b, Sin[Plus[c, Times[d, x]]]]], Plus[n, -1]]], x], x]]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], RationalQ[m, n], Greater[n, 1], Less[m, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Dist[Plus[Power[a, 2], Power[b, 2]], Int[Times[Power[Cos[Plus[c, Times[d, x]]], Plus[m, 2]], Power[Plus[Times[a, Cos[Plus[c, Times[d, x]]]], Times[b, Sin[Plus[c, Times[d, x]]]]], Plus[n, -2]]], x], x]], Plus[Dist[Times[2, a], Int[Times[Power[Cos[Plus[c, Times[d, x]]], Plus[m, 1]], Power[Plus[Times[a, Cos[Plus[c, Times[d, x]]]], Times[b, Sin[Plus[c, Times[d, x]]]]], Plus[n, -1]]], x], x], Dist[Power[b, 2], Int[Times[Power[Cos[Plus[c, Times[d, x]]], m], Power[Plus[Times[a, Cos[Plus[c, Times[d, x]]]], Times[b, Sin[Plus[c, Times[d, x]]]]], Plus[n, -2]]], x], x]]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], RationalQ[m, n], Greater[n, 1], Less[m, -1]]]], RuleDelayed[HoldPattern[Int[Times[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, x, Power[Plus[Power[a, 2], Power[b, 2]], -1]], x], Times[-1, Dist[Times[a, Power[Plus[Power[a, 2], Power[b, 2]], -1]], Int[Times[Plus[Times[b, Cos[Plus[c, Times[d, x]]]], Times[-1, a, Sin[Plus[c, Times[d, x]]]]], Power[Plus[Times[a, Cos[Plus[c, Times[d, x]]]], Times[b, Sin[Plus[c, Times[d, x]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[a, x, Power[Plus[Power[a, 2], Power[b, 2]], -1]], x], Dist[Times[b, Power[Plus[Power[a, 2], Power[b, 2]], -1]], Int[Times[Plus[Times[b, Cos[Plus[c, Times[d, x]]]], Times[-1, a, Sin[Plus[c, Times[d, x]]]]], Power[Plus[Times[a, Cos[Plus[c, Times[d, x]]]], Times[b, Sin[Plus[c, Times[d, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[a, Power[Sin[Plus[c, Times[d, x]]], Plus[m, -1]], Power[Times[d, Plus[Power[a, 2], Power[b, 2]], Plus[m, -1]], -1]], x]], Plus[Dist[Times[Power[a, 2], Power[Plus[Power[a, 2], Power[b, 2]], -1]], Int[Times[Power[Sin[Plus[c, Times[d, x]]], Plus[m, -2]], Power[Plus[Times[a, Cos[Plus[c, Times[d, x]]]], Times[b, Sin[Plus[c, Times[d, x]]]]], -1]], x], x], Dist[Times[b, Power[Plus[Power[a, 2], Power[b, 2]], -1]], Int[Power[Sin[Plus[c, Times[d, x]]], Plus[m, -1]], x], x]]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], RationalQ[m], Greater[m, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, Power[Cos[Plus[c, Times[d, x]]], Plus[m, -1]], Power[Times[d, Plus[Power[a, 2], Power[b, 2]], Plus[m, -1]], -1]], x], Plus[Dist[Times[a, Power[Plus[Power[a, 2], Power[b, 2]], -1]], Int[Power[Cos[Plus[c, Times[d, x]]], Plus[m, -1]], x], x], Dist[Times[Power[b, 2], Power[Plus[Power[a, 2], Power[b, 2]], -1]], Int[Times[Power[Cos[Plus[c, Times[d, x]]], Plus[m, -2]], Power[Plus[Times[a, Cos[Plus[c, Times[d, x]]]], Times[b, Sin[Plus[c, Times[d, x]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], RationalQ[m], Greater[m, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[a, -1]], Int[Cot[Plus[c, Times[d, x]]], x], x], Times[-1, Dist[Times[1, Power[a, -1]], Int[Times[Plus[Times[b, Cos[Plus[c, Times[d, x]]]], Times[-1, a, Sin[Plus[c, Times[d, x]]]]], Power[Plus[Times[a, Cos[Plus[c, Times[d, x]]]], Times[b, Sin[Plus[c, Times[d, x]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], -1], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[b, -1]], Int[Tan[Plus[c, Times[d, x]]], x], x], Dist[Times[1, Power[b, -1]], Int[Times[Plus[Times[b, Cos[Plus[c, Times[d, x]]]], Times[-1, a, Sin[Plus[c, Times[d, x]]]]], Power[Plus[Times[a, Cos[Plus[c, Times[d, x]]]], Times[b, Sin[Plus[c, Times[d, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Sin[Plus[c, Times[d, x]]], Plus[m, 1]], Power[Times[a, d, Plus[m, 1]], -1]], x], Plus[Times[-1, Dist[Times[b, Power[Power[a, 2], -1]], Int[Power[Sin[Plus[c, Times[d, x]]], Plus[m, 1]], x], x]], Dist[Times[Plus[Power[a, 2], Power[b, 2]], Power[Power[a, 2], -1]], Int[Times[Power[Sin[Plus[c, Times[d, x]]], Plus[m, 2]], Power[Plus[Times[a, Cos[Plus[c, Times[d, x]]]], Times[b, Sin[Plus[c, Times[d, x]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], RationalQ[m], Less[m, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Cos[Plus[c, Times[d, x]]], Plus[m, 1]], Power[Times[b, d, Plus[m, 1]], -1]], x]], Plus[Times[-1, Dist[Times[a, Power[Power[b, 2], -1]], Int[Power[Cos[Plus[c, Times[d, x]]], Plus[m, 1]], x], x]], Dist[Times[Plus[Power[a, 2], Power[b, 2]], Power[Power[b, 2], -1]], Int[Times[Power[Cos[Plus[c, Times[d, x]]], Plus[m, 2]], Power[Plus[Times[a, Cos[Plus[c, Times[d, x]]]], Times[b, Sin[Plus[c, Times[d, x]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], RationalQ[m], Less[m, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Plus[Power[a, 2], Power[b, 2]], Power[Power[a, 2], -1]], Int[Times[Power[Sin[Plus[c, Times[d, x]]], Plus[m, 2]], Power[Plus[Times[a, Cos[Plus[c, Times[d, x]]]], Times[b, Sin[Plus[c, Times[d, x]]]]], n]], x], 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RuleDelayed[HoldPattern[Int[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]]], Power[Sqrt[Times[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]], Power[Plus[a, Sqrt[Plus[Power[b, 2], Power[c, 2]]]], -1]]], -1]], Int[Sqrt[Plus[Times[a, Power[Plus[a, Sqrt[Plus[Power[b, 2], Power[c, 2]]]], -1]], Times[Sqrt[Plus[Power[b, 2], Power[c, 2]]], Power[Plus[a, Sqrt[Plus[Power[b, 2], Power[c, 2]]]], -1], Cos[Plus[d, Times[e, x], Times[-1, ArcTan[b, c]]]]]]], x], x], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]], 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RuleDelayed[HoldPattern[Int[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[f, FreeFactors[Tan[Plus[Times[Plus[d, Times[e, x]], Power[2, -1]], Times[Pi, Power[4, -1]]]], x]]], Dist[Times[f, Power[e, -1]], Subst[Int[Times[1, Power[Plus[a, Times[b, f, x]], -1]], x], x, Times[Tan[Plus[Times[Plus[d, Times[e, x]], Power[2, -1]], Times[Pi, Power[4, -1]]]], Power[f, -1]]], x]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[a, c]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[f, FreeFactors[Cot[Plus[Times[Plus[d, Times[e, x]], Power[2, -1]], Times[Pi, Power[4, -1]]]], x]]], Times[-1, Dist[Times[f, Power[e, -1]], Subst[Int[Times[1, Power[Plus[a, Times[b, f, x]], -1]], x], x, Times[Cot[Plus[Times[Plus[d, Times[e, x]], Power[2, -1]], Times[Pi, Power[4, -1]]]], Power[f, -1]]], x]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[a, Times[-1, c]]], NonzeroQ[Plus[a, Times[-1, b]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[f, 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And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[Power[b, 2], Power[c, 2]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[1, Power[Sqrt[Plus[a, Times[Sqrt[Plus[Power[b, 2], Power[c, 2]]], Cos[Plus[d, Times[e, x], Times[-1, ArcTan[b, c]]]]]]], -1]], x], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[b, 2], Power[c, 2]]], PositiveQ[Plus[a, Sqrt[Plus[Power[b, 2], Power[c, 2]]]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Times[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]], Power[Plus[a, Sqrt[Plus[Power[b, 2], Power[c, 2]]]], -1]]], Power[Sqrt[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]]], -1]], Int[Times[1, Power[Sqrt[Plus[Times[a, Power[Plus[a, Sqrt[Plus[Power[b, 2], Power[c, 2]]]], -1]], Times[Sqrt[Plus[Power[b, 2], Power[c, 2]]], Power[Plus[a, Sqrt[Plus[Power[b, 2], Power[c, 2]]]], -1], Cos[Plus[d, Times[e, x], Times[-1, ArcTan[b, c]]]]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]], Times[-1, Power[c, 2]]]], NonzeroQ[Plus[Power[b, 2], Power[c, 2]]], Not[PositiveQ[Plus[a, Sqrt[Plus[Power[b, 2], Power[c, 2]]]]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-3, 2]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[2, Plus[Times[c, Cos[Plus[d, Times[e, x]]]], Times[-1, b, Sin[Plus[d, Times[e, x]]]]], Power[Times[e, Plus[Power[a, 2], Times[-1, Power[b, 2]], Times[-1, Power[c, 2]]], Sqrt[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]]]], -1]], x], Dist[Times[1, Power[Plus[Power[a, 2], Times[-1, Power[b, 2]], Times[-1, Power[c, 2]]], -1]], Int[Sqrt[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]]], x], x]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]], Times[-1, Power[c, 2]]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[-1, c, Cos[Plus[d, Times[e, x]]]], Times[b, Sin[Plus[d, Times[e, x]]]]], Power[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]], Plus[n, 1]], Power[Times[e, Plus[n, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]], Times[-1, Power[c, 2]]]], -1]], x], Dist[Times[1, Power[Times[Plus[n, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]], Times[-1, Power[c, 2]]]], -1]], Int[Times[Plus[Times[a, Plus[n, 1]], Times[-1, b, Plus[n, 2], Cos[Plus[d, Times[e, x]]]], Times[-1, c, Plus[n, 2], Sin[Plus[d, Times[e, x]]]]], Power[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]], Plus[n, 1]]], x], x]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]], Times[-1, Power[c, 2]]]], RationalQ[n], Less[n, -1], Unequal[n, Times[-3, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[2, a, A], Times[-1, b, B], Times[-1, c, C]], x, Power[Times[2, Power[a, 2]], -1]], x], Plus[Times[-1, Simp[Times[Plus[Times[b, B], Times[c, C]], Plus[Times[b, Cos[Plus[d, Times[e, x]]]], Times[-1, c, Sin[Plus[d, Times[e, x]]]]], Power[Times[2, a, b, c, e], -1]], x]], Simp[Times[Plus[Times[Power[a, 2], Plus[Times[b, B], Times[-1, c, C]]], Times[-1, 2, a, A, Power[b, 2]], Times[Power[b, 2], Plus[Times[b, B], Times[c, C]]]], Log[RemoveContent[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]], x]], Power[Times[2, Power[a, 2], b, c, e], -1]], x]]], And[FreeQ[List[a, b, c, d, e, A, B, C], x], ZeroQ[Plus[Power[b, 2], Power[c, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[2, a, A], Times[-1, c, C]], x, Power[Times[2, Power[a, 2]], -1]], x], Plus[Times[-1, Simp[Times[C, Cos[Plus[d, Times[e, x]]], Power[Times[2, a, e], -1]], x]], Simp[Times[c, C, Sin[Plus[d, Times[e, x]]], Power[Times[2, a, b, e], -1]], x], Simp[Times[Plus[Times[-1, Power[a, 2], C], Times[2, a, c, A], Times[Power[b, 2], C]], Log[RemoveContent[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]], x]], Power[Times[2, Power[a, 2], b, e], -1]], x]]], And[FreeQ[List[a, b, c, d, e, A, C], x], ZeroQ[Plus[Power[b, 2], Power[c, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[2, a, A], Times[-1, b, B]], x, Power[Times[2, Power[a, 2]], -1]], x], Plus[Simp[Times[B, Sin[Plus[d, Times[e, x]]], Power[Times[2, a, e], -1]], x], Times[-1, Simp[Times[b, B, Cos[Plus[d, Times[e, x]]], Power[Times[2, a, c, e], -1]], x]], Simp[Times[Plus[Times[Power[a, 2], B], Times[-1, 2, a, b, A], Times[Power[b, 2], B]], Log[RemoveContent[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]], x]], Power[Times[2, Power[a, 2], c, e], -1]], x]]], And[FreeQ[List[a, b, c, d, e, A, B], x], ZeroQ[Plus[Power[b, 2], Power[c, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[b, B], Times[c, C]], x, Power[Plus[Power[b, 2], Power[c, 2]], -1]], x], Simp[Times[Plus[Times[c, B], Times[-1, b, C]], Log[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]]], Power[Times[e, Plus[Power[b, 2], Power[c, 2]]], -1]], x]], And[FreeQ[List[a, b, c, d, e, A, B, C], x], NonzeroQ[Plus[Power[b, 2], Power[c, 2]]], ZeroQ[Plus[Times[A, Plus[Power[b, 2], Power[c, 2]]], Times[-1, a, Plus[Times[b, B], Times[c, C]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[c, C, x, Power[Plus[Power[b, 2], Power[c, 2]], -1]], x], Times[-1, Simp[Times[b, C, Log[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]]], Power[Times[e, Plus[Power[b, 2], Power[c, 2]]], -1]], x]]], And[FreeQ[List[a, b, c, d, e, A, C], x], NonzeroQ[Plus[Power[b, 2], Power[c, 2]]], ZeroQ[Plus[Times[A, Plus[Power[b, 2], Power[c, 2]]], Times[-1, a, c, C]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, B, x, Power[Plus[Power[b, 2], Power[c, 2]], -1]], x], Simp[Times[c, B, Log[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]]], Power[Times[e, Plus[Power[b, 2], Power[c, 2]]], -1]], x]], And[FreeQ[List[a, b, c, d, e, A, B], x], NonzeroQ[Plus[Power[b, 2], Power[c, 2]]], ZeroQ[Plus[Times[A, Plus[Power[b, 2], Power[c, 2]]], Times[-1, a, b, B]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[b, B], Times[c, C]], x, Power[Plus[Power[b, 2], Power[c, 2]], -1]], x], Plus[Dist[Times[Plus[Times[A, Plus[Power[b, 2], Power[c, 2]]], Times[-1, a, Plus[Times[b, B], Times[c, C]]]], Power[Plus[Power[b, 2], Power[c, 2]], -1]], Int[Times[1, Power[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]], -1]], x], x], Simp[Times[Plus[Times[c, B], Times[-1, b, C]], Log[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]]], Power[Times[e, Plus[Power[b, 2], Power[c, 2]]], -1]], x]]], And[FreeQ[List[a, b, c, d, e, A, B, C], x], NonzeroQ[Plus[Power[b, 2], Power[c, 2]]], NonzeroQ[Plus[Times[A, Plus[Power[b, 2], Power[c, 2]]], Times[-1, a, Plus[Times[b, B], Times[c, C]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[c, C, Plus[d, Times[e, x]], Power[Times[e, Plus[Power[b, 2], Power[c, 2]]], -1]], x], Plus[Dist[Times[Plus[Times[A, Plus[Power[b, 2], Power[c, 2]]], Times[-1, a, c, C]], Power[Plus[Power[b, 2], Power[c, 2]], -1]], Int[Times[1, Power[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]], -1]], x], x], Times[-1, Simp[Times[b, C, Log[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]]], Power[Times[e, Plus[Power[b, 2], Power[c, 2]]], -1]], x]]]], And[FreeQ[List[a, b, c, d, e, A, C], x], NonzeroQ[Plus[Power[b, 2], Power[c, 2]]], NonzeroQ[Plus[Times[A, Plus[Power[b, 2], Power[c, 2]]], Times[-1, a, c, C]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, B, Plus[d, Times[e, x]], Power[Times[e, Plus[Power[b, 2], Power[c, 2]]], -1]], x], Plus[Dist[Times[Plus[Times[A, Plus[Power[b, 2], Power[c, 2]]], Times[-1, a, b, B]], Power[Plus[Power[b, 2], Power[c, 2]], -1]], Int[Times[1, Power[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]], -1]], x], x], Simp[Times[c, B, Log[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]]], Power[Times[e, Plus[Power[b, 2], Power[c, 2]]], -1]], x]]], And[FreeQ[List[a, b, c, d, e, A, B], x], NonzeroQ[Plus[Power[b, 2], Power[c, 2]]], NonzeroQ[Plus[Times[A, Plus[Power[b, 2], Power[c, 2]]], Times[-1, a, b, B]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Plus[Times[B, c], Times[-1, b, C], Times[-1, a, C, Cos[Plus[d, Times[e, x]]]], Times[a, B, Sin[Plus[d, Times[e, x]]]]], Power[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]], n], Power[Times[a, e, Plus[n, 1]], -1]], x], And[FreeQ[List[a, b, c, d, e, A, B, C, n], x], NonzeroQ[Plus[n, 1]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]], Times[-1, Power[c, 2]]]], ZeroQ[Plus[Times[Plus[Times[b, B], Times[c, C]], n], Times[a, A, Plus[n, 1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Simp[Times[Plus[Times[b, C], Times[a, C, Cos[Plus[d, Times[e, x]]]]], Power[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]], n], Power[Times[a, e, Plus[n, 1]], -1]], x]], And[FreeQ[List[a, b, c, d, e, A, C, n], x], NonzeroQ[Plus[n, 1]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]], Times[-1, Power[c, 2]]]], ZeroQ[Plus[Times[c, C, n], Times[a, A, Plus[n, 1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Plus[Times[B, c], Times[a, B, Sin[Plus[d, Times[e, x]]]]], Power[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]], n], Power[Times[a, e, Plus[n, 1]], -1]], x], And[FreeQ[List[a, b, c, d, e, A, B, n], x], NonzeroQ[Plus[n, 1]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]], Times[-1, Power[c, 2]]]], ZeroQ[Plus[Times[b, B, n], Times[a, A, Plus[n, 1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[B, c], Times[-1, b, C], Times[-1, a, C, Cos[Plus[d, Times[e, x]]]], Times[a, B, Sin[Plus[d, Times[e, x]]]]], Power[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]], n], Power[Times[a, e, Plus[n, 1]], -1]], x], Dist[Times[Plus[Times[Plus[Times[b, B], Times[c, C]], n], Times[a, A, Plus[n, 1]]], Power[Times[a, Plus[n, 1]], -1]], Int[Power[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]], n], x], x]], And[FreeQ[List[a, b, c, d, e, A, B, C, n], x], NonzeroQ[Plus[n, 1]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]], Times[-1, Power[c, 2]]]], NonzeroQ[Plus[Times[Plus[Times[b, B], Times[c, C]], n], Times[a, A, Plus[n, 1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[b, C], Times[a, C, Cos[Plus[d, Times[e, x]]]]], Power[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]], n], Power[Times[a, e, Plus[n, 1]], -1]], x]], Dist[Times[Plus[Times[c, C, n], Times[a, A, Plus[n, 1]]], Power[Times[a, Plus[n, 1]], -1]], Int[Power[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]], n], x], x]], And[FreeQ[List[a, b, c, d, e, A, C, n], x], NonzeroQ[Plus[n, 1]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]], Times[-1, Power[c, 2]]]], NonzeroQ[Plus[Times[c, C, n], Times[a, A, Plus[n, 1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[B, c], Times[a, B, Sin[Plus[d, Times[e, x]]]]], Power[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]], n], Power[Times[a, e, Plus[n, 1]], -1]], x], Dist[Times[Plus[Times[b, B, n], Times[a, A, Plus[n, 1]]], Power[Times[a, Plus[n, 1]], -1]], Int[Power[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]], n], x], x]], And[FreeQ[List[a, b, c, d, e, A, B, n], x], NonzeroQ[Plus[n, 1]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]], Times[-1, Power[c, 2]]]], NonzeroQ[Plus[Times[b, B, n], Times[a, A, Plus[n, 1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Plus[Times[c, B], Times[-1, b, C]], Power[Plus[Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]], Plus[n, 1]], Power[Times[e, Plus[n, 1], Plus[Power[b, 2], Power[c, 2]]], -1]], x], And[FreeQ[List[b, c, d, e, B, C], x], NonzeroQ[Plus[n, 1]], NonzeroQ[Plus[Power[b, 2], Power[c, 2]]], ZeroQ[Plus[Times[b, B], Times[c, C]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[B, c], Times[-1, b, C], Times[-1, a, C, Cos[Plus[d, Times[e, x]]]], Times[a, B, Sin[Plus[d, Times[e, x]]]]], Power[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]], n], Power[Times[a, e, Plus[n, 1]], -1]], x], Dist[Times[1, Power[Times[a, Plus[n, 1]], -1]], Int[Times[Power[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]], Plus[n, -1]], Simp[Plus[Times[a, Plus[Times[b, B], Times[c, C]], n], Times[Power[a, 2], A, Plus[n, 1]], Times[Plus[Times[n, Plus[Times[Power[a, 2], B], Times[-1, B, Power[c, 2]], Times[b, c, C]]], Times[a, b, A, Plus[n, 1]]], Cos[Plus[d, Times[e, x]]]], Times[Plus[Times[n, Plus[Times[b, B, c], Times[Power[a, 2], C], Times[-1, Power[b, 2], C]]], Times[a, c, A, Plus[n, 1]]], Sin[Plus[d, Times[e, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, A, B, C], x], RationalQ[n], Greater[n, 0], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]], Times[-1, Power[c, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[b, C], Times[a, C, Cos[Plus[d, Times[e, x]]]]], Power[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]], n], Power[Times[a, e, Plus[n, 1]], -1]], x]], Dist[Times[1, Power[Times[a, Plus[n, 1]], -1]], Int[Times[Power[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]], Plus[n, -1]], Simp[Plus[Times[a, c, C, n], Times[Power[a, 2], A, Plus[n, 1]], Times[Plus[Times[c, b, C, n], Times[a, b, A, Plus[n, 1]]], Cos[Plus[d, Times[e, x]]]], Times[Plus[Times[Power[a, 2], C, n], Times[-1, Power[b, 2], C, n], Times[a, c, A, Plus[n, 1]]], Sin[Plus[d, Times[e, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, A, C], x], RationalQ[n], Greater[n, 0], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]], Times[-1, Power[c, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]]], Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[B, c], Times[a, B, Sin[Plus[d, Times[e, x]]]]], Power[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]], n], Power[Times[a, e, Plus[n, 1]], -1]], x], Dist[Times[1, Power[Times[a, Plus[n, 1]], -1]], Int[Times[Power[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]], Plus[n, -1]], Simp[Plus[Times[a, b, B, n], Times[Power[a, 2], A, Plus[n, 1]], Times[Plus[Times[Power[a, 2], B, n], Times[-1, Power[c, 2], B, n], Times[a, b, A, Plus[n, 1]]], Cos[Plus[d, Times[e, x]]]], Times[Plus[Times[b, c, B, n], Times[a, c, A, Plus[n, 1]]], Sin[Plus[d, Times[e, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, A, B], x], RationalQ[n], Greater[n, 0], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]], Times[-1, Power[c, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[B, Power[b, -1]], Int[Sqrt[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]]], x], x], Dist[Times[Plus[Times[A, b], Times[-1, a, B]], Power[b, -1]], Int[Times[1, Power[Sqrt[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, A, B, C], x], ZeroQ[Plus[Times[B, c], Times[-1, b, C]]], NonzeroQ[Plus[Times[A, b], Times[-1, a, B]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -2], Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Plus[Times[c, B], Times[-1, b, C], Times[-1, Plus[Times[a, C], Times[-1, c, A]], Cos[Plus[d, Times[e, x]]]], Times[Plus[Times[a, B], Times[-1, b, A]], Sin[Plus[d, Times[e, x]]]]], Power[Times[e, Plus[Power[a, 2], Times[-1, Power[b, 2]], Times[-1, Power[c, 2]]], Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]]], -1]], x], And[FreeQ[List[a, b, c, d, e, A, B, C], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]], Times[-1, Power[c, 2]]]], ZeroQ[Plus[Times[a, A], Times[-1, b, B], Times[-1, c, C]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -2], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Simp[Times[Plus[Times[b, C], Times[Plus[Times[a, C], Times[-1, c, A]], Cos[Plus[d, Times[e, x]]]], Times[b, A, Sin[Plus[d, Times[e, x]]]]], Power[Times[e, Plus[Power[a, 2], Times[-1, Power[b, 2]], Times[-1, Power[c, 2]]], Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]]], -1]], x]], And[FreeQ[List[a, b, c, d, e, A, C], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]], Times[-1, Power[c, 2]]]], ZeroQ[Plus[Times[a, A], Times[-1, c, C]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -2]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Plus[Times[c, B], Times[c, A, Cos[Plus[d, Times[e, x]]]], Times[Plus[Times[a, B], Times[-1, b, A]], Sin[Plus[d, Times[e, x]]]]], Power[Times[e, Plus[Power[a, 2], Times[-1, Power[b, 2]], Times[-1, Power[c, 2]]], Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]]], -1]], x], And[FreeQ[List[a, b, c, d, e, A, B], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]], Times[-1, Power[c, 2]]]], ZeroQ[Plus[Times[a, A], Times[-1, b, B]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -2], Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[c, B], Times[-1, b, C], Times[-1, Plus[Times[a, C], Times[-1, c, A]], Cos[Plus[d, Times[e, x]]]], Times[Plus[Times[a, B], Times[-1, b, A]], Sin[Plus[d, Times[e, x]]]]], Power[Times[e, Plus[Power[a, 2], Times[-1, Power[b, 2]], Times[-1, Power[c, 2]]], Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]]], -1]], x], Dist[Times[Plus[Times[a, A], Times[-1, b, B], Times[-1, c, C]], Power[Plus[Power[a, 2], Times[-1, Power[b, 2]], Times[-1, Power[c, 2]]], -1]], Int[Times[1, Power[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, A, B, C], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]], Times[-1, Power[c, 2]]]], NonzeroQ[Plus[Times[a, A], Times[-1, b, B], Times[-1, c, C]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -2], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[b, C], Times[Plus[Times[a, C], Times[-1, c, A]], Cos[Plus[d, Times[e, x]]]], Times[b, A, Sin[Plus[d, Times[e, x]]]]], Power[Times[e, Plus[Power[a, 2], Times[-1, Power[b, 2]], Times[-1, Power[c, 2]]], Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]]], -1]], x]], Dist[Times[Plus[Times[a, A], Times[-1, c, C]], Power[Plus[Power[a, 2], Times[-1, Power[b, 2]], Times[-1, Power[c, 2]]], -1]], Int[Times[1, Power[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Sin[Plus[d, Times[e, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, A, C], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]], Times[-1, Power[c, 2]]]], NonzeroQ[Plus[Times[a, A], Times[-1, c, C]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -2]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[c, B], Times[c, A, Cos[Plus[d, Times[e, x]]]], Times[Plus[Times[a, B], 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RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[p, Blank[]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[a, Power[Sin[Plus[c, Times[d, x]]], 2]], p], Power[Power[Sin[Plus[c, Times[d, x]]], Times[2, p]], -1]], Int[Times[ActivateTrig[u], Power[Sin[Plus[c, Times[d, x]]], Times[2, p]]], x], x], And[FreeQ[List[a, b, c, d, p], x], ZeroQ[Plus[a, b]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], 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Power[Plus[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrig[Times[Power[cos[Plus[c, Times[d, x]]], m], Power[Plus[a, Times[b, Power[cos[Plus[c, Times[d, x]]], n]]], p]], x], x], And[FreeQ[List[a, b, c, d], x], IntegersQ[m, n, p]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[f, FreeFactors[Tan[Plus[c, Times[d, x]]], x]]], Dist[Times[f, Power[d, -1]], 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Power[Plus[Times[Optional[Pattern[b, Blank[]]], Power[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Pattern[n, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[f, FreeFactors[Cos[Plus[c, Times[d, x]]], x]]], Times[-1, Dist[Times[Power[f, Plus[m, 1]], Power[d, -1]], Subst[Int[Times[Power[x, m], Power[Plus[a, Times[b, Power[Times[e, f, x], n]]], p], Power[Power[Plus[1, Times[-1, Power[f, 2], Power[x, 2]]], Times[Plus[m, 1], Power[2, -1]]], -1]], x], x, Times[Cos[Plus[c, Times[d, x]]], Power[f, -1]]], x]]], And[FreeQ[List[a, b, c, d, e, n], x], OddQ[m], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[f, FreeFactors[Tan[Plus[c, Times[d, x]]], x]]], Dist[Times[Power[f, Plus[m, 1]], Power[d, -1]], Subst[Int[Times[Power[x, m], Power[ExpandToSum[Plus[Times[b, Power[f, n], Power[x, n]], Times[a, Power[Plus[1, Times[Power[f, 2], Power[x, 2]]], Times[n, Power[2, -1]]]]], x], p], Power[Power[Plus[1, Times[Power[f, 2], Power[x, 2]]], Plus[Times[n, p, Power[2, -1]], 1]], -1]], x], x, Times[Tan[Plus[c, Times[d, x]]], Power[f, -1]]], x]], And[FreeQ[List[a, b, c, d, m], x], Not[OddQ[m]], EvenQ[n], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Times[Power[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, 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Optional[Pattern[c, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Times[Power[4, p], Power[c, p]], -1]], Int[Power[Plus[b, Times[2, c, Power[Cos[Plus[d, Times[e, x]]], n]]], Times[2, p]], x], x], And[FreeQ[List[a, b, c, d, e, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, Power[Sin[Plus[d, Times[e, x]]], n]], Times[c, Power[Sin[Plus[d, Times[e, x]]], Times[2, 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x]]], n]]], Times[2, p]]], x], x], And[FreeQ[List[a, b, c, d, e, m, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Times[Power[4, p], Power[c, p]], -1]], Int[Times[Power[Cos[Plus[d, Times[e, x]]], m], Power[Plus[b, Times[2, c, Power[Cos[Plus[d, Times[e, x]]], n]]], Times[2, p]]], x], x], And[FreeQ[List[a, b, c, d, e, m, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, Power[Sin[Plus[d, Times[e, x]]], n]], Times[c, Power[Sin[Plus[d, Times[e, x]]], Times[2, n]]]], p], Power[Power[Plus[b, Times[2, c, Power[Sin[Plus[d, Times[e, x]]], n]]], Times[2, p]], -1]], Int[Times[Power[Sin[Plus[d, Times[e, x]]], m], Power[Plus[b, Times[2, c, Power[Sin[Plus[d, Times[e, x]]], n]]], Times[2, p]]], x], x], And[FreeQ[List[a, b, c, d, e, m, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, Power[Cos[Plus[d, Times[e, x]]], n]], Times[c, 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Blank[Symbol]]]], Condition[Module[List[Set[f, FreeFactors[Cot[Plus[d, Times[e, x]]], x]]], Times[-1, Dist[Times[f, Power[e, -1]], Subst[Int[Times[Power[ExpandToSum[Plus[c, Times[b, Power[Plus[1, Power[x, 2]], Times[n, Power[2, -1]]]], Times[a, Power[Plus[1, Power[x, 2]], n]]], x], p], Power[Power[Plus[1, Times[Power[f, 2], Power[x, 2]]], Plus[Times[m, Power[2, -1]], Times[n, p], 1]], -1]], x], x, Times[Cot[Plus[d, Times[e, x]]], Power[f, -1]]], x]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], EvenQ[m], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], EvenQ[n], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]], Optional[Pattern[c, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[f, FreeFactors[Tan[Plus[d, Times[e, x]]], x]]], Dist[Times[f, Power[e, -1]], Subst[Int[Times[Power[ExpandToSum[Plus[c, Times[b, Power[Plus[1, Power[x, 2]], Times[n, Power[2, -1]]]], Times[a, Power[Plus[1, Power[x, 2]], n]]], x], p], Power[Power[Plus[1, Times[Power[f, 2], Power[x, 2]]], Plus[Times[m, Power[2, -1]], Times[n, p], 1]], -1]], x], x, Times[Tan[Plus[d, Times[e, x]]], Power[f, -1]]], x]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], EvenQ[m], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], EvenQ[n], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrig[Times[Power[sin[Plus[d, Times[e, x]]], m], Power[Plus[a, Times[b, Power[sin[Plus[d, Times[e, x]]], n]], Times[c, Power[sin[Plus[d, Times[e, x]]], Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegersQ[m, n, p]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrig[Times[Power[cos[Plus[d, Times[e, x]]], m], Power[Plus[a, Times[b, Power[cos[Plus[d, Times[e, x]]], n]], Times[c, Power[cos[Plus[d, Times[e, x]]], Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegersQ[m, n, p]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[g, FreeFactors[Sin[Plus[d, Times[e, x]]], x]]], Dist[Times[g, Power[e, -1]], Subst[Int[Times[Power[Plus[1, Times[-1, Power[g, 2], Power[x, 2]]], Times[Plus[m, -1], Power[2, -1]]], Power[Plus[a, Times[b, Power[Times[f, g, x], n]], Times[c, Power[Times[f, g, x], Times[2, n]]]], p]], x], x, Times[Sin[Plus[d, Times[e, x]]], Power[g, -1]]], x]], And[FreeQ[List[a, b, c, d, e, f, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], OddQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[f, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[f, Blank[]]]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[g, FreeFactors[Cos[Plus[d, Times[e, x]]], x]]], Times[-1, Dist[Times[g, Power[e, -1]], Subst[Int[Times[Power[Plus[1, Times[-1, Power[g, 2], Power[x, 2]]], Times[Plus[m, -1], Power[2, -1]]], Power[Plus[a, Times[b, Power[Times[f, g, x], n]], Times[c, Power[Times[f, g, x], Times[2, n]]]], p]], x], x, Times[Cos[Plus[d, Times[e, x]]], Power[g, -1]]], x]]], And[FreeQ[List[a, b, c, d, e, f, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], OddQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Times[Power[4, p], Power[c, p]], -1]], Int[Times[Power[Cos[Plus[d, Times[e, x]]], m], Power[Plus[b, Times[2, c, Power[Sin[Plus[d, Times[e, x]]], n]]], Times[2, p]]], x], x], And[FreeQ[List[a, b, c, d, e, m, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], Not[OddQ[m]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Times[Power[4, p], Power[c, p]], -1]], Int[Times[Power[Sin[Plus[d, Times[e, x]]], m], Power[Plus[b, Times[2, c, Power[Cos[Plus[d, Times[e, x]]], n]]], Times[2, p]]], x], x], And[FreeQ[List[a, b, c, d, e, m, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], Not[OddQ[m]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, Power[Sin[Plus[d, Times[e, x]]], n]], Times[c, Power[Sin[Plus[d, Times[e, x]]], Times[2, n]]]], p], Power[Power[Plus[b, Times[2, c, Power[Sin[Plus[d, Times[e, x]]], n]]], Times[2, p]], -1]], Int[Times[Power[Cos[Plus[d, Times[e, x]]], m], Power[Plus[b, Times[2, c, Power[Sin[Plus[d, Times[e, x]]], n]]], Times[2, p]]], x], x], And[FreeQ[List[a, b, c, d, e, m, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], Not[OddQ[m]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Pattern[p, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, Power[Cos[Plus[d, Times[e, x]]], n]], Times[c, Power[Cos[Plus[d, Times[e, x]]], Times[2, n]]]], p], Power[Power[Plus[b, Times[2, c, Power[Cos[Plus[d, Times[e, x]]], n]]], Times[2, p]], -1]], Int[Times[Power[Sin[Plus[d, Times[e, x]]], m], Power[Plus[b, Times[2, c, Power[Cos[Plus[d, Times[e, x]]], n]]], Times[2, p]]], x], x], And[FreeQ[List[a, b, c, d, e, m, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], Not[OddQ[m]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[f, FreeFactors[Cot[Plus[d, Times[e, x]]], x]]], Times[-1, Dist[Times[Power[f, Plus[m, 1]], Power[e, -1]], Subst[Int[Times[Power[x, m], Power[ExpandToSum[Plus[c, Times[b, Power[Plus[1, Power[x, 2]], Times[n, Power[2, -1]]]], Times[a, Power[Plus[1, Power[x, 2]], n]]], x], p], Power[Power[Plus[1, Times[Power[f, 2], Power[x, 2]]], Plus[Times[m, Power[2, -1]], Times[n, p], 1]], -1]], x], x, Times[Cot[Plus[d, Times[e, x]]], Power[f, -1]]], x]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], EvenQ[m], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], EvenQ[n], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]], Optional[Pattern[c, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[f, FreeFactors[Tan[Plus[d, Times[e, x]]], x]]], Dist[Times[Power[f, Plus[m, 1]], Power[e, -1]], Subst[Int[Times[Power[x, m], Power[ExpandToSum[Plus[c, Times[b, Power[Plus[1, Power[x, 2]], Times[n, Power[2, -1]]]], Times[a, Power[Plus[1, Power[x, 2]], n]]], x], p], Power[Power[Plus[1, Times[Power[f, 2], Power[x, 2]]], Plus[Times[m, Power[2, -1]], Times[n, p], 1]], -1]], x], x, Times[Tan[Plus[d, Times[e, x]]], Power[f, -1]]], x]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], EvenQ[m], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], EvenQ[n], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrig[Times[Power[Plus[1, Times[-1, Power[sin[Plus[d, Times[e, x]]], 2]]], Times[m, Power[2, -1]]], Power[Plus[a, Times[b, Power[sin[Plus[d, Times[e, x]]], n]], Times[c, Power[sin[Plus[d, Times[e, x]]], Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], EvenQ[m], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegersQ[n, p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrig[Times[Power[Plus[1, Times[-1, Power[cos[Plus[d, Times[e, x]]], 2]]], Times[m, Power[2, -1]]], Power[Plus[a, Times[b, Power[cos[Plus[d, Times[e, x]]], n]], Times[c, Power[cos[Plus[d, Times[e, x]]], Times[2, n]]]], p]], x], x], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], EvenQ[m], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegersQ[n, p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[g, FreeFactors[Sin[Plus[d, Times[e, x]]], x]]], Dist[Times[Power[g, Plus[m, 1]], Power[e, -1]], Subst[Int[Times[Power[x, m], Power[Plus[a, Times[b, Power[Times[f, g, x], n]], Times[c, Power[Times[f, g, x], Times[2, n]]]], p], Power[Power[Plus[1, Times[-1, Power[g, 2], Power[x, 2]]], Times[Plus[m, 1], Power[2, -1]]], -1]], x], x, Times[Sin[Plus[d, Times[e, x]]], Power[g, 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Power[Plus[b, Times[2, c, Power[Cos[Plus[d, Times[e, x]]], n]]], Times[2, p]]], x], x], And[FreeQ[List[a, b, c, d, e, m, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], Not[OddQ[m]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, Power[Sin[Plus[d, Times[e, x]]], n]], Times[c, Power[Sin[Plus[d, Times[e, x]]], Times[2, n]]]], p], Power[Power[Plus[b, Times[2, c, Power[Sin[Plus[d, Times[e, x]]], n]]], Times[2, p]], -1]], Int[Times[Power[Tan[Plus[d, Times[e, x]]], m], Power[Plus[b, Times[2, c, Power[Sin[Plus[d, Times[e, x]]], n]]], Times[2, p]]], x], x], And[FreeQ[List[a, b, c, d, e, m, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], Not[OddQ[m]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, Power[Cos[Plus[d, Times[e, x]]], n]], Times[c, Power[Cos[Plus[d, Times[e, x]]], Times[2, n]]]], p], Power[Power[Plus[b, Times[2, c, Power[Cos[Plus[d, Times[e, x]]], n]]], Times[2, p]], -1]], Int[Times[Power[Cot[Plus[d, Times[e, x]]], m], Power[Plus[b, Times[2, c, Power[Cos[Plus[d, Times[e, x]]], n]]], Times[2, p]]], x], x], And[FreeQ[List[a, b, c, d, e, m, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], Not[OddQ[m]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[f, FreeFactors[Tan[Plus[d, Times[e, x]]], x]]], Dist[Times[Power[f, Plus[m, 1]], Power[e, -1]], Subst[Int[Times[Power[x, m], Power[ExpandToSum[Plus[Times[c, Power[x, Times[2, n]]], Times[b, Power[x, n], Power[Plus[1, Power[x, 2]], Times[n, Power[2, -1]]]], Times[a, Power[Plus[1, Power[x, 2]], n]]], x], p], Power[Power[Plus[1, Times[Power[f, 2], Power[x, 2]]], Plus[Times[n, p], 1]], -1]], x], x, Times[Tan[Plus[d, Times[e, x]]], Power[f, -1]]], x]], And[FreeQ[List[a, b, c, d, e, m], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], Not[OddQ[m]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], EvenQ[n], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]], Optional[Pattern[c, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[f, FreeFactors[Cot[Plus[d, Times[e, x]]], x]]], Times[-1, Dist[Times[Power[f, Plus[m, 1]], Power[e, -1]], Subst[Int[Times[Power[x, m], Power[ExpandToSum[Plus[Times[c, Power[x, Times[2, n]]], Times[b, Power[x, n], Power[Plus[1, Power[x, 2]], Times[n, Power[2, -1]]]], Times[a, Power[Plus[1, Power[x, 2]], n]]], x], p], Power[Power[Plus[1, Times[Power[f, 2], Power[x, 2]]], Plus[Times[n, p], 1]], -1]], x], x, Times[Cot[Plus[d, Times[e, x]]], Power[f, -1]]], x]]], And[FreeQ[List[a, b, c, d, e, m], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], Not[OddQ[m]], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], EvenQ[n], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrig[Times[Power[sin[Plus[d, Times[e, x]]], m], Power[Plus[a, Times[b, Power[sin[Plus[d, Times[e, x]]], n]], Times[c, Power[sin[Plus[d, Times[e, x]]], Times[2, n]]]], p], Power[Power[Plus[1, Times[-1, Power[sin[Plus[d, Times[e, x]]], 2]]], Times[m, Power[2, -1]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], EvenQ[m], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegersQ[n, p]]]], RuleDelayed[HoldPattern[Int[Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrig[Times[Power[cos[Plus[d, Times[e, x]]], m], Power[Plus[a, Times[b, Power[cos[Plus[d, Times[e, x]]], n]], Times[c, Power[cos[Plus[d, Times[e, x]]], Times[2, n]]]], p], Power[Power[Plus[1, Times[-1, Power[cos[Plus[d, Times[e, x]]], 2]]], Times[m, Power[2, -1]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], EvenQ[m], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegersQ[n, p]]]], RuleDelayed[HoldPattern[Int[Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[g, FreeFactors[Sin[Plus[d, Times[e, x]]], x]]], Dist[Times[Power[g, Plus[m, 1]], Power[e, -1]], Subst[Int[Times[Power[Plus[1, Times[-1, Power[g, 2], Power[x, 2]]], Times[Plus[m, -1], Power[2, -1]]], Power[Plus[a, Times[b, Power[Times[f, g, x], n]], Times[c, Power[Times[f, g, x], Times[2, n]]]], p], Power[Power[x, m], -1]], x], x, Times[Sin[Plus[d, Times[e, x]]], Power[g, -1]]], x]], And[FreeQ[List[a, b, c, d, e, f, n], x], OddQ[m], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[Optional[Pattern[c, Blank[]]], Power[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[f, Blank[]]]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[f, Blank[]]]], Pattern[n, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[g, FreeFactors[Cos[Plus[d, Times[e, x]]], x]]], Times[-1, Dist[Times[Power[g, Plus[m, 1]], Power[e, -1]], Subst[Int[Times[Power[Plus[1, Times[-1, Power[g, 2], Power[x, 2]]], Times[Plus[m, -1], Power[2, -1]]], Power[Plus[a, Times[b, Power[Times[f, g, x], n]], Times[c, Power[Times[f, g, x], Times[2, n]]]], p], Power[Power[x, m], -1]], x], x, Times[Cos[Plus[d, Times[e, x]]], Power[g, -1]]], x]]], And[FreeQ[List[a, b, c, d, e, f, n], x], OddQ[m], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Times[Power[4, p], Power[c, p]], -1]], Int[Times[Power[Cot[Plus[d, Times[e, x]]], m], Power[Plus[b, Times[2, c, Power[Sin[Plus[d, Times[e, x]]], n]]], Times[2, p]]], x], x], And[FreeQ[List[a, b, c, d, e, m, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], Not[OddQ[m]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Times[Power[4, p], Power[c, p]], -1]], Int[Times[Power[Tan[Plus[d, Times[e, x]]], m], Power[Plus[b, Times[2, c, Power[Cos[Plus[d, Times[e, x]]], n]]], Times[2, p]]], x], x], And[FreeQ[List[a, b, c, d, e, m, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], Not[OddQ[m]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, Power[Sin[Plus[d, Times[e, x]]], n]], Times[c, Power[Sin[Plus[d, Times[e, x]]], Times[2, n]]]], p], Power[Power[Plus[b, Times[2, c, Power[Sin[Plus[d, Times[e, x]]], n]]], Times[2, p]], -1]], Int[Times[Power[Cot[Plus[d, Times[e, x]]], m], Power[Plus[b, Times[2, c, Power[Sin[Plus[d, Times[e, x]]], n]]], Times[2, p]]], x], x], And[FreeQ[List[a, b, c, d, e, m, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], Not[OddQ[m]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Pattern[p, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, Power[Cos[Plus[d, Times[e, x]]], n]], Times[c, Power[Cos[Plus[d, Times[e, x]]], Times[2, n]]]], p], Power[Power[Plus[b, Times[2, c, Power[Cos[Plus[d, Times[e, x]]], n]]], Times[2, p]], -1]], Int[Times[Power[Tan[Plus[d, Times[e, x]]], m], Power[Plus[b, Times[2, c, Power[Cos[Plus[d, Times[e, x]]], n]]], Times[2, p]]], x], x], And[FreeQ[List[a, b, c, d, e, m, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], Not[OddQ[m]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[f, FreeFactors[Cot[Plus[d, Times[e, x]]], x]]], Times[-1, Dist[Times[Power[f, Plus[m, 1]], Power[e, -1]], Subst[Int[Times[Power[x, m], Power[ExpandToSum[Plus[c, Times[b, Power[Plus[1, Times[Power[f, 2], Power[x, 2]]], Times[n, Power[2, -1]]]], Times[a, Power[Plus[1, Times[Power[f, 2], Power[x, 2]]], n]]], x], p], Power[Power[Plus[1, Times[Power[f, 2], Power[x, 2]]], Plus[Times[n, p], 1]], -1]], x], x, Times[Cot[Plus[d, Times[e, x]]], Power[f, -1]]], x]]], And[FreeQ[List[a, b, c, d, e, m], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], EvenQ[n], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]], Optional[Pattern[c, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[f, FreeFactors[Tan[Plus[d, Times[e, x]]], x]]], Dist[Times[Power[f, Plus[m, 1]], Power[e, -1]], Subst[Int[Times[Power[x, m], Power[ExpandToSum[Plus[c, Times[b, Power[Plus[1, Times[Power[f, 2], Power[x, 2]]], Times[n, Power[2, -1]]]], Times[a, Power[Plus[1, Times[Power[f, 2], Power[x, 2]]], n]]], x], p], Power[Power[Plus[1, Times[Power[f, 2], Power[x, 2]]], Plus[Times[n, p], 1]], -1]], x], x, Times[Tan[Plus[d, Times[e, x]]], Power[f, -1]]], x]], And[FreeQ[List[a, b, c, d, e, m], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], EvenQ[n], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrig[Times[Power[Plus[1, Times[-1, Power[sin[Plus[d, Times[e, x]]], 2]]], Times[m, Power[2, -1]]], Power[Plus[a, Times[b, Power[sin[Plus[d, Times[e, x]]], n]], Times[c, Power[sin[Plus[d, Times[e, x]]], Times[2, n]]]], p], Power[Power[sin[Plus[d, Times[e, x]]], m], -1]], x], x], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], EvenQ[m], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegersQ[n, p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n2, Blank[]]]], Optional[Pattern[c, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrig[Times[Power[Plus[1, Times[-1, Power[cos[Plus[d, Times[e, x]]], 2]]], Times[m, Power[2, -1]]], Power[Plus[a, Times[b, Power[cos[Plus[d, Times[e, x]]], n]], Times[c, Power[cos[Plus[d, Times[e, x]]], Times[2, n]]]], p], Power[Power[cos[Plus[d, Times[e, x]]], m], -1]], x], x], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], EvenQ[m], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegersQ[n, p]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Times[Power[4, n], Power[c, n]], -1]], Int[Times[Plus[A, Times[B, Sin[Plus[d, Times[e, x]]]]], Power[Plus[b, Times[2, c, Sin[Plus[d, Times[e, x]]]]], Times[2, n]]], x], x], And[FreeQ[List[a, b, c, d, e, A, B], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[c, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]], Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Times[Power[4, n], Power[c, n]], -1]], Int[Times[Plus[A, Times[B, Cos[Plus[d, Times[e, x]]]]], Power[Plus[b, Times[2, c, Cos[Plus[d, Times[e, x]]]]], Times[2, n]]], x], x], And[FreeQ[List[a, b, c, d, e, A, B], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[c, Blank[]]], Power[sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, Sin[Plus[d, Times[e, x]]]], Times[c, Power[Sin[Plus[d, Times[e, x]]], 2]]], n], Power[Power[Plus[b, Times[2, c, Sin[Plus[d, Times[e, x]]]]], Times[2, n]], -1]], Int[Times[Plus[A, Times[B, Sin[Plus[d, Times[e, x]]]]], Power[Plus[b, Times[2, c, Sin[Plus[d, Times[e, x]]]]], Times[2, n]]], x], x], And[FreeQ[List[a, b, c, d, e, A, B], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[c, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]], Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, Cos[Plus[d, Times[e, x]]]], Times[c, Power[Cos[Plus[d, Times[e, x]]], 2]]], n], Power[Power[Plus[b, Times[2, c, Cos[Plus[d, Times[e, x]]]]], 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Blank[Symbol]]]], Condition[Int[ExpandTrig[Times[Plus[A, Times[B, sin[Plus[d, Times[e, x]]]]], Power[Plus[a, Times[b, sin[Plus[d, Times[e, x]]]], Times[c, Power[sin[Plus[d, Times[e, x]]], 2]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, A, B], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[c, Blank[]]]]], Pattern[n, Blank[]]], Plus[Times[cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrig[Times[Plus[A, Times[B, cos[Plus[d, Times[e, x]]]]], Power[Plus[a, Times[b, 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RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[SinIntegral[Plus[e, Times[f, x]]], Power[d, -1]], x], And[FreeQ[List[c, d, e, f], x], ZeroQ[Plus[Times[d, e], Times[-1, c, f]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Complex[0, Pattern[fz, Blank[]]], Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[CoshIntegral[Plus[Times[-1, c, f, fz, Power[d, -1]], Times[-1, f, fz, x]]], Power[d, -1]], x], And[FreeQ[List[c, d, e, f, fz], x], ZeroQ[Plus[Times[d, Plus[e, Times[-1, Pi, Power[2, -1]]]], Times[-1, c, f, fz, I]]], NegQ[Times[c, f, fz, Power[d, 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Times[-1, I, k, Pi]], Power[E, Times[-1, I, Plus[e, Times[f, x]]]]], x], x], Times[-1, Dist[Times[I, Power[2, -1]], Int[Times[Power[Plus[c, Times[d, x]], m], Power[E, Times[I, k, Pi]], Power[E, Times[I, Plus[e, Times[f, x]]]]], x], x]]], And[FreeQ[List[c, d, e, f, m], x], IntegerQ[Times[2, k]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[I, Power[2, -1]], Int[Times[Power[Plus[c, Times[d, x]], m], Power[E, Times[-1, I, Plus[e, Times[f, x]]]]], x], x], Times[-1, Dist[Times[I, Power[2, -1]], Int[Times[Power[Plus[c, Times[d, x]], m], Power[E, Times[I, Plus[e, Times[f, x]]]]], x], x]]], FreeQ[List[c, d, e, f, m], x]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[c, Blank[]]], 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Int[ExpandTrigReduce[Power[Plus[c, Times[d, x]], Plus[m, 1]], Times[Cos[Plus[e, Times[f, x]]], Power[Sin[Plus[e, Times[f, x]]], Plus[n, -1]]], x], x], x]]], And[FreeQ[List[c, d, e, f, m], x], IntegerQ[n], Greater[n, 1], RationalQ[m], Inequality[-2, LessEqual, m, Less, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[c, Times[d, x]], Plus[m, 1]], Power[Times[b, Sin[Plus[e, Times[f, x]]]], n], Power[Times[d, Plus[m, 1]], -1]], x], Plus[Dist[Times[Power[b, 2], Power[f, 2], n, Plus[n, -1], Power[Times[Power[d, 2], Plus[m, 1], Plus[m, 2]], -1]], Int[Times[Power[Plus[c, Times[d, x]], Plus[m, 2]], Power[Times[b, Sin[Plus[e, Times[f, x]]]], Plus[n, 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Plus[Dist[Times[Plus[n, 2], Power[Times[Power[b, 2], Plus[n, 1]], -1]], Int[Times[Power[Plus[c, Times[d, x]], m], Power[Times[b, Sin[Plus[e, Times[f, x]]]], Plus[n, 2]]], x], x], Dist[Times[Power[d, 2], m, Plus[m, -1], Power[Times[Power[b, 2], Power[f, 2], Plus[n, 1], Plus[n, 2]], -1]], Int[Times[Power[Plus[c, Times[d, x]], Plus[m, -2]], Power[Times[b, Sin[Plus[e, Times[f, x]]]], Plus[n, 2]]], x], x], Times[-1, Simp[Times[d, m, Power[Plus[c, Times[d, x]], Plus[m, -1]], Power[Times[b, Sin[Plus[e, Times[f, x]]]], Plus[n, 2]], Power[Times[Power[b, 2], Power[f, 2], Plus[n, 1], Plus[n, 2]], -1]], x]]]], And[FreeQ[List[b, c, d, e, f], x], RationalQ[m, n], Less[n, -1], Unequal[n, -2], Greater[m, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], 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x], And[FreeQ[List[a, b, c, d, e, f, m], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegerQ[Plus[n, Times[1, Power[2, -1]]]], Or[Greater[n, 0], PositiveIntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Pi, Optional[Pattern[k, Blank[]]]], Times[Complex[0, Pattern[fz, Blank[]]], Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[2, Int[Times[Power[Plus[c, Times[d, x]], m], Power[E, Times[-1, I, Pi, Plus[k, Times[-1, 1, Power[2, -1]]]]], Power[E, Plus[Times[-1, I, e], Times[f, fz, x]]], Power[Plus[b, Times[2, a, Power[E, Times[-1, I, Pi, Plus[k, Times[-1, 1, Power[2, -1]]]]], Power[E, Plus[Times[-1, I, e], Times[f, fz, x]]]], Times[-1, b, Power[E, Times[-2, I, k, Pi]], 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Times[f, x]]]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f], x], IntegerQ[Times[2, k]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], PositiveIntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Complex[0, Pattern[fz, Blank[]]], Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[2, Int[Times[Power[Plus[c, Times[d, x]], m], Power[E, Plus[Times[-1, I, e], Times[f, fz, x]]], Power[Plus[Times[-1, I, b], Times[2, a, Power[E, Plus[Times[-1, I, e], Times[f, fz, x]]]], Times[I, b, Power[E, Times[2, Plus[Times[-1, I, e], Times[f, fz, x]]]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, fz], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], PositiveIntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[2, Int[Times[Power[Plus[c, Times[d, x]], m], Power[E, Times[I, Plus[e, Times[f, x]]]], Power[Plus[Times[I, b], Times[2, a, Power[E, Times[I, Plus[e, Times[f, x]]]]], Times[-1, I, b, Power[E, Times[2, I, Plus[e, Times[f, x]]]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], PositiveIntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -2]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, Power[Plus[c, Times[d, x]], m], Cos[Plus[e, Times[f, x]]], Power[Times[f, Plus[Power[a, 2], Times[-1, Power[b, 2]]], Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]]], -1]], x], Plus[Dist[Times[a, Power[Plus[Power[a, 2], Times[-1, Power[b, 2]]], -1]], Int[Times[Power[Plus[c, Times[d, x]], m], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], -1]], x], x], Times[-1, Dist[Times[b, d, m, Power[Times[f, Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], Int[Times[Power[Plus[c, Times[d, x]], Plus[m, -1]], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], PositiveIntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], 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Power[Times[f, Plus[n, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], Int[Times[Power[Plus[c, Times[d, x]], Plus[m, -1]], Cos[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], Plus[n, 1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NegativeIntegerQ[Plus[n, 2]], PositiveIntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Plus[c, Times[d, x]], m], Power[Plus[a, Times[b, Sin[Plus[e, Times[f, x]]]]], n]], x], FreeQ[List[a, b, c, d, e, f, m, n], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Pattern[v, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[ExpandToSum[u, x], m], Power[Plus[a, Times[b, Sin[ExpandToSum[v, x]]]], n]], x], And[FreeQ[List[a, b, m, n], x], LinearQ[List[u, v], x], Not[LinearMatchQ[List[u, v], x]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Pattern[v, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[ExpandToSum[u, x], m], Power[Plus[a, Times[b, Cos[ExpandToSum[v, x]]]], n]], x], And[FreeQ[List[a, b, m, n], x], LinearQ[List[u, v], x], Not[LinearMatchQ[List[u, v], x]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Sin[Plus[c, Times[d, x]]], Power[Plus[a, Times[b, Power[x, n]]], p], x], x], And[FreeQ[List[a, b, c, d, n], x], PositiveIntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Cos[Plus[c, Times[d, x]]], Power[Plus[a, Times[b, Power[x, n]]], p], x], x], And[FreeQ[List[a, b, c, d, n], x], PositiveIntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[Times[-1, n], 1]], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Sin[Plus[c, Times[d, x]]], Power[Times[b, n, Plus[p, 1]], -1]], x], Plus[Times[-1, Dist[Times[Plus[Times[-1, n], 1], Power[Times[b, n, Plus[p, 1]], -1]], Int[Times[Power[x, Times[-1, n]], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Sin[Plus[c, Times[d, x]]]], x], x]], Times[-1, Dist[Times[d, Power[Times[b, n, Plus[p, 1]], -1]], Int[Times[Power[x, Plus[Times[-1, n], 1]], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Cos[Plus[c, Times[d, x]]]], x], x]]]], And[FreeQ[List[a, b, c, d], x], IntegerQ[p], PositiveIntegerQ[n], Less[p, -1], Greater[n, 2]]]], RuleDelayed[HoldPattern[Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[Times[-1, n], 1]], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Cos[Plus[c, Times[d, x]]], Power[Times[b, n, Plus[p, 1]], -1]], x], Plus[Times[-1, Dist[Times[Plus[Times[-1, n], 1], Power[Times[b, n, Plus[p, 1]], -1]], Int[Times[Power[x, Times[-1, n]], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Cos[Plus[c, Times[d, x]]]], x], x]], Dist[Times[d, Power[Times[b, n, Plus[p, 1]], -1]], Int[Times[Power[x, Plus[Times[-1, n], 1]], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Sin[Plus[c, Times[d, x]]]], x], x]]], And[FreeQ[List[a, b, c, d], x], IntegerQ[p], PositiveIntegerQ[n], Less[p, -1], Greater[n, 2]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Sin[Plus[c, Times[d, x]]], Power[Plus[a, Times[b, Power[x, n]]], p], x], x], And[FreeQ[List[a, b, c, d], x], NegativeIntegerQ[p], PositiveIntegerQ[n], Or[Equal[n, 2], Equal[p, -1]]]]], RuleDelayed[HoldPattern[Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Cos[Plus[c, Times[d, x]]], Power[Plus[a, Times[b, Power[x, n]]], p], x], x], And[FreeQ[List[a, b, c, d], x], NegativeIntegerQ[p], PositiveIntegerQ[n], Or[Equal[n, 2], Equal[p, -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[x, Times[n, p]], Power[Plus[b, Times[a, Power[x, Times[-1, n]]]], p], Sin[Plus[c, Times[d, x]]]], x], And[FreeQ[List[a, b, c, d], x], NegativeIntegerQ[p], NegativeIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[x, Times[n, p]], Power[Plus[b, Times[a, Power[x, Times[-1, n]]]], p], Cos[Plus[c, Times[d, x]]]], x], And[FreeQ[List[a, b, c, d], x], NegativeIntegerQ[p], NegativeIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Plus[a, Times[b, Power[x, n]]], p], Sin[Plus[c, Times[d, x]]]], x], FreeQ[List[a, b, c, d, n, p], x]]], RuleDelayed[HoldPattern[Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Plus[a, Times[b, Power[x, n]]], p], Cos[Plus[c, Times[d, x]]]], x], FreeQ[List[a, b, c, d, n, p], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Sin[Plus[c, Times[d, x]]], Times[Power[Times[e, x], m], Power[Plus[a, Times[b, Power[x, n]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, m, n], x], PositiveIntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Cos[Plus[c, Times[d, x]]], Times[Power[Times[e, x], m], Power[Plus[a, Times[b, Power[x, n]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, m, n], x], PositiveIntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[e, m], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Sin[Plus[c, Times[d, x]]], Power[Times[b, n, Plus[p, 1]], -1]], x], Times[-1, Dist[Times[d, Power[e, m], Power[Times[b, n, Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Cos[Plus[c, Times[d, x]]]], x], x]]], And[FreeQ[List[a, b, c, d, e, m, n], x], IntegerQ[p], ZeroQ[Plus[m, Times[-1, n], 1]], RationalQ[p], Less[p, -1], Or[IntegerQ[n], PositiveQ[e]]]]], RuleDelayed[HoldPattern[Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[e, m], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Cos[Plus[c, Times[d, x]]], Power[Times[b, n, Plus[p, 1]], -1]], x], Dist[Times[d, Power[e, m], Power[Times[b, n, Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Sin[Plus[c, Times[d, x]]]], x], x]], And[FreeQ[List[a, b, c, d, e, m, n], x], IntegerQ[p], ZeroQ[Plus[m, Times[-1, n], 1]], RationalQ[p], Less[p, -1], Or[IntegerQ[n], PositiveQ[e]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[m, Times[-1, n], 1]], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Sin[Plus[c, Times[d, x]]], Power[Times[b, n, Plus[p, 1]], -1]], x], Plus[Times[-1, Dist[Times[Plus[m, Times[-1, n], 1], Power[Times[b, n, Plus[p, 1]], -1]], Int[Times[Power[x, Plus[m, Times[-1, n]]], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Sin[Plus[c, Times[d, x]]]], x], x]], Times[-1, Dist[Times[d, Power[Times[b, n, Plus[p, 1]], -1]], Int[Times[Power[x, Plus[m, Times[-1, n], 1]], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Cos[Plus[c, Times[d, x]]]], x], x]]]], And[FreeQ[List[a, b, c, d], x], IntegerQ[p], PositiveIntegerQ[n], RationalQ[m], Less[p, -1], Or[Greater[Plus[m, Times[-1, n], 1], 0], Greater[n, 2]]]]], RuleDelayed[HoldPattern[Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[m, Times[-1, n], 1]], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Cos[Plus[c, Times[d, x]]], Power[Times[b, n, Plus[p, 1]], -1]], x], Plus[Times[-1, Dist[Times[Plus[m, Times[-1, n], 1], Power[Times[b, n, Plus[p, 1]], -1]], Int[Times[Power[x, Plus[m, Times[-1, n]]], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Cos[Plus[c, Times[d, x]]]], x], x]], Dist[Times[d, Power[Times[b, n, Plus[p, 1]], -1]], Int[Times[Power[x, Plus[m, Times[-1, n], 1]], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Sin[Plus[c, Times[d, x]]]], x], x]]], And[FreeQ[List[a, b, c, d], x], IntegerQ[p], PositiveIntegerQ[n], RationalQ[m], Less[p, -1], Or[Greater[Plus[m, Times[-1, n], 1], 0], Greater[n, 2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Sin[Plus[c, Times[d, x]]], Times[Power[x, m], Power[Plus[a, Times[b, Power[x, n]]], p]], x], x], And[FreeQ[List[a, b, c, d], x], NegativeIntegerQ[p], IntegerQ[m], PositiveIntegerQ[n], Or[Equal[n, 2], Equal[p, -1]]]]], RuleDelayed[HoldPattern[Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Cos[Plus[c, Times[d, x]]], Times[Power[x, m], Power[Plus[a, Times[b, Power[x, n]]], p]], x], x], And[FreeQ[List[a, b, c, d], x], NegativeIntegerQ[p], IntegerQ[m], PositiveIntegerQ[n], Or[Equal[n, 2], Equal[p, -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[x, Plus[m, Times[n, p]]], Power[Plus[b, Times[a, Power[x, Times[-1, n]]]], p], Sin[Plus[c, Times[d, x]]]], x], And[FreeQ[List[a, b, c, d, m], x], NegativeIntegerQ[p], NegativeIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[x, Plus[m, Times[n, p]]], Power[Plus[b, Times[a, Power[x, Times[-1, n]]]], p], Cos[Plus[c, Times[d, x]]]], x], And[FreeQ[List[a, b, c, d, m], x], NegativeIntegerQ[p], NegativeIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Times[e, x], m], Power[Plus[a, Times[b, Power[x, n]]], p], Sin[Plus[c, Times[d, x]]]], x], FreeQ[List[a, b, c, d, e, m, n, p], x]]], RuleDelayed[HoldPattern[Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Times[e, x], m], Power[Plus[a, Times[b, Power[x, n]]], p], Cos[Plus[c, Times[d, x]]]], x], FreeQ[List[a, b, c, d, e, m, n, p], x]]], RuleDelayed[HoldPattern[Int[Sin[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Sqrt[Times[Pi, Power[2, -1]]], Power[Rt[d, 2], -1], FresnelS[Times[Sqrt[Times[2, Power[Pi, -1]]], Rt[d, 2], x]]], x], FreeQ[d, x]]], RuleDelayed[HoldPattern[Int[Cos[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Sqrt[Times[Pi, Power[2, -1]]], Power[Rt[d, 2], -1], FresnelC[Times[Sqrt[Times[2, Power[Pi, -1]]], Rt[d, 2], x]]], x], FreeQ[d, x]]], RuleDelayed[HoldPattern[Int[Sin[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Sin[c], Int[Cos[Times[d, Power[x, 2]]], x], x], Dist[Cos[c], Int[Sin[Times[d, Power[x, 2]]], x], x]], FreeQ[List[c, d], x]]], RuleDelayed[HoldPattern[Int[Cos[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Cos[c], Int[Cos[Times[d, Power[x, 2]]], x], x], Times[-1, Dist[Sin[c], Int[Sin[Times[d, Power[x, 2]]], x], x]]], FreeQ[List[c, d], x]]], RuleDelayed[HoldPattern[Int[Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[I, Power[2, -1]], Int[Power[E, Plus[Times[-1, c, I], Times[-1, d, I, Power[x, n]]]], x], x], Times[-1, Dist[Times[I, Power[2, -1]], Int[Power[E, Plus[Times[c, I], Times[d, I, Power[x, n]]]], x], x]]], And[FreeQ[List[c, d], x], IntegerQ[n], Greater[n, 2]]]], RuleDelayed[HoldPattern[Int[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[2, -1]], Int[Power[E, Plus[Times[-1, c, I], Times[-1, d, I, Power[x, n]]]], x], x], Dist[Times[1, Power[2, -1]], Int[Power[E, Plus[Times[c, I], Times[d, I, Power[x, n]]]], x], x]], And[FreeQ[List[c, d], x], IntegerQ[n], Greater[n, 2]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Power[Plus[a, Times[b, Sin[Plus[c, Times[d, Power[x, n]]]]]], p], x], x], And[FreeQ[List[a, b, c, d], x], IntegersQ[n, p], Greater[n, 1], Greater[p, 1]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Power[Plus[a, Times[b, Cos[Plus[c, Times[d, Power[x, n]]]]]], p], x], x], And[FreeQ[List[a, b, c, d], x], IntegersQ[n, p], Greater[n, 1], Greater[p, 1]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Subst[Int[Times[Power[Plus[a, Times[b, Sin[Plus[c, Times[d, Power[x, Times[-1, n]]]]]]], p], Power[Power[x, 2], -1]], x], x, Times[1, Power[x, -1]]]], And[FreeQ[List[a, b, c, d], x], IntegerQ[p], NegativeIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Subst[Int[Times[Power[Plus[a, Times[b, Cos[Plus[c, Times[d, Power[x, Times[-1, n]]]]]]], p], Power[Power[x, 2], -1]], x], x, Times[1, Power[x, -1]]]], And[FreeQ[List[a, b, c, d], x], IntegerQ[p], NegativeIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[k, Denominator[n]]], Dist[k, Subst[Int[Times[Power[x, Plus[k, -1]], Power[Plus[a, Times[b, Sin[Plus[c, Times[d, Power[x, Times[k, n]]]]]]], p]], x], x, Power[x, Times[1, Power[k, -1]]]], x]], And[FreeQ[List[a, b, c, d], x], IntegerQ[p], FractionQ[n]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[k, Denominator[n]]], Dist[k, Subst[Int[Times[Power[x, Plus[k, -1]], Power[Plus[a, Times[b, Cos[Plus[c, Times[d, Power[x, Times[k, n]]]]]]], p]], x], x, Power[x, Times[1, Power[k, -1]]]], x]], And[FreeQ[List[a, b, c, d], x], IntegerQ[p], FractionQ[n]]]], RuleDelayed[HoldPattern[Int[Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[I, Power[2, -1]], Int[Power[E, Plus[Times[-1, c, I], Times[-1, d, I, Power[x, n]]]], x], x], Times[-1, Dist[Times[I, Power[2, -1]], Int[Power[E, Plus[Times[c, I], Times[d, I, Power[x, n]]]], x], x]]], FreeQ[List[c, d, n], x]]], RuleDelayed[HoldPattern[Int[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[2, -1]], Int[Power[E, Plus[Times[-1, c, I], Times[-1, d, I, Power[x, n]]]], x], x], Dist[Times[1, Power[2, -1]], Int[Power[E, Plus[Times[c, I], Times[d, I, Power[x, n]]]], x], x]], FreeQ[List[c, d, n], x]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Power[Plus[a, Times[b, Sin[Plus[c, Times[d, Power[x, n]]]]]], p], x], x], And[FreeQ[List[a, b, c, d, n], x], PositiveIntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Power[Plus[a, Times[b, Cos[Plus[c, Times[d, Power[x, n]]]]]], p], x], x], And[FreeQ[List[a, b, c, d, n], x], PositiveIntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Coefficient[u, x, 1], -1]], Subst[Int[Power[Plus[a, Times[b, Sin[Plus[c, Times[d, Power[x, n]]]]]], p], x], x, u], x], And[FreeQ[List[a, b, c, d, n], x], IntegerQ[p], LinearQ[u, x], NonzeroQ[Plus[u, Times[-1, x]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Coefficient[u, x, 1], -1]], Subst[Int[Power[Plus[a, Times[b, Cos[Plus[c, Times[d, Power[x, n]]]]]], p], x], x, u], x], And[FreeQ[List[a, b, c, d, n], x], IntegerQ[p], LinearQ[u, x], NonzeroQ[Plus[u, Times[-1, x]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Power[Plus[a, Times[b, Sin[Plus[c, Times[d, Power[u, n]]]]]], p], x], And[FreeQ[List[a, b, c, d, n, p], x], LinearQ[u, x]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Power[Plus[a, Times[b, Cos[Plus[c, Times[d, Power[u, n]]]]]], p], x], And[FreeQ[List[a, b, c, d, n, p], x], LinearQ[u, x]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Pattern[u, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Power[Plus[a, Times[b, Sin[ExpandToSum[u, x]]]], p], x], And[FreeQ[List[a, b, p], x], BinomialQ[u, x], Not[BinomialMatchQ[u, x]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Power[Plus[a, Times[b, Cos[ExpandToSum[u, x]]]], p], x], And[FreeQ[List[a, b, p], x], BinomialQ[u, x], Not[BinomialMatchQ[u, x]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], -1], Sin[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[SinIntegral[Times[d, Power[x, n]]], Power[n, -1]], x], FreeQ[List[d, n], x]]], RuleDelayed[HoldPattern[Int[Times[Cos[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[CosIntegral[Times[d, Power[x, n]]], Power[n, -1]], x], FreeQ[List[d, n], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], -1], Sin[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Sin[c], Int[Times[Cos[Times[d, Power[x, n]]], Power[x, -1]], x], x], Dist[Cos[c], Int[Times[Sin[Times[d, Power[x, n]]], Power[x, -1]], x], x]], FreeQ[List[c, d, n], x]]], RuleDelayed[HoldPattern[Int[Times[Cos[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Cos[c], Int[Times[Cos[Times[d, Power[x, n]]], Power[x, -1]], x], x], Times[-1, Dist[Sin[c], Int[Times[Sin[Times[d, Power[x, n]]], Power[x, -1]], x], x]]], 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x], x], Times[-1, Simp[Times[n, Power[Sin[Plus[a, Times[b, Power[x, n]]]], Plus[p, 2]], Power[Times[Power[b, 2], Power[n, 2], Plus[p, 1], Plus[p, 2]], -1]], x]]]], And[FreeQ[List[a, b, m, n], x], ZeroQ[Plus[m, Times[-1, 2, n], 1]], RationalQ[p], Less[p, -1], Unequal[p, -2]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[x, n], Sin[Plus[a, Times[b, Power[x, n]]]], Power[Cos[Plus[a, Times[b, Power[x, n]]]], Plus[p, 1]], Power[Times[b, n, Plus[p, 1]], -1]], x]], Plus[Dist[Times[Plus[p, 2], Power[Plus[p, 1], -1]], Int[Times[Power[x, m], Power[Cos[Plus[a, Times[b, Power[x, n]]]], Plus[p, 2]]], x], x], Times[-1, Simp[Times[n, Power[Cos[Plus[a, Times[b, Power[x, n]]]], Plus[p, 2]], Power[Times[Power[b, 2], Power[n, 2], Plus[p, 1], Plus[p, 2]], -1]], x]]]], And[FreeQ[List[a, b, m, n], x], ZeroQ[Plus[m, Times[-1, 2, n], 1]], RationalQ[p], Less[p, -1], Unequal[p, -2]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[m, Times[-1, n], 1]], Cos[Plus[a, Times[b, Power[x, n]]]], Power[Sin[Plus[a, Times[b, Power[x, n]]]], Plus[p, 1]], Power[Times[b, n, Plus[p, 1]], -1]], x], Plus[Dist[Times[Plus[p, 2], Power[Plus[p, 1], -1]], Int[Times[Power[x, m], Power[Sin[Plus[a, Times[b, Power[x, n]]]], Plus[p, 2]]], x], x], Dist[Times[Plus[m, Times[-1, n], 1], Plus[m, Times[-1, 2, n], 1], Power[Times[Power[b, 2], Power[n, 2], Plus[p, 1], Plus[p, 2]], -1]], Int[Times[Power[x, Plus[m, Times[-1, 2, n]]], Power[Sin[Plus[a, Times[b, Power[x, n]]]], Plus[p, 2]]], x], x], Times[-1, Simp[Times[Plus[m, Times[-1, n], 1], Power[x, Plus[m, Times[-1, 2, n], 1]], Power[Sin[Plus[a, Times[b, Power[x, n]]]], Plus[p, 2]], Power[Times[Power[b, 2], Power[n, 2], Plus[p, 1], Plus[p, 2]], -1]], x]]]], And[FreeQ[List[a, b], x], IntegersQ[m, n], RationalQ[p], Less[p, -1], Unequal[p, -2], Less[0, Times[2, n], Plus[m, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[x, Plus[m, Times[-1, n], 1]], Sin[Plus[a, Times[b, Power[x, n]]]], Power[Cos[Plus[a, Times[b, Power[x, n]]]], Plus[p, 1]], Power[Times[b, n, Plus[p, 1]], -1]], x]], Plus[Dist[Times[Plus[p, 2], Power[Plus[p, 1], -1]], Int[Times[Power[x, m], Power[Cos[Plus[a, Times[b, Power[x, n]]]], Plus[p, 2]]], x], x], Dist[Times[Plus[m, Times[-1, n], 1], Plus[m, Times[-1, 2, n], 1], Power[Times[Power[b, 2], Power[n, 2], Plus[p, 1], Plus[p, 2]], -1]], Int[Times[Power[x, Plus[m, Times[-1, 2, n]]], Power[Cos[Plus[a, Times[b, Power[x, n]]]], Plus[p, 2]]], x], x], Times[-1, Simp[Times[Plus[m, Times[-1, n], 1], Power[x, Plus[m, Times[-1, 2, n], 1]], Power[Cos[Plus[a, Times[b, Power[x, n]]]], Plus[p, 2]], Power[Times[Power[b, 2], Power[n, 2], Plus[p, 1], Plus[p, 2]], -1]], x]]]], And[FreeQ[List[a, b], x], IntegersQ[m, n], RationalQ[p], Less[p, -1], Unequal[p, -2], Less[0, Times[2, n], Plus[m, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Subst[Int[Times[Power[Plus[a, Times[b, 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Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[k, Denominator[m]]], Times[-1, Dist[Times[k, Power[e, -1]], Subst[Int[Times[Power[Plus[a, Times[b, Sin[Plus[c, Times[d, Power[Times[Power[e, n], Power[x, Times[k, n]]], -1]]]]]], p], Power[Power[x, Plus[Times[k, Plus[m, 1]], 1]], -1]], x], x, Times[1, Power[Power[Times[e, x], Times[1, Power[k, -1]]], -1]]], x]]], And[FreeQ[List[a, b, c, d, e], x], IntegerQ[p], NegativeIntegerQ[n], FractionQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[k, Denominator[m]]], Times[-1, Dist[Times[k, Power[e, -1]], Subst[Int[Times[Power[Plus[a, Times[b, Cos[Plus[c, Times[d, Power[Times[Power[e, n], Power[x, Times[k, n]]], -1]]]]]], p], Power[Power[x, Plus[Times[k, Plus[m, 1]], 1]], -1]], x], x, Times[1, Power[Power[Times[e, x], Times[1, Power[k, -1]]], -1]]], x]]], And[FreeQ[List[a, b, c, d, e], x], IntegerQ[p], NegativeIntegerQ[n], FractionQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Times[Power[Times[e, x], m], Power[Power[x, -1], m]], Subst[Int[Times[Power[Plus[a, Times[b, Sin[Plus[c, 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Denominator[n]]], Dist[k, Subst[Int[Times[Power[x, Plus[Times[k, Plus[m, 1]], -1]], Power[Plus[a, Times[b, Cos[Plus[c, Times[d, Power[x, Times[k, n]]]]]]], p]], x], x, Power[x, Times[1, Power[k, -1]]]], x]], And[FreeQ[List[a, b, c, d, m], x], IntegerQ[p], FractionQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[e, IntPart[m]], Power[Times[e, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Plus[a, Times[b, Sin[Plus[c, Times[d, Power[x, n]]]]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, m], x], IntegerQ[p], FractionQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[e, IntPart[m]], Power[Times[e, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Plus[a, Times[b, Cos[Plus[c, Times[d, Power[x, n]]]]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, m], x], IntegerQ[p], FractionQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Plus[m, 1], -1]], Subst[Int[Power[Plus[a, Times[b, Sin[Plus[c, Times[d, Power[x, Simplify[Times[n, Power[Plus[m, 1], -1]]]]]]]]], p], x], x, Power[x, Plus[m, 1]]], x], And[FreeQ[List[a, b, c, d, m, n], x], IntegerQ[p], NonzeroQ[Plus[m, 1]], PositiveIntegerQ[Simplify[Times[n, Power[Plus[m, 1], -1]]]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Plus[m, 1], -1]], Subst[Int[Power[Plus[a, Times[b, Cos[Plus[c, Times[d, Power[x, Simplify[Times[n, Power[Plus[m, 1], -1]]]]]]]]], p], x], x, Power[x, Plus[m, 1]]], x], And[FreeQ[List[a, b, c, d, m, n], x], IntegerQ[p], NonzeroQ[Plus[m, 1]], PositiveIntegerQ[Simplify[Times[n, Power[Plus[m, 1], -1]]]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[e, IntPart[m]], Power[Times[e, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Plus[a, Times[b, Sin[Plus[c, Times[d, Power[x, n]]]]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, m, n], x], IntegerQ[p], NonzeroQ[Plus[m, 1]], PositiveIntegerQ[Simplify[Times[n, Power[Plus[m, 1], -1]]]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[e, IntPart[m]], Power[Times[e, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Plus[a, Times[b, Cos[Plus[c, Times[d, Power[x, n]]]]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, m, n], x], IntegerQ[p], NonzeroQ[Plus[m, 1]], PositiveIntegerQ[Simplify[Times[n, Power[Plus[m, 1], -1]]]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[I, Power[2, -1]], Int[Times[Power[Times[e, x], m], Power[E, Plus[Times[-1, c, I], Times[-1, d, I, Power[x, n]]]]], x], x], Times[-1, Dist[Times[I, Power[2, -1]], Int[Times[Power[Times[e, x], m], Power[E, Plus[Times[c, I], Times[d, I, Power[x, n]]]]], x], x]]], FreeQ[List[c, d, e, m, n], x]]], RuleDelayed[HoldPattern[Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[2, -1]], Int[Times[Power[Times[e, x], m], Power[E, Plus[Times[-1, c, I], Times[-1, d, I, Power[x, n]]]]], x], x], Dist[Times[1, Power[2, -1]], Int[Times[Power[Times[e, x], m], Power[E, Plus[Times[c, I], Times[d, I, Power[x, n]]]]], x], x]], FreeQ[List[c, d, e, m, n], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, 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Condition[Dist[Times[1, Power[Power[Coefficient[u, x, 1], Plus[m, 1]], -1]], Subst[Int[Times[Power[Plus[x, Times[-1, Coefficient[u, x, 0]]], m], Power[Plus[a, Times[b, Cos[Plus[c, Times[d, Power[x, n]]]]]], p]], x], x, u], x], And[FreeQ[List[a, b, c, d, n, p], x], LinearQ[u, x], NonzeroQ[Plus[u, Times[-1, x]]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Times[e, x], m], Power[Plus[a, Times[b, Sin[Plus[c, Times[d, Power[u, n]]]]]], p]], x], And[FreeQ[List[a, b, c, d, e, m, n, p], x], LinearQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Times[e, x], m], Power[Plus[a, Times[b, Cos[Plus[c, Times[d, Power[u, n]]]]]], p]], x], And[FreeQ[List[a, b, c, d, e, m, n, p], x], LinearQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sin[Pattern[u, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Times[e, x], m], Power[Plus[a, Times[b, Sin[ExpandToSum[u, x]]]], p]], x], And[FreeQ[List[a, b, e, m, p], x], BinomialQ[u, x], Not[BinomialMatchQ[u, x]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cos[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Times[e, x], m], Power[Plus[a, Times[b, Cos[ExpandToSum[u, x]]]], p]], x], And[FreeQ[List[a, b, e, m, p], x], BinomialQ[u, x], Not[BinomialMatchQ[u, x]]]]], RuleDelayed[HoldPattern[Int[Times[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[Sin[Plus[a, Times[b, Power[x, n]]]], Plus[p, 1]], Power[Times[b, n, Plus[p, 1]], -1]], x], And[FreeQ[List[a, b, m, n, p], x], ZeroQ[Plus[m, Times[-1, n], 1]], NonzeroQ[Plus[p, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Simp[Times[Power[Cos[Plus[a, Times[b, Power[x, n]]]], Plus[p, 1]], Power[Times[b, n, Plus[p, 1]], -1]], x]], And[FreeQ[List[a, b, m, n, p], x], ZeroQ[Plus[m, Times[-1, n], 1]], NonzeroQ[Plus[p, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[m, Times[-1, n], 1]], Power[Sin[Plus[a, Times[b, Power[x, n]]]], Plus[p, 1]], Power[Times[b, n, Plus[p, 1]], -1]], x], Times[-1, Dist[Times[Plus[m, Times[-1, n], 1], Power[Times[b, n, Plus[p, 1]], -1]], Int[Times[Power[x, Plus[m, Times[-1, n]]], Power[Sin[Plus[a, Times[b, Power[x, n]]]], Plus[p, 1]]], x], x]]], And[FreeQ[List[a, b, p], x], RationalQ[m, n], Less[0, n, Plus[m, 1]], NonzeroQ[Plus[p, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[x, Plus[m, Times[-1, n], 1]], Power[Cos[Plus[a, Times[b, Power[x, n]]]], Plus[p, 1]], Power[Times[b, n, Plus[p, 1]], -1]], x]], Dist[Times[Plus[m, Times[-1, n], 1], Power[Times[b, n, Plus[p, 1]], -1]], Int[Times[Power[x, Plus[m, Times[-1, n]]], Power[Cos[Plus[a, Times[b, Power[x, n]]]], Plus[p, 1]]], x], x]], And[FreeQ[List[a, b, p], x], RationalQ[m, n], Less[0, n, Plus[m, 1]], NonzeroQ[Plus[p, 1]]]]], RuleDelayed[HoldPattern[Int[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Sin[Times[Power[Plus[b, Times[2, c, x]], 2], Power[Times[4, c], -1]]], x], And[FreeQ[List[a, b, c], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Cos[Times[Power[Plus[b, Times[2, c, x]], 2], Power[Times[4, c], -1]]], x], And[FreeQ[List[a, b, c], x], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Cos[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Power[Times[4, c], -1]]], Int[Sin[Times[Power[Plus[b, Times[2, c, x]], 2], Power[Times[4, c], -1]]], x], x], Times[-1, Dist[Sin[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Power[Times[4, c], -1]]], Int[Cos[Times[Power[Plus[b, Times[2, c, x]], 2], Power[Times[4, c], -1]]], x], x]]], And[FreeQ[List[a, b, c], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Cos[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Power[Times[4, c], -1]]], Int[Cos[Times[Power[Plus[b, Times[2, c, x]], 2], Power[Times[4, c], -1]]], x], x], Dist[Sin[Times[Plus[Power[b, 2], Times[-1, 4, a, c]], Power[Times[4, c], -1]]], Int[Sin[Times[Power[Plus[b, Times[2, c, x]], 2], Power[Times[4, c], -1]]], x], x]], And[FreeQ[List[a, b, c], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Power[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Power[Sin[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], n], x], x], And[FreeQ[List[a, b, c], x], IntegerQ[n], Greater[n, 1]]]], RuleDelayed[HoldPattern[Int[Power[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Power[Cos[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], n], x], x], And[FreeQ[List[a, b, c], x], IntegerQ[n], Greater[n, 1]]]], RuleDelayed[HoldPattern[Int[Power[Sin[Pattern[v, Blank[]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Power[Sin[ExpandToSum[v, x]], n], x], And[PositiveIntegerQ[n], QuadraticQ[v, x], Not[QuadraticMatchQ[v, x]]]]], RuleDelayed[HoldPattern[Int[Power[Cos[Pattern[v, Blank[]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Power[Cos[ExpandToSum[v, x]], n], x], And[PositiveIntegerQ[n], QuadraticQ[v, x], Not[QuadraticMatchQ[v, x]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Simp[Times[e, Cos[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], Power[Times[2, c], -1]], x]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[Times[2, c, d], Times[-1, b, e]]]]]], RuleDelayed[HoldPattern[Int[Times[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[e, Sin[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], Power[Times[2, c], -1]], x], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[Times[2, c, d], Times[-1, b, e]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[e, Cos[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], Power[Times[2, c], -1]], x]], Dist[Times[Plus[Times[2, c, d], Times[-1, b, e]], Power[Times[2, c], -1]], Int[Sin[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], x], x]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]]]]], RuleDelayed[HoldPattern[Int[Times[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[e, Sin[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], Power[Times[2, c], -1]], x], Dist[Times[Plus[Times[2, c, d], Times[-1, b, e]], Power[Times[2, c], -1]], Int[Cos[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], x], x]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Times[2, c, d], Times[-1, b, e]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[e, Power[Plus[d, Times[e, x]], Plus[m, -1]], Cos[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], Power[Times[2, c], -1]], x]], Dist[Times[Power[e, 2], Plus[m, -1], Power[Times[2, c], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -2]], Cos[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]], x], x]], And[FreeQ[List[a, b, c, d, e], x], RationalQ[m], Greater[m, 1], ZeroQ[Plus[Times[b, e], Times[-1, 2, c, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[e, Power[Plus[d, Times[e, x]], Plus[m, -1]], Sin[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], Power[Times[2, c], -1]], x], Times[-1, Dist[Times[Power[e, 2], Plus[m, -1], Power[Times[2, c], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -2]], Sin[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], RationalQ[m], Greater[m, 1], ZeroQ[Plus[Times[b, e], Times[-1, 2, c, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[e, Power[Plus[d, Times[e, x]], Plus[m, -1]], Cos[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], Power[Times[2, c], -1]], x]], Plus[Dist[Times[Power[e, 2], Plus[m, -1], Power[Times[2, c], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -2]], Cos[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]], x], x], Times[-1, Dist[Times[Plus[Times[b, e], Times[-1, 2, c, d]], Power[Times[2, c], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Sin[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]], x], x]]]], And[FreeQ[List[a, b, c, d, e], x], RationalQ[m], Greater[m, 1], NonzeroQ[Plus[Times[b, e], Times[-1, 2, c, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[e, Power[Plus[d, Times[e, x]], Plus[m, -1]], Sin[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], Power[Times[2, c], -1]], x], Plus[Times[-1, Dist[Times[Power[e, 2], Plus[m, -1], Power[Times[2, c], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -2]], Sin[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]], x], x]], Times[-1, Dist[Times[Plus[Times[b, e], Times[-1, 2, c, d]], Power[Times[2, c], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, -1]], Cos[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]], x], x]]]], And[FreeQ[List[a, b, c, d, e], x], RationalQ[m], Greater[m, 1], NonzeroQ[Plus[Times[b, e], Times[-1, 2, c, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Sin[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], Power[Times[e, Plus[m, 1]], -1]], x], Times[-1, Dist[Times[2, c, Power[Times[Power[e, 2], Plus[m, 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 2]], Cos[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], RationalQ[m], Less[m, -1], ZeroQ[Plus[Times[b, e], Times[-1, 2, c, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Cos[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], Power[Times[e, Plus[m, 1]], -1]], x], Dist[Times[2, c, Power[Times[Power[e, 2], Plus[m, 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 2]], Sin[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]], x], x]], And[FreeQ[List[a, b, c, d, e], x], RationalQ[m], Less[m, -1], ZeroQ[Plus[Times[b, e], Times[-1, 2, c, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Sin[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], Power[Times[e, Plus[m, 1]], -1]], x], Plus[Times[-1, Dist[Times[2, c, Power[Times[Power[e, 2], Plus[m, 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 2]], Cos[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]], x], x]], Times[-1, Dist[Times[Plus[Times[b, e], Times[-1, 2, c, d]], Power[Times[Power[e, 2], Plus[m, 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Cos[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]], x], x]]]], And[FreeQ[List[a, b, c, d, e], x], RationalQ[m], Less[m, -1], NonzeroQ[Plus[Times[b, e], Times[-1, 2, c, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Cos[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], Power[Times[e, Plus[m, 1]], -1]], x], Plus[Dist[Times[2, c, Power[Times[Power[e, 2], Plus[m, 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 2]], Sin[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]], x], x], Dist[Times[Plus[Times[b, e], Times[-1, 2, c, d]], Power[Times[Power[e, 2], Plus[m, 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Sin[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], RationalQ[m], Less[m, -1], NonzeroQ[Plus[Times[b, e], Times[-1, 2, c, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Plus[d, Times[e, x]], m], Sin[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]], x], FreeQ[List[a, b, c, d, e, m], x]]], RuleDelayed[HoldPattern[Int[Times[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Plus[d, Times[e, x]], m], Cos[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]], x], FreeQ[List[a, b, c, d, e, m], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Power[Plus[d, Times[e, x]], m], Power[Sin[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], n], x], x], And[FreeQ[List[a, b, c, d, e, m], x], IntegerQ[n], Greater[n, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Power[Plus[d, Times[e, x]], m], Power[Cos[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], n], x], x], And[FreeQ[List[a, b, c, d, e, m], x], 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x]], And[FreeQ[List[e, f], x], IntegersQ[m, n, Times[Plus[m, n, -1], Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Power[b, m], -1]], Int[Times[Power[Times[b, Tan[Plus[e, Times[f, x]]]], Plus[m, n]], Power[Power[Sec[Plus[e, Times[f, x]]], m], -1]], x], x], And[FreeQ[List[b, e, f, n], x], IntegerQ[m], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], 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Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[a, Sin[Plus[e, Times[f, x]]]], m], Power[Times[b, Tan[Plus[e, Times[f, x]]]], Plus[n, 1]], Power[Times[b, f, m], -1]], x], Times[-1, Dist[Times[Power[a, 2], Plus[n, 1], Power[Times[Power[b, 2], m], -1]], Int[Times[Power[Times[a, Sin[Plus[e, Times[f, x]]]], Plus[m, -2]], Power[Times[b, Tan[Plus[e, Times[f, x]]]], Plus[n, 2]]], x], x]]], And[FreeQ[List[a, b, e, f], x], RationalQ[m, n], Less[n, -1], Greater[m, 1], IntegersQ[Times[2, m], Times[2, n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[a, Blank[]]], sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[a, Sin[Plus[e, 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Power[Times[a, Sec[Plus[e, Times[f, x]]]], m], Power[Times[b, Tan[Plus[e, Times[f, x]]]], Plus[n, -1]], Power[Times[f, Plus[m, n, -1]], -1]], x], Times[-1, Dist[Times[Power[b, 2], Plus[n, -1], Power[Plus[m, n, -1], -1]], Int[Times[Power[Times[a, Sec[Plus[e, Times[f, x]]]], m], Power[Times[b, Tan[Plus[e, Times[f, x]]]], Plus[n, -2]]], x], x]]], And[FreeQ[List[a, b, e, f, m], x], RationalQ[n], Greater[n, 1], NonzeroQ[Plus[m, n, -1]], IntegersQ[Times[2, m], Times[2, n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[a, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Times[a, Sec[Plus[e, Times[f, x]]]], m], Power[Times[b, Tan[Plus[e, Times[f, x]]]], 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x]]]], n], Power[Power[Times[Sin[Plus[e, Times[f, x]]], Power[a, -1]], m], -1]], x], x], And[FreeQ[List[a, b, e, f, m, n], x], Not[IntegerQ[m]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[a, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, Power[Times[a, Power[Times[d, Sec[Plus[e, Times[f, x]]]], p]], m], Power[Times[b, Tan[Plus[e, Times[f, x]]]], Plus[n, -1]], Power[Times[f, Plus[Times[m, p], n, -1]], -1]], x], Times[-1, Dist[Times[Power[b, 2], Plus[n, -1], Power[Plus[Times[m, p], n, -1], -1]], Int[Times[Power[Times[a, Power[Times[d, Sec[Plus[e, Times[f, x]]]], 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1], Power[Times[Power[b, 2], Plus[n, 1]], -1]], Int[Times[Power[Times[a, Power[Times[d, Sec[Plus[e, Times[f, x]]]], p]], m], Power[Times[b, Tan[Plus[e, Times[f, x]]]], Plus[n, 2]]], x], x]]], And[FreeQ[List[a, b, d, e, f, m, p], x], RationalQ[n], Less[n, -1], NonzeroQ[Plus[Times[m, p], n, 1]], IntegersQ[Times[2, m, p], Times[2, n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[a, Blank[]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[p, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[a, Power[Times[d, Csc[Plus[e, Times[f, x]]]], p]], m], Power[Times[b, Cot[Plus[e, Times[f, x]]]], Plus[n, 1]], Power[Times[b, f, Plus[n, 1]], -1]], 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Blank[]]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, Power[Plus[a, Times[b, Tan[Plus[c, Times[d, x]]]]], Plus[n, -1]], Power[Times[d, Plus[n, -1]], -1]], x], Int[Times[Plus[Power[a, 2], Times[-1, Power[b, 2]], Times[2, a, b, Tan[Plus[c, Times[d, x]]]]], Power[Plus[a, Times[b, Tan[Plus[c, Times[d, x]]]]], Plus[n, -2]]], x]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], RationalQ[n], Greater[n, 1]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, Power[Plus[a, Times[b, Tan[Plus[c, Times[d, x]]]]], Plus[n, 1]], Power[Times[d, Plus[n, 1], Plus[Power[a, 2], Power[b, 2]]], -1]], x], Dist[Times[1, Power[Plus[Power[a, 2], Power[b, 2]], -1]], Int[Times[Plus[a, Times[-1, b, Tan[Plus[c, Times[d, x]]]]], Power[Plus[a, Times[b, Tan[Plus[c, Times[d, x]]]]], Plus[n, 1]]], x], x]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], RationalQ[n], Less[n, -1]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[a, x, Power[Plus[Power[a, 2], Power[b, 2]], -1]], x], Dist[Times[b, Power[Plus[Power[a, 2], Power[b, 2]], -1]], Int[Times[Plus[b, Times[-1, a, Tan[Plus[c, Times[d, x]]]]], Power[Plus[a, Times[b, Tan[Plus[c, Times[d, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[b, Power[d, -1]], Subst[Int[Times[Power[Plus[a, x], n], Power[Plus[Power[b, 2], Power[x, 2]], -1]], x], x, Times[b, Tan[Plus[c, Times[d, x]]]]], x], And[FreeQ[List[a, b, c, d, n], x], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, Power[Times[d, Sec[Plus[e, Times[f, x]]]], m], Power[Times[f, m], -1]], x], Dist[a, Int[Power[Times[d, Sec[Plus[e, Times[f, x]]]], m], x], x]], And[FreeQ[List[a, b, d, e, f, m], x], Or[IntegerQ[Times[2, m]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Times[Power[a, Plus[m, -2]], b, f], -1]], Subst[Int[Times[Power[Plus[a, Times[-1, x]], Plus[Times[m, Power[2, -1]], -1]], Power[Plus[a, x], Plus[n, Times[m, Power[2, -1]], -1]]], x], x, Times[b, Tan[Plus[e, Times[f, x]]]]], x], And[FreeQ[List[a, b, e, f, n], x], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], IntegerQ[Times[m, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], 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RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[2, b, Power[Times[d, Sec[Plus[e, Times[f, x]]]], m], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[n, -1]], Power[Times[f, m], -1]], x], And[FreeQ[List[a, b, d, e, f, m, n], x], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], ZeroQ[Simplify[Plus[Times[m, Power[2, -1]], n, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, Power[Times[d, Sec[Plus[e, Times[f, x]]]], m], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[n, -1]], Power[Times[f, Plus[m, n, -1]], -1]], x], Dist[Times[a, Plus[m, Times[2, n], -2], Power[Plus[m, n, -1], -1]], Int[Times[Power[Times[d, Sec[Plus[e, Times[f, x]]]], m], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[n, -1]]], x], x]], And[FreeQ[List[a, b, d, e, f, m, n], x], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], PositiveIntegerQ[Simplify[Plus[Times[m, Power[2, -1]], n, -1]]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, 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Condition[Plus[Simp[Times[Power[b, 2], ArcTanh[Sin[Plus[e, Times[f, x]]]], Power[f, -1]], x], Plus[Times[-1, Simp[Times[2, a, b, Cos[Plus[e, Times[f, x]]], Power[f, -1]], x]], Simp[Times[Plus[Power[a, 2], Times[-1, Power[b, 2]]], Sin[Plus[e, Times[f, x]]], Power[f, -1]], x]]], And[FreeQ[List[a, b, e, f], x], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], 2]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, Power[Times[d, Sec[Plus[e, Times[f, x]]]], m], Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Power[Times[f, Plus[m, 1]], -1]], x], Dist[Times[1, Power[Plus[m, 1], -1]], Int[Times[Power[Times[d, Sec[Plus[e, Times[f, x]]]], m], Plus[Times[Power[a, 2], Plus[m, 1]], Times[-1, Power[b, 2]], Times[a, b, Plus[m, 2], Tan[Plus[e, Times[f, x]]]]]], x], x]], And[FreeQ[List[a, b, d, e, f, m], x], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[m, 1]]]]], RuleDelayed[HoldPattern[Int[Times[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Power[f, -1], Subst[Int[Times[1, Power[Plus[Power[a, 2], Power[b, 2], Times[-1, Power[x, 2]]], -1]], x], x, Times[Plus[b, Times[-1, a, Tan[Plus[e, Times[f, x]]]]], Power[Sec[Plus[e, Times[f, x]]], -1]]], x]], And[FreeQ[List[a, b, e, f], x], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Dist[Times[Power[d, 2], Power[Power[b, 2], -1]], Int[Times[Power[Times[d, Sec[Plus[e, Times[f, x]]]], Plus[m, -2]], Plus[a, Times[-1, b, Tan[Plus[e, Times[f, x]]]]]], x], x]], Dist[Times[Power[d, 2], Plus[Power[a, 2], Power[b, 2]], Power[Power[b, 2], -1]], Int[Times[Power[Times[d, Sec[Plus[e, Times[f, x]]]], Plus[m, -2]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, d, e, f], x], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], PositiveIntegerQ[Plus[m, -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[Plus[Power[a, 2], Power[b, 2]], -1]], Int[Times[Power[Times[d, Sec[Plus[e, Times[f, x]]]], m], Plus[a, Times[-1, b, Tan[Plus[e, Times[f, x]]]]]], x], x], Dist[Times[Power[b, 2], Power[Times[Power[d, 2], Plus[Power[a, 2], Power[b, 2]]], -1]], Int[Times[Power[Times[d, Sec[Plus[e, Times[f, x]]]], Plus[m, 2]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, d, e, f], x], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], NegativeIntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[d, Times[2, IntPart[Times[m, Power[2, -1]]]]], Power[Times[d, Sec[Plus[e, Times[f, x]]]], Times[2, FracPart[Times[m, Power[2, -1]]]]], Power[Times[b, f, Power[Power[Sec[Plus[e, Times[f, x]]], 2], FracPart[Times[m, Power[2, -1]]]]], -1]], Subst[Int[Times[Power[Plus[a, x], n], Power[Plus[1, Times[Power[x, 2], Power[Power[b, 2], -1]]], Plus[Times[m, Power[2, -1]], -1]]], x], x, Times[b, Tan[Plus[e, Times[f, x]]]]], x], And[FreeQ[List[a, b, d, e, f, m, n], x], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], Not[IntegerQ[Times[m, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[-4, b, Power[f, -1]], Subst[Int[Times[Power[x, 2], Power[Plus[Times[Power[a, 2], Power[d, 2]], Power[x, 4]], -1]], x], x, Times[Sqrt[Times[d, Cos[Plus[e, Times[f, x]]]]], Sqrt[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]]]]], x], And[FreeQ[List[a, b, d, e, f], x], ZeroQ[Plus[Power[a, 2], Power[b, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[-3, 2]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Times[d, Cos[Plus[e, Times[f, x]]], Sqrt[Plus[a, Times[-1, b, Tan[Plus[e, Times[f, x]]]]]], Sqrt[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]]]], -1]], Int[Times[Sqrt[Plus[a, Times[-1, b, Tan[Plus[e, Times[f, x]]]]]], Power[Sqrt[Times[d, Cos[Plus[e, Times[f, x]]]]], -1]], x], x], And[FreeQ[List[a, b, d, e, f], x], ZeroQ[Plus[Power[a, 2], Power[b, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[d, Cos[Plus[e, Times[f, x]]]], m], Power[Times[d, Sec[Plus[e, Times[f, x]]]], m]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], n], Power[Power[Times[d, Sec[Plus[e, Times[f, x]]]], m], -1]], x], x], And[FreeQ[List[a, b, d, e, f, m, n], x], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[b, Power[f, -1]], Subst[Int[Times[Power[x, m], Power[Plus[a, x], n], Power[Power[Plus[Power[b, 2], Power[x, 2]], Plus[Times[m, Power[2, -1]], 1]], -1]], x], x, Times[b, Tan[Plus[e, Times[f, x]]]]], x], And[FreeQ[List[a, b, e, f, n], x], IntegerQ[Times[m, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Expand[Times[Power[Sin[Plus[e, Times[f, x]]], m], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], n]], x], x], And[FreeQ[List[a, b, e, f], x], IntegerQ[Times[Plus[m, -1], Power[2, -1]]], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Sin[Plus[e, Times[f, x]]], m], Power[Plus[Times[a, Cos[Plus[e, Times[f, x]]]], Times[b, Sin[Plus[e, Times[f, x]]]]], n], Power[Power[Cos[Plus[e, Times[f, x]]], n], -1]], x], And[FreeQ[List[a, b, e, f], x], IntegerQ[Times[Plus[m, -1], Power[2, -1]]], NegativeIntegerQ[n], Or[And[Less[m, 5], Greater[n, -4]], And[Equal[m, 5], Equal[n, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[d, Csc[Plus[e, Times[f, x]]]], FracPart[m]], Power[Times[Sin[Plus[e, Times[f, x]]], Power[d, -1]], FracPart[m]]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], n], Power[Power[Times[Sin[Plus[e, Times[f, x]]], Power[d, -1]], m], -1]], x], x], And[FreeQ[List[a, b, d, e, f, m, n], x], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Cos[Plus[e, Times[f, x]]], Plus[m, Times[-1, n]]], Power[Sin[Plus[e, Times[f, x]]], p], Power[Plus[Times[a, Cos[Plus[e, Times[f, x]]]], Times[b, Sin[Plus[e, Times[f, x]]]]], n]], x], And[FreeQ[List[a, b, e, f, m, p], x], IntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[sin[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Sin[Plus[e, Times[f, x]]], Plus[m, Times[-1, n]]], Power[Cos[Plus[e, Times[f, x]]], p], Power[Plus[Times[a, Sin[Plus[e, Times[f, x]]]], Times[b, Cos[Plus[e, Times[f, x]]]]], n]], x], And[FreeQ[List[a, b, e, f, m, p], x], IntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[a, m], Power[c, m]], Int[Times[Power[Sec[Plus[e, Times[f, x]]], Times[2, m]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, Times[-1, m]]]], x], x], And[FreeQ[List[a, b, c, d, e, f, n], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], IntegerQ[m], Not[And[IntegerQ[n], Greater[n, 0], Or[Less[m, 0], Less[n, m]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[a, c, Power[f, -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], Plus[m, -1]], Power[Plus[c, Times[d, x]], Plus[n, -1]]], x], x, Tan[Plus[e, Times[f, x]]]], x], And[FreeQ[List[a, b, c, d, e, f, m, n], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[a, c], Times[-1, b, d]], x], x], Simp[Times[b, d, Tan[Plus[e, Times[f, x]]], Power[f, -1]], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Times[b, c], Times[a, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[a, c], Times[-1, b, d]], x], x], Plus[Dist[Plus[Times[b, c], Times[a, d]], Int[Tan[Plus[e, Times[f, x]]], x], x], Simp[Times[b, d, Tan[Plus[e, Times[f, x]]], Power[f, -1]], x]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Times[b, c], Times[a, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[b, c], Times[-1, a, d]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Times[2, a, f, m], -1]], x]], Dist[Times[Plus[Times[b, c], Times[a, d]], Power[Times[2, a, b], -1]], Int[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], RationalQ[m], Less[m, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[d, Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Times[f, m], -1]], x], Dist[Times[Plus[Times[b, c], Times[a, d]], Power[b, -1]], Int[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], x], x]], And[FreeQ[List[a, b, c, d, e, f, m], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], Not[And[RationalQ[m], Less[m, 0]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[d, Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Times[f, m], -1]], x], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, -1]], Simp[Plus[Times[a, c], Times[-1, b, d], Times[Plus[Times[b, c], Times[a, d]], Tan[Plus[e, Times[f, x]]]]], x]], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], RationalQ[m], Greater[m, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[b, c], Times[-1, a, d]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[f, Plus[m, 1], Plus[Power[a, 2], Power[b, 2]]], -1]], x], Dist[Times[1, Power[Plus[Power[a, 2], Power[b, 2]], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Simp[Plus[Times[a, c], Times[b, d], Times[-1, Plus[Times[b, c], Times[-1, a, d]], Tan[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], RationalQ[m], Less[m, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[c, Power[Times[b, f], -1], Log[RemoveContent[Plus[Times[a, Cos[Plus[e, Times[f, x]]]], Times[b, Sin[Plus[e, Times[f, x]]]]], x]]], x], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], ZeroQ[Plus[Times[a, c], Times[b, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[a, c], Times[b, d]], x, Power[Plus[Power[a, 2], Power[b, 2]], -1]], x], Dist[Times[Plus[Times[b, c], Times[-1, a, d]], Power[Plus[Power[a, 2], Power[b, 2]], -1]], Int[Times[Plus[b, Times[-1, a, Tan[Plus[e, Times[f, x]]]]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Times[a, c], Times[b, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[-2, Power[d, 2], Power[f, -1]], Subst[Int[Times[1, Power[Plus[Times[2, c, d], Times[b, Power[x, 2]]], -1]], x], x, Times[Plus[c, Times[-1, d, Tan[Plus[e, Times[f, x]]]]], Power[Sqrt[Times[b, Tan[Plus[e, Times[f, x]]]]], -1]]], x], And[FreeQ[List[b, c, d, e, f], x], ZeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[2, Power[c, 2], Power[f, -1]], Subst[Int[Times[1, Power[Plus[Times[b, c], Times[-1, d, Power[x, 2]]], -1]], x], x, Sqrt[Times[b, Tan[Plus[e, Times[f, x]]]]]], x], And[FreeQ[List[b, c, d, e, f], x], ZeroQ[Plus[Power[c, 2], Power[d, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Rational[-1, 2]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[2, Power[f, -1]], Subst[Int[Times[Plus[Times[b, c], Times[d, Power[x, 2]]], Power[Plus[Power[b, 2], Power[x, 4]], -1]], x], x, Sqrt[Times[b, Tan[Plus[e, Times[f, x]]]]]], x], And[FreeQ[List[b, c, d, e, f], x], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[-2, Power[d, 2], Power[f, -1]], Subst[Int[Times[1, Power[Plus[Times[2, b, c, d], Times[-1, 4, a, Power[d, 2]], Power[x, 2]], -1]], x], x, Times[Plus[Times[b, c], Times[-1, 2, a, d], Times[-1, b, d, Tan[Plus[e, Times[f, x]]]]], Power[Sqrt[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]]], -1]]], x], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], ZeroQ[Plus[Times[2, a, c, d], Times[-1, b, Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[q, Rt[Plus[Power[a, 2], Power[b, 2]], 2]]], Plus[Dist[Times[1, Power[Times[2, q], -1]], Int[Times[Plus[Times[a, c], Times[b, d], Times[c, q], Times[Plus[Times[b, c], Times[-1, a, d], Times[d, q]], Tan[Plus[e, Times[f, x]]]]], Power[Sqrt[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]]], -1]], x], x], Times[-1, Dist[Times[1, Power[Times[2, q], -1]], Int[Times[Plus[Times[a, c], Times[b, d], Times[-1, c, q], Times[Plus[Times[b, c], Times[-1, a, d], Times[-1, d, q]], Tan[Plus[e, Times[f, x]]]]], Power[Sqrt[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], NonzeroQ[Plus[Times[2, a, c, d], Times[-1, b, Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]], Or[PerfectSquareQ[Plus[Power[a, 2], Power[b, 2]]], RationalQ[a, b, c, d]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[c, d, Power[f, -1]], Subst[Int[Times[Power[Plus[a, Times[b, Power[d, -1], x]], m], Power[Plus[Power[d, 2], Times[c, x]], -1]], x], x, Times[d, Tan[Plus[e, Times[f, x]]]]], x], And[FreeQ[List[a, b, c, d, e, f, m], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], ZeroQ[Plus[Power[c, 2], Power[d, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[c, Int[Power[Times[b, Tan[Plus[e, Times[f, x]]]], m], x], x], Dist[Times[d, Power[b, -1]], Int[Power[Times[b, Tan[Plus[e, Times[f, x]]]], Plus[m, 1]], x], x]], And[FreeQ[List[b, c, d, e, f, m], x], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], Not[IntegerQ[Times[2, m]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Plus[c, Times[I, d]], Power[2, -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Plus[1, Times[-1, I, Tan[Plus[e, Times[f, x]]]]]], x], x], Dist[Times[Plus[c, Times[-1, I, d]], Power[2, -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Plus[1, Times[I, Tan[Plus[e, Times[f, x]]]]]], x], x]], And[FreeQ[List[a, b, c, d, e, f, m], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], 2]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[b, Power[Plus[Times[a, c], Times[b, d]], 2], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Times[2, Power[a, 3], f, m], -1]], x]], Dist[Times[1, Power[Times[2, Power[a, 2]], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Simp[Plus[Times[a, Power[c, 2]], Times[-1, 2, b, c, d], Times[a, Power[d, 2]], Times[-1, 2, b, Power[d, 2], Tan[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[m], LessEqual[m, -1], ZeroQ[Plus[Power[a, 2], Power[b, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], 2]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[d, Plus[Times[2, b, c], Times[-1, a, d]], x, Power[Power[b, 2], -1]], x], Plus[Dist[Times[Power[d, 2], Power[b, -1]], Int[Tan[Plus[e, Times[f, x]]], x], x], Dist[Times[Power[Plus[Times[b, c], Times[-1, a, d]], 2], Power[Power[b, 2], -1]], Int[Times[1, Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], 2]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[Times[b, c], Times[-1, a, d]], 2], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[b, f, Plus[m, 1], Plus[Power[a, 2], Power[b, 2]]], -1]], x], Dist[Times[1, Power[Plus[Power[a, 2], Power[b, 2]], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Simp[Plus[Times[a, Power[c, 2]], Times[2, b, c, d], Times[-1, a, Power[d, 2]], Times[-1, Plus[Times[b, Power[c, 2]], Times[-1, 2, a, c, d], Times[-1, b, Power[d, 2]]], Tan[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[m], Less[m, -1], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], 2]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[d, 2], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[b, f, Plus[m, 1]], -1]], x], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Simp[Plus[Power[c, 2], Times[-1, Power[d, 2]], Times[2, c, d, Tan[Plus[e, Times[f, x]]]]], x]], x]], And[FreeQ[List[a, b, c, d, e, f, m], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], Not[And[RationalQ[m], LessEqual[m, -1]]], Not[And[ZeroQ[Plus[m, -2]], ZeroQ[a]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[-2, a, b, Power[f, -1]], Subst[Int[Times[1, Power[Plus[Times[a, c], Times[-1, b, d], Times[-1, 2, Power[a, 2], Power[x, 2]]], -1]], x], x, Times[Sqrt[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]]], Power[Sqrt[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]]], -1]]], x], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[a, b, Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[f, Plus[m, -1], Plus[Times[a, c], Times[-1, b, d]]], -1]], x], Dist[Times[2, Power[a, 2], Power[Plus[Times[a, c], Times[-1, b, d]], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], RationalQ[m, n], Equal[Plus[m, n], 0], Greater[m, Times[1, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[a, Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], n], Power[Times[2, b, f, m], -1]], x], Times[-1, Dist[Times[Plus[Times[a, c], Times[-1, b, d]], Power[Times[2, Power[b, 2]], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, -1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], RationalQ[m, n], Equal[Plus[m, n], 0], LessEqual[m, Times[-1, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[a, Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[2, f, m, Plus[Times[b, c], Times[-1, a, d]]], -1]], x], Dist[Times[1, Power[Times[2, a], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], n]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], RationalQ[m, n], Equal[Plus[m, n, 1], 0], Less[m, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[d, Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[f, m, Plus[Power[c, 2], Power[d, 2]]], -1]], x]], Dist[Times[a, Power[Plus[Times[a, c], Times[-1, b, d]], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]]], x], x]], And[FreeQ[List[a, b, c, d, e, f, m, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], ZeroQ[Plus[m, n, 1]], Not[And[RationalQ[m], Less[m, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[a, c], Times[b, d]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], n], Power[Times[2, Plus[Times[b, c], Times[-1, a, d]], f, Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]]], -1]], x]], Dist[Times[1, Power[Times[2, a, Plus[Times[b, c], Times[-1, a, d]]], -1]], Int[Times[Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, -1]], Simp[Plus[Times[a, c, d, Plus[n, -1]], Times[b, Power[c, 2]], Times[b, Power[d, 2], n], Times[-1, d, Plus[Times[b, c], Times[-1, a, d]], Plus[n, -1], Tan[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], RationalQ[n], Less[0, n, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[b, c], Times[-1, a, d]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, -1]], Power[Times[2, a, f, Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]]], -1]], x], Dist[Times[1, Power[Times[2, Power[a, 2]], -1]], Int[Times[Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, -2]], Simp[Plus[Times[a, Power[c, 2]], Times[a, Power[d, 2], Plus[n, -1]], Times[-1, b, c, d, n], Times[-1, d, Plus[Times[a, c, Plus[n, -2]], Times[b, d, n]], Tan[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], RationalQ[n], Greater[n, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[b, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Int[Times[1, Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], -1]], x], x], Times[-1, Dist[Times[d, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Int[Times[1, Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[a, Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[2, f, Plus[Times[b, c], Times[-1, a, d]], Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]]], -1]], x]], Dist[Times[1, Power[Times[2, a, Plus[Times[b, c], Times[-1, a, d]]], -1]], Int[Times[Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], n], Simp[Plus[Times[b, c], Times[a, d, Plus[n, -1]], Times[-1, b, d, n, Tan[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], Not[And[RationalQ[n], Greater[n, 0]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[a, 2], Plus[Times[b, c], Times[-1, a, d]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, -2]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[d, f, Plus[Times[b, c], Times[a, d]], Plus[n, 1]], -1]], x]], Dist[Times[a, Power[Times[d, Plus[Times[b, c], Times[a, d]], Plus[n, 1]], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, -2]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Simp[Plus[Times[b, Plus[Times[b, c, Plus[m, -2]], Times[-1, a, d, Plus[m, Times[-1, 2, n], -4]]]], Times[Plus[Times[a, b, c, Plus[m, -2]], Times[Power[b, 2], d, Plus[n, 1]], Times[-1, Power[a, 2], d, Plus[m, n, -1]]], Tan[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], RationalQ[m, n], Greater[m, 1], Less[n, -1], Or[IntegerQ[m], IntegersQ[Times[2, m], Times[2, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[3, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[2, Power[a, 2], Power[Plus[Times[a, c], Times[-1, b, d]], -1]], Int[Sqrt[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]]], x], x], Times[-1, Dist[Times[Plus[Times[2, b, c, d], Times[a, Plus[Power[c, 2], Times[-1, Power[d, 2]]]]], Power[Times[a, Plus[Power[c, 2], Power[d, 2]]], -1]], Int[Times[Plus[a, Times[-1, b, Tan[Plus[e, Times[f, x]]]]], Sqrt[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[3, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[2, a], Int[Times[Sqrt[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]]], Power[Sqrt[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]]], -1]], x], x], Dist[Times[b, Power[a, -1]], Int[Times[Plus[b, Times[a, Tan[Plus[e, Times[f, x]]]]], Sqrt[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]]], Power[Sqrt[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[b, 2], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, -2]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[d, f, Plus[m, n, -1]], -1]], x], Dist[Times[a, Power[Times[d, Plus[m, n, -1]], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, -2]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], n], Simp[Plus[Times[b, c, Plus[m, -2]], Times[a, d, Plus[m, Times[2, n]]], Times[Plus[Times[a, c, Plus[m, -2]], Times[b, d, Plus[Times[3, m], Times[2, n], -4]]], Tan[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], IntegerQ[Times[2, m]], Greater[m, 1], NonzeroQ[Plus[m, n, -1]], Or[IntegerQ[m], IntegersQ[Times[2, m], Times[2, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[b, Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Sqrt[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]]], Power[Times[2, a, f, m], -1]], x]], Dist[Times[1, Power[Times[4, Power[a, 2], m], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Simp[Plus[Times[2, a, c, m], Times[b, d], Times[a, d, Plus[Times[2, m], 1], Tan[Plus[e, Times[f, x]]]]], x], Power[Sqrt[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], RationalQ[m], Less[m, 0], IntegersQ[Times[2, m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[b, c], Times[-1, a, d]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, -1]], Power[Times[2, a, f, m], -1]], x]], Dist[Times[1, Power[Times[2, Power[a, 2], m], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, -2]], Simp[Plus[Times[c, Plus[Times[a, c, m], Times[b, d, Plus[n, -1]]]], Times[-1, d, Plus[Times[b, c, m], Times[a, d, Plus[n, -1]]]], Times[-1, d, Plus[Times[b, d, Plus[m, Times[-1, n], 1]], Times[-1, a, c, Plus[m, n, -1]]], Tan[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], RationalQ[m, n], Less[m, 0], Greater[n, 1], Or[IntegerQ[m], IntegersQ[Times[2, m], Times[2, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[a, Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[2, f, m, Plus[Times[b, c], Times[-1, a, d]]], -1]], x], Dist[Times[1, Power[Times[2, a, m, Plus[Times[b, c], Times[-1, a, d]]], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], n], Simp[Plus[Times[b, c, m], Times[-1, a, d, Plus[Times[2, m], n, 1]], Times[b, d, Plus[m, n, 1], Tan[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], RationalQ[m], Less[m, 0], Or[IntegerQ[m], IntegersQ[Times[2, m], Times[2, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[d, Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, -1]], Power[Times[f, Plus[m, n, -1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[a, Plus[m, n, -1]], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, -2]], Simp[Plus[Times[d, Plus[Times[b, c, m], Times[a, d, Plus[-1, n]]]], Times[-1, a, Power[c, 2], Plus[m, n, -1]], Times[d, Plus[Times[b, d, m], Times[-1, a, c, Plus[m, Times[2, n], -2]]], Tan[Plus[e, Times[f, x]]]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, m], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], RationalQ[n], Greater[n, 1], NonzeroQ[Plus[m, n, -1]], Or[IntegerQ[n], IntegersQ[Times[2, m], Times[2, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[d, Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[f, Plus[n, 1], Plus[Power[c, 2], Power[d, 2]]], -1]], x], Times[-1, Dist[Times[1, Power[Times[a, Plus[Power[c, 2], Power[d, 2]], Plus[n, 1]], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Simp[Plus[Times[b, d, m], Times[-1, a, c, Plus[n, 1]], Times[a, d, Plus[m, n, 1], Tan[Plus[e, Times[f, x]]]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, m], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], RationalQ[n], Less[n, -1], Or[IntegerQ[n], IntegersQ[Times[2, m], Times[2, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[a, Power[Plus[Times[a, c], Times[-1, b, d]], -1]], Int[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], x], x], Times[-1, Dist[Times[d, Power[Plus[Times[a, c], Times[-1, b, d]], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Plus[b, Times[a, Tan[Plus[e, Times[f, x]]]]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, m], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Plus[Times[a, c], Times[-1, b, d]], Power[a, -1]], Int[Times[Sqrt[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]]], Power[Sqrt[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]]], -1]], x], x], Dist[Times[d, Power[a, -1]], Int[Times[Sqrt[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]]], Plus[b, Times[a, Tan[Plus[e, Times[f, x]]]]], Power[Sqrt[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[a, b, Power[f, -1]], Subst[Int[Times[Power[Plus[a, x], Plus[m, -1]], Power[Plus[c, Times[d, Power[b, -1], x]], n], Power[Plus[Power[b, 2], Times[a, x]], -1]], x], x, Times[b, Tan[Plus[e, Times[f, x]]]]], x], And[FreeQ[List[a, b, c, d, e, f, m, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[Times[b, c], Times[-1, a, d]], 2], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, -2]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[d, f, Plus[n, 1], Plus[Power[c, 2], Power[d, 2]]], -1]], x], Times[-1, Dist[Times[1, Power[Times[d, Plus[n, 1], Plus[Power[c, 2], Power[d, 2]]], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, -3]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Simp[Plus[Times[Power[a, 2], d, Plus[Times[b, d, Plus[m, -2]], Times[-1, a, c, Plus[n, 1]]]], Times[b, Plus[Times[b, c], Times[-1, 2, a, d]], Plus[Times[b, c, Plus[m, -2]], Times[a, d, Plus[n, 1]]]], Times[-1, d, Plus[n, 1], Plus[Times[3, Power[a, 2], b, c], Times[-1, Power[b, 3], c], Times[-1, Power[a, 3], d], Times[3, a, Power[b, 2], d]], Tan[Plus[e, Times[f, x]]]], Times[-1, b, Plus[Times[a, d, Plus[Times[2, b, c], Times[-1, a, d]], Plus[m, n, -1]], Times[-1, Power[b, 2], Plus[Times[Power[c, 2], Plus[m, -2]], Times[-1, Power[d, 2], Plus[n, 1]]]]], Power[Tan[Plus[e, Times[f, x]]], 2]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], RationalQ[m, n], Greater[m, 2], Less[n, -1], IntegerQ[Times[2, m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[b, 2], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, -2]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[d, f, Plus[m, n, -1]], -1]], x], Dist[Times[1, Power[Times[d, Plus[m, n, -1]], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, -3]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], n], Simp[Plus[Times[Power[a, 3], d, Plus[m, n, -1]], Times[-1, Power[b, 2], Plus[Times[b, c, Plus[m, -2]], Times[a, d, Plus[1, n]]]], Times[b, d, Plus[m, n, -1], Plus[Times[3, Power[a, 2]], Times[-1, Power[b, 2]]], Tan[Plus[e, Times[f, x]]]], Times[-1, Power[b, 2], Plus[Times[b, c, Plus[m, -2]], Times[-1, a, d, Plus[Times[3, m], Times[2, n], -4]]], Power[Tan[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], IntegerQ[Times[2, m]], Greater[m, 2], Or[And[RationalQ[n], GreaterEqual[n, -1]], IntegerQ[m]], Not[And[IntegerQ[n], Greater[n, 2], Or[Not[IntegerQ[m]], And[ZeroQ[c], NonzeroQ[a]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[b, c], Times[-1, a, d]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, -1]], Power[Times[f, Plus[m, 1], Plus[Power[a, 2], Power[b, 2]]], -1]], x], Dist[Times[1, Power[Times[Plus[m, 1], Plus[Power[a, 2], Power[b, 2]]], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, -2]], Simp[Plus[Times[a, Power[c, 2], Plus[m, 1]], Times[a, Power[d, 2], Plus[n, -1]], Times[b, c, d, Plus[m, Times[-1, n], 2]], Times[-1, Plus[Times[b, Power[c, 2]], Times[-1, 2, a, c, d], Times[-1, b, Power[d, 2]]], Plus[m, 1], Tan[Plus[e, Times[f, x]]]], Times[-1, d, Plus[Times[b, c], Times[-1, a, d]], Plus[m, n], Power[Tan[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], RationalQ[m, n], Less[m, -1], Less[1, n, 2], IntegerQ[Times[2, m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], n], Power[Times[f, Plus[m, 1], Plus[Power[a, 2], Power[b, 2]]], -1]], x], Dist[Times[1, Power[Times[Plus[m, 1], Plus[Power[a, 2], Power[b, 2]]], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, -1]], Simp[Plus[Times[a, c, Plus[m, 1]], Times[-1, b, d, n], Times[-1, Plus[Times[b, c], Times[-1, a, d]], Plus[m, 1], Tan[Plus[e, Times[f, x]]]], Times[-1, b, d, Plus[m, n, 1], Power[Tan[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], RationalQ[m, n], Less[m, -1], Greater[n, 0], IntegerQ[Times[2, m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[b, 2], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[f, Plus[m, 1], Plus[Power[a, 2], Power[b, 2]], Plus[Times[b, c], Times[-1, a, d]]], -1]], x], Dist[Times[1, Power[Times[Plus[m, 1], Plus[Power[a, 2], Power[b, 2]], Plus[Times[b, c], Times[-1, a, d]]], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], n], Simp[Plus[Times[a, Plus[Times[b, c], Times[-1, a, d]], Plus[m, 1]], Times[-1, Power[b, 2], d, Plus[m, n, 2]], Times[-1, b, Plus[Times[b, c], Times[-1, a, d]], Plus[m, 1], Tan[Plus[e, Times[f, x]]]], Times[-1, Power[b, 2], d, Plus[m, n, 2], Power[Tan[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], IntegerQ[Times[2, m]], Less[m, -1], Or[And[RationalQ[n], Less[n, 0]], IntegerQ[m]], Not[And[IntegerQ[n], Less[n, -1], Or[Not[IntegerQ[m]], And[ZeroQ[c], NonzeroQ[a]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], 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2], Power[d, 2]]], RationalQ[m, n], Greater[m, 1], Greater[n, 0], IntegerQ[Times[2, n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[a, c], Times[-1, b, d]], x, Power[Times[Plus[Power[a, 2], Power[b, 2]], Plus[Power[c, 2], Power[d, 2]]], -1]], x], Plus[Dist[Times[Power[b, 2], Power[Times[Plus[Times[b, c], Times[-1, a, d]], Plus[Power[a, 2], Power[b, 2]]], -1]], Int[Times[Plus[b, Times[-1, a, Tan[Plus[e, Times[f, x]]]]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], -1]], x], x], Times[-1, Dist[Times[Power[d, 2], Power[Times[Plus[Times[b, c], Times[-1, a, d]], 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Power[Sqrt[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]]], -1]], x], x], Times[-1, Dist[Times[d, Plus[Times[b, c], Times[-1, a, d]], Power[Plus[Power[c, 2], Power[d, 2]], -1]], Int[Times[Plus[1, Power[Tan[Plus[e, Times[f, x]]], 2]], Power[Times[Sqrt[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]]], Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[3, 2]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[Plus[Power[c, 2], Power[d, 2]], -1]], Int[Times[Simp[Plus[Times[Power[a, 2], c], Times[-1, Power[b, 2], c], Times[2, a, b, d], Times[Plus[Times[2, a, b, c], Times[-1, Power[a, 2], d], Times[Power[b, 2], d]], Tan[Plus[e, Times[f, x]]]]], x], Power[Sqrt[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]]], -1]], x], x], Dist[Times[Power[Plus[Times[b, c], Times[-1, a, d]], 2], Power[Plus[Power[c, 2], Power[d, 2]], -1]], Int[Times[Plus[1, Power[Tan[Plus[e, Times[f, x]]], 2]], Power[Times[Sqrt[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]]], Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], 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Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[b, Power[f, -1]], Subst[Int[Times[Power[Plus[a, x], m], Power[Plus[c, Times[d, x, Power[b, -1]]], n], Power[Plus[Power[b, 2], Power[x, 2]], -1]], x], x, Times[b, Tan[Plus[e, Times[f, x]]]]], x], And[FreeQ[List[a, b, c, d, e, f, m, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[d, m], Int[Times[Power[Plus[b, Times[a, Cot[Plus[e, Times[f, x]]]]], m], Power[Times[d, Cot[Plus[e, Times[f, x]]]], Plus[n, Times[-1, m]]]], x], x], And[FreeQ[List[a, b, d, e, f, n], x], Not[IntegerQ[n]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Power[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[d, m], Int[Times[Power[Plus[b, Times[a, Tan[Plus[e, Times[f, x]]]]], m], Power[Times[d, Tan[Plus[e, Times[f, x]]]], Plus[n, Times[-1, m]]]], x], x], And[FreeQ[List[a, b, d, e, f, n], x], Not[IntegerQ[n]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[c, Blank[]]], Power[Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[c, IntPart[n]], Power[Times[c, Power[Times[d, Tan[Plus[e, Times[f, x]]]], p]], FracPart[n]], Power[Power[Times[d, Tan[Plus[e, Times[f, x]]]], Times[p, FracPart[n]]], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Times[d, Tan[Plus[e, 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m, n, p], x], Not[IntegerQ[n]], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Times[g, Tan[Plus[e, Times[f, x]]]], p], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], n]], x], FreeQ[List[a, b, c, d, e, f, g, m, n, p], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[g, Plus[m, n]], Int[Times[Power[Times[g, Cot[Plus[e, Times[f, x]]]], Plus[p, Times[-1, m], Times[-1, n]]], Power[Plus[b, Times[a, Cot[Plus[e, Times[f, x]]]]], m], Power[Plus[d, Times[c, Cot[Plus[e, Times[f, x]]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, p], x], Not[IntegerQ[p]], IntegerQ[m], IntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Power[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Times[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[g, Plus[m, n]], Int[Times[Power[Times[g, Tan[Plus[e, Times[f, x]]]], Plus[p, Times[-1, m], Times[-1, n]]], Power[Plus[b, Times[a, Tan[Plus[e, Times[f, x]]]]], m], Power[Plus[d, Times[c, Tan[Plus[e, Times[f, x]]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, p], x], Not[IntegerQ[p]], IntegerQ[m], IntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[g, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[q, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[g, Power[Tan[Plus[e, Times[f, x]]], q]], p], Power[Power[Times[g, Tan[Plus[e, Times[f, x]]]], Times[p, q]], -1]], Int[Times[Power[Times[g, Tan[Plus[e, Times[f, x]]]], Times[p, q]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, m, n, p, q], x], Not[IntegerQ[p]], Not[And[IntegerQ[m], IntegerQ[n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Power[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Pattern[q, Blank[]]], Optional[Pattern[g, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Times[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[g, Power[Cot[Plus[e, Times[f, x]]], q]], p], Power[Power[Times[g, Cot[Plus[e, Times[f, x]]]], Times[p, q]], -1]], Int[Times[Power[Times[g, Cot[Plus[e, Times[f, x]]]], Times[p, q]], Power[Plus[a, Times[b, Cot[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Cot[Plus[e, Times[f, x]]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, m, n, p, q], x], Not[IntegerQ[p]], Not[And[IntegerQ[m], IntegerQ[n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[g, n], Int[Times[Power[Times[g, Tan[Plus[e, Times[f, x]]]], Plus[p, Times[-1, n]]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Plus[d, Times[c, Tan[Plus[e, Times[f, x]]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, m, p], x], IntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[b, Times[a, Cot[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Cot[Plus[e, Times[f, x]]]]], n], Power[Power[Cot[Plus[e, Times[f, x]]], Plus[m, p]], -1]], x], And[FreeQ[List[a, b, c, d, e, f, n], x], Not[IntegerQ[n]], IntegerQ[m], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Cot[Plus[e, Times[f, x]]], p], Power[Times[g, Tan[Plus[e, Times[f, x]]]], p]], Int[Times[Power[Plus[b, Times[a, Cot[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Cot[Plus[e, Times[f, x]]]]], n], Power[Power[Cot[Plus[e, Times[f, x]]], Plus[m, p]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, n, p], x], Not[IntegerQ[n]], IntegerQ[m], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[g, Tan[Plus[e, Times[f, x]]]], n], Power[Plus[c, Times[d, Cot[Plus[e, Times[f, x]]]]], n], Power[Power[Plus[d, Times[c, Tan[Plus[e, Times[f, x]]]]], n], -1]], Int[Times[Power[Times[g, Tan[Plus[e, Times[f, x]]]], Plus[p, Times[-1, n]]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Plus[d, Times[c, Tan[Plus[e, Times[f, x]]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, m, n, p], x], Not[IntegerQ[n]], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[a, c, Power[f, -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], Plus[m, -1]], Power[Plus[c, Times[d, x]], Plus[n, -1]], Plus[A, Times[B, x]]], x], x, Tan[Plus[e, Times[f, x]]]], x], And[FreeQ[List[a, b, c, d, e, f, A, B, m, n], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[B, d, Power[b, -1]], Int[Tan[Plus[e, Times[f, x]]], x], x], Dist[Times[1, Power[b, -1]], Int[Times[Simp[Plus[Times[A, b, c], Times[Plus[Times[A, b, d], Times[B, Plus[Times[b, c], Times[-1, a, d]]]], Tan[Plus[e, Times[f, x]]]]], x], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[A, b], Times[-1, a, B]], Plus[Times[a, c], Times[b, d]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Times[2, Power[a, 2], f, m], -1]], x]], Dist[Times[1, Power[Times[2, a, b], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Simp[Plus[Times[A, b, c], Times[a, B, c], Times[a, A, d], Times[b, B, d], Times[2, a, B, d, Tan[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[m], Less[m, -1], ZeroQ[Plus[Power[a, 2], Power[b, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[b, c], Times[-1, a, d]], Plus[Times[A, b], Times[-1, a, B]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[b, f, Plus[m, 1], Plus[Power[a, 2], Power[b, 2]]], -1]], x], Dist[Times[1, Power[Plus[Power[a, 2], Power[b, 2]], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Simp[Plus[Times[a, A, c], Times[b, B, c], Times[A, b, d], Times[-1, a, B, d], Times[-1, Plus[Times[A, b, c], Times[-1, a, B, c], Times[-1, a, A, d], Times[-1, b, B, d]], Tan[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[m], Less[m, -1], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[B, d, Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[b, f, Plus[m, 1]], -1]], x], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Simp[Plus[Times[A, c], Times[-1, B, d], Times[Plus[Times[B, c], Times[A, d]], Tan[Plus[e, Times[f, x]]]]], x]], x]], And[FreeQ[List[a, b, c, d, e, f, A, B, m], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], Not[And[RationalQ[m], LessEqual[m, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[a, 2], Plus[Times[B, c], Times[-1, A, d]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[d, f, Plus[Times[b, c], Times[a, d]], Plus[n, 1]], -1]], x]], Times[-1, Dist[Times[a, Power[Times[d, Plus[Times[b, c], Times[a, d]], Plus[n, 1]], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Simp[Plus[Times[A, b, d, Plus[m, Times[-1, n], -2]], Times[-1, B, Plus[Times[b, c, Plus[m, -1]], Times[a, d, Plus[n, 1]]]], Times[Plus[Times[a, A, d, Plus[m, n]], Times[-1, B, Plus[Times[a, c, Plus[m, -1]], Times[b, d, Plus[n, 1]]]]], Tan[Plus[e, Times[f, x]]]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, A, B], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], RationalQ[m, n], Greater[m, 1], Less[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, B, Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[d, f, Plus[m, n]], -1]], x], Dist[Times[1, Power[Times[d, Plus[m, n]], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], n], Simp[Plus[Times[a, A, d, Plus[m, n]], Times[B, Plus[Times[a, c, Plus[m, -1]], Times[-1, b, d, Plus[n, 1]]]], Times[-1, Plus[Times[B, Plus[Times[b, c], Times[-1, a, d]], Plus[m, -1]], Times[-1, d, Plus[Times[A, b], Times[a, B]], Plus[m, n]]], Tan[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], RationalQ[m], Greater[m, 1], Not[And[RationalQ[n], Less[n, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[A, b], Times[-1, a, B]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], n], Power[Times[2, a, f, m], -1]], x]], Dist[Times[1, Power[Times[2, Power[a, 2], m], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, 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RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[b, B, Power[f, -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], Plus[m, -1]], Power[Plus[c, Times[d, x]], n]], x], x, Tan[Plus[e, Times[f, x]]]], x], And[FreeQ[List[a, b, c, d, e, f, A, B, m, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], ZeroQ[Plus[Times[A, b], Times[a, B]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Plus[Times[A, b], Times[a, B]], Power[Plus[Times[b, c], Times[a, d]], -1]], Int[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], x], x], Times[-1, Dist[Times[Plus[Times[B, c], Times[-1, A, d]], Power[Plus[Times[b, c], Times[a, d]], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Plus[a, Times[-1, b, Tan[Plus[e, Times[f, 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2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[b, c], Times[-1, a, d]], Plus[Times[B, c], Times[-1, A, d]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[d, f, Plus[n, 1], Plus[Power[c, 2], Power[d, 2]]], -1]], x], Times[-1, Dist[Times[1, Power[Times[d, 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Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, -1]], Simp[Plus[Times[b, B, Plus[Times[b, c, Plus[m, 1]], Times[a, d, n]]], Times[A, b, Plus[Times[a, c, Plus[m, 1]], Times[-1, b, d, n]]], Times[-1, b, Plus[Times[A, Plus[Times[b, c], Times[-1, a, d]]], Times[-1, B, Plus[Times[a, c], Times[b, d]]]], Plus[m, 1], Tan[Plus[e, Times[f, x]]]], Times[-1, b, d, Plus[Times[A, b], Times[-1, a, B]], Plus[m, n, 1], Power[Tan[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], RationalQ[m, n], Less[m, -1], Less[0, n, 1], Or[IntegerQ[m], IntegersQ[Times[2, m], Times[2, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, 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Times[f, x]]]]], n], Simp[Plus[Times[b, B, Plus[Times[b, c, Plus[m, 1]], Times[a, d, Plus[n, 1]]]], Times[A, Plus[Times[a, Plus[Times[b, c], Times[-1, a, d]], Plus[m, 1]], Times[-1, Power[b, 2], d, Plus[m, n, 2]]]], Times[-1, Plus[Times[A, b], Times[-1, a, B]], Plus[Times[b, c], Times[-1, a, d]], Plus[m, 1], Tan[Plus[e, Times[f, x]]]], Times[-1, b, d, Plus[Times[A, b], Times[-1, a, B]], Plus[m, n, 2], Power[Tan[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], RationalQ[m], Less[m, -1], Or[IntegerQ[m], IntegersQ[Times[2, m], Times[2, n]]], Not[And[IntegerQ[n], Less[n, -1], Or[Not[IntegerQ[m]], And[ZeroQ[c], NonzeroQ[a]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[B, Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], n], Power[Times[f, Plus[m, n]], -1]], x], Dist[Times[1, Power[Plus[m, n], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, -1]], Simp[Plus[Times[a, A, c, Plus[m, n]], Times[-1, B, Plus[Times[b, c, m], Times[a, d, n]]], Times[Plus[Times[A, b, c], Times[a, B, c], Times[a, A, d], Times[-1, b, B, d]], Plus[m, n], Tan[Plus[e, Times[f, x]]]], Times[Plus[Times[A, b, d, Plus[m, n]], Times[B, Plus[Times[a, d, m], Times[b, c, n]]]], Power[Tan[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], RationalQ[m, n], Less[0, m, 1], Less[0, n, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[B, Plus[Times[b, c], Times[a, d]]], Times[A, Plus[Times[a, c], Times[-1, b, d]]]], x, Power[Times[Plus[Power[a, 2], Power[b, 2]], Plus[Power[c, 2], Power[d, 2]]], -1]], x], Plus[Dist[Times[b, Plus[Times[A, b], Times[-1, a, B]], Power[Times[Plus[Times[b, c], Times[-1, a, d]], Plus[Power[a, 2], Power[b, 2]]], -1]], Int[Times[Plus[b, Times[-1, a, Tan[Plus[e, Times[f, x]]]]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], -1]], x], x], Dist[Times[d, Plus[Times[B, c], Times[-1, A, d]], Power[Times[Plus[Times[b, c], Times[-1, a, d]], Plus[Power[c, 2], Power[d, 2]]], -1]], Int[Times[Plus[d, Times[-1, c, Tan[Plus[e, Times[f, x]]]]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, A, B], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[Plus[Power[a, 2], Power[b, 2]], -1]], Int[Times[Simp[Plus[Times[A, Plus[Times[a, c], Times[b, d]]], Times[B, Plus[Times[b, c], Times[-1, a, d]]], Times[-1, Plus[Times[A, Plus[Times[b, c], Times[-1, a, d]]], Times[-1, B, Plus[Times[a, c], Times[b, d]]]], Tan[Plus[e, Times[f, x]]]]], x], Power[Sqrt[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]]], -1]], x], x], Times[-1, Dist[Times[Plus[Times[b, c], Times[-1, a, d]], Plus[Times[B, a], Times[-1, A, b]], Power[Plus[Power[a, 2], Power[b, 2]], -1]], Int[Times[Plus[1, Power[Tan[Plus[e, Times[f, x]]], 2]], Power[Times[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Sqrt[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, A, B], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[Plus[Power[a, 2], Power[b, 2]], -1]], Int[Times[Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], n], Simp[Plus[Times[a, A], Times[b, B], Times[-1, Plus[Times[A, b], Times[-1, a, B]], Tan[Plus[e, Times[f, x]]]]], x]], x], x], Dist[Times[b, Plus[Times[A, b], Times[-1, a, B]], Power[Plus[Power[a, 2], Power[b, 2]], -1]], Int[Times[Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], n], Plus[1, Power[Tan[Plus[e, Times[f, x]]], 2]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[1, 2]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Int[Times[Simp[Plus[Times[a, A], Times[-1, b, B], Times[Plus[Times[A, b], Times[a, B]], Tan[Plus[e, Times[f, x]]]]], x], Power[Times[Sqrt[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]]]], -1]], x], Dist[Times[b, B], Int[Times[Plus[1, Power[Tan[Plus[e, Times[f, x]]], 2]], Power[Times[Sqrt[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[A, 2], Power[f, -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], m], Power[Plus[c, Times[d, x]], n], Power[Plus[A, Times[-1, B, x]], -1]], x], x, Tan[Plus[e, Times[f, x]]]], x], And[FreeQ[List[a, b, c, d, e, f, A, B, m, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], ZeroQ[Plus[Power[A, 2], Power[B, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Plus[A, Times[I, B]], Power[2, -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], n], Plus[1, Times[-1, I, Tan[Plus[e, Times[f, x]]]]]], x], x], Dist[Times[Plus[A, Times[-1, I, B]], Power[2, -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], n], Plus[1, Times[I, Tan[Plus[e, Times[f, x]]]]]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B, m, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[A, 2], Power[B, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[A, Power[Times[b, f], -1]], Subst[Int[Power[Plus[a, x], m], x], x, Times[b, Tan[Plus[e, Times[f, x]]]]], x], And[FreeQ[List[a, b, e, f, A, C, m], x], ZeroQ[Plus[A, Times[-1, C]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Plus[Times[Power[cot[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Times[A, Power[Times[b, f], -1]], Subst[Int[Power[Plus[a, x], m], x], x, Times[b, Cot[Plus[e, Times[f, x]]]]], x]], And[FreeQ[List[a, b, e, f, A, C, m], x], ZeroQ[Plus[A, Times[-1, C]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Power[b, 2], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Simp[Plus[Times[b, B], Times[-1, a, C], Times[b, C, Tan[Plus[e, Times[f, x]]]]], x]], x], x], And[FreeQ[List[a, b, e, f, A, B, C, m], x], ZeroQ[Plus[Times[A, Power[b, 2]], Times[-1, a, b, B], Times[Power[a, 2], C]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Times[C, Power[Power[b, 2], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Plus[a, Times[-1, b, Tan[Plus[e, Times[f, x]]]]]], x], x]], And[FreeQ[List[a, b, e, f, A, C, m], x], ZeroQ[Plus[Times[A, Power[b, 2]], Times[Power[a, 2], C]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[a, A], Times[b, B], Times[-1, a, C]], Tan[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Times[2, a, f, m], -1]], x]], Dist[Times[1, Power[Times[2, Power[a, 2], m], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Simp[Plus[Plus[Times[b, B], Times[-1, a, C]], Times[a, A, Plus[Times[2, m], 1]], Times[-1, Plus[Times[b, C, Plus[m, -1]], Times[Plus[Times[A, b], Times[-1, a, B]], Plus[m, 1]]], Tan[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, e, f, A, B, C], x], NonzeroQ[Plus[Times[A, Power[b, 2]], Times[-1, a, b, B], Times[Power[a, 2], C]]], RationalQ[m], LessEqual[m, -1], ZeroQ[Plus[Power[a, 2], Power[b, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[a, A], Times[-1, a, C]], Tan[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Times[2, a, f, m], -1]], x]], Dist[Times[1, Power[Times[2, Power[a, 2], m], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Simp[Plus[Times[-1, a, C], Times[a, A, Plus[Times[2, m], 1]], Times[-1, Plus[Times[b, C, Plus[m, -1]], Times[A, b, Plus[m, 1]]], Tan[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, e, f, A, C], x], NonzeroQ[Plus[Times[A, Power[b, 2]], Times[Power[a, 2], C]]], RationalQ[m], LessEqual[m, -1], ZeroQ[Plus[Power[a, 2], Power[b, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[a, A], Times[b, B], Times[-1, a, C]], x, Power[Plus[Power[a, 2], Power[b, 2]], -1]], x], Dist[Times[Plus[Times[A, Power[b, 2]], Times[-1, a, b, B], Times[Power[a, 2], C]], Power[Plus[Power[a, 2], Power[b, 2]], -1]], Int[Times[Plus[1, Power[Tan[Plus[e, Times[f, x]]], 2]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, e, f, A, B, C], x], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], ZeroQ[Plus[Times[A, b], Times[-1, a, B], Times[-1, b, C]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[B, x], x], Plus[Dist[A, Int[Times[1, Power[Tan[Plus[e, Times[f, x]]], -1]], x], x], Dist[C, Int[Tan[Plus[e, Times[f, x]]], x], x]]], And[FreeQ[List[e, f, A, B, C], x], NonzeroQ[Plus[A, Times[-1, C]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[A, Int[Times[1, Power[Tan[Plus[e, Times[f, x]]], -1]], x], x], Dist[C, Int[Tan[Plus[e, Times[f, x]]], x], x]], And[FreeQ[List[e, f, A, C], x], NonzeroQ[Plus[A, Times[-1, C]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[a, A], Times[b, B], Times[-1, a, C]], x, Power[Plus[Power[a, 2], Power[b, 2]], -1]], x], Plus[Dist[Times[Plus[Times[A, Power[b, 2]], Times[-1, a, b, B], Times[Power[a, 2], C]], Power[Plus[Power[a, 2], Power[b, 2]], -1]], Int[Times[Plus[1, Power[Tan[Plus[e, Times[f, x]]], 2]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], -1]], x], x], Times[-1, Dist[Times[Plus[Times[A, b], Times[-1, a, B], Times[-1, b, C]], Power[Plus[Power[a, 2], Power[b, 2]], -1]], Int[Tan[Plus[e, Times[f, x]]], x], x]]]], And[FreeQ[List[a, b, e, f, A, B, C], x], NonzeroQ[Plus[Times[A, Power[b, 2]], Times[-1, a, b, B], Times[Power[a, 2], C]]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Times[A, b], Times[-1, a, B], Times[-1, b, C]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[a, Plus[A, Times[-1, C]], x, Power[Plus[Power[a, 2], Power[b, 2]], -1]], x], Plus[Dist[Times[Plus[Times[Power[a, 2], C], Times[A, Power[b, 2]]], Power[Plus[Power[a, 2], Power[b, 2]], -1]], Int[Times[Plus[1, Power[Tan[Plus[e, Times[f, x]]], 2]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], -1]], x], x], Times[-1, Dist[Times[b, Plus[A, Times[-1, C]], Power[Plus[Power[a, 2], Power[b, 2]], -1]], Int[Tan[Plus[e, Times[f, x]]], x], x]]]], And[FreeQ[List[a, b, e, f, A, C], x], NonzeroQ[Plus[Times[Power[a, 2], C], Times[A, Power[b, 2]]]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[A, Times[-1, C]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[A, Power[b, 2]], Times[-1, a, b, B], Times[Power[a, 2], C]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[b, f, Plus[m, 1], Plus[Power[a, 2], Power[b, 2]]], -1]], x], Dist[Times[1, Power[Plus[Power[a, 2], Power[b, 2]], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Simp[Plus[Times[b, B], Times[a, Plus[A, Times[-1, C]]], Times[-1, Plus[Times[A, b], Times[-1, a, B], Times[-1, b, C]], Tan[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, e, f, A, B, C], x], NonzeroQ[Plus[Times[A, Power[b, 2]], Times[-1, a, b, B], Times[Power[a, 2], C]]], RationalQ[m], Less[m, -1], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[A, Power[b, 2]], Times[Power[a, 2], C]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[b, f, Plus[m, 1], Plus[Power[a, 2], Power[b, 2]]], -1]], x], Dist[Times[1, Power[Plus[Power[a, 2], Power[b, 2]], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Simp[Plus[Times[a, Plus[A, 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1]], -1]], x], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Simp[Plus[A, Times[-1, C], Times[B, Tan[Plus[e, Times[f, x]]]]], x]], x]], And[FreeQ[List[a, b, e, f, A, B, C, m], x], NonzeroQ[Plus[Times[A, Power[b, 2]], Times[-1, a, b, B], Times[Power[a, 2], C]]], Not[And[RationalQ[m], LessEqual[m, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[C, Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[b, f, Plus[m, 1]], -1]], x], Dist[Plus[A, Times[-1, C]], Int[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], x], x]], And[FreeQ[List[a, b, e, f, A, C, m], x], NonzeroQ[Plus[Times[A, Power[b, 2]], Times[Power[a, 2], C]]], Not[And[RationalQ[m], LessEqual[m, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[A, Power[f, -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], m], Power[Plus[c, Times[d, x]], n]], x], x, Tan[Plus[e, Times[f, x]]]], x], And[FreeQ[List[a, b, c, d, e, f, A, C, m, n], x], ZeroQ[Plus[A, Times[-1, C]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[b, c], Times[-1, a, d]], Plus[Times[Power[c, 2], C], Times[-1, B, c, d], Times[A, Power[d, 2]]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[Power[d, 2], f, Plus[n, 1], Plus[Power[c, 2], Power[d, 2]]], -1]], x]], Dist[Times[1, Power[Times[d, Plus[Power[c, 2], Power[d, 2]]], -1]], Int[Times[Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Simp[Plus[Times[a, d, Plus[Times[A, c], Times[-1, c, C], Times[B, d]]], Times[b, Plus[Times[Power[c, 2], C], Times[-1, B, c, d], Times[A, Power[d, 2]]]], Times[d, Plus[Times[A, b, c], Times[a, B, c], Times[-1, b, c, C], Times[-1, a, A, d], Times[b, B, d], Times[a, C, d]], Tan[Plus[e, Times[f, x]]]], Times[b, C, Plus[Power[c, 2], Power[d, 2]], Power[Tan[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B, C], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], RationalQ[n], Less[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[b, c], Times[-1, a, d]], Plus[Times[Power[c, 2], C], Times[A, Power[d, 2]]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[Power[d, 2], f, Plus[n, 1], Plus[Power[c, 2], Power[d, 2]]], -1]], x]], Dist[Times[1, Power[Times[d, Plus[Power[c, 2], Power[d, 2]]], -1]], Int[Times[Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Simp[Plus[Times[a, d, Plus[Times[A, c], Times[-1, c, C]]], Times[b, Plus[Times[Power[c, 2], C], Times[A, Power[d, 2]]]], Times[d, Plus[Times[A, b, c], Times[-1, b, c, C], Times[-1, a, A, d], Times[a, C, d]], Tan[Plus[e, Times[f, x]]]], Times[b, C, Plus[Power[c, 2], Power[d, 2]], Power[Tan[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, C], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], RationalQ[n], Less[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, C, Tan[Plus[e, Times[f, x]]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[d, f, Plus[n, 2]], -1]], x], Times[-1, Dist[Times[1, Power[Times[d, Plus[n, 2]], -1]], Int[Times[Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], n], Simp[Plus[Times[b, c, C], Times[-1, a, A, d, Plus[n, 2]], Times[-1, Plus[Times[A, b], Times[a, B], Times[-1, b, C]], d, Plus[n, 2], Tan[Plus[e, Times[f, x]]]], Times[-1, Plus[Times[a, C, d, Plus[n, 2]], Times[-1, b, Plus[Times[c, C], Times[-1, B, d, Plus[n, 2]]]]], Power[Tan[Plus[e, Times[f, x]]], 2]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, A, B, C, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], Not[And[RationalQ[n], Less[n, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, C, Tan[Plus[e, Times[f, x]]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[d, f, Plus[n, 2]], -1]], x], Times[-1, Dist[Times[1, Power[Times[d, Plus[n, 2]], -1]], Int[Times[Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], n], Simp[Plus[Times[b, c, C], Times[-1, a, A, d, Plus[n, 2]], Times[-1, Plus[Times[A, b], Times[-1, b, C]], d, Plus[n, 2], Tan[Plus[e, Times[f, x]]]], Times[-1, Plus[Times[a, C, d, Plus[n, 2]], Times[-1, b, c, C]], Power[Tan[Plus[e, Times[f, x]]], 2]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, A, C, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], Not[And[RationalQ[n], Less[n, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[a, A], Times[b, B], Times[-1, a, C]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[2, f, m, Plus[Times[b, c], Times[-1, a, d]]], -1]], x], Dist[Times[1, Power[Times[2, a, m, Plus[Times[b, c], Times[-1, a, d]]], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], n], Simp[Plus[Times[b, Plus[Times[c, Plus[A, C], m], Times[-1, B, d, Plus[n, 1]]]], Times[a, Plus[Times[B, c, m], Times[C, d, Plus[n, 1]], Times[-1, A, d, Plus[Times[2, m], n, 1]]]], Times[Plus[Times[b, C, d, Plus[m, Times[-1, n], -1]], Times[A, b, d, Plus[m, n, 1]], Times[a, Plus[Times[2, c, C, m], Times[-1, B, d, Plus[m, n, 1]]]]], Tan[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B, C, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], Or[And[RationalQ[m], Less[m, 0]], ZeroQ[Plus[m, n, 1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[a, Plus[A, Times[-1, C]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Tan[Plus[e, 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Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[Power[c, 2], C], Times[-1, B, c, d], Times[A, Power[d, 2]]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[d, f, Plus[n, 1], Plus[Power[c, 2], Power[d, 2]]], -1]], x], Times[-1, Dist[Times[1, Power[Times[a, d, Plus[n, 1], Plus[Power[c, 2], Power[d, 2]]], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Simp[Plus[Times[b, Plus[Times[Power[c, 2], C], Times[-1, B, c, d], Times[A, Power[d, 2]]], m], Times[-1, a, d, Plus[n, 1], Plus[Times[A, c], Times[-1, c, C], Times[B, d]]], Times[-1, a, Plus[Times[d, Plus[Times[B, c], Times[-1, A, d]], Plus[m, n, 1]], Times[-1, C, Plus[Times[Power[c, 2], m], Times[-1, Power[d, 2], Plus[n, 1]]]]], Tan[Plus[e, Times[f, x]]]]], x]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, A, B, C, m], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], Not[And[RationalQ[m], Less[m, 0]]], RationalQ[n], Less[n, -1], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[Power[c, 2], C], Times[A, Power[d, 2]]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[d, f, Plus[n, 1], Plus[Power[c, 2], Power[d, 2]]], -1]], x], Times[-1, Dist[Times[1, Power[Times[a, d, Plus[n, 1], Plus[Power[c, 2], Power[d, 2]]], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Simp[Plus[Times[b, Plus[Times[Power[c, 2], C], Times[A, Power[d, 2]]], m], Times[-1, a, d, Plus[n, 1], Plus[Times[A, c], 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Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[C, Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[d, f, Plus[m, n, 1]], -1]], x], Dist[Times[1, Power[Times[b, d, Plus[m, n, 1]], -1]], Int[Times[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], n], Simp[Plus[Times[A, b, d, Plus[m, n, 1]], Times[C, Plus[Times[a, c, m], Times[-1, b, d, Plus[n, 1]]]], Times[-1, Plus[Times[C, m, Plus[Times[b, c], Times[-1, a, d]]], Times[-1, b, B, d, Plus[m, n, 1]]], Tan[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B, C, m, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], Not[And[RationalQ[m], Less[m, 0]]], NonzeroQ[Plus[m, n, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[C, Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Tan[Plus[e, Times[f, x]]]]], Plus[n, 1]], Power[Times[d, f, Plus[m, n, 1]], -1]], x], Dist[Times[1, Power[Times[b, d, Plus[m, n, 1]], -1]], 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x]], x], x]], And[FreeQ[List[a, b, c, d, e, f, A, B, C, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], NonzeroQ[Plus[Power[c, 2], Power[d, 2]]], RationalQ[m], Greater[m, 0], Not[And[IntegerQ[n], Greater[n, 0], Or[Not[IntegerQ[m]], And[ZeroQ[c], NonzeroQ[a]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], 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RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[c, Blank[]]]]], -1], Plus[Times[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[q, Rt[Plus[Power[b, 2], Times[-1, 4, a, c]], 2]]], Plus[Dist[Plus[B, Times[Plus[Times[b, B], Times[-1, 2, A, c]], Power[q, -1]]], Int[Times[1, Power[Simp[Plus[b, q, Times[2, c, Cot[Plus[d, Times[e, x]]]]], x], -1]], x], x], Dist[Plus[B, Times[-1, Plus[Times[b, B], Times[-1, 2, A, c]], Power[q, -1]]], Int[Times[1, Power[Simp[Plus[b, Times[-1, q], Times[2, c, Cot[Plus[d, Times[e, x]]]]], x], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, A, B], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[c, Blank[]]], Power[tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrig[Times[Plus[A, Times[B, tan[Plus[d, Times[e, x]]]]], Power[Plus[a, Times[b, tan[Plus[d, Times[e, x]]]], Times[c, Power[tan[Plus[d, Times[e, x]]], 2]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, A, B], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Times[Power[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[c, Blank[]]]]], Pattern[n, Blank[]]], Plus[Times[cot[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrig[Times[Plus[A, Times[B, cot[Plus[d, Times[e, x]]]]], Power[Plus[a, Times[b, cot[Plus[d, Times[e, x]]]], Times[c, Power[cot[Plus[d, Times[e, x]]], 2]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, A, B], x], NonzeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Pi, Optional[Pattern[k, Blank[]]]], Times[Complex[0, Pattern[fz, Blank[]]], Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[I, Power[Plus[c, Times[d, x]], Plus[m, 1]], Power[Times[d, Plus[m, 1]], -1]], x]], Dist[Times[2, I], Int[Times[Power[Plus[c, Times[d, x]], m], Power[E, Times[-2, I, k, Pi]], Power[E, Times[2, Plus[Times[-1, I, e], Times[f, fz, x]]]], Power[Plus[1, Times[Power[E, Times[-2, I, k, Pi]], Power[E, Times[2, Plus[Times[-1, I, e], Times[f, fz, x]]]]]], -1]], x], x]], And[FreeQ[List[c, d, e, f, fz], x], IntegerQ[Times[4, k]], PositiveIntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Pi, Optional[Pattern[k, Blank[]]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[I, Power[Plus[c, Times[d, x]], Plus[m, 1]], Power[Times[d, Plus[m, 1]], -1]], x], Times[-1, Dist[Times[2, I], Int[Times[Power[Plus[c, Times[d, x]], m], Power[E, Times[2, I, k, Pi]], Power[E, Times[2, I, Plus[e, Times[f, x]]]], Power[Plus[1, Times[Power[E, Times[2, I, k, Pi]], Power[E, Times[2, I, Plus[e, Times[f, x]]]]]], -1]], x], x]]], And[FreeQ[List[c, d, e, f], x], IntegerQ[Times[4, k]], PositiveIntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Complex[0, Pattern[fz, Blank[]]], Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[I, Power[Plus[c, Times[d, x]], Plus[m, 1]], Power[Times[d, Plus[m, 1]], -1]], x]], Dist[Times[2, I], Int[Times[Power[Plus[c, Times[d, x]], m], Power[E, Times[2, Plus[Times[-1, I, e], Times[f, fz, x]]]], Power[Plus[1, Power[E, Times[2, Plus[Times[-1, I, e], Times[f, fz, x]]]]], -1]], x], x]], And[FreeQ[List[c, d, e, f, fz], x], PositiveIntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[I, Power[Plus[c, Times[d, x]], Plus[m, 1]], Power[Times[d, Plus[m, 1]], -1]], x], Times[-1, Dist[Times[2, I], Int[Times[Power[Plus[c, Times[d, x]], m], Power[E, Times[2, I, Plus[e, Times[f, x]]]], Power[Plus[1, Power[E, Times[2, I, Plus[e, Times[f, x]]]]], -1]], x], x]]], And[FreeQ[List[c, d, e, f], x], PositiveIntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, Power[Plus[c, Times[d, x]], m], Power[Times[b, Tan[Plus[e, Times[f, x]]]], Plus[n, -1]], Power[Times[f, Plus[n, -1]], -1]], x], Plus[Times[-1, Dist[Times[b, d, m, Power[Times[f, Plus[n, -1]], -1]], Int[Times[Power[Plus[c, Times[d, x]], Plus[m, -1]], Power[Times[b, Tan[Plus[e, Times[f, x]]]], Plus[n, -1]]], x], x]], Times[-1, Dist[Power[b, 2], Int[Times[Power[Plus[c, Times[d, x]], m], Power[Times[b, Tan[Plus[e, Times[f, x]]]], Plus[n, -2]]], x], x]]]], And[FreeQ[List[b, c, d, e, f], x], RationalQ[m, n], Greater[n, 1], Greater[m, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[c, Times[d, x]], m], Power[Times[b, Tan[Plus[e, Times[f, x]]]], Plus[n, 1]], Power[Times[b, f, Plus[n, 1]], -1]], x], Plus[Times[-1, Dist[Times[d, m, Power[Times[b, f, Plus[n, 1]], -1]], Int[Times[Power[Plus[c, Times[d, x]], Plus[m, -1]], Power[Times[b, Tan[Plus[e, Times[f, x]]]], Plus[n, 1]]], x], x]], Times[-1, Dist[Times[1, Power[Power[b, 2], -1]], Int[Times[Power[Plus[c, Times[d, x]], m], Power[Times[b, Tan[Plus[e, Times[f, x]]]], Plus[n, 2]]], x], x]]]], And[FreeQ[List[b, c, d, e, f], x], RationalQ[m, n], Less[n, -1], Greater[m, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Power[Plus[c, Times[d, x]], m], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], n], x], x], And[FreeQ[List[a, b, c, d, e, f, m], x], PositiveIntegerQ[m, n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[c, Times[d, x]], Plus[m, 1]], Power[Times[2, a, d, Plus[m, 1]], -1]], x], Plus[Dist[Times[a, d, m, Power[Times[2, b, f], -1]], Int[Times[Power[Plus[c, Times[d, x]], Plus[m, -1]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], -1]], x], x], Times[-1, Simp[Times[a, Power[Plus[c, Times[d, x]], m], Power[Times[2, b, f, Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]]], -1]], x]]]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], RationalQ[m], Greater[m, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Power[Times[d, Plus[c, Times[d, x]], Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]]], -1], x]], Plus[Times[-1, Dist[Times[f, Power[Times[a, d], -1]], Int[Times[Sin[Plus[Times[2, e], Times[2, f, x]]], Power[Plus[c, Times[d, x]], -1]], x], x]], Dist[Times[f, Power[Times[b, d], -1]], Int[Times[Cos[Plus[Times[2, e], Times[2, f, x]]], Power[Plus[c, Times[d, x]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[Power[a, 2], Power[b, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[f, Power[Plus[c, Times[d, x]], Plus[m, 2]], Power[Times[b, Power[d, 2], Plus[m, 1], Plus[m, 2]], -1]], x], Plus[Dist[Times[2, b, f, Power[Times[a, d, Plus[m, 1]], -1]], Int[Times[Power[Plus[c, Times[d, x]], Plus[m, 1]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], -1]], x], x], Simp[Times[Power[Plus[c, Times[d, x]], Plus[m, 1]], Power[Times[d, Plus[m, 1], Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]]], -1]], x]]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], RationalQ[m], Less[m, -1], Unequal[m, -2]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Log[Plus[c, Times[d, x]]], Power[Times[2, a, d], -1]], x], Plus[Dist[Times[1, Power[Times[2, a], -1]], Int[Times[Cos[Plus[Times[2, e], Times[2, f, x]]], Power[Plus[c, Times[d, x]], -1]], x], x], Dist[Times[1, Power[Times[2, b], -1]], Int[Times[Sin[Plus[Times[2, e], Times[2, f, x]]], Power[Plus[c, Times[d, x]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[Power[a, 2], Power[b, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[c, Times[d, x]], Plus[m, 1]], Power[Times[2, a, d, Plus[m, 1]], -1]], x], Dist[Times[1, Power[Times[2, a], -1]], Int[Times[Power[Plus[c, Times[d, x]], m], Power[E, Times[2, a, Power[b, -1], Plus[e, Times[f, x]]]]], x], x]], And[FreeQ[List[a, b, c, d, e, f, m], x], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Power[Plus[c, Times[d, x]], m], Power[Plus[Times[1, Power[Times[2, a], -1]], Times[Cos[Plus[Times[2, e], Times[2, f, x]]], Power[Times[2, a], -1]], Times[Sin[Plus[Times[2, e], Times[2, f, x]]], Power[Times[2, b], -1]]], Times[-1, n]], x], x], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], NegativeIntegerQ[m, n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Power[Plus[c, Times[d, x]], m], Power[Plus[Times[1, Power[Times[2, a], -1]], Times[Power[E, Times[2, a, Power[b, -1], Plus[e, Times[f, x]]]], Power[Times[2, a], -1]]], Times[-1, n]], x], x], And[FreeQ[List[a, b, c, d, e, f, m], x], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], NegativeIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, IntHide[Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], n], x]]], Plus[Dist[Power[Plus[c, Times[d, x]], m], u, x], Times[-1, Dist[Times[d, m], Int[Dist[Power[Plus[c, Times[d, x]], Plus[m, -1]], u, x], x], x]]]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[Power[a, 2], Power[b, 2]]], NegativeIntegerQ[Plus[n, 1]], RationalQ[m], Greater[m, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Pi, Optional[Pattern[k, Blank[]]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[c, Times[d, x]], Plus[m, 1]], Power[Times[d, Plus[m, 1], Plus[a, Times[-1, I, b]]], -1]], x], Times[-1, Dist[Times[2, I, b], Int[Times[Power[Plus[c, Times[d, x]], m], Power[Plus[Power[a, 2], Power[b, 2], Times[Power[Plus[a, Times[-1, I, b]], 2], Power[E, Times[2, I, k, Pi]], Power[E, Times[2, I, Plus[e, Times[f, x]]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], IntegerQ[Times[4, k]], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], PositiveIntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[c, Times[d, x]], Plus[m, 1]], Power[Times[d, Plus[m, 1], Plus[a, Times[-1, I, b]]], -1]], x], Times[-1, Dist[Times[2, I, b], Int[Times[Power[Plus[c, Times[d, x]], m], Power[Plus[Power[a, 2], Power[b, 2], Times[Power[Plus[a, Times[-1, I, b]], 2], Power[E, Times[2, I, Plus[e, Times[f, x]]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], PositiveIntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], -2]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Plus[c, Times[d, x]], 2], Power[Times[2, d, Plus[Power[a, 2], Power[b, 2]]], -1]], x]], Plus[Dist[Times[1, Power[Times[f, Plus[Power[a, 2], Power[b, 2]]], -1]], Int[Times[Plus[Times[b, d], Times[2, a, c, f], Times[2, a, d, f, x]], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], -1]], x], x], Times[-1, Simp[Times[b, Plus[c, Times[d, x]], Power[Times[f, Plus[Power[a, 2], Power[b, 2]], Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]]], -1]], x]]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Power[Plus[c, Times[d, x]], m], Power[Plus[Times[1, Power[Plus[a, Times[-1, I, b]], -1]], Times[-1, 2, I, b, Power[Plus[Power[a, 2], Power[b, 2], Times[Power[Plus[a, Times[-1, I, b]], 2], Power[E, Times[2, I, Plus[e, Times[f, x]]]]]], -1]]], Times[-1, n]], x], x], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Power[a, 2], Power[b, 2]]], NegativeIntegerQ[n], PositiveIntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[If[MatchQ[f, Times[Optional[Pattern[f1, Blank[]]], Complex[0, Pattern[j, Blank[]]]]], If[MatchQ[e, Plus[Optional[Pattern[e1, Blank[]]], Times[Pi, Power[2, -1]]]], Times[Power[I, n], Defer[Int][Times[Power[Plus[c, Times[d, x]], m], Power[Coth[Plus[Times[-1, I, Plus[e, Times[-1, Pi, Power[2, -1]]]], Times[-1, I, f, x]]], n]], x]], Times[Power[I, n], Defer[Int][Times[Power[Plus[c, Times[d, x]], m], Power[Tanh[Plus[Times[-1, I, e], Times[-1, I, f, x]]], n]], x]]], If[MatchQ[e, Plus[Optional[Pattern[e1, Blank[]]], Times[Pi, Power[2, -1]]]], Times[Power[-1, n], Defer[Int][Times[Power[Plus[c, Times[d, x]], m], Power[Cot[Plus[e, Times[-1, Pi, Power[2, -1]], Times[f, x]]], n]], x]], Defer[Int][Times[Power[Plus[c, Times[d, x]], m], Power[Tan[Plus[e, Times[f, x]]], n]], x]]], x], And[FreeQ[List[c, d, e, f, m, n], x], IntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], tan[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Plus[c, Times[d, x]], m], Power[Plus[a, Times[b, Tan[Plus[e, Times[f, x]]]]], n]], x], FreeQ[List[a, b, c, d, e, f, m, n], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Tan[Pattern[v, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[ExpandToSum[u, x], m], Power[Plus[a, Times[b, Tan[ExpandToSum[v, x]]]], n]], x], And[FreeQ[List[a, b, m, n], x], LinearQ[List[u, v], x], Not[LinearMatchQ[List[u, v], x]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cot[Pattern[v, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[ExpandToSum[u, x], m], Power[Plus[a, Times[b, Cot[ExpandToSum[v, 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RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[a, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[a, b, Power[Times[a, Csc[Plus[e, Times[f, x]]]], Plus[m, -1]], Power[Times[b, Sec[Plus[e, Times[f, x]]]], Plus[n, -1]], Power[Times[f, Plus[n, -1]], -1]], x], And[FreeQ[List[a, b, e, f, m, n], x], ZeroQ[Plus[m, n, -2]], NonzeroQ[Plus[n, -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, 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Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[a, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Power[Times[f, m], -1]], x]], Dist[Times[b, Plus[Times[2, m], -1], Power[Times[d, m], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, 1]]], x], x]], And[FreeQ[List[a, b, d, e, f, m, n], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[m, n]], RationalQ[m], Greater[m, Times[1, Power[2, -1]]], IntegerQ[Times[2, m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, 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Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[b, 2], Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, -2]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Power[Times[f, n], -1]], x], Times[-1, Dist[Times[a, Power[Times[d, n], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, -2]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, 1]], Plus[Times[b, Plus[m, Times[-1, 2, n], -2]], Times[-1, a, Plus[m, Times[2, n], -1], Csc[Plus[e, Times[f, x]]]]]], x], x]]], And[FreeQ[List[a, b, d, e, f], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m, n], Greater[m, 1], Or[Less[n, -1], And[Equal[m, Times[3, Power[2, -1]]], Equal[n, Times[-1, Power[2, -1]]]]], IntegerQ[Times[2, m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[b, 2], Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, -2]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Power[Times[f, Plus[m, n, -1]], -1]], x]], Dist[Times[b, Power[Plus[m, n, -1], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, -2]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Plus[Times[b, Plus[m, Times[2, n], -1]], Times[a, Plus[Times[3, m], Times[2, n], -4], Csc[Plus[e, Times[f, x]]]]]], x], x]], And[FreeQ[List[a, b, d, e, f, n], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m], Greater[m, 1], NonzeroQ[Plus[m, n, -1]], IntegerQ[Times[2, m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, d, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, -1]], Power[Times[a, f, Plus[Times[2, m], 1]], -1]], x], Times[-1, Dist[Times[d, Power[Times[a, b, Plus[Times[2, m], 1]], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, -1]], Plus[Times[a, Plus[n, -1]], Times[-1, b, Plus[m, n], Csc[Plus[e, Times[f, x]]]]]], x], x]]], And[FreeQ[List[a, b, d, e, f], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m, n], Less[m, -1], Less[1, n, 2], Or[IntegersQ[Times[2, m], Times[2, n]], IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[d, 2], Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, -2]], Power[Times[f, Plus[Times[2, m], 1]], -1]], x]], Dist[Times[Power[d, 2], 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Times[b, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x]]], And[FreeQ[List[a, b, d, e, f], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[-2, Power[d, 2], Cot[Plus[e, Times[f, x]]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, -2]], Power[Times[f, Plus[Times[2, n], -3], Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]]], -1]], x], Dist[Times[Power[d, 2], Power[Times[b, Plus[Times[2, n], -3]], -1]], Int[Times[Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, -2]], Plus[Times[2, b, Plus[n, -2]], Times[-1, a, Csc[Plus[e, Times[f, x]]]]], Power[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x]], And[FreeQ[List[a, b, d, e, f], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[n], Greater[n, 2], IntegerQ[Times[2, n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Cot[Plus[e, Times[f, x]]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Power[Times[f, n, Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]]], -1]], x], Dist[Times[1, Power[Times[2, b, d, n], -1]], Int[Times[Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, 1]], Plus[a, Times[b, Plus[Times[2, n], 1], Csc[Plus[e, Times[f, x]]]]], Power[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x]], And[FreeQ[List[a, b, d, e, f], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[n], Less[n, 0], IntegerQ[Times[2, n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[d, 2], Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, -2]], Power[Times[f, Plus[m, n, -1]], -1]], x]], Dist[Times[Power[d, 2], Power[Times[b, Plus[m, n, -1]], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, -2]], Plus[Times[b, Plus[n, -2]], Times[a, m, Csc[Plus[e, Times[f, x]]]]]], x], x]], And[FreeQ[List[a, b, d, e, f, m], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[n], Greater[n, 2], NonzeroQ[Plus[m, n, -1]], IntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Times[Power[Times[a, d, Power[b, -1]], n], Cot[Plus[e, Times[f, x]]], Power[Times[Power[a, Plus[n, -2]], f, Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Sqrt[Plus[a, Times[-1, b, Csc[Plus[e, Times[f, x]]]]]]], -1]], Subst[Int[Times[Power[Plus[a, 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Times[f, x]]]]]], Sqrt[Plus[a, Times[-1, b, Csc[Plus[e, Times[f, x]]]]]]], -1]], Subst[Int[Times[Power[Times[d, x], Plus[n, -1]], Power[Plus[a, Times[b, x]], Plus[m, Times[-1, 1, Power[2, -1]]]], Power[Sqrt[Plus[a, Times[-1, b, x]]], -1]], x], x, Csc[Plus[e, Times[f, x]]]], x], And[FreeQ[List[a, b, d, e, f, m, n], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], Not[IntegerQ[m]], PositiveQ[a]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[a, IntPart[m]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], FracPart[m]], Power[Power[Plus[1, Times[b, Power[a, -1], 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f], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[b, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Times[f, m], -1]], x]], Dist[Times[1, Power[m, -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, -2]], Plus[Times[Power[b, 2], Plus[m, -1]], Times[Power[a, 2], m], Times[a, b, Plus[Times[2, m], -1], Csc[Plus[e, Times[f, x]]]]]], x], x]], And[FreeQ[List[a, b, e, f], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m], Greater[m, 1], IntegerQ[Times[2, m]]]]], 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x]]]], -1], Sqrt[Times[Times[b, Plus[1, Times[-1, Csc[Plus[e, Times[f, x]]]]]], Power[Plus[a, b], -1]]], Sqrt[Times[-1, b, Plus[1, Csc[Plus[e, Times[f, x]]]], Power[Plus[a, Times[-1, b]], -1]]], EllipticF[ArcSin[Times[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Power[Rt[Plus[a, b], 2], -1]]], Times[Plus[a, b], Power[Plus[a, Times[-1, b]], -1]]]], x], And[FreeQ[List[a, b, e, f], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[b, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[f, Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], x]], Dist[Times[1, Power[Times[Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Plus[Times[a, Plus[m, 1]], Times[-1, b, Plus[m, 2], Csc[Plus[e, Times[f, x]]]]]], x], x]], And[FreeQ[List[a, b, e, f], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m], Less[m, -1], IntegerQ[Times[2, m]]]]], RuleDelayed[HoldPattern[Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Cot[Plus[e, Times[f, x]]], Power[Times[f, Sqrt[Plus[1, Csc[Plus[e, Times[f, x]]]]], Sqrt[Plus[1, Times[-1, Csc[Plus[e, Times[f, x]]]]]]], -1]], Subst[Int[Times[Power[Plus[a, 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Times[-1, Dist[Times[1, Power[Times[Power[e, 2], Plus[m, 1]], -1]], Int[Times[Power[Times[e, Cot[Plus[c, Times[d, x]]]], Plus[m, 2]], Plus[Times[a, Plus[m, 1]], Times[b, Plus[m, 2], Csc[Plus[c, Times[d, x]]]]]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], RationalQ[m], Less[m, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], -1], Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Plus[b, Times[a, Sin[Plus[c, Times[d, x]]]]], Power[Cos[Plus[c, Times[d, x]]], -1]], x], FreeQ[List[a, b, c, d], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Optional[Pattern[m, Blank[]]]], 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Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[-2, Power[a, Plus[Times[m, Power[2, -1]], n, Times[1, Power[2, -1]]]], Power[d, -1]], Subst[Int[Times[Power[x, m], Power[Plus[2, Times[a, Power[x, 2]]], Plus[Times[m, Power[2, -1]], n, Times[-1, 1, Power[2, -1]]]], Power[Plus[1, Times[a, Power[x, 2]]], -1]], x], x, Times[Cot[Plus[c, Times[d, x]]], Power[Sqrt[Plus[a, Times[b, Csc[Plus[c, Times[d, x]]]]]], -1]]], x], And[FreeQ[List[a, b, c, d], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegerQ[Times[m, Power[2, -1]]], IntegerQ[Plus[n, Times[-1, 1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[a, Times[2, n]], Power[e, Times[-2, n]]], Int[Times[Power[Times[e, Cot[Plus[c, Times[d, x]]]], Plus[m, Times[2, n]]], Power[Power[Plus[Times[-1, a], Times[b, Csc[Plus[c, Times[d, x]]]]], n], -1]], x], x], And[FreeQ[List[a, b, c, d, e, m], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NegativeIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Simp[Times[Power[2, Plus[m, n, 1]], Power[Times[e, Cot[Plus[c, Times[d, x]]]], Plus[m, 1]], Power[Plus[a, Times[b, Csc[Plus[c, Times[d, x]]]]], n], Power[Times[d, e, Plus[m, 1]], -1], Power[Times[a, Power[Plus[a, Times[b, Csc[Plus[c, Times[d, x]]]]], -1]], Plus[m, n, 1]], AppellF1[Times[Plus[m, 1], Power[2, -1]], Plus[m, n], 1, Times[Plus[m, 3], Power[2, -1]], Times[-1, Plus[a, Times[-1, b, Csc[Plus[c, Times[d, x]]]]], Power[Plus[a, Times[b, Csc[Plus[c, Times[d, x]]]]], -1]], Times[Plus[a, Times[-1, b, Csc[Plus[c, Times[d, x]]]]], Power[Plus[a, Times[b, Csc[Plus[c, Times[d, x]]]]], -1]]]], x]], And[FreeQ[List[a, b, c, d, e, m, n], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[a, -1]], Int[Sqrt[Times[e, Cot[Plus[c, Times[d, x]]]]], x], x], Times[-1, Dist[Times[b, Power[a, -1]], Int[Times[Sqrt[Times[e, Cot[Plus[c, Times[d, x]]]]], Power[Plus[b, Times[a, Sin[Plus[c, Times[d, x]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Dist[Times[Power[e, 2], Power[Power[b, 2], -1]], Int[Times[Power[Times[e, Cot[Plus[c, Times[d, x]]]], Plus[m, -2]], Plus[a, Times[-1, b, Csc[Plus[c, Times[d, x]]]]]], x], x]], Dist[Times[Power[e, 2], Plus[Power[a, 2], Times[-1, Power[b, 2]]], Power[Power[b, 2], -1]], Int[Times[Power[Times[e, Cot[Plus[c, Times[d, x]]]], Plus[m, -2]], Power[Plus[a, Times[b, Csc[Plus[c, Times[d, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], PositiveIntegerQ[Plus[m, Times[-1, 1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[a, -1]], Int[Times[1, Power[Sqrt[Times[e, Cot[Plus[c, Times[d, x]]]]], -1]], x], x], Times[-1, Dist[Times[b, Power[a, -1]], Int[Times[1, Power[Times[Sqrt[Times[e, Cot[Plus[c, Times[d, x]]]]], Plus[b, Times[a, Sin[Plus[c, Times[d, x]]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[Plus[Power[a, 2], Times[-1, Power[b, 2]]], -1]], Int[Times[Power[Times[e, Cot[Plus[c, Times[d, x]]]], m], Plus[a, Times[-1, b, Csc[Plus[c, Times[d, x]]]]]], x], x], Dist[Times[Power[b, 2], Power[Times[Power[e, 2], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], Int[Times[Power[Times[e, Cot[Plus[c, Times[d, x]]]], Plus[m, 2]], Power[Plus[a, Times[b, Csc[Plus[c, Times[d, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NegativeIntegerQ[Plus[m, Times[1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Plus[-1, Power[Csc[Plus[c, Times[d, x]]], 2]], Power[Plus[a, Times[b, Csc[Plus[c, Times[d, x]]]]], n]], x], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Power[Times[e, Cot[Plus[c, Times[d, x]]]], m], Power[Plus[a, Times[b, Csc[Plus[c, Times[d, x]]]]], n], x], x], And[FreeQ[List[a, b, c, d, e, m], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Cos[Plus[c, Times[d, x]]], m], Power[Plus[b, Times[a, Sin[Plus[c, Times[d, x]]]]], n], Power[Power[Sin[Plus[c, Times[d, x]]], Plus[m, n]], -1]], x], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegerQ[n], IntegerQ[m], Or[IntegerQ[Times[m, Power[2, -1]]], LessEqual[m, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Times[e, Cot[Plus[c, Times[d, x]]]], m], Power[Plus[a, Times[b, Csc[Plus[c, Times[d, x]]]]], n]], x], FreeQ[List[a, b, c, d, e, m, n], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[e, Blank[]]], Power[tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[e, Power[Tan[Plus[c, Times[d, x]]], p]], m], Power[Power[Times[e, Tan[Plus[c, Times[d, x]]]], Times[m, p]], -1]], Int[Times[Power[Times[e, Tan[Plus[c, Times[d, x]]]], Times[m, p]], Power[Plus[a, Times[b, Sec[Plus[c, Times[d, x]]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, m, n, p], x], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Power[cot[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]], Optional[Pattern[e, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[e, Power[Cot[Plus[c, Times[d, x]]], p]], m], Power[Power[Times[e, Cot[Plus[c, Times[d, x]]]], Times[m, p]], -1]], Int[Times[Power[Times[e, Cot[Plus[c, Times[d, x]]]], Times[m, p]], Power[Plus[a, Times[b, Csc[Plus[c, Times[d, x]]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, m, n, p], x], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, n], Int[ExpandTrig[Power[Plus[1, Times[d, Power[c, -1], csc[Plus[e, Times[f, x]]]]], n], Power[Plus[a, Times[b, csc[Plus[e, Times[f, x]]]]], m], x], x], x], And[FreeQ[List[a, b, c, d, e, f, n], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], PositiveIntegerQ[m], NegativeIntegerQ[n], Less[Plus[m, n], 2]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[Times[-1, a, c], m], Int[Times[Power[Cot[Plus[e, Times[f, x]]], Times[2, m]], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], Plus[n, Times[-1, m]]]], x], x], And[FreeQ[List[a, b, c, d, e, f, n], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegerQ[m], RationalQ[n], Not[And[IntegerQ[n], Greater[Plus[m, Times[-1, n]], 0]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[-1, a, c], Plus[m, Times[1, Power[2, -1]]]], Cot[Plus[e, Times[f, x]]], Power[Times[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]]]], -1]], Int[Power[Cot[Plus[e, Times[f, x]]], Times[2, m]], x], x], And[FreeQ[List[a, b, c, d, e, f, m], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegerQ[Plus[m, Times[1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[2, a, c, Cot[Plus[e, Times[f, x]]], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], Plus[n, -1]], Power[Times[f, Plus[Times[2, n], -1], Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]]], -1]], x], Dist[c, Int[Times[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], Plus[n, -1]]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[n], Greater[n, Times[1, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[-2, a, Cot[Plus[e, Times[f, x]]], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], n], Power[Times[f, Plus[Times[2, n], 1], Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]]], -1]], x], Dist[Times[1, Power[c, -1]], Int[Times[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], Plus[n, 1]]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[n], Less[n, Times[-1, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[3, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[-4, Power[a, 2], Cot[Plus[e, Times[f, x]]], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], n], Power[Times[f, Plus[Times[2, n], 1], Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]]], -1]], x], Dist[Times[a, Power[c, -1]], Int[Times[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], Plus[n, 1]]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[n], Less[n, Times[-1, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[3, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[-2, Power[a, 2], Cot[Plus[e, Times[f, x]]], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], n], Power[Times[f, Plus[Times[2, n], 1], Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]]], -1]], x], Dist[a, Int[Times[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], n]], x], x]], And[FreeQ[List[a, b, c, d, e, f, n], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], Not[And[RationalQ[n], LessEqual[n, Times[-1, Power[2, -1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[5, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[-8, Power[a, 3], Cot[Plus[e, Times[f, x]]], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], n], Power[Times[f, Plus[Times[2, n], 1], Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]]], -1]], x], Dist[Times[Power[a, 2], Power[Power[c, 2], -1]], Int[Times[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], Plus[n, 2]]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[n], Less[n, Times[-1, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Times[a, c, Cot[Plus[e, Times[f, x]]], Power[Times[f, Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]]]], -1]], Subst[Int[Times[Power[Plus[b, Times[a, x]], Plus[m, Times[-1, 1, Power[2, -1]]]], Power[Plus[d, Times[c, x]], Plus[n, Times[-1, 1, Power[2, -1]]]], Power[Power[x, Plus[m, n]], -1]], x], x, Sin[Plus[e, Times[f, x]]]], x]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegerQ[Plus[m, Times[-1, 1, Power[2, -1]]]], ZeroQ[Plus[m, n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[a, c, Cot[Plus[e, Times[f, x]]], Power[Times[f, Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]]]], -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], Plus[m, Times[-1, 1, Power[2, -1]]]], Power[Plus[c, Times[d, x]], Plus[n, Times[-1, 1, Power[2, -1]]]], Power[x, -1]], x], x, Csc[Plus[e, Times[f, x]]]], x], And[FreeQ[List[a, b, c, d, e, f, m, n], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[a, c, x], x], Dist[Times[b, d], Int[Power[Csc[Plus[e, Times[f, x]]], 2], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[Times[b, c], Times[a, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[a, c, x], x], Plus[Dist[Times[b, d], Int[Power[Csc[Plus[e, Times[f, x]]], 2], x], x], Dist[Plus[Times[b, c], Times[a, d]], Int[Csc[Plus[e, Times[f, x]]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Times[b, c], Times[a, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[c, Int[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], x], x], Dist[d, Int[Times[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Csc[Plus[e, Times[f, x]]]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[a, c], Int[Times[1, Power[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x], Int[Times[Csc[Plus[e, Times[f, x]]], Plus[Times[b, c], Times[a, d], Times[b, d, Csc[Plus[e, Times[f, x]]]]], Power[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], -1]], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[b, d, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Times[f, m], -1]], x]], Dist[Times[1, Power[m, -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, -1]], Simp[Plus[Times[a, c, m], Times[Plus[Times[b, c, m], Times[a, d, Plus[Times[2, m], -1]]], Csc[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[m], Greater[m, 1], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegerQ[Times[2, m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[b, d, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Times[f, m], -1]], x]], Dist[Times[1, Power[m, -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, -2]], Simp[Plus[Times[Power[a, 2], c, m], Times[Plus[Times[Power[b, 2], d, Plus[m, -1]], Times[2, a, b, c, m], Times[Power[a, 2], d, m]], Csc[Plus[e, Times[f, x]]]], Times[b, Plus[Times[b, c, m], Times[a, d, Plus[Times[2, m], -1]]], Power[Csc[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[m], Greater[m, 1], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegerQ[Times[2, m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[c, x, Power[a, -1]], x], Times[-1, Dist[Times[Plus[Times[b, c], Times[-1, a, d]], Power[a, -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[c, Power[a, -1]], Int[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], x], x], Times[-1, Dist[Times[Plus[Times[b, c], Times[-1, a, d]], Power[a, -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Power[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[c, Int[Times[1, Power[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x], Dist[d, Int[Times[Csc[Plus[e, Times[f, x]]], Power[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[b, c], Times[-1, a, d]], Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[b, f, Plus[Times[2, m], 1]], -1]], x]], Dist[Times[1, Power[Times[Power[a, 2], Plus[Times[2, m], 1]], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Simp[Plus[Times[a, c, Plus[Times[2, m], 1]], Times[-1, Plus[Times[b, c], Times[-1, a, d]], Plus[m, 1], Csc[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[m], Less[m, -1], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegerQ[Times[2, m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, Plus[Times[b, c], Times[-1, a, d]], Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[a, f, Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], x], Dist[Times[1, Power[Times[a, Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Simp[Plus[Times[c, Plus[Power[a, 2], Times[-1, Power[b, 2]]], Plus[m, 1]], Times[-1, Times[a, Plus[Times[b, c], Times[-1, a, d]], Plus[m, 1]], Csc[Plus[e, Times[f, x]]]], Times[b, Plus[Times[b, c], Times[-1, a, d]], Plus[m, 2], Power[Csc[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], RationalQ[m], Less[m, -1], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegerQ[Times[2, m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[c, Int[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], x], x], Dist[d, Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Csc[Plus[e, Times[f, x]]]], x], x]], And[FreeQ[List[a, b, c, d, e, f, m], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], Not[IntegerQ[Times[2, m]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[c, -1]], Int[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], x], x], Times[-1, Dist[Times[d, Power[c, -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], Or[ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[a, Power[c, -1]], Int[Times[1, Power[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x], Dist[Times[Plus[Times[b, c], Times[-1, a, d]], Power[c, -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Power[Times[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[3, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[a, Power[c, -1]], Int[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], x], x], Dist[Times[Plus[Times[b, c], Times[-1, a, d]], Power[c, -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], Or[ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[3, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[Times[c, d], -1]], Int[Times[Plus[Times[Power[a, 2], d], Times[Power[b, 2], c, Csc[Plus[e, Times[f, x]]]]], Power[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x], Times[-1, Dist[Times[Power[Plus[Times[b, c], Times[-1, a, d]], 2], Power[Times[c, d], -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Power[Times[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[Times[c, Plus[Times[b, c], Times[-1, a, d]]], -1]], Int[Times[Plus[Times[b, c], Times[-1, a, d], Times[-1, b, d, Csc[Plus[e, Times[f, x]]]]], Power[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x], Dist[Times[Power[d, 2], Power[Times[c, Plus[Times[b, c], Times[-1, a, d]]], -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Power[Plus[c, Times[d, Csc[Plus[e, 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Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[d, m], Int[Times[Power[Plus[b, Times[a, Cos[Plus[e, Times[f, x]]]]], m], Power[Times[d, Cos[Plus[e, Times[f, x]]]], Plus[n, Times[-1, m]]]], x], x], And[FreeQ[List[a, b, d, e, f, n], x], Not[IntegerQ[n]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[d, m], Int[Times[Power[Plus[b, Times[a, Sin[Plus[e, Times[f, x]]]]], m], Power[Times[d, Sin[Plus[e, Times[f, x]]]], Plus[n, Times[-1, m]]]], x], x], And[FreeQ[List[a, b, d, e, f, n], x], Not[IntegerQ[n]], IntegerQ[m]]]], 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Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], n], Power[Times[a, f, Plus[Times[2, m], 1]], -1]], x], Dist[Times[Plus[m, n, 1], Power[Times[a, Plus[Times[2, m], 1]], -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], n]], x], x]], And[FreeQ[List[a, b, c, d, e, f, m, n], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], 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Times[d, Csc[Plus[e, Times[f, x]]]]]]], -1]], x], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[2, a, c, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[b, f, Plus[Times[2, m], 1], Sqrt[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]]]], -1]], x], And[FreeQ[List[a, b, c, d, e, f, m], x], 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RuleDelayed[HoldPattern[Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[2, a, c, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], Plus[n, -1]], Power[Times[b, f, Plus[Times[2, m], 1]], -1]], x], Times[-1, Dist[Times[d, Plus[Times[2, n], -1], Power[Times[b, Plus[Times[2, m], 1]], -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], Plus[n, -1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], PositiveIntegerQ[n], Less[m, Times[-1, Power[2, -1]]], IntegerQ[Times[2, m]]]]], RuleDelayed[HoldPattern[Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[Times[-1, a, c], m], Int[ExpandTrig[Times[csc[Plus[e, Times[f, x]]], Power[cot[Plus[e, Times[f, x]]], Times[2, m]]], 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Blank[]]]]], Optional[Pattern[g, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[-1, a, c], Plus[m, Times[1, Power[2, -1]]]], Cot[Plus[e, Times[f, x]]], Power[Times[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]]]], -1]], Int[Times[Power[Times[g, Csc[Plus[e, Times[f, x]]]], p], Power[Cot[Plus[e, Times[f, x]]], Times[2, m]]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, p], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegerQ[Plus[m, Times[1, 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Power[2, -1]]]]], x], x, Csc[Plus[e, Times[f, x]]]], x], And[FreeQ[List[a, b, c, d, e, f, g, m, n, p], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[-2, b, g, Power[f, -1]], Subst[Int[Times[1, Power[Plus[Times[b, c], Times[a, d], Times[-1, c, g, Power[x, 2]]], -1]], x], x, Times[b, Cot[Plus[e, Times[f, x]]], Power[Times[Sqrt[Times[g, Csc[Plus[e, Times[f, x]]]]], Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]]], -1]]], x], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[a, Power[c, -1]], Int[Times[Sqrt[Times[g, Csc[Plus[e, Times[f, x]]]]], Power[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x], Dist[Times[Plus[Times[b, c], Times[-1, a, d]], Power[Times[c, g], -1]], Int[Times[Power[Times[g, Csc[Plus[e, Times[f, x]]]], Times[3, Power[2, -1]]], Power[Times[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[-2, b, Power[f, -1]], Subst[Int[Times[1, Power[Plus[Times[b, c], Times[a, d], Times[d, Power[x, 2]]], -1]], x], x, Times[b, Cot[Plus[e, Times[f, x]]], Power[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], -1]]], x], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Simp[Times[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Sqrt[Times[c, Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], -1]]], Power[Times[d, f, Sqrt[Times[c, d, Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Power[Times[Plus[Times[b, c], Times[a, d]], Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]]], -1]]]], -1], EllipticE[ArcSin[Times[c, Cot[Plus[e, Times[f, x]]], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], -1]]], Times[-1, Plus[Times[b, c], Times[-1, a, d]], Power[Plus[Times[b, c], Times[a, d]], -1]]]], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[b, Power[d, -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Power[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x], Times[-1, Dist[Times[Plus[Times[b, c], Times[-1, a, d]], Power[d, -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Power[Times[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Rational[3, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[g, Power[d, -1]], Int[Times[Sqrt[Times[g, Csc[Plus[e, Times[f, x]]]]], Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]]], x], x], Times[-1, Dist[Times[c, g, Power[d, -1]], Int[Times[Sqrt[Times[g, Csc[Plus[e, Times[f, x]]]]], Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Rational[3, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[b, Power[d, -1]], Int[Times[Power[Times[g, Csc[Plus[e, Times[f, x]]]], Times[3, Power[2, -1]]], Power[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x], Times[-1, Dist[Times[Plus[Times[b, c], Times[-1, a, d]], Power[d, -1]], Int[Times[Power[Times[g, Csc[Plus[e, Times[f, x]]]], Times[3, Power[2, -1]]], Power[Times[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[b, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Power[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x], Times[-1, Dist[Times[d, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], Or[ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]]], RuleDelayed[HoldPattern[Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[-2, Cot[Plus[e, Times[f, x]]], Power[Times[f, Plus[c, d], Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Sqrt[Times[-1, Power[Cot[Plus[e, Times[f, x]]], 2]]]], -1], Sqrt[Times[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Power[Plus[a, b], -1]]], EllipticPi[Times[2, d, Power[Plus[c, d], -1]], ArcSin[Times[Sqrt[Plus[1, Times[-1, Csc[Plus[e, Times[f, x]]]]]], Power[Sqrt[2], -1]]], Times[2, b, Power[Plus[a, b], -1]]]], x], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Rational[3, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Dist[Times[a, g, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Int[Times[Sqrt[Times[g, Csc[Plus[e, Times[f, x]]]]], Power[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x]], Dist[Times[c, g, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Int[Times[Sqrt[Times[g, Csc[Plus[e, Times[f, x]]]]], Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Rational[3, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[g, Sqrt[Times[g, Csc[Plus[e, Times[f, x]]]]], Sqrt[Plus[b, Times[a, Sin[Plus[e, Times[f, x]]]]]], Power[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], -1]], Int[Times[1, Power[Times[Sqrt[Plus[b, Times[a, Sin[Plus[e, Times[f, x]]]]]], Plus[d, Times[c, Sin[Plus[e, Times[f, x]]]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Dist[Times[a, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Power[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x]], Dist[Times[c, Power[Plus[Times[b, c], Times[-1, a, d]], -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], Or[ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[d, -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Power[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x], Times[-1, Dist[Times[c, Power[d, -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Power[Times[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Rational[5, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Dist[Times[Power[c, 2], Power[g, 2], Power[Times[d, Plus[Times[b, c], Times[-1, a, d]]], -1]], Int[Times[Sqrt[Times[g, Csc[Plus[e, Times[f, x]]]]], Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], -1]], x], x]], Dist[Times[Power[g, 2], Power[Times[d, Plus[Times[b, c], Times[-1, a, d]]], -1]], Int[Times[Sqrt[Times[g, Csc[Plus[e, Times[f, x]]]]], Plus[Times[a, c], Times[Plus[Times[b, c], Times[-1, a, d]], Csc[Plus[e, Times[f, x]]]]], Power[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Rational[5, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[g, Power[d, -1]], Int[Times[Power[Times[g, Csc[Plus[e, Times[f, x]]]], Times[3, Power[2, -1]]], Power[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x], Times[-1, Dist[Times[c, g, Power[d, -1]], Int[Times[Power[Times[g, Csc[Plus[e, Times[f, x]]]], Times[3, Power[2, -1]]], Power[Times[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[-2, b, Power[f, -1]], Subst[Int[Times[1, Power[Plus[1, Times[-1, b, d, Power[x, 2]]], -1]], x], x, Times[Cot[Plus[e, Times[f, x]]], Power[Times[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]]]], -1]]], x], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Dist[Times[Plus[Times[b, c], Times[-1, a, d]], Power[d, -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Power[Times[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]]]], -1]], x], x]], Dist[Times[b, Power[d, -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Sqrt[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]]], Power[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, 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Csc[Plus[e, Times[f, x]]]]]], -1]]], Times[Plus[a, Times[-1, b]], Plus[c, d], Power[Times[Plus[a, b], Plus[c, Times[-1, d]]], -1]]]], x], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[-2, a, Power[Times[b, f], -1]], Subst[Int[Times[1, Power[Plus[2, Times[Plus[Times[a, c], Times[-1, b, d]], Power[x, 2]]], -1]], x], x, Times[Cot[Plus[e, Times[f, x]]], Power[Times[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]]]], -1]]], x], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[-2, Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], Power[Times[f, Plus[Times[b, c], Times[-1, a, d]], Rt[Times[Plus[c, d], Power[Plus[a, b], -1]], 2], Cot[Plus[e, Times[f, x]]]], -1], Sqrt[Times[Plus[Times[b, c], Times[-1, a, d]], Plus[1, Times[-1, Csc[Plus[e, Times[f, x]]]]], Power[Times[Plus[a, b], Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]]], -1]]], Sqrt[Times[-1, Plus[Times[b, c], Times[-1, a, d]], Plus[1, Csc[Plus[e, Times[f, x]]]], Power[Times[Plus[a, Times[-1, b]], Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]]], -1]]], EllipticF[ArcSin[Times[Rt[Times[Plus[c, d], Power[Plus[a, b], -1]], 2], Times[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Power[Sqrt[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]]], -1]]]], Times[Plus[a, b], Plus[c, Times[-1, d]], Power[Times[Plus[a, Times[-1, b]], Plus[c, d]], -1]]]], x], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Dist[Times[a, Power[b, -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Power[Times[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]]]], -1]], x], x]], Dist[Times[1, Power[b, -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Power[Sqrt[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]]]]], RuleDelayed[HoldPattern[Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Plus[a, Times[-1, b]], Power[Plus[c, Times[-1, d]], -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Power[Times[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]]]], -1]], x], x], Dist[Times[Plus[Times[b, c], Times[-1, a, d]], Power[Plus[c, Times[-1, d]], -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Plus[1, Csc[Plus[e, Times[f, x]]]], Power[Times[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], Times[3, Power[2, -1]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[a, 2], g, Cot[Plus[e, Times[f, x]]], Power[Times[f, Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Sqrt[Plus[a, Times[-1, b, Csc[Plus[e, Times[f, x]]]]]]], -1]], Subst[Int[Times[Power[Times[g, x], Plus[p, -1]], Power[Plus[a, Times[b, x]], Plus[m, Times[-1, 1, Power[2, -1]]]], Power[Plus[c, Times[d, x]], n], Power[Sqrt[Plus[a, Times[-1, b, x]]], -1]], x], x, Csc[Plus[e, Times[f, x]]]], x], And[FreeQ[List[a, b, c, d, e, f, g, m, n, p], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Power[c, 2], Times[-1, Power[d, 2]]]], Or[ZeroQ[Plus[p, -1]], IntegerQ[Plus[m, Times[-1, 1, Power[2, -1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Power[g, Plus[m, n]], -1]], Int[Times[Power[Times[g, Csc[Plus[e, Times[f, x]]]], Plus[m, n, p]], Power[Plus[b, Times[a, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[d, Times[c, Sin[Plus[e, Times[f, x]]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, p], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], IntegerQ[m], IntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, 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Optional[Pattern[p, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[g, Csc[Plus[e, Times[f, x]]]], p], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]], n], Power[Times[Power[Plus[b, Times[a, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[d, Times[c, Sin[Plus[e, Times[f, x]]]]], n]], -1]], Int[Times[Power[Plus[b, Times[a, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[d, Times[c, Sin[Plus[e, Times[f, x]]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, m, n, p], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[m, n, p]], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Plus[d, Times[c, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Power[Times[Sqrt[Plus[b, Times[a, Sin[Plus[e, Times[f, x]]]]]], Sqrt[Plus[c, Times[d, Csc[Plus[e, Times[f, x]]]]]]], -1]], Int[Times[Power[Plus[b, Times[a, Sin[Plus[e, Times[f, x]]]]], m], Power[Plus[d, Times[c, Sin[Plus[e, Times[f, x]]]]], n], Power[Power[Sin[Plus[e, Times[f, x]]], Plus[m, n, p]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, m, n], x], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]], IntegerQ[Plus[m, Times[-1, 1, Power[2, -1]]]], IntegerQ[Plus[n, Times[-1, 1, Power[2, -1]]]], IntegerQ[p], MemberQ[List[-1, -2], Plus[m, n, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrig[Times[Power[Times[g, csc[Plus[e, 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Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[B, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[b, f, Plus[m, 2]], -1]], x]], Dist[Times[1, Power[Times[b, Plus[m, 2]], -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Simp[Plus[Times[b, B, Plus[m, 1]], Times[Plus[Times[A, b, Plus[m, 2]], Times[-1, a, B]], Csc[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, e, f, A, B, m], x], NonzeroQ[Plus[Times[A, b], Times[-1, a, B]]], Not[And[RationalQ[m], Less[m, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[A, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Power[Times[f, n], -1]], x], And[FreeQ[List[a, b, d, e, f, A, B, m, n], x], NonzeroQ[Plus[Times[A, b], Times[-1, a, B]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[m, n, 1]], ZeroQ[Plus[Times[a, A, m], Times[-1, b, B, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[A, b], Times[-1, a, B]], Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Power[Times[b, f, Plus[Times[2, m], 1]], -1]], x]], Dist[Times[Plus[Times[a, A, m], Times[b, B, Plus[m, 1]]], Power[Times[Power[a, 2], Plus[Times[2, m], 1]], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n]], x], x]], And[FreeQ[List[a, b, d, e, f, A, B, n], x], NonzeroQ[Plus[Times[A, b], Times[-1, a, B]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[m, n, 1]], RationalQ[m], LessEqual[m, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[A, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Power[Times[f, n], -1]], x], Times[-1, Dist[Times[Plus[Times[a, A, m], Times[-1, b, B, n]], Power[Times[b, d, n], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, 1]]], x], x]]], And[FreeQ[List[a, b, d, e, f, A, B, m, n], x], NonzeroQ[Plus[Times[A, b], Times[-1, a, B]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[m, n, 1]], Not[And[RationalQ[m], LessEqual[m, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[-2, b, B, Cot[Plus[e, Times[f, x]]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Power[Times[f, Plus[Times[2, n], 1], Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]]], -1]], x], And[FreeQ[List[a, b, d, e, f, A, B, n], x], NonzeroQ[Plus[Times[A, b], Times[-1, a, B]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], ZeroQ[Plus[Times[A, b, Plus[Times[2, n], 1]], Times[2, a, B, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[A, Power[b, 2], Cot[Plus[e, Times[f, x]]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Power[Times[a, f, n, Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]]], -1]], x], Dist[Times[Plus[Times[A, b, Plus[Times[2, n], 1]], Times[2, a, B, n]], Power[Times[2, a, d, n], -1]], Int[Times[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, 1]]], x], x]], And[FreeQ[List[a, b, d, e, f, A, B], x], NonzeroQ[Plus[Times[A, b], Times[-1, a, B]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Times[A, b, Plus[Times[2, n], 1]], Times[2, a, B, n]]], RationalQ[n], Less[n, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[-2, b, B, Cot[Plus[e, Times[f, x]]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Power[Times[f, Plus[Times[2, n], 1], Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]]], -1]], x], Dist[Times[Plus[Times[A, b, Plus[Times[2, n], 1]], Times[2, a, B, n]], Power[Times[b, Plus[Times[2, n], 1]], -1]], Int[Times[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n]], x], x]], And[FreeQ[List[a, b, d, e, f, A, B, n], x], NonzeroQ[Plus[Times[A, b], Times[-1, a, B]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], NonzeroQ[Plus[Times[A, b, Plus[Times[2, n], 1]], Times[2, a, B, n]]], Not[And[RationalQ[n], Less[n, 0]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[a, A, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Power[Times[f, n], -1]], x], Times[-1, Dist[Times[b, Power[Times[a, d, n], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, 1]], Simp[Plus[Times[a, A, Plus[m, Times[-1, n], -1]], Times[-1, b, B, n], Times[-1, Plus[Times[a, B, n], Times[A, b, Plus[m, n]]], Csc[Plus[e, Times[f, x]]]]], x]], x], x]]], And[FreeQ[List[a, b, d, e, f, A, B], x], NonzeroQ[Plus[Times[A, b], Times[-1, a, B]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m, n], Greater[m, Times[1, Power[2, -1]]], Less[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[b, B, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Power[Times[f, Plus[m, n]], -1]], x]], Dist[Times[1, Power[Times[d, Plus[m, n]], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Simp[Plus[Times[a, A, d, Plus[m, n]], Times[B, Times[b, d, n]], Times[Plus[Times[A, b, d, Plus[m, n]], Times[a, B, d, Plus[Times[2, m], n, -1]]], Csc[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, d, e, f, A, B, n], x], NonzeroQ[Plus[Times[A, b], Times[-1, a, B]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m], Greater[m, Times[1, Power[2, -1]]], Not[And[RationalQ[n], Less[n, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[d, Plus[Times[A, b], Times[-1, a, B]], Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, -1]], Power[Times[a, f, Plus[Times[2, m], 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[a, b, Plus[Times[2, m], 1]], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, -1]], Simp[Plus[Times[A, Times[a, d, Plus[n, -1]]], Times[-1, B, Times[b, d, Plus[n, -1]]], Times[-1, d, Plus[Times[a, B, Plus[m, Times[-1, n], 1]], Times[A, b, Plus[m, n]]], Csc[Plus[e, Times[f, x]]]]], x]], x], x]]], And[FreeQ[List[a, b, d, e, f, A, B], x], NonzeroQ[Plus[Times[A, b], Times[-1, a, B]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m, n], Less[m, Times[-1, Power[2, -1]]], Greater[n, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[A, b], Times[-1, a, B]], Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Power[Times[b, f, Plus[Times[2, m], 1]], -1]], x]], Times[-1, Dist[Times[1, Power[Times[Power[a, 2], Plus[Times[2, m], 1]], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Simp[Plus[Times[b, B, n], Times[-1, a, A, Plus[Times[2, m], n, 1]], Times[Plus[Times[A, b], Times[-1, a, B]], Plus[m, n, 1], Csc[Plus[e, Times[f, x]]]]], x]], x], x]]], And[FreeQ[List[a, b, d, e, f, A, B, n], x], NonzeroQ[Plus[Times[A, b], Times[-1, a, B]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m], Less[m, Times[-1, Power[2, -1]]], Not[And[RationalQ[n], Greater[n, 0]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[B, d, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, -1]], Power[Times[f, Plus[m, n]], -1]], x]], Dist[Times[d, Power[Times[b, Plus[m, n]], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, -1]], Simp[Plus[Times[b, B, Plus[n, -1]], Times[Plus[Times[A, b, Plus[m, n]], Times[a, B, m]], Csc[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, d, e, f, A, B, m], x], NonzeroQ[Plus[Times[A, b], Times[-1, a, B]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[n], Greater[n, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[A, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Power[Times[f, n], -1]], x], Times[-1, Dist[Times[1, Power[Times[b, d, n], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, 1]], Simp[Plus[Times[a, A, m], Times[-1, b, B, n], Times[-1, A, b, Plus[m, n, 1], Csc[Plus[e, Times[f, x]]]]], x]], x], x]]], And[FreeQ[List[a, b, d, e, f, A, B, m], x], NonzeroQ[Plus[Times[A, b], Times[-1, a, B]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[n], Less[n, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Plus[Times[A, b], Times[-1, a, B]], Power[b, -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n]], x], x], Dist[Times[B, Power[b, -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n]], x], x]], And[FreeQ[List[a, b, d, e, f, A, B, m], x], NonzeroQ[Plus[Times[A, b], Times[-1, a, B]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[a, A, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Power[Times[f, n], -1]], x], Dist[Times[1, Power[Times[d, n], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, -2]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, 1]], Simp[Plus[Times[a, Plus[Times[a, B, n], Times[-1, A, b, Plus[m, Times[-1, n], -1]]]], Times[Plus[Times[2, a, b, B, n], Times[A, Plus[Times[Power[b, 2], n], Times[Power[a, 2], Plus[1, n]]]]], Csc[Plus[e, Times[f, x]]]], Times[b, Plus[Times[b, B, n], Times[a, A, Plus[m, n]]], Power[Csc[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, d, e, f, A, B], x], NonzeroQ[Plus[Times[A, b], Times[-1, a, B]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m, n], Greater[m, 1], LessEqual[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[b, B, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Power[Times[f, Plus[m, n]], -1]], x]], Dist[Times[1, Power[Plus[m, n], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, -2]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Simp[Plus[Times[Power[a, 2], A, Plus[m, n]], Times[a, b, B, n], Times[Plus[Times[a, Plus[Times[2, A, b], Times[a, B]], Plus[m, n]], Times[Power[b, 2], B, Plus[m, n, -1]]], Csc[Plus[e, Times[f, x]]]], Times[b, Plus[Times[A, b, Plus[m, n]], Times[a, B, Plus[Times[2, m], n, -1]]], Power[Csc[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, d, e, f, A, B, n], x], NonzeroQ[Plus[Times[A, b], Times[-1, a, B]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m], Greater[m, 1], Not[And[IntegerQ[n], Greater[n, 1], Not[IntegerQ[m]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[d, Plus[Times[A, b], Times[-1, a, B]], Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, -1]], Power[Times[f, Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], x]], Dist[Times[1, Power[Times[Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, -1]], Simp[Plus[Times[d, Plus[n, -1], Plus[Times[A, b], Times[-1, a, B]]], Times[d, Plus[Times[a, A], Times[-1, b, B]], Plus[m, 1], Csc[Plus[e, Times[f, x]]]], Times[-1, d, Plus[Times[A, b], Times[-1, a, B]], Plus[m, n, 1], Power[Csc[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, d, e, f, A, B], x], NonzeroQ[Plus[Times[A, b], Times[-1, a, B]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m, n], Less[m, -1], Less[0, n, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[a, Power[d, 2], Plus[Times[A, b], Times[-1, a, B]], Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, -2]], Power[Times[b, f, Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], x], Times[-1, Dist[Times[d, Power[Times[b, Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, -2]], Simp[Plus[Times[a, d, Plus[Times[A, b], Times[-1, a, B]], Plus[n, -2]], Times[b, d, Plus[Times[A, b], Times[-1, a, B]], Plus[m, 1], Csc[Plus[e, Times[f, x]]]], Times[-1, Plus[Times[a, A, b, d, Plus[m, n]], Times[-1, d, B, Plus[Times[Power[a, 2], Plus[n, -1]], Times[Power[b, 2], Plus[m, 1]]]]], Power[Csc[Plus[e, Times[f, x]]], 2]]], x]], x], x]]], And[FreeQ[List[a, b, d, e, f, A, B], x], NonzeroQ[Plus[Times[A, b], Times[-1, a, B]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m, n], Less[m, -1], Greater[n, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, Plus[Times[A, b], Times[-1, a, B]], Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Power[Times[a, f, Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], x], Dist[Times[1, Power[Times[a, Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Simp[Plus[Times[A, Plus[Times[Power[a, 2], Plus[m, 1]], Times[-1, Power[b, 2], Plus[m, n, 1]]]], Times[a, b, B, n], Times[-1, a, Plus[Times[A, b], Times[-1, a, B]], Plus[m, 1], Csc[Plus[e, Times[f, x]]]], Times[b, Plus[Times[A, b], Times[-1, a, B]], Plus[m, n, 2], Power[Csc[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, d, e, f, A, B, n], x], NonzeroQ[Plus[Times[A, b], Times[-1, a, B]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m], Less[m, -1], Not[NegativeIntegerQ[Plus[m, Times[1, Power[2, -1]]], n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[B, d, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, -1]], Power[Times[f, Plus[m, n]], -1]], x]], Dist[Times[d, Power[Plus[m, n], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, -1]], Simp[Plus[Times[a, B, Plus[n, -1]], Times[Plus[Times[b, B, Plus[m, n, -1]], Times[a, A, Plus[m, n]]], Csc[Plus[e, Times[f, x]]]], Times[Plus[Times[a, B, m], Times[A, b, Plus[m, n]]], Power[Csc[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, d, e, f, A, B], x], NonzeroQ[Plus[Times[A, b], Times[-1, a, B]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m, n], Less[0, m, 1], Greater[n, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[A, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Power[Times[f, n], -1]], x], Times[-1, Dist[Times[1, Power[Times[d, n], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, 1]], Simp[Plus[Times[A, b, m], Times[-1, a, B, n], Times[-1, Plus[Times[b, B, n], Times[a, A, Plus[n, 1]]], Csc[Plus[e, Times[f, x]]]], Times[-1, A, b, Plus[m, n, 1], Power[Csc[Plus[e, Times[f, x]]], 2]]], x]], x], x]]], And[FreeQ[List[a, b, d, e, f, A, B], x], NonzeroQ[Plus[Times[A, b], Times[-1, a, B]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m, n], Less[0, m, 1], LessEqual[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[B, Power[d, 2], Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, -2]], Power[Times[b, f, Plus[m, n]], -1]], x]], Dist[Times[Power[d, 2], Power[Times[b, Plus[m, n]], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, -2]], Simp[Plus[Times[a, B, Plus[n, -2]], Times[B, b, Plus[m, n, -1], Csc[Plus[e, Times[f, x]]]], Times[Plus[Times[A, b, Plus[m, n]], Times[-1, a, B, Plus[n, -1]]], Power[Csc[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, d, e, f, A, B, m], x], NonzeroQ[Plus[Times[A, b], Times[-1, a, B]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[n], Greater[n, 1], NonzeroQ[Plus[m, n]], Not[And[IntegerQ[m], Greater[m, 1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[A, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Power[Times[a, f, n], -1]], x], Dist[Times[1, Power[Times[a, d, n], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, 1]], Simp[Plus[Times[a, B, n], Times[-1, A, b, Plus[m, n, 1]], Times[A, a, Plus[n, 1], Csc[Plus[e, Times[f, x]]]], Times[A, b, Plus[m, n, 2], Power[Csc[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, d, e, f, A, B, m], x], NonzeroQ[Plus[Times[A, b], Times[-1, a, B]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[n], LessEqual[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[A, Power[a, -1]], Int[Times[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Power[Sqrt[Times[d, Csc[Plus[e, Times[f, x]]]]], -1]], x], x], Times[-1, Dist[Times[Plus[Times[A, b], Times[-1, a, B]], Power[Times[a, d], -1]], Int[Times[Sqrt[Times[d, Csc[Plus[e, Times[f, x]]]]], Power[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x]]], And[FreeQ[List[a, b, d, e, f, A, B], x], NonzeroQ[Plus[Times[A, b], Times[-1, a, B]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[A, Int[Times[Sqrt[Times[d, Csc[Plus[e, Times[f, x]]]]], Power[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x], Dist[Times[B, Power[d, -1]], Int[Times[Power[Times[d, Csc[Plus[e, Times[f, x]]]], Times[3, Power[2, -1]]], Power[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x]], And[FreeQ[List[a, b, d, e, f, A, B], x], NonzeroQ[Plus[Times[A, b], Times[-1, a, B]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[1, 2]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[B, Power[d, -1]], Int[Times[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Sqrt[Times[d, Csc[Plus[e, Times[f, x]]]]]], x], x], Dist[A, Int[Times[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], Power[Sqrt[Times[d, Csc[Plus[e, Times[f, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, d, e, f, A, B], x], NonzeroQ[Plus[Times[A, b], Times[-1, a, B]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[A, Power[a, -1]], Int[Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], x], x], Times[-1, Dist[Times[Plus[Times[A, b], Times[-1, a, B]], Power[Times[a, d], -1]], Int[Times[Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, 1]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], -1]], x], x]]], And[FreeQ[List[a, b, d, e, f, A, B, n], x], NonzeroQ[Plus[Times[A, b], Times[-1, a, B]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Plus[A, Times[B, Csc[Plus[e, Times[f, x]]]]]], x], And[FreeQ[List[a, b, d, e, f, A, B, m, n], x], NonzeroQ[Plus[Times[A, b], Times[-1, a, B]]], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[c, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[Times[-1, a, c], m], Int[Times[Power[Cos[Plus[e, Times[f, x]]], Times[2, m]], Power[Plus[d, Times[c, Sin[Plus[e, Times[f, x]]]]], Plus[n, Times[-1, m]]], Power[Plus[B, Times[A, Sin[Plus[e, Times[f, x]]]]], p], Power[Power[Sin[Plus[e, Times[f, x]]], Plus[m, n, p]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, A, B, n, p], x], ZeroQ[Plus[Times[b, c], Times[a, d]]], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], IntegersQ[m, n, p]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Power[b, 2], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Simp[Plus[Times[b, B], Times[-1, a, C], Times[b, C, Csc[Plus[e, Times[f, x]]]]], x]], x], x], And[FreeQ[List[a, b, e, f, A, B, C, m], x], ZeroQ[Plus[Times[A, Power[b, 2]], Times[-1, a, b, B], Times[Power[a, 2], C]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[C, Power[Power[b, 2], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Simp[Plus[Times[-1, a], Times[b, Csc[Plus[e, Times[f, x]]]]], x]], x], x], And[FreeQ[List[a, b, e, f, A, C, m], x], ZeroQ[Plus[Times[A, Power[b, 2]], Times[Power[a, 2], C]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[A, Cot[Plus[e, Times[f, x]]], Power[Times[b, Csc[Plus[e, Times[f, x]]]], m], Power[Times[f, m], -1]], x], And[FreeQ[List[b, e, f, A, C, m], x], ZeroQ[Plus[Times[C, m], Times[A, Plus[m, 1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Plus[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Plus[C, Times[A, Power[Sin[Plus[e, Times[f, x]]], 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RationalQ[m], LessEqual[m, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]], Pattern[A, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[C, Cot[Plus[e, Times[f, x]]], Power[Times[b, Csc[Plus[e, Times[f, x]]]], m], Power[Times[f, Plus[m, 1]], -1]], x]], Dist[Times[Plus[Times[C, m], Times[A, Plus[m, 1]]], Power[Plus[m, 1], -1]], Int[Power[Times[b, Csc[Plus[e, Times[f, x]]]], m], x], x]], And[FreeQ[List[b, e, f, A, C, m], x], NonzeroQ[Plus[Times[C, m], Times[A, Plus[m, 1]]]], Not[And[RationalQ[m], LessEqual[m, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[B, Power[b, -1]], Int[Power[Times[b, Csc[Plus[e, Times[f, x]]]], Plus[m, 1]], x], x], Int[Times[Power[Times[b, Csc[Plus[e, Times[f, x]]]], m], Plus[A, Times[C, Power[Csc[Plus[e, Times[f, x]]], 2]]]], x]], FreeQ[List[b, e, f, A, B, C, m], x]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[b, C, Csc[Plus[e, Times[f, x]]], Cot[Plus[e, Times[f, x]]], Power[Times[2, f], -1]], x]], Dist[Times[1, Power[2, -1]], Int[Simp[Plus[Times[2, A, a], Times[Plus[Times[2, B, a], Times[b, Plus[Times[2, A], C]]], Csc[Plus[e, Times[f, x]]]], Times[2, Plus[Times[a, C], Times[B, b]], Power[Csc[Plus[e, Times[f, x]]], 2]]], x], x], x]], FreeQ[List[a, b, e, f, A, B, C], x]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[b, C, Csc[Plus[e, Times[f, x]]], Cot[Plus[e, Times[f, x]]], Power[Times[2, f], -1]], x]], Dist[Times[1, Power[2, -1]], Int[Simp[Plus[Times[2, A, a], Times[b, Plus[Times[2, A], C], Csc[Plus[e, Times[f, x]]]], Times[2, a, C, Power[Csc[Plus[e, Times[f, x]]], 2]]], x], x], x]], FreeQ[List[a, b, e, f, A, C], x]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[C, Power[b, -1]], Int[Csc[Plus[e, Times[f, x]]], x], x], Dist[Times[1, Power[b, -1]], Int[Times[Plus[Times[A, b], Times[Plus[Times[b, B], Times[-1, a, C]], Csc[Plus[e, Times[f, x]]]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], -1]], x], x]], FreeQ[List[a, b, e, f, A, B, C], x]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[C, Power[b, -1]], Int[Csc[Plus[e, Times[f, x]]], x], x], Dist[Times[1, Power[b, -1]], Int[Times[Plus[Times[A, b], Times[-1, a, C, Csc[Plus[e, Times[f, x]]]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], -1]], x], x]], FreeQ[List[a, b, e, f, A, C], x]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[a, A], Times[-1, b, B], Times[a, C]], Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[a, f, Plus[Times[2, m], 1]], -1]], x]], Dist[Times[1, Power[Times[a, b, Plus[Times[2, m], 1]], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Simp[Plus[Times[A, b, Plus[Times[2, m], 1]], Times[Plus[Times[b, B, Plus[m, 1]], Times[-1, a, Plus[Times[A, Plus[m, 1]], Times[-1, C, m]]]], Csc[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, e, f, A, B, C], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m], Less[m, Times[-1, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[a, Plus[A, C], Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[a, f, Plus[Times[2, m], 1]], -1]], x]], Dist[Times[1, Power[Times[a, b, Plus[Times[2, m], 1]], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Simp[Plus[Times[A, b, Plus[Times[2, m], 1]], Times[-1, a, Plus[Times[A, Plus[m, 1]], Times[-1, C, m]], Csc[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, e, f, A, C], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m], Less[m, Times[-1, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[C, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[f, Plus[m, 1]], -1]], x]], Dist[Times[1, Power[Times[b, Plus[m, 1]], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Simp[Plus[Times[A, b, Plus[m, 1]], Times[Plus[Times[a, C, m], Times[b, B, Plus[m, 1]]], Csc[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, e, f, A, B, C, m], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], Not[And[RationalQ[m], Less[m, Times[-1, Power[2, -1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[C, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[f, Plus[m, 1]], -1]], x]], Dist[Times[1, Power[Times[b, Plus[m, 1]], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Simp[Plus[Times[A, b, Plus[m, 1]], Times[a, C, m, Csc[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, e, f, A, C, m], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], Not[And[RationalQ[m], Less[m, Times[-1, Power[2, -1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], 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Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[C, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[f, Plus[m, 1]], -1]], x]], Dist[Times[1, Power[Plus[m, 1], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, -1]], Simp[Plus[Times[a, A, Plus[m, 1]], Times[Plus[Times[A, b, Plus[m, 1]], Times[b, C, m]], Csc[Plus[e, Times[f, x]]]], Times[a, C, m, Power[Csc[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, e, f, A, C], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], PositiveIntegerQ[Times[2, m]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Int[Times[Plus[A, Times[Plus[B, Times[-1, C]], Csc[Plus[e, Times[f, x]]]]], Power[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], -1]], x], Dist[C, Int[Times[Csc[Plus[e, Times[f, x]]], Plus[1, Csc[Plus[e, Times[f, x]]]], Power[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x]], And[FreeQ[List[a, b, e, f, A, B, C], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, 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2]]], x]], x], x]], And[FreeQ[List[a, b, e, f, A, B, C], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m], Less[m, -1]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[A, Power[b, 2]], Times[Power[a, 2], C]], Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[a, f, Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], x], Dist[Times[1, Power[Times[a, Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], 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Csc[Plus[e, Times[f, x]]]]], m], Power[Times[a, f, Plus[Times[2, m], 1]], -1]], x]], Times[-1, Dist[Times[1, Power[Times[a, b, Plus[Times[2, m], 1]], -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Simp[Plus[Times[a, B], Times[-1, b, C], Times[-1, 2, A, b, Plus[m, 1]], Times[-1, Plus[Times[b, B, Plus[m, 2]], Times[-1, a, Plus[Times[A, Plus[m, 2]], Times[-1, C, Plus[m, -1]]]]], Csc[Plus[e, Times[f, x]]]]], x]], x], x]]], And[FreeQ[List[a, b, e, f, A, B, C], x], RationalQ[m], Less[m, -1], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], 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1]], Simp[Plus[Times[b, Plus[Times[a, A], Times[-1, b, B], Times[a, C]], Plus[m, 1]], Times[-1, Plus[Times[A, Power[b, 2]], Times[-1, a, b, B], Times[Power[a, 2], C], Times[b, Plus[Times[A, b], Times[-1, a, B], Times[b, C]], Plus[m, 1]]], Csc[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, e, f, A, B, C], x], RationalQ[m], Less[m, -1], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[A, 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Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[C, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[b, f, Plus[m, 2]], -1]], x]], Dist[Times[1, Power[Times[b, Plus[m, 2]], -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Simp[Plus[Times[b, A, Plus[m, 2]], Times[b, C, Plus[m, 1]], Times[Plus[Times[b, B, Plus[m, 2]], Times[-1, a, C]], Csc[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, e, f, A, B, C, m], x], Not[And[RationalQ[m], Less[m, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[C, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[b, f, Plus[m, 2]], -1]], x]], Dist[Times[1, Power[Times[b, Plus[m, 2]], -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Simp[Plus[Times[b, A, Plus[m, 2]], Times[b, C, Plus[m, 1]], Times[-1, a, C, Csc[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, e, f, A, C, m], x], Not[And[RationalQ[m], Less[m, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[Times[a, A], Times[-1, b, B], Times[a, C]], Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[d, 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2]]]], RationalQ[m], Less[m, Times[-1, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[A, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Power[Times[f, n], -1]], x], Times[-1, Dist[Times[1, Power[Times[b, d, n], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, 1]], Simp[Plus[Times[a, A, m], Times[-1, b, B, n], Times[-1, b, Plus[Times[A, Plus[m, n, 1]], Times[C, n]], Csc[Plus[e, Times[f, x]]]]], x]], x], x]]], And[FreeQ[List[a, b, d, e, f, A, B, C, m], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], Not[And[RationalQ[m], Less[m, Times[-1, Power[2, -1]]]]], Or[And[RationalQ[n], Less[n, Times[-1, Power[2, -1]]]], ZeroQ[Plus[m, n, 1]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, 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Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[C, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Power[Times[f, Plus[m, n, 1]], -1]], x]], Dist[Times[1, Power[Times[b, Plus[m, n, 1]], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Simp[Plus[Times[A, b, Plus[m, n, 1]], Times[b, C, n], Times[Plus[Times[a, C, m], Times[b, B, Plus[m, n, 1]]], Csc[Plus[e, Times[f, x]]]]], x]], x], x]], And[FreeQ[List[a, b, d, e, f, A, B, C, m, n], x], ZeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], Not[And[RationalQ[m], Less[m, Times[-1, Power[2, -1]]]]], Not[And[RationalQ[n], Less[n, Times[-1, Power[2, -1]]]]], NonzeroQ[Plus[m, n, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], 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Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[a, Plus[Times[A, Power[b, 2]], Times[-1, a, b, B], Times[Power[a, 2], C]], Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[Power[b, 2], f, Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], x], Times[-1, Dist[Times[1, Power[Times[Power[b, 2], Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Simp[Plus[Times[b, Plus[m, 1], Plus[Times[-1, a, Plus[Times[b, B], Times[-1, a, C]]], Times[A, Power[b, 2]]]], Times[Plus[Times[b, B, Plus[Power[a, 2], Times[Power[b, 2], Plus[m, 1]]]], Times[-1, a, Plus[Times[A, Power[b, 2], Plus[m, 2]], Times[C, Plus[Power[a, 2], Times[Power[b, 2], Plus[m, 1]]]]]]], Csc[Plus[e, Times[f, x]]]], Times[-1, b, C, Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]], Power[Csc[Plus[e, Times[f, x]]], 2]]], x]], x], x]]], And[FreeQ[List[a, b, e, f, A, B, C], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m], Less[m, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[a, Plus[Times[A, Power[b, 2]], Times[Power[a, 2], C]], Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[Power[b, 2], f, Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], x], Times[-1, Dist[Times[1, Power[Times[Power[b, 2], Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Simp[Plus[Times[b, Plus[m, 1], Plus[Times[Power[a, 2], C], Times[A, Power[b, 2]]]], Times[-1, a, Plus[Times[A, Power[b, 2], Plus[m, 2]], Times[C, Plus[Power[a, 2], Times[Power[b, 2], Plus[m, 1]]]]], Csc[Plus[e, Times[f, x]]]], Times[-1, b, C, Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]], Power[Csc[Plus[e, Times[f, x]]], 2]]], x]], x], x]]], And[FreeQ[List[a, b, e, f, A, C], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m], Less[m, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[C, Csc[Plus[e, Times[f, x]]], Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[b, f, Plus[m, 3]], -1]], x]], Dist[Times[1, Power[Times[b, Plus[m, 3]], -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Simp[Plus[Times[a, C], Times[b, Plus[Times[C, Plus[m, 2]], Times[A, Plus[m, 3]]], Csc[Plus[e, Times[f, x]]]], Times[-1, Plus[Times[2, a, C], Times[-1, b, B, Plus[m, 3]]], Power[Csc[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, e, f, A, B, C, m], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], Not[And[RationalQ[m], Less[m, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[C, Csc[Plus[e, Times[f, x]]], Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[b, f, Plus[m, 3]], -1]], x]], Dist[Times[1, Power[Times[b, Plus[m, 3]], -1]], Int[Times[Csc[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Simp[Plus[Times[a, C], Times[b, Plus[Times[C, Plus[m, 2]], Times[A, Plus[m, 3]]], Csc[Plus[e, Times[f, x]]]], Times[-1, 2, a, C, Power[Csc[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, e, f, A, C, m], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], Not[And[RationalQ[m], Less[m, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[A, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Power[Times[f, n], -1]], x], Times[-1, Dist[Times[1, Power[Times[d, n], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, 1]], Simp[Plus[Times[A, b, m], Times[-1, a, B, n], Times[-1, Plus[Times[b, B, n], Times[a, Plus[Times[C, n], Times[A, Plus[n, 1]]]]], Csc[Plus[e, Times[f, x]]]], Times[-1, b, Plus[Times[C, n], Times[A, Plus[m, n, 1]]], Power[Csc[Plus[e, Times[f, x]]], 2]]], x]], x], x]]], And[FreeQ[List[a, b, d, e, f, A, B, C], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m, n], Greater[m, 0], LessEqual[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[A, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Power[Times[f, n], -1]], x], Times[-1, Dist[Times[1, Power[Times[d, n], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, 1]], Simp[Plus[Times[A, b, m], Times[-1, a, Plus[Times[C, n], Times[A, Plus[n, 1]]], Csc[Plus[e, Times[f, x]]]], Times[-1, b, Plus[Times[C, n], Times[A, Plus[m, n, 1]]], Power[Csc[Plus[e, Times[f, x]]], 2]]], x]], x], x]]], And[FreeQ[List[a, b, d, e, f, A, C], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m, n], Greater[m, 0], LessEqual[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[C, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Power[Times[f, Plus[m, n, 1]], -1]], x]], Dist[Times[1, Power[Plus[m, n, 1], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Simp[Plus[Times[a, A, Plus[m, n, 1]], Times[a, C, n], Times[Plus[Times[Plus[Times[A, b], Times[a, B]], Plus[m, n, 1]], Times[b, C, Plus[m, n]]], Csc[Plus[e, Times[f, x]]]], Times[Plus[Times[b, B, Plus[m, n, 1]], Times[a, C, m]], Power[Csc[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, d, e, f, A, B, C, n], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m], Greater[m, 0], Not[And[RationalQ[n], LessEqual[n, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[C, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Power[Times[f, Plus[m, n, 1]], -1]], x]], Dist[Times[1, Power[Plus[m, n, 1], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, -1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Simp[Plus[Times[a, A, Plus[m, n, 1]], Times[a, C, n], Times[b, Plus[Times[A, Plus[m, n, 1]], Times[C, Plus[m, n]]], Csc[Plus[e, Times[f, x]]]], Times[a, C, m, Power[Csc[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, d, e, f, A, C, n], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m], Greater[m, 0], Not[And[RationalQ[n], LessEqual[n, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[d, Plus[Times[A, Power[b, 2]], Times[-1, a, b, B], Times[Power[a, 2], C]], Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, -1]], Power[Times[b, f, Plus[Power[a, 2], Times[-1, Power[b, 2]]], Plus[m, 1]], -1]], x]], Dist[Times[d, Power[Times[b, Plus[Power[a, 2], Times[-1, Power[b, 2]]], Plus[m, 1]], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, -1]], Simp[Plus[Times[A, Power[b, 2], Plus[n, -1]], Times[-1, a, Plus[Times[b, B], Times[-1, a, C]], Plus[n, -1]], Times[b, Plus[Times[a, A], Times[-1, b, B], Times[a, C]], Plus[m, 1], Csc[Plus[e, Times[f, x]]]], Times[-1, Plus[Times[b, Plus[Times[A, b], Times[-1, a, B]], Plus[m, n, 1]], Times[C, Plus[Times[Power[a, 2], n], Times[Power[b, 2], Plus[m, 1]]]]], Power[Csc[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, d, e, f, A, B, C], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m, n], Less[m, -1], Greater[n, 0]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, 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NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[m, n], Less[m, -1], Greater[n, 0]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[A, Power[b, 2]], Times[-1, a, b, B], Times[Power[a, 2], C]], Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, 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n]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[A, Power[b, 2]], Times[Power[a, 2], C]], Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Power[Times[a, f, Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 2]]]], -1]], x], Dist[Times[1, Power[Times[a, Plus[m, 1], Plus[Power[a, 2], Times[-1, Power[b, 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n]], Power[Csc[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, d, e, f, A, B, C, m], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[n], Greater[n, 0]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[C, d, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, -1]], 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LessEqual[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Pattern[n, Blank[]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[A, Cot[Plus[e, Times[f, x]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], Plus[m, 1]], Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Power[Times[a, f, n], -1]], x], Dist[Times[1, Power[Times[a, d, n], -1]], Int[Times[Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Power[Times[d, Csc[Plus[e, Times[f, x]]]], Plus[n, 1]], Simp[Plus[Times[-1, A, b, Plus[m, n, 1]], Times[a, Plus[A, Times[A, n], Times[C, n]], Csc[Plus[e, Times[f, x]]]], Times[A, b, Plus[m, n, 2], Power[Csc[Plus[e, Times[f, x]]], 2]]], x]], x], x]], And[FreeQ[List[a, b, d, e, f, A, C, m], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]], RationalQ[n], LessEqual[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[Plus[Times[A, Power[b, 2]], Times[-1, a, b, B], Times[Power[a, 2], C]], Power[Times[Power[a, 2], Power[d, 2]], -1]], Int[Times[Power[Times[d, Csc[Plus[e, Times[f, x]]]], Times[3, Power[2, -1]]], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], -1]], x], x], Dist[Times[1, Power[Power[a, 2], -1]], Int[Times[Plus[Times[a, A], Times[-1, Plus[Times[A, b], Times[-1, a, B]], Csc[Plus[e, Times[f, x]]]]], Power[Sqrt[Times[d, Csc[Plus[e, Times[f, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, d, e, f, A, B, C], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 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Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[C, Power[Power[d, 2], -1]], Int[Times[Power[Times[d, Csc[Plus[e, Times[f, x]]]], Times[3, Power[2, -1]]], Power[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x], Int[Times[Plus[A, Times[B, Csc[Plus[e, Times[f, x]]]]], Power[Times[Sqrt[Times[d, Csc[Plus[e, Times[f, x]]]]], Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]]], -1]], x]], And[FreeQ[List[a, b, d, e, f, A, B, C], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Rational[-1, 2]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[C, Power[Power[d, 2], -1]], Int[Times[Power[Times[d, Csc[Plus[e, Times[f, x]]]], Times[3, Power[2, -1]]], Power[Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]], -1]], x], x], Dist[A, Int[Times[1, Power[Times[Sqrt[Times[d, Csc[Plus[e, Times[f, x]]]]], Sqrt[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]]]], -1]], x], x]], And[FreeQ[List[a, b, d, e, f, A, C], x], NonzeroQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Times[d, Csc[Plus[e, Times[f, x]]]], n], Power[Plus[a, Times[b, Csc[Plus[e, Times[f, x]]]]], m], Plus[A, Times[B, 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RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[d, Plus[m, 2]], Int[Times[Power[Plus[b, Times[a, Cos[Plus[e, Times[f, x]]]]], m], Power[Times[d, Cos[Plus[e, Times[f, x]]]], Plus[n, Times[-1, m], -2]], Plus[C, Times[B, Cos[Plus[e, Times[f, x]]]], Times[A, 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m], Power[Times[d, Sin[Plus[e, Times[f, x]]]], Plus[n, Times[-1, m], -2]], Plus[C, Times[B, Sin[Plus[e, Times[f, x]]]], Times[A, Power[Sin[Plus[e, Times[f, x]]], 2]]]], x], x], And[FreeQ[List[a, b, d, e, f, A, B, C, n], x], Not[IntegerQ[n]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[d, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], -1]], Pattern[n, Blank[]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[m, Blank[]]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[sec[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[d, Plus[m, 2]], Int[Times[Power[Plus[b, Times[a, Cos[Plus[e, Times[f, x]]]]], m], 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Condition[Dist[Times[Power[c, IntPart[n]], Power[Times[c, Power[Times[d, Sec[Plus[e, Times[f, x]]]], p]], FracPart[n]], Power[Power[Times[d, Sec[Plus[e, Times[f, x]]]], Times[p, FracPart[n]]], -1]], Int[Times[Power[Plus[a, Times[b, Sec[Plus[e, Times[f, x]]]]], m], Power[Times[d, Sec[Plus[e, Times[f, x]]]], Times[n, p]], Plus[A, Times[B, Sec[Plus[e, Times[f, x]]]], Times[C, Power[Sec[Plus[e, Times[f, x]]], 2]]]], x], x], And[FreeQ[List[a, b, c, d, e, f, A, B, C, m, n, p], x], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Power[Times[Optional[Pattern[c, Blank[]]], Power[Times[csc[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], 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Subst[Int[Times[Power[x, Plus[Times[1, Power[n, -1]], -1]], Power[Plus[a, Times[b, Sec[Plus[c, Times[d, x]]]]], p]], x], x, Power[x, n]], x], And[FreeQ[List[a, b, c, d, p], x], PositiveIntegerQ[Times[1, Power[n, -1]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[n, -1]], Subst[Int[Times[Power[x, Plus[Times[1, Power[n, -1]], -1]], Power[Plus[a, Times[b, Csc[Plus[c, Times[d, x]]]]], p]], x], x, Power[x, n]], x], And[FreeQ[List[a, b, c, d, p], x], PositiveIntegerQ[Times[1, Power[n, -1]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sec[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Power[Plus[a, Times[b, Sec[Plus[c, Times[d, Power[x, n]]]]]], p], x], FreeQ[List[a, b, c, d, n, p], x]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Power[Plus[a, Times[b, Csc[Plus[c, Times[d, Power[x, n]]]]]], p], x], FreeQ[List[a, b, c, d, n, p], x]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sec[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, 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Sec[ExpandToSum[u, x]]]], p], x], And[FreeQ[List[a, b, p], x], BinomialQ[u, x], Not[BinomialMatchQ[u, x]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Csc[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Power[Plus[a, Times[b, Csc[ExpandToSum[u, x]]]], p], x], And[FreeQ[List[a, b, p], x], BinomialQ[u, x], Not[BinomialMatchQ[u, x]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sec[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[n, -1]], Subst[Int[Times[Power[x, Plus[Simplify[Times[Plus[m, 1], Power[n, -1]]], -1]], Power[Plus[a, Times[b, Sec[Plus[c, Times[d, x]]]]], p]], x], x, Power[x, n]], x], And[FreeQ[List[a, b, c, d, m, n, p], x], PositiveIntegerQ[Simplify[Times[Plus[m, 1], Power[n, -1]]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[n, -1]], Subst[Int[Times[Power[x, Plus[Simplify[Times[Plus[m, 1], Power[n, -1]]], -1]], Power[Plus[a, Times[b, Csc[Plus[c, Times[d, x]]]]], p]], x], x, Power[x, n]], x], And[FreeQ[List[a, b, c, d, m, n, p], x], PositiveIntegerQ[Simplify[Times[Plus[m, 1], Power[n, -1]]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sec[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[x, m], Power[Plus[a, Times[b, Sec[Plus[c, Times[d, Power[x, n]]]]]], p]], x], FreeQ[List[a, b, c, d, m, n, p], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[x, m], Power[Plus[a, Times[b, Csc[Plus[c, Times[d, Power[x, n]]]]]], p]], x], FreeQ[List[a, b, c, d, m, n, p], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sec[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[e, IntPart[m]], Power[Times[e, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Plus[a, Times[b, Sec[Plus[c, Times[d, Power[x, n]]]]]], p]], x], x], FreeQ[List[a, b, c, d, e, m, n, p], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Csc[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[e, IntPart[m]], Power[Times[e, x], FracPart[m]], 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Power[Times[Optional[Pattern[d, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[c, Tan[Plus[a, Times[b, x]]]], m], Power[Times[d, Cos[Plus[a, Times[b, x]]]], m], Power[Power[Times[d, Sin[Plus[a, Times[b, x]]]], m], -1]], Int[Times[ActivateTrig[u], Power[Times[d, Sin[Plus[a, Times[b, x]]]], Plus[m, n]], Power[Power[Times[d, Cos[Plus[a, Times[b, x]]]], m], -1]], x], x], And[FreeQ[List[a, b, c, d, m, n], x], KnownSineIntegrandQ[u, x], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[n, 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Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[c, Cot[Plus[a, Times[b, x]]]], m], Power[Times[d, Sin[Plus[a, Times[b, x]]]], m], Power[Power[Times[d, Cos[Plus[a, Times[b, x]]]], m], -1]], Int[Times[ActivateTrig[u], Power[Times[d, Cos[Plus[a, Times[b, x]]]], m], Power[Power[Times[d, Sin[Plus[a, Times[b, x]]]], Plus[m, Times[-1, n]]], -1]], x], x], And[FreeQ[List[a, b, c, d, m, n], x], KnownSineIntegrandQ[u, x], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[c, Cot[Plus[a, Times[b, x]]]], m], Power[Times[d, Sin[Plus[a, Times[b, x]]]], m], Power[Power[Times[d, Cos[Plus[a, Times[b, x]]]], m], -1]], Int[Times[ActivateTrig[u], Power[Times[d, Cos[Plus[a, Times[b, x]]]], Plus[m, n]], Power[Power[Times[d, Sin[Plus[a, Times[b, x]]]], m], -1]], x], x], And[FreeQ[List[a, b, c, d, m, n], x], KnownSineIntegrandQ[u, x], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[u, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[c, Csc[Plus[a, Times[b, x]]]], m], Power[Times[d, Sin[Plus[a, Times[b, x]]]], m]], Int[Times[ActivateTrig[u], 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KnownSineIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[m, Blank[]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[c, Csc[Plus[a, Times[b, x]]]], m], Power[Times[c, Sin[Plus[a, Times[b, x]]]], m]], Int[Times[ActivateTrig[u], Power[Power[Times[c, Sin[Plus[a, Times[b, x]]]], m], -1]], x], x], And[FreeQ[List[a, b, c, m], x], Not[IntegerQ[m]], KnownSineIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Times[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]], Pattern[u, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[c, Int[Times[ActivateTrig[u], Power[Times[c, Sin[Plus[a, Times[b, x]]]], Plus[n, -1]], Plus[B, Times[A, Sin[Plus[a, Times[b, x]]]]]], x], x], And[FreeQ[List[a, b, c, A, B, n], x], KnownSineIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[u, Blank[]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[c, Int[Times[ActivateTrig[u], Power[Times[c, Cos[Plus[a, Times[b, x]]]], Plus[n, -1]], Plus[B, Times[A, Cos[Plus[a, Times[b, x]]]]]], x], x], And[FreeQ[List[a, b, c, A, B, n], x], KnownSineIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Times[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[ActivateTrig[u], Plus[B, Times[A, Sin[Plus[a, Times[b, x]]]]], Power[Sin[Plus[a, Times[b, x]]], -1]], x], And[FreeQ[List[a, b, A, B], x], KnownSineIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[ActivateTrig[u], Plus[B, Times[A, Cos[Plus[a, Times[b, x]]]]], Power[Cos[Plus[a, Times[b, x]]], -1]], x], And[FreeQ[List[a, b, A, B], x], KnownSineIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, 2], Int[Times[ActivateTrig[u], Power[Times[c, Sin[Plus[a, Times[b, x]]]], Plus[n, -2]], Plus[C, Times[B, Sin[Plus[a, Times[b, x]]]], Times[A, Power[Sin[Plus[a, Times[b, x]]], 2]]]], x], x], And[FreeQ[List[a, b, c, A, B, C, n], x], KnownSineIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, 2], Int[Times[ActivateTrig[u], Power[Times[c, Cos[Plus[a, Times[b, x]]]], Plus[n, -2]], Plus[C, Times[B, Cos[Plus[a, Times[b, x]]]], Times[A, Power[Cos[Plus[a, Times[b, x]]], 2]]]], x], x], And[FreeQ[List[a, b, c, A, B, C, n], x], KnownSineIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Optional[Pattern[u, Blank[]]], Plus[Times[Power[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]], Pattern[A, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, 2], Int[Times[ActivateTrig[u], Power[Times[c, Sin[Plus[a, Times[b, x]]]], Plus[n, -2]], Plus[C, Times[A, Power[Sin[Plus[a, Times[b, x]]], 2]]]], x], x], And[FreeQ[List[a, b, c, A, C, n], x], KnownSineIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, 2], Int[Times[ActivateTrig[u], Power[Times[c, Cos[Plus[a, Times[b, x]]]], Plus[n, -2]], Plus[C, Times[A, Power[Cos[Plus[a, Times[b, x]]], 2]]]], x], x], And[FreeQ[List[a, b, c, A, C, n], x], KnownSineIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[ActivateTrig[u], Plus[C, Times[B, Sin[Plus[a, Times[b, x]]]], Times[A, Power[Sin[Plus[a, Times[b, x]]], 2]]], Power[Power[Sin[Plus[a, Times[b, x]]], 2], -1]], x], And[FreeQ[List[a, b, A, B, C], x], KnownSineIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[ActivateTrig[u], Plus[C, Times[B, Cos[Plus[a, Times[b, x]]]], Times[A, Power[Cos[Plus[a, Times[b, x]]], 2]]], Power[Power[Cos[Plus[a, Times[b, x]]], 2], -1]], x], And[FreeQ[List[a, b, A, B, C], x], KnownSineIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Times[Power[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]], Pattern[A, Blank[]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[ActivateTrig[u], Plus[C, Times[A, Power[Sin[Plus[a, Times[b, x]]], 2]]], Power[Power[Sin[Plus[a, Times[b, x]]], 2], -1]], x], And[FreeQ[List[a, b, A, C], x], KnownSineIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[ActivateTrig[u], Plus[C, Times[A, Power[Cos[Plus[a, Times[b, x]]], 2]]], Power[Power[Cos[Plus[a, Times[b, x]]], 2], -1]], x], And[FreeQ[List[a, b, A, C], x], KnownSineIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Plus[Optional[Pattern[A, Blank[]]], Times[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[C, Blank[]]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[ActivateTrig[u], Plus[C, Times[A, Sin[Plus[a, Times[b, x]]]], Times[B, Power[Sin[Plus[a, Times[b, x]]], 2]]], Power[Sin[Plus[a, Times[b, x]]], -1]], x], FreeQ[List[a, b, A, B, C], x]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Optional[Pattern[C, Blank[]]], sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[ActivateTrig[u], Plus[C, Times[A, Cos[Plus[a, Times[b, x]]]], Times[B, Power[Cos[Plus[a, Times[b, x]]], 2]]], Power[Cos[Plus[a, Times[b, x]]], -1]], x], FreeQ[List[a, b, A, B, C], x]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Plus[Times[Optional[Pattern[A, Blank[]]], Power[sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[B, Blank[]]], Power[sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n1, Blank[]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[ActivateTrig[u], Power[Sin[Plus[a, Times[b, x]]], n], Plus[A, Times[B, Sin[Plus[a, Times[b, x]]]], Times[C, Power[Sin[Plus[a, Times[b, x]]], 2]]]], x], And[FreeQ[List[a, b, A, B, C, n], x], ZeroQ[Plus[n1, Times[-1, n], -1]], ZeroQ[Plus[n2, Times[-1, n], -2]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Times[Power[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[A, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n1, Blank[]]], Optional[Pattern[B, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]], Optional[Pattern[C, Blank[]]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[ActivateTrig[u], Power[Cos[Plus[a, Times[b, x]]], n], Plus[A, Times[B, Cos[Plus[a, Times[b, x]]]], Times[C, Power[Cos[Plus[a, Times[b, x]]], 2]]]], x], And[FreeQ[List[a, b, A, B, C, n], x], ZeroQ[Plus[n1, Times[-1, n], -1]], ZeroQ[Plus[n2, Times[-1, n], -2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[m, Blank[]]]], Pattern[u, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[c, Cot[Plus[a, Times[b, x]]]], m], Power[Times[d, Tan[Plus[a, Times[b, x]]]], m]], Int[Times[ActivateTrig[u], Power[Times[d, Tan[Plus[a, Times[b, x]]]], Plus[n, Times[-1, m]]]], x], x], And[FreeQ[List[a, b, c, d, m, n], x], KnownTangentIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[u, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[c, Tan[Plus[a, Times[b, x]]]], m], Power[Times[d, Cos[Plus[a, Times[b, x]]]], m], Power[Power[Times[d, Sin[Plus[a, Times[b, x]]]], m], -1]], Int[Times[ActivateTrig[u], Power[Times[d, Sin[Plus[a, Times[b, x]]]], m], Power[Power[Times[d, Cos[Plus[a, Times[b, x]]]], Plus[m, Times[-1, n]]], -1]], x], x], And[FreeQ[List[a, b, c, d, m, n], x], KnownCotangentIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[m, Blank[]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[c, Cot[Plus[a, Times[b, x]]]], m], Power[Times[c, Tan[Plus[a, Times[b, x]]]], m]], Int[Times[ActivateTrig[u], Power[Power[Times[c, Tan[Plus[a, Times[b, x]]]], m], -1]], x], x], And[FreeQ[List[a, b, c, m], x], Not[IntegerQ[m]], KnownTangentIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[c, Cot[Plus[a, Times[b, x]]]], m], Power[Times[c, Tan[Plus[a, Times[b, x]]]], m]], Int[Times[ActivateTrig[u], Power[Power[Times[c, Cot[Plus[a, Times[b, x]]]], m], -1]], x], x], And[FreeQ[List[a, b, c, m], x], Not[IntegerQ[m]], KnownCotangentIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Times[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]], Pattern[u, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[c, Int[Times[ActivateTrig[u], Power[Times[c, Tan[Plus[a, Times[b, x]]]], Plus[n, -1]], Plus[B, Times[A, Tan[Plus[a, Times[b, x]]]]]], x], x], And[FreeQ[List[a, b, c, A, B, n], x], KnownTangentIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[u, Blank[]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[c, Int[Times[ActivateTrig[u], Power[Times[c, Cot[Plus[a, Times[b, x]]]], Plus[n, -1]], Plus[B, Times[A, Cot[Plus[a, Times[b, x]]]]]], x], x], And[FreeQ[List[a, b, c, A, B, n], x], KnownCotangentIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Times[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[ActivateTrig[u], Plus[B, Times[A, Tan[Plus[a, Times[b, x]]]]], Power[Tan[Plus[a, Times[b, x]]], -1]], x], And[FreeQ[List[a, b, A, B], x], KnownTangentIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[ActivateTrig[u], Plus[B, Times[A, Cot[Plus[a, Times[b, x]]]]], Power[Cot[Plus[a, Times[b, x]]], -1]], x], And[FreeQ[List[a, b, A, B], x], KnownCotangentIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, 2], Int[Times[ActivateTrig[u], Power[Times[c, Tan[Plus[a, Times[b, x]]]], Plus[n, -2]], Plus[C, Times[B, Tan[Plus[a, Times[b, x]]]], Times[A, Power[Tan[Plus[a, Times[b, x]]], 2]]]], x], x], And[FreeQ[List[a, b, c, A, B, C, n], x], KnownTangentIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, 2], Int[Times[ActivateTrig[u], Power[Times[c, Cot[Plus[a, Times[b, x]]]], Plus[n, -2]], Plus[C, Times[B, Cot[Plus[a, Times[b, x]]]], Times[A, Power[Cot[Plus[a, Times[b, x]]], 2]]]], x], x], And[FreeQ[List[a, b, c, A, B, C, n], x], KnownCotangentIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Optional[Pattern[u, Blank[]]], Plus[Times[Power[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]], Pattern[A, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, 2], Int[Times[ActivateTrig[u], Power[Times[c, Tan[Plus[a, Times[b, x]]]], Plus[n, -2]], Plus[C, Times[A, Power[Tan[Plus[a, Times[b, x]]], 2]]]], x], x], And[FreeQ[List[a, b, c, A, C, n], x], KnownTangentIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, 2], Int[Times[ActivateTrig[u], Power[Times[c, Cot[Plus[a, Times[b, x]]]], Plus[n, -2]], Plus[C, Times[A, Power[Cot[Plus[a, Times[b, x]]], 2]]]], x], x], And[FreeQ[List[a, b, c, A, C, n], x], KnownCotangentIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[ActivateTrig[u], Plus[C, Times[B, Tan[Plus[a, Times[b, x]]]], Times[A, Power[Tan[Plus[a, Times[b, x]]], 2]]], Power[Power[Tan[Plus[a, Times[b, x]]], 2], -1]], x], And[FreeQ[List[a, b, A, B, C], x], KnownTangentIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[ActivateTrig[u], Plus[C, Times[B, Cot[Plus[a, Times[b, x]]]], Times[A, Power[Cot[Plus[a, Times[b, x]]], 2]]], Power[Power[Cot[Plus[a, Times[b, x]]], 2], -1]], x], And[FreeQ[List[a, b, A, B, C], x], KnownCotangentIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Times[Power[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]], Pattern[A, Blank[]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[ActivateTrig[u], Plus[C, Times[A, Power[Tan[Plus[a, Times[b, x]]], 2]]], Power[Power[Tan[Plus[a, Times[b, x]]], 2], -1]], x], And[FreeQ[List[a, b, A, C], x], KnownTangentIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[ActivateTrig[u], Plus[C, Times[A, Power[Cot[Plus[a, Times[b, x]]], 2]]], Power[Power[Cot[Plus[a, Times[b, x]]], 2], -1]], x], And[FreeQ[List[a, b, A, C], x], KnownCotangentIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Plus[Optional[Pattern[A, Blank[]]], Times[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[C, Blank[]]]], Times[Optional[Pattern[B, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[ActivateTrig[u], Plus[C, Times[A, Tan[Plus[a, Times[b, x]]]], Times[B, Power[Tan[Plus[a, Times[b, x]]], 2]]], Power[Tan[Plus[a, Times[b, x]]], -1]], x], FreeQ[List[a, b, A, B, C], x]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Plus[Times[Optional[Pattern[A, Blank[]]], Power[tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[B, Blank[]]], Power[tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n1, Blank[]]]], Times[Optional[Pattern[C, Blank[]]], Power[tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[ActivateTrig[u], Power[Tan[Plus[a, Times[b, x]]], n], Plus[A, Times[B, Tan[Plus[a, Times[b, x]]]], Times[C, Power[Tan[Plus[a, Times[b, x]]], 2]]]], x], And[FreeQ[List[a, b, A, B, C, n], x], ZeroQ[Plus[n1, Times[-1, n], -1]], ZeroQ[Plus[n2, Times[-1, n], -2]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Times[Power[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[A, Blank[]]]], Times[Power[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n1, Blank[]]], Optional[Pattern[B, Blank[]]]], Times[Power[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]], Optional[Pattern[C, Blank[]]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[ActivateTrig[u], Power[Cot[Plus[a, Times[b, x]]], n], Plus[A, Times[B, Cot[Plus[a, Times[b, x]]]], Times[C, Power[Cot[Plus[a, Times[b, x]]], 2]]]], x], And[FreeQ[List[a, b, A, B, C, n], x], ZeroQ[Plus[n1, Times[-1, n], -1]], ZeroQ[Plus[n2, Times[-1, n], -2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[u, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[c, Sin[Plus[a, Times[b, x]]]], m], Power[Times[d, Csc[Plus[a, Times[b, x]]]], m]], Int[Times[ActivateTrig[u], Power[Times[d, Csc[Plus[a, Times[b, x]]]], Plus[n, Times[-1, m]]]], x], x], And[FreeQ[List[a, b, c, d, m, n], x], KnownSecantIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[m, Blank[]]]], Pattern[u, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[c, Cos[Plus[a, Times[b, x]]]], m], Power[Times[d, Sec[Plus[a, Times[b, x]]]], m]], Int[Times[ActivateTrig[u], Power[Times[d, Sec[Plus[a, Times[b, x]]]], Plus[n, Times[-1, m]]]], x], x], And[FreeQ[List[a, b, c, d, m, n], x], KnownSecantIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[c, Tan[Plus[a, Times[b, x]]]], m], Power[Times[d, Csc[Plus[a, Times[b, x]]]], m], Power[Power[Times[d, Sec[Plus[a, Times[b, x]]]], m], -1]], Int[Times[ActivateTrig[u], Power[Times[d, Sec[Plus[a, Times[b, x]]]], Plus[m, n]], Power[Power[Times[d, Csc[Plus[a, Times[b, x]]]], m], -1]], x], x], And[FreeQ[List[a, b, c, d, m, n], x], KnownSecantIntegrandQ[u, x], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[u, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[c, Tan[Plus[a, Times[b, x]]]], m], Power[Times[d, Csc[Plus[a, Times[b, x]]]], m], Power[Power[Times[d, Sec[Plus[a, Times[b, x]]]], m], -1]], Int[Times[ActivateTrig[u], Power[Times[d, Sec[Plus[a, Times[b, x]]]], m], Power[Power[Times[d, Csc[Plus[a, Times[b, x]]]], Plus[m, Times[-1, n]]], -1]], x], x], And[FreeQ[List[a, b, c, d, m, n], x], KnownSecantIntegrandQ[u, x], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[m, Blank[]]]], Pattern[u, Blank[]], Power[Times[Optional[Pattern[d, Blank[]]], sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[c, Cot[Plus[a, Times[b, x]]]], m], Power[Times[d, Sec[Plus[a, Times[b, x]]]], m], Power[Power[Times[d, Csc[Plus[a, Times[b, x]]]], m], -1]], Int[Times[ActivateTrig[u], Power[Times[d, Csc[Plus[a, Times[b, x]]]], m], Power[Power[Times[d, Sec[Plus[a, Times[b, x]]]], Plus[m, Times[-1, n]]], -1]], x], x], And[FreeQ[List[a, b, c, d, m, n], x], KnownSecantIntegrandQ[u, x], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Times[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[c, Cot[Plus[a, Times[b, x]]]], m], Power[Times[d, Sec[Plus[a, Times[b, x]]]], m], Power[Power[Times[d, Csc[Plus[a, Times[b, x]]]], m], -1]], Int[Times[ActivateTrig[u], Power[Times[d, Csc[Plus[a, Times[b, x]]]], Plus[m, n]], Power[Power[Times[d, Sec[Plus[a, Times[b, x]]]], m], -1]], x], x], And[FreeQ[List[a, b, c, d, m, n], x], KnownSecantIntegrandQ[u, x], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[c, Csc[Plus[a, Times[b, x]]]], m], Power[Times[c, Sin[Plus[a, Times[b, x]]]], m]], Int[Times[ActivateTrig[u], Power[Power[Times[c, Csc[Plus[a, Times[b, x]]]], m], -1]], x], x], And[FreeQ[List[a, b, c, m], x], Not[IntegerQ[m]], KnownSecantIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[m, Blank[]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[c, Cos[Plus[a, Times[b, x]]]], m], Power[Times[c, Sec[Plus[a, Times[b, x]]]], m]], Int[Times[ActivateTrig[u], Power[Power[Times[c, Sec[Plus[a, Times[b, x]]]], m], -1]], x], x], And[FreeQ[List[a, b, c, m], x], Not[IntegerQ[m]], KnownSecantIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[c, Tan[Plus[a, Times[b, x]]]], m], Power[Times[c, Csc[Plus[a, Times[b, x]]]], m], Power[Power[Times[c, Sec[Plus[a, Times[b, x]]]], m], -1]], Int[Times[ActivateTrig[u], Power[Times[c, Sec[Plus[a, Times[b, x]]]], m], Power[Power[Times[c, Csc[Plus[a, Times[b, x]]]], m], -1]], x], x], And[FreeQ[List[a, b, c, m], x], Not[IntegerQ[m]], KnownSecantIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[m, Blank[]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[c, Cot[Plus[a, Times[b, x]]]], m], Power[Times[c, Sec[Plus[a, Times[b, x]]]], m], Power[Power[Times[c, Csc[Plus[a, Times[b, x]]]], m], -1]], Int[Times[ActivateTrig[u], Power[Times[c, Csc[Plus[a, Times[b, x]]]], m], Power[Power[Times[c, Sec[Plus[a, Times[b, x]]]], m], -1]], x], x], And[FreeQ[List[a, b, c, m], x], Not[IntegerQ[m]], KnownSecantIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]], Pattern[u, Blank[]], Power[Times[Optional[Pattern[c, Blank[]]], sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[c, Int[Times[ActivateTrig[u], Power[Times[c, Sec[Plus[a, Times[b, x]]]], Plus[n, -1]], Plus[B, Times[A, Sec[Plus[a, Times[b, x]]]]]], x], x], And[FreeQ[List[a, b, c, A, B, n], x], KnownSecantIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[u, Blank[]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[c, Int[Times[ActivateTrig[u], Power[Times[c, Csc[Plus[a, Times[b, x]]]], Plus[n, -1]], Plus[B, Times[A, Csc[Plus[a, Times[b, x]]]]]], x], x], And[FreeQ[List[a, b, c, A, B, n], x], KnownSecantIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Pattern[A, Blank[]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[ActivateTrig[u], Plus[B, Times[A, Sec[Plus[a, Times[b, x]]]]], Power[Sec[Plus[a, Times[b, x]]], -1]], x], And[FreeQ[List[a, b, A, B], x], KnownSecantIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[ActivateTrig[u], Plus[B, Times[A, Csc[Plus[a, Times[b, x]]]]], Power[Csc[Plus[a, Times[b, x]]], -1]], x], And[FreeQ[List[a, b, A, B], x], KnownSecantIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, 2], Int[Times[ActivateTrig[u], Power[Times[c, Sec[Plus[a, Times[b, x]]]], Plus[n, -2]], Plus[C, Times[B, Sec[Plus[a, Times[b, x]]]], Times[A, Power[Sec[Plus[a, Times[b, x]]], 2]]]], x], x], And[FreeQ[List[a, b, c, A, B, C, n], x], KnownSecantIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, 2], Int[Times[ActivateTrig[u], Power[Times[c, Csc[Plus[a, Times[b, x]]]], Plus[n, -2]], Plus[C, Times[B, Csc[Plus[a, Times[b, x]]]], Times[A, Power[Csc[Plus[a, Times[b, x]]], 2]]]], x], x], And[FreeQ[List[a, b, c, A, B, C, n], x], KnownSecantIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Optional[Pattern[u, Blank[]]], Plus[Times[Power[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]], Pattern[A, Blank[]]], Power[Times[Optional[Pattern[c, Blank[]]], sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, 2], Int[Times[ActivateTrig[u], Power[Times[c, Sec[Plus[a, Times[b, x]]]], Plus[n, -2]], Plus[C, Times[A, Power[Sec[Plus[a, Times[b, x]]], 2]]]], x], x], And[FreeQ[List[a, b, c, A, C, n], x], KnownSecantIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[c, Blank[]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, 2], Int[Times[ActivateTrig[u], Power[Times[c, Csc[Plus[a, Times[b, x]]]], Plus[n, -2]], Plus[C, Times[A, Power[Csc[Plus[a, Times[b, x]]], 2]]]], x], x], And[FreeQ[List[a, b, c, A, C, n], x], KnownSecantIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[A, Blank[]]], Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[B, Blank[]]]], Times[Power[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[ActivateTrig[u], Plus[C, Times[B, Sec[Plus[a, Times[b, x]]]], Times[A, Power[Sec[Plus[a, Times[b, x]]], 2]]], Power[Power[Sec[Plus[a, Times[b, x]]], 2], -1]], x], And[FreeQ[List[a, b, A, B, C], x], KnownSecantIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[ActivateTrig[u], Plus[C, Times[B, Csc[Plus[a, Times[b, x]]]], Times[A, Power[Csc[Plus[a, Times[b, x]]], 2]]], Power[Power[Csc[Plus[a, Times[b, x]]], 2], -1]], x], And[FreeQ[List[a, b, A, B, C], x], KnownSecantIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Times[Power[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[C, Blank[]]]], Pattern[A, Blank[]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[ActivateTrig[u], Plus[C, Times[A, Power[Sec[Plus[a, Times[b, x]]], 2]]], Power[Power[Sec[Plus[a, Times[b, x]]], 2], -1]], x], And[FreeQ[List[a, b, A, C], x], KnownSecantIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Plus[Pattern[A, Blank[]], Times[Optional[Pattern[C, Blank[]]], Power[sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[ActivateTrig[u], Plus[C, Times[A, Power[Csc[Plus[a, Times[b, x]]], 2]]], Power[Power[Csc[Plus[a, Times[b, x]]], 2], -1]], x], And[FreeQ[List[a, b, A, C], x], KnownSecantIntegrandQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Plus[Times[Optional[Pattern[A, Blank[]]], Power[sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[B, Blank[]]], Power[sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n1, Blank[]]]], Times[Optional[Pattern[C, Blank[]]], Power[sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[ActivateTrig[u], Power[Sec[Plus[a, Times[b, x]]], n], Plus[A, Times[B, Sec[Plus[a, Times[b, x]]]], Times[C, Power[Sec[Plus[a, Times[b, x]]], 2]]]], x], And[FreeQ[List[a, b, A, B, C, n], x], ZeroQ[Plus[n1, Times[-1, n], -1]], ZeroQ[Plus[n2, Times[-1, n], -2]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Times[Power[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[A, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n1, Blank[]]], Optional[Pattern[B, Blank[]]]], Times[Power[csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n2, Blank[]]], Optional[Pattern[C, Blank[]]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[ActivateTrig[u], Power[Csc[Plus[a, Times[b, x]]], n], Plus[A, Times[B, Csc[Plus[a, Times[b, x]]]], Times[C, Power[Csc[Plus[a, Times[b, x]]], 2]]]], x], And[FreeQ[List[a, b, A, B, C, n], x], ZeroQ[Plus[n1, Times[-1, n], -1]], ZeroQ[Plus[n2, Times[-1, n], -2]]]]], RuleDelayed[HoldPattern[Int[Times[sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Sin[Plus[a, Times[-1, c], Times[Plus[b, Times[-1, d]], x]]], Power[Times[2, Plus[b, Times[-1, d]]], -1]], x], Times[-1, Simp[Times[Sin[Plus[a, c, Times[Plus[b, d], x]]], Power[Times[2, Plus[b, d]], -1]], x]]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Power[b, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Sin[Plus[a, Times[-1, c], Times[Plus[b, Times[-1, d]], x]]], Power[Times[2, Plus[b, Times[-1, d]]], -1]], x], Simp[Times[Sin[Plus[a, c, Times[Plus[b, d], x]]], Power[Times[2, Plus[b, d]], -1]], x]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Power[b, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Cos[Plus[a, Times[-1, c], Times[Plus[b, Times[-1, d]], x]]], Power[Times[2, Plus[b, Times[-1, d]]], -1]], x]], Times[-1, Simp[Times[Cos[Plus[a, c, Times[Plus[b, d], x]]], Power[Times[2, Plus[b, d]], -1]], x]]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Power[b, 2], Times[-1, Power[d, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[2, -1]], Int[Power[Times[g, Sin[Plus[c, Times[d, x]]]], p], x], x], Dist[Times[1, Power[2, -1]], Int[Times[Cos[Plus[c, Times[d, x]]], Power[Times[g, Sin[Plus[c, Times[d, x]]]], p]], x], x]], And[FreeQ[List[a, b, c, d, g], x], ZeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Times[d, Power[b, -1]], -2]], PositiveIntegerQ[Times[p, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[2, -1]], Int[Power[Times[g, Sin[Plus[c, Times[d, x]]]], p], x], x], Times[-1, Dist[Times[1, Power[2, -1]], Int[Times[Cos[Plus[c, Times[d, x]]], Power[Times[g, Sin[Plus[c, Times[d, x]]]], p]], x], x]]], 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IntegersQ[Times[2, m], Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[e, 2], Power[Times[e, Cos[Plus[a, Times[b, x]]]], Plus[m, -2]], Power[Times[g, Sin[Plus[c, Times[d, x]]]], Plus[p, 1]], Power[Times[2, b, g, Plus[m, Times[2, p]]], -1]], x], Dist[Times[Power[e, 2], Plus[m, p, -1], Power[Plus[m, Times[2, p]], -1]], Int[Times[Power[Times[e, Cos[Plus[a, Times[b, x]]]], Plus[m, -2]], Power[Times[g, Sin[Plus[c, Times[d, x]]]], p]], x], x]], And[FreeQ[List[a, b, c, d, e, g, p], x], ZeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Times[d, Power[b, -1]], -2]], Not[IntegerQ[p]], RationalQ[m], Greater[m, 1], NonzeroQ[Plus[m, Times[2, p]]], IntegersQ[Times[2, m], Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[e, 2], Power[Times[e, Sin[Plus[a, Times[b, x]]]], Plus[m, -2]], Power[Times[g, Sin[Plus[c, Times[d, x]]]], Plus[p, 1]], Power[Times[2, b, g, Plus[m, Times[2, p]]], -1]], x]], Dist[Times[Power[e, 2], Plus[m, p, -1], Power[Plus[m, Times[2, p]], -1]], Int[Times[Power[Times[e, Sin[Plus[a, Times[b, x]]]], Plus[m, -2]], Power[Times[g, Sin[Plus[c, Times[d, x]]]], p]], x], x]], And[FreeQ[List[a, b, c, d, e, g, p], x], ZeroQ[Plus[Times[b, c], Times[-1, a, d]]], 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x], x]], And[FreeQ[List[a, b, c, d, e, g, p], x], ZeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Times[d, Power[b, -1]], -2]], Not[IntegerQ[p]], RationalQ[m], Less[m, -1], NonzeroQ[Plus[m, Times[2, p], 2]], NonzeroQ[Plus[m, p, 1]], IntegersQ[Times[2, m], Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[2, Sin[Plus[a, Times[b, x]]], Power[Times[g, Sin[Plus[c, Times[d, x]]]], p], Power[Times[d, Plus[Times[2, p], 1]], -1]], x], Dist[Times[2, p, g, Power[Plus[Times[2, p], 1], -1]], Int[Times[Sin[Plus[a, Times[b, x]]], Power[Times[g, Sin[Plus[c, Times[d, x]]]], Plus[p, -1]]], x], x]], And[FreeQ[List[a, b, c, d, g], x], ZeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Times[d, Power[b, -1]], -2]], Not[IntegerQ[p]], RationalQ[p], Greater[p, 0], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[-2, Cos[Plus[a, Times[b, x]]], Power[Times[g, Sin[Plus[c, Times[d, x]]]], p], Power[Times[d, Plus[Times[2, p], 1]], -1]], x], Dist[Times[2, p, g, Power[Plus[Times[2, p], 1], -1]], Int[Times[Cos[Plus[a, Times[b, x]]], Power[Times[g, Sin[Plus[c, Times[d, x]]]], Plus[p, -1]]], x], x]], And[FreeQ[List[a, b, c, d, g], x], ZeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Times[d, Power[b, -1]], -2]], Not[IntegerQ[p]], RationalQ[p], Greater[p, 0], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Cos[Plus[a, Times[b, x]]], Power[Times[g, Sin[Plus[c, Times[d, x]]]], Plus[p, 1]], Power[Times[2, b, g, Plus[p, 1]], -1]], x], Dist[Times[Plus[Times[2, p], 3], Power[Times[2, g, Plus[p, 1]], -1]], Int[Times[Sin[Plus[a, Times[b, x]]], Power[Times[g, Sin[Plus[c, Times[d, x]]]], Plus[p, 1]]], x], x]], And[FreeQ[List[a, b, c, d, g], x], ZeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Times[d, Power[b, -1]], -2]], Not[IntegerQ[p]], RationalQ[p], Less[p, -1], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Sin[Plus[a, Times[b, x]]], Power[Times[g, Sin[Plus[c, Times[d, x]]]], Plus[p, 1]], Power[Times[2, b, g, Plus[p, 1]], -1]], x]], Dist[Times[Plus[Times[2, p], 3], Power[Times[2, g, Plus[p, 1]], -1]], Int[Times[Cos[Plus[a, Times[b, x]]], Power[Times[g, Sin[Plus[c, Times[d, x]]]], Plus[p, 1]]], x], x]], And[FreeQ[List[a, b, c, d, g], x], ZeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Times[d, Power[b, -1]], -2]], Not[IntegerQ[p]], RationalQ[p], Less[p, -1], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[ArcSin[Plus[Cos[Plus[a, Times[b, x]]], Times[-1, Sin[Plus[a, Times[b, x]]]]]], Power[d, -1]], x]], Simp[Times[Log[Plus[Cos[Plus[a, Times[b, x]]], Sin[Plus[a, Times[b, x]]], Sqrt[Sin[Plus[c, Times[d, x]]]]]], Power[d, -1]], x]], And[FreeQ[List[a, b, c, d], x], ZeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Times[d, Power[b, -1]], -2]]]]], RuleDelayed[HoldPattern[Int[Times[sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[ArcSin[Plus[Cos[Plus[a, Times[b, x]]], Times[-1, Sin[Plus[a, Times[b, x]]]]]], Power[d, -1]], x]], Times[-1, Simp[Times[Log[Plus[Cos[Plus[a, Times[b, x]]], Sin[Plus[a, Times[b, x]]], Sqrt[Sin[Plus[c, Times[d, x]]]]]], Power[d, -1]], x]]], And[FreeQ[List[a, b, c, d], x], 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x]]]], Plus[m, p]], Power[Sin[Plus[a, Times[b, x]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, g, m, p], x], ZeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Times[d, Power[b, -1]], -2]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[f, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[g, Sin[Plus[c, Times[d, x]]]], p], Power[Times[Power[Cos[Plus[a, Times[b, x]]], p], Power[Times[f, Sin[Plus[a, Times[b, x]]]], p]], -1]], Int[Times[Power[Cos[Plus[a, Times[b, x]]], p], Power[Times[f, Sin[Plus[a, Times[b, x]]]], Plus[n, p]]], x], x], And[FreeQ[List[a, b, c, d, f, g, n, p], x], ZeroQ[Plus[Times[b, c], Times[-1, a, d]]], 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x], ZeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Times[d, Power[b, -1]], -2]], Not[IntegerQ[p]], RationalQ[m, p], Greater[m, 3], Less[p, -1], NonzeroQ[Plus[n, p, 1]], IntegersQ[Times[2, m], Times[2, n], Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[f, Blank[]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[e, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[e, 2], Power[Times[e, Sin[Plus[a, Times[b, x]]]], Plus[m, -2]], Power[Times[f, Cos[Plus[a, Times[b, x]]]], n], Power[Times[g, Sin[Plus[c, Times[d, x]]]], Plus[p, 1]], Power[Times[2, b, g, Plus[n, p, 1]], -1]], x]], Dist[Times[Power[e, 4], Plus[m, p, -1], Power[Times[4, Power[g, 2], Plus[n, p, 1]], -1]], Int[Times[Power[Times[e, Sin[Plus[a, Times[b, x]]]], Plus[m, -4]], Power[Times[f, Cos[Plus[a, Times[b, x]]]], n], Power[Times[g, Sin[Plus[c, Times[d, x]]]], Plus[p, 2]]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, n], x], ZeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Times[d, Power[b, -1]], -2]], Not[IntegerQ[p]], RationalQ[m, p], Greater[m, 3], Less[p, -1], NonzeroQ[Plus[n, p, 1]], IntegersQ[Times[2, m], Times[2, n], Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[e, Blank[]]]], Pattern[m, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[g, Blank[]]], sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[e, Cos[Plus[a, Times[b, x]]]], m], Power[Times[f, Sin[Plus[a, Times[b, x]]]], n], Power[Times[g, Sin[Plus[c, Times[d, x]]]], Plus[p, 1]], Power[Times[2, b, g, Plus[n, p, 1]], -1]], x], Dist[Times[Power[e, 2], Plus[m, n, Times[2, p], 2], Power[Times[4, Power[g, 2], Plus[n, p, 1]], -1]], Int[Times[Power[Times[e, Cos[Plus[a, Times[b, x]]]], Plus[m, -2]], Power[Times[f, Sin[Plus[a, Times[b, x]]]], n], Power[Times[g, Sin[Plus[c, Times[d, x]]]], Plus[p, 2]]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, n], x], ZeroQ[Plus[Times[b, c], Times[-1, a, d]]], ZeroQ[Plus[Times[d, Power[b, -1]], -2]], Not[IntegerQ[p]], RationalQ[m, p], Greater[m, 1], Less[p, -1], NonzeroQ[Plus[m, n, Times[2, p], 2]], NonzeroQ[Plus[n, p, 1]], IntegersQ[Times[2, m], Times[2, n], Times[2, 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Power[Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[c, Times[d, x]], m], Power[Tan[Plus[a, Times[b, x]]], Plus[n, 1]], Power[Times[b, Plus[n, 1]], -1]], x], Times[-1, Dist[Times[d, m, Power[Times[b, Plus[n, 1]], -1]], Int[Times[Power[Plus[c, Times[d, x]], Plus[m, -1]], Power[Tan[Plus[a, Times[b, x]]], Plus[n, 1]]], x], x]]], And[FreeQ[List[a, b, c, d, n], x], PositiveIntegerQ[m], NonzeroQ[Plus[n, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Plus[c, Times[d, x]], m], Power[Cot[Plus[a, Times[b, x]]], Plus[n, 1]], Power[Times[b, Plus[n, 1]], -1]], x]], Dist[Times[d, m, Power[Times[b, Plus[n, 1]], -1]], Int[Times[Power[Plus[c, Times[d, x]], Plus[m, -1]], Power[Cot[Plus[a, Times[b, x]]], Plus[n, 1]]], x], x]], And[FreeQ[List[a, b, c, d, n], x], PositiveIntegerQ[m], NonzeroQ[Plus[n, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Power[Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Int[Times[Power[Plus[c, Times[d, x]], m], Sec[Plus[a, Times[b, x]]], Power[Tan[Plus[a, Times[b, x]]], Plus[p, -2]]], x]], Int[Times[Power[Plus[c, Times[d, x]], m], Power[Sec[Plus[a, Times[b, x]]], 3], Power[Tan[Plus[a, Times[b, x]]], Plus[p, -2]]], x]], And[FreeQ[List[a, b, c, d, m], x], PositiveIntegerQ[Times[p, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Int[Times[Power[Plus[c, Times[d, x]], m], Power[Sec[Plus[a, Times[b, x]]], n], Power[Tan[Plus[a, Times[b, x]]], Plus[p, -2]]], x]], Int[Times[Power[Plus[c, Times[d, x]], m], Power[Sec[Plus[a, Times[b, x]]], Plus[n, 2]], Power[Tan[Plus[a, Times[b, x]]], Plus[p, -2]]], x]], And[FreeQ[List[a, b, c, 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Blank[]]], Power[Csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Int[Times[Power[Plus[c, Times[d, x]], m], Power[Csc[Plus[a, Times[b, x]]], n], Power[Cot[Plus[a, Times[b, x]]], Plus[p, -2]]], x]], Int[Times[Power[Plus[c, Times[d, x]], m], Power[Csc[Plus[a, Times[b, x]]], Plus[n, 2]], Power[Cot[Plus[a, Times[b, x]]], Plus[p, -2]]], x]], And[FreeQ[List[a, b, c, d, m, n], x], PositiveIntegerQ[Times[p, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Tan[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[u, IntHide[Times[Power[Sec[Plus[a, Times[b, x]]], n], Power[Tan[Plus[a, Times[b, x]]], p]], x]]], Plus[Dist[Power[Plus[c, Times[d, x]], m], u, x], Times[-1, Dist[Times[d, m], Int[Times[Power[Plus[c, Times[d, x]], Plus[m, -1]], u], x], x]]]], And[FreeQ[List[a, b, c, d, n, p], x], PositiveIntegerQ[m], Or[EvenQ[n], OddQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cot[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[u, IntHide[Times[Power[Csc[Plus[a, Times[b, x]]], n], Power[Cot[Plus[a, Times[b, x]]], p]], x]]], Plus[Dist[Power[Plus[c, Times[d, x]], m], u, x], Times[-1, Dist[Times[d, m], Int[Times[Power[Plus[c, Times[d, x]], Plus[m, -1]], u], x], x]]]], And[FreeQ[List[a, b, c, d, n, p], x], PositiveIntegerQ[m], Or[EvenQ[n], OddQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[2, n], Int[Times[Power[Plus[c, Times[d, x]], m], Power[Csc[Plus[Times[2, a], Times[2, b, x]]], n]], x], x], And[FreeQ[List[a, b, c, d], x], RationalQ[m], IntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Csc[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sec[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[u, IntHide[Times[Power[Csc[Plus[a, Times[b, x]]], n], Power[Sec[Plus[a, Times[b, x]]], p]], x]]], Plus[Dist[Power[Plus[c, Times[d, x]], m], u, x], Times[-1, Dist[Times[d, m], Int[Times[Power[Plus[c, Times[d, x]], Plus[m, -1]], u], x], x]]]], And[FreeQ[List[a, b, c, d], x], IntegersQ[n, p], RationalQ[m], Greater[m, 0], Unequal[n, p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[u, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Pattern[F, Blank[]][Pattern[v, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[G, Blank[]][Pattern[w, Blank[]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[ExpandToSum[u, x], m], Power[F[ExpandToSum[v, x]], n], Power[G[ExpandToSum[v, x]], p]], x], And[FreeQ[List[m, n, p], x], TrigQ[F], TrigQ[G], ZeroQ[Plus[v, Times[-1, w]]], LinearQ[List[u, v, w], x], Not[LinearMatchQ[List[u, v, w], x]]]]], RuleDelayed[HoldPattern[Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[e, Times[f, x]], m], 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x]]]]], Plus[n, 1]], Power[Times[b, d, Plus[n, 1]], -1]], x]], Dist[Times[f, m, Power[Times[b, d, Plus[n, 1]], -1]], Int[Times[Power[Plus[e, Times[f, x]], Plus[m, -1]], Power[Plus[a, Times[b, Cos[Plus[c, Times[d, x]]]]], Plus[n, 1]]], x], x]], And[FreeQ[List[a, b, c, d, e, f, n], x], PositiveIntegerQ[m], NonzeroQ[Plus[n, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sec[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[e, Times[f, x]], m], Power[Plus[a, Times[b, Tan[Plus[c, Times[d, x]]]]], Plus[n, 1]], Power[Times[b, d, Plus[n, 1]], 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Power[Times[b, d, Plus[n, 1]], -1]], Int[Times[Power[Plus[e, Times[f, x]], Plus[m, -1]], Power[Plus[a, Times[b, Cot[Plus[c, Times[d, x]]]]], Plus[n, 1]]], x], x]], And[FreeQ[List[a, b, c, d, e, f, n], x], PositiveIntegerQ[m], NonzeroQ[Plus[n, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Sec[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sec[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Tan[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[e, Times[f, x]], m], Power[Plus[a, Times[b, Sec[Plus[c, Times[d, x]]]]], Plus[n, 1]], Power[Times[b, d, 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c, d, e, f], x], PositiveIntegerQ[p, q], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Power[Plus[e, Times[f, x]], m], Times[Power[Cos[Plus[a, Times[b, x]]], p], Power[Cos[Plus[c, Times[d, x]]], q]], x], x], And[FreeQ[List[a, b, c, d, e, f], x], PositiveIntegerQ[p, q], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], 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Condition[Simp[Times[Power[2, n], Power[E, Times[I, k, n, Pi]], Power[E, Times[I, n, Plus[d, Times[e, x]]]], Power[F, Times[c, Plus[a, Times[b, x]]]], Power[Plus[Times[I, e, n], Times[b, c, Log[F]]], -1], Hypergeometric2F1[n, Plus[Times[n, Power[2, -1]], Times[-1, I, b, c, Log[F], Power[Times[2, e], -1]]], Plus[1, Times[n, Power[2, -1]], Times[-1, I, b, c, Log[F], Power[Times[2, e], -1]]], Times[-1, Power[E, Times[2, I, k, Pi]], Power[E, Times[2, I, Plus[d, Times[e, x]]]]]]], x], And[FreeQ[List[F, a, b, c, d, e], x], IntegerQ[Times[4, k]], IntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[2, n], Power[E, Times[I, n, Plus[d, Times[e, x]]]], Power[F, Times[c, Plus[a, Times[b, x]]]], Power[Plus[Times[I, e, n], Times[b, c, Log[F]]], -1], Hypergeometric2F1[n, Plus[Times[n, Power[2, -1]], Times[-1, I, b, c, Log[F], Power[Times[2, e], -1]]], Plus[1, Times[n, Power[2, -1]], Times[-1, I, b, c, Log[F], Power[Times[2, e], -1]]], Times[-1, Power[E, Times[2, I, Plus[d, Times[e, x]]]]]]], x], And[FreeQ[List[F, a, b, c, d, e], x], IntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Csc[Plus[Optional[Pattern[d, Blank[]]], Times[Pi, Optional[Pattern[k, Blank[]]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[Times[-2, I], n], Power[E, Times[I, k, n, Pi]], Power[E, Times[I, n, Plus[d, Times[e, x]]]], Times[Power[F, Times[c, Plus[a, Times[b, x]]]], Power[Plus[Times[I, e, n], Times[b, c, Log[F]]], -1]], Hypergeometric2F1[n, Plus[Times[n, Power[2, -1]], Times[-1, I, b, c, Log[F], Power[Times[2, e], -1]]], Plus[1, Times[n, Power[2, -1]], Times[-1, I, b, c, Log[F], Power[Times[2, e], -1]]], Times[Power[E, Times[2, I, k, Pi]], Power[E, Times[2, I, Plus[d, Times[e, x]]]]]]], x], And[FreeQ[List[F, a, b, c, d, e], x], IntegerQ[Times[4, k]], IntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[Times[-2, I], n], Power[E, Times[I, n, Plus[d, Times[e, x]]]], Times[Power[F, Times[c, Plus[a, Times[b, x]]]], Power[Plus[Times[I, e, n], Times[b, c, Log[F]]], -1]], Hypergeometric2F1[n, Plus[Times[n, Power[2, -1]], Times[-1, I, b, c, Log[F], Power[Times[2, e], -1]]], Plus[1, Times[n, Power[2, -1]], Times[-1, I, b, c, Log[F], Power[Times[2, e], -1]]], Power[E, Times[2, I, Plus[d, Times[e, x]]]]]], x], And[FreeQ[List[F, a, b, c, d, e], x], IntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[1, Power[E, Times[2, I, Plus[d, Times[e, x]]]]], n], Power[Sec[Plus[d, Times[e, x]]], n], Power[Power[E, Times[I, n, Plus[d, Times[e, x]]]], -1]], Int[SimplifyIntegrand[Times[Power[F, Times[c, Plus[a, Times[b, x]]]], Power[E, Times[I, n, Plus[d, Times[e, x]]]], Power[Power[Plus[1, Power[E, Times[2, I, Plus[d, Times[e, x]]]]], n], -1]], x], x], x], And[FreeQ[List[F, a, b, c, d, e], x], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[1, Times[-1, Power[E, Times[-2, I, Plus[d, Times[e, x]]]]]], n], Power[Csc[Plus[d, Times[e, x]]], n], Power[Power[E, Times[-1, I, n, Plus[d, Times[e, x]]]], -1]], Int[SimplifyIntegrand[Times[Power[F, Times[c, Plus[a, Times[b, x]]]], Power[E, Times[-1, I, n, Plus[d, Times[e, x]]]], Power[Power[Plus[1, Times[-1, Power[E, Times[-2, I, Plus[d, Times[e, x]]]]]], n], -1]], x], x], x], And[FreeQ[List[F, a, b, c, d, e], x], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[2, n], Power[f, n]], Int[Times[Power[F, Times[c, Plus[a, Times[b, x]]]], Power[Cos[Plus[Times[d, Power[2, -1]], Times[e, x, Power[2, -1]], Times[-1, f, Pi, Power[Times[4, g], -1]]]], Times[2, n]]], x], x], And[FreeQ[List[F, a, b, c, d, e, f, g], x], ZeroQ[Plus[Power[f, 2], Times[-1, Power[g, 2]]]], NegativeIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[Cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[f, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[2, n], Power[f, n]], Int[Times[Power[F, Times[c, Plus[a, Times[b, x]]]], Power[Cos[Plus[Times[d, Power[2, -1]], Times[e, x, Power[2, -1]]]], Times[2, n]]], x], x], And[FreeQ[List[F, a, b, c, d, e, f, g], x], ZeroQ[Plus[f, Times[-1, g]]], NegativeIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[Cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[f, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[2, n], Power[f, n]], Int[Times[Power[F, Times[c, Plus[a, Times[b, x]]]], Power[Sin[Plus[Times[d, Power[2, -1]], Times[e, x, 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IntegersQ[m, n], Equal[Plus[m, n], 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[Cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[f, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[f, n], Int[Times[Power[F, Times[c, Plus[a, Times[b, x]]]], Power[Tan[Plus[Times[d, Power[2, -1]], Times[e, x, Power[2, -1]]]], m]], x], x], And[FreeQ[List[F, a, b, c, d, e, f, g], x], ZeroQ[Plus[f, Times[-1, g]]], IntegersQ[m, n], Equal[Plus[m, n], 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[Cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[g, Blank[]]]], Pattern[f, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[f, n], Int[Times[Power[F, Times[c, Plus[a, Times[b, x]]]], Power[Cot[Plus[Times[d, Power[2, -1]], Times[e, x, Power[2, -1]]]], m]], x], x], And[FreeQ[List[F, a, b, c, d, e, f, g], x], ZeroQ[Plus[f, g]], IntegersQ[m, n], Equal[Plus[m, n], 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Plus[Times[Cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[i, Blank[]]]], Pattern[h, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[2, i], Int[Times[Power[F, Times[c, Plus[a, Times[b, x]]]], Times[Cos[Plus[d, Times[e, x]]], Power[Plus[f, Times[g, Sin[Plus[d, Times[e, x]]]]], -1]]], x], x], Int[Times[Power[F, Times[c, Plus[a, Times[b, x]]]], Times[Plus[h, Times[-1, i, Cos[Plus[d, Times[e, x]]]]], Power[Plus[f, Times[g, Sin[Plus[d, Times[e, x]]]]], -1]]], x]], And[FreeQ[List[F, a, b, c, d, e, f, g, h, i], x], ZeroQ[Plus[Power[f, 2], Times[-1, Power[g, 2]]]], ZeroQ[Plus[Power[h, 2], Times[-1, Power[i, 2]]]], ZeroQ[Plus[Times[g, h], Times[-1, f, i]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[Cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, 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RuleDelayed[HoldPattern[Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Pattern[u, Blank[]]]], Power[Pattern[G, Blank[]][Pattern[v, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[F, Times[c, ExpandToSum[u, x]]], Power[G[ExpandToSum[v, x]], n]], x], And[FreeQ[List[F, c, n], x], TrigQ[G], LinearQ[List[u, v], x], Not[LinearMatchQ[List[u, v], x]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[u, IntHide[Times[Power[F, Times[c, Plus[a, Times[b, x]]]], Power[Sin[Plus[d, Times[e, x]]], n]], x]]], Plus[Dist[Power[x, m], u, x], Times[-1, Dist[m, Int[Times[Power[x, Plus[m, -1]], u], x], x]]]], And[FreeQ[List[F, a, b, c, d, e], x], RationalQ[m], Greater[m, 0], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[u, IntHide[Times[Power[F, Times[c, Plus[a, Times[b, x]]]], Power[Cos[Plus[d, Times[e, x]]], n]], x]]], Plus[Dist[Power[x, m], u, x], Times[-1, Dist[m, Int[Times[Power[x, Plus[m, -1]], u], x], x]]]], And[FreeQ[List[F, a, b, c, d, e], x], RationalQ[m], Greater[m, 0], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cos[Plus[Optional[Pattern[f, Blank[]]], 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Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigToExp[Power[F, Times[c, Plus[a, Times[b, x]]]], Times[Power[G[Plus[d, Times[e, x]]], m], Power[H[Plus[d, Times[e, x]]], n]], x], x], And[FreeQ[List[F, a, b, c, d, e], x], PositiveIntegerQ[m, n], TrigQ[G], TrigQ[H]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[F, Blank[]], Pattern[u, Blank[]]], Power[Sin[Pattern[v, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigToExp[Power[F, u], Power[Sin[v], n], x], x], And[FreeQ[F, x], Or[LinearQ[u, x], PolyQ[u, x, 2]], Or[LinearQ[v, x], PolyQ[v, x, 2]], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cos[Pattern[v, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigToExp[Power[F, u], Power[Cos[v], n], x], x], And[FreeQ[F, x], Or[LinearQ[u, x], PolyQ[u, x, 2]], Or[LinearQ[v, x], PolyQ[v, x, 2]], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cos[Pattern[v, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Pattern[u, Blank[]]], Power[Sin[Pattern[v, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigToExp[Power[F, u], Times[Power[Sin[v], m], Power[Cos[v], n]], x], x], And[FreeQ[F, x], Or[LinearQ[u, x], PolyQ[u, x, 2]], Or[LinearQ[v, x], PolyQ[v, x, 2]], PositiveIntegerQ[m, n]]]], RuleDelayed[HoldPattern[Int[Power[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[x, Plus[p, 2], Power[Sin[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 2]], Power[Plus[p, 1], -1]], x], Simp[Times[x, Cot[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Power[Sin[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 2]], 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Plus[Dist[Times[Plus[Times[Power[b, 2], Power[n, 2], Power[Plus[p, 2], 2]], 1], Power[Times[Power[b, 2], Power[n, 2], Plus[p, 1], Plus[p, 2]], -1]], Int[Power[Sin[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 2]], x], x], Times[-1, Simp[Times[x, Power[Sin[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 2]], Power[Times[Power[b, 2], Power[n, 2], Plus[p, 1], Plus[p, 2]], -1]], x]]]], And[FreeQ[List[a, b, c, n], x], RationalQ[p], Less[p, -1], Unequal[p, -2], NonzeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[Plus[p, 2], 2]], 1]]]]], RuleDelayed[HoldPattern[Int[Power[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[x, Tan[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Power[Cos[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 2]], Power[Times[b, 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2], Power[p, 2]], Power[Plus[m, 1], 2]], -1]], x]]], And[FreeQ[List[a, b, c, m, n], x], RationalQ[p], Greater[p, 1], NonzeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], Power[Plus[m, 1], 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[m, 1]], Cot[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Power[Sin[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 2]], Power[Times[b, n, Plus[p, 1]], -1]], x], Plus[Dist[Times[Plus[Times[Power[b, 2], Power[n, 2], Power[Plus[p, 2], 2]], Power[Plus[m, 1], 2]], Power[Times[Power[b, 2], Power[n, 2], Plus[p, 1], Plus[p, 2]], -1]], Int[Times[Power[x, m], Power[Sin[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], 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Power[n, 2], Power[Plus[p, 2], 2]], Power[Plus[m, 1], 2]], Power[Times[Power[b, 2], Power[n, 2], Plus[p, 1], Plus[p, 2]], -1]], Int[Times[Power[x, m], Power[Cos[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 2]]], x], x], Times[-1, Simp[Times[Plus[m, 1], Power[x, Plus[m, 1]], Power[Cos[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 2]], Power[Times[Power[b, 2], Power[n, 2], Plus[p, 1], Plus[p, 2]], -1]], x]]]], And[FreeQ[List[a, b, c, m, n], x], RationalQ[p], Less[p, -1], Unequal[p, -2], NonzeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[Plus[p, 2], 2]], Power[Plus[m, 1], 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[x, Plus[m, 1]], Power[Plus[Times[I, Power[E, Times[-1, I, a]], Power[Times[c, Power[x, n]], Times[-1, I, b]]], Times[-1, I, Power[E, Times[I, a]], Power[Times[c, Power[x, n]], Times[I, b]]]], p], Power[Times[Plus[m, 1, Times[-1, I, b, n, p]], Power[Plus[2, Times[-1, 2, Power[E, Times[2, I, a]], Power[Times[c, Power[x, n]], Times[2, I, b]]]], p]], -1], Hypergeometric2F1[Times[-1, p], Times[Plus[m, 1, Times[-1, I, b, n, p]], Power[Times[2, I, b, n], -1]], Plus[1, Times[Plus[m, 1, Times[-1, I, b, n, p]], Power[Times[2, I, b, n], -1]]], Times[Power[E, Times[2, I, a]], Power[Times[c, Power[x, n]], Times[2, I, b]]]]], x], And[FreeQ[List[a, b, c, m, n, p], x], NonzeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], Power[Plus[m, 1], 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[x, Plus[m, 1]], Power[Plus[Times[Power[E, Times[-1, I, a]], Power[Times[c, Power[x, n]], Times[-1, I, b]]], Times[Power[E, Times[I, a]], Power[Times[c, Power[x, n]], Times[I, b]]]], p], Power[Times[Plus[m, 1, Times[-1, I, b, n, p]], Power[Plus[2, Times[2, Power[E, Times[2, I, a]], Power[Times[c, Power[x, n]], Times[2, I, b]]]], p]], -1], Hypergeometric2F1[Times[-1, p], Times[Plus[m, 1, Times[-1, I, b, n, p]], Power[Times[2, I, b, n], -1]], Plus[1, Times[Plus[m, 1, Times[-1, I, b, n, p]], Power[Times[2, I, b, n], -1]]], Times[-1, Power[E, Times[2, I, a]], Power[Times[c, Power[x, n]], Times[2, I, b]]]]], x], And[FreeQ[List[a, b, c, m, n, p], x], NonzeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], Power[Plus[m, 1], 2]]]]]], RuleDelayed[HoldPattern[Int[Sec[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[2, Power[E, Times[a, b, n]]], Int[Times[Power[Times[c, Power[x, n]], Times[1, Power[n, -1]]], Power[Plus[Power[E, Times[2, a, b, n]], Power[Times[c, Power[x, n]], Times[2, Power[n, -1]]]], -1]], x], x], And[FreeQ[List[a, b, c, n], x], ZeroQ[Plus[Times[Power[b, 2], Power[n, 2]], 1]]]]], RuleDelayed[HoldPattern[Int[Csc[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[2, b, n, Power[E, Times[a, b, n]]], Int[Times[Power[Times[c, Power[x, n]], Times[1, Power[n, -1]]], Power[Plus[Power[E, Times[2, a, b, n]], Times[-1, Power[Times[c, Power[x, n]], Times[2, Power[n, -1]]]]], -1]], x], x], And[FreeQ[List[a, b, c, n], x], ZeroQ[Plus[Times[Power[b, 2], Power[n, 2]], 1]]]]], RuleDelayed[HoldPattern[Int[Power[Sec[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[p, -2], x, Power[Sec[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, -2]], Power[Plus[p, -1], -1]], x], Simp[Times[x, Tan[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Power[Sec[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, -2]], Power[Times[b, n, Plus[p, -1]], -1]], x]], And[FreeQ[List[a, b, c, n, p], x], ZeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[Plus[p, -2], 2]], 1]], NonzeroQ[Plus[p, -1]]]]], RuleDelayed[HoldPattern[Int[Power[Csc[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[p, -2], x, Power[Csc[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, -2]], Power[Plus[p, -1], -1]], x], Times[-1, Simp[Times[x, Cot[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Power[Csc[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, -2]], Power[Times[b, n, Plus[p, -1]], -1]], x]]], And[FreeQ[List[a, b, c, n, p], x], ZeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[Plus[p, -2], 2]], 1]], NonzeroQ[Plus[p, -1]]]]], RuleDelayed[HoldPattern[Int[Power[Sec[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[x, Tan[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Power[Sec[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, -2]], Power[Times[b, n, Plus[p, -1]], -1]], x], Plus[Dist[Times[Plus[Times[Power[b, 2], Power[n, 2], Power[Plus[p, 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Plus[Dist[Times[Plus[Times[Power[b, 2], Power[n, 2], Power[Plus[p, -2], 2]], 1], Power[Times[Power[b, 2], Power[n, 2], Plus[p, -1], Plus[p, -2]], -1]], Int[Power[Csc[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, -2]], x], x], Times[-1, Simp[Times[x, Power[Csc[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, -2]], Power[Times[Power[b, 2], Power[n, 2], Plus[p, -1], Plus[p, -2]], -1]], x]]]], And[FreeQ[List[a, b, c, n], x], RationalQ[p], Greater[p, 1], Unequal[p, 2], NonzeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[Plus[p, -2], 2]], 1]]]]], RuleDelayed[HoldPattern[Int[Power[Sec[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[b, n, p, x, Sin[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Power[Sec[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 1]], Power[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], 1], -1]], x]], Plus[Dist[Times[Power[b, 2], Power[n, 2], p, Plus[p, 1], Power[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], 1], -1]], Int[Power[Sec[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 2]], x], x], Simp[Times[x, Power[Sec[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], p], Power[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], 1], -1]], x]]], And[FreeQ[List[a, b, c, n], x], RationalQ[p], Less[p, -1], NonzeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], 1]]]]], RuleDelayed[HoldPattern[Int[Power[Csc[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, n, p, x, Cos[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Power[Csc[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 1]], Power[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], 1], -1]], x], Plus[Dist[Times[Power[b, 2], Power[n, 2], p, Plus[p, 1], Power[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], 1], -1]], Int[Power[Csc[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 2]], x], x], Simp[Times[x, Power[Csc[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], p], Power[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], 1], -1]], x]]], And[FreeQ[List[a, b, c, n], x], RationalQ[p], Less[p, -1], NonzeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], 1]]]]], RuleDelayed[HoldPattern[Int[Power[Sec[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[x, Power[Plus[2, Times[2, Power[E, Times[2, I, a]], Power[Times[c, Power[x, n]], Times[2, I, b]]]], p], Power[Plus[1, Times[I, b, n, p]], -1], Power[Times[Power[E, Times[I, a]], Power[Times[c, Power[x, n]], Times[I, b]], Power[Plus[1, Times[Power[E, Times[2, I, a]], Power[Times[c, Power[x, n]], Times[2, I, b]]]], -1]], p], Hypergeometric2F1[p, Times[Plus[1, Times[I, b, n, p]], Power[Times[2, I, b, n], -1]], Plus[1, Times[Plus[1, Times[I, b, n, p]], Power[Times[2, I, b, n], -1]]], Times[-1, Power[E, Times[2, I, a]], Power[Times[c, Power[x, n]], Times[2, I, b]]]]], x], And[FreeQ[List[a, b, c, n, p], x], NonzeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], 1]]]]], RuleDelayed[HoldPattern[Int[Power[Csc[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[x, Power[Plus[2, Times[-1, 2, Power[E, Times[2, I, a]], Power[Times[c, Power[x, n]], Times[2, I, b]]]], p], Power[Plus[1, Times[I, b, n, p]], -1], Power[Times[-1, I, Power[E, Times[I, a]], Power[Times[c, Power[x, n]], Times[I, b]], Power[Plus[1, Times[-1, Power[E, Times[2, I, a]], Power[Times[c, Power[x, n]], Times[2, I, b]]]], -1]], p], Hypergeometric2F1[p, Times[Plus[1, Times[I, b, n, p]], Power[Times[2, I, b, n], -1]], Plus[1, Times[Plus[1, Times[I, b, n, p]], Power[Times[2, I, b, n], -1]]], Times[Power[E, Times[2, I, a]], Power[Times[c, Power[x, n]], Times[2, I, b]]]]], x], And[FreeQ[List[a, b, c, n, p], x], NonzeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Sec[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[2, Power[E, Times[a, b, n, Power[Plus[m, 1], -1]]]], Int[Times[Power[x, m], Power[Times[c, Power[x, n]], Times[Plus[m, 1], Power[n, -1]]], Power[Plus[Power[E, Times[2, a, b, n, Power[Plus[m, 1], -1]]], Power[Times[c, Power[x, n]], Times[2, Plus[m, 1], Power[n, -1]]]], -1]], x], x], And[FreeQ[List[a, b, c, m, n], x], ZeroQ[Plus[Times[Power[b, 2], Power[n, 2]], Power[Plus[m, 1], 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Csc[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[2, b, n, Power[Plus[m, 1], -1], Power[E, Times[a, b, n, Power[Plus[m, 1], -1]]]], Int[Times[Power[x, m], Power[Times[c, Power[x, n]], Times[Plus[m, 1], Power[n, -1]]], Power[Plus[Power[E, Times[2, a, b, n, Power[Plus[m, 1], -1]]], Times[-1, Power[Times[c, Power[x, n]], Times[2, Plus[m, 1], Power[n, -1]]]]], -1]], x], x], And[FreeQ[List[a, b, c, m, n], x], ZeroQ[Plus[Times[Power[b, 2], Power[n, 2]], Power[Plus[m, 1], 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sec[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[p, -2], Power[x, Plus[m, 1]], Power[Sec[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, -2]], Power[Times[Plus[m, 1], Plus[p, -1]], -1]], x], Simp[Times[Power[x, Plus[m, 1]], Tan[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Power[Sec[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, -2]], Power[Times[b, n, Plus[p, -1]], -1]], x]], And[FreeQ[List[a, b, c, m, n, p], x], ZeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[Plus[p, -2], 2]], Power[Plus[m, 1], 2]]], NonzeroQ[Plus[m, 1]], NonzeroQ[Plus[p, -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Csc[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[p, -2], Power[x, Plus[m, 1]], Power[Csc[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, -2]], Power[Times[Plus[m, 1], Plus[p, -1]], -1]], x], Times[-1, Simp[Times[Power[x, Plus[m, 1]], Cot[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Power[Csc[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, -2]], Power[Times[b, n, Plus[p, -1]], -1]], x]]], And[FreeQ[List[a, b, c, m, n, p], x], ZeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[Plus[p, -2], 2]], Power[Plus[m, 1], 2]]], NonzeroQ[Plus[m, 1]], NonzeroQ[Plus[p, -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sec[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[m, 1]], Tan[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Power[Sec[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, -2]], Power[Times[b, n, Plus[p, -1]], -1]], x], Plus[Dist[Times[Plus[Times[Power[b, 2], Power[n, 2], Power[Plus[p, -2], 2]], Power[Plus[m, 1], 2]], Power[Times[Power[b, 2], Power[n, 2], Plus[p, -1], Plus[p, -2]], -1]], Int[Times[Power[x, m], Power[Sec[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, -2]]], x], x], Times[-1, Simp[Times[Plus[m, 1], Power[x, Plus[m, 1]], Power[Sec[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, -2]], Power[Times[Power[b, 2], Power[n, 2], Plus[p, -1], Plus[p, -2]], -1]], x]]]], And[FreeQ[List[a, b, c, m, n], x], RationalQ[p], Greater[p, 1], Unequal[p, 2], NonzeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[Plus[p, -2], 2]], Power[Plus[m, 1], 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Csc[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[x, Plus[m, 1]], Cot[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Power[Csc[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, -2]], Power[Times[b, n, Plus[p, -1]], -1]], x]], Plus[Dist[Times[Plus[Times[Power[b, 2], Power[n, 2], Power[Plus[p, -2], 2]], Power[Plus[m, 1], 2]], Power[Times[Power[b, 2], Power[n, 2], Plus[p, -1], Plus[p, -2]], -1]], Int[Times[Power[x, m], Power[Csc[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, -2]]], x], x], Times[-1, Simp[Times[Plus[m, 1], Power[x, Plus[m, 1]], Power[Csc[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, -2]], Power[Times[Power[b, 2], Power[n, 2], Plus[p, -1], Plus[p, -2]], -1]], x]]]], And[FreeQ[List[a, b, c, m, n], x], RationalQ[p], Greater[p, 1], Unequal[p, 2], NonzeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[Plus[p, -2], 2]], Power[Plus[m, 1], 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sec[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[b, n, p, Power[x, Plus[m, 1]], Sin[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Power[Sec[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 1]], Power[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], Power[Plus[m, 1], 2]], -1]], x]], Plus[Dist[Times[Power[b, 2], Power[n, 2], p, Plus[p, 1], Power[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], Power[Plus[m, 1], 2]], -1]], Int[Times[Power[x, m], Power[Sec[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 2]]], x], x], Simp[Times[Plus[m, 1], Power[x, Plus[m, 1]], Power[Sec[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], p], Power[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], Power[Plus[m, 1], 2]], -1]], x]]], And[FreeQ[List[a, b, c, m, n], x], RationalQ[p], Less[p, -1], NonzeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], Power[Plus[m, 1], 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Csc[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, n, p, Power[x, Plus[m, 1]], Cos[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Power[Csc[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 1]], Power[Plus[Times[Power[b, 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Condition[Simp[Times[Power[x, Plus[m, 1]], Power[Plus[2, Times[2, Power[E, Times[2, I, a]], Power[Times[c, Power[x, n]], Times[2, I, b]]]], p], Power[Plus[m, 1, Times[I, b, n, p]], -1], Power[Times[Power[E, Times[I, a]], Power[Times[c, Power[x, n]], Times[I, b]], Power[Plus[1, Times[Power[E, Times[2, I, a]], Power[Times[c, Power[x, n]], Times[2, I, b]]]], -1]], p], Hypergeometric2F1[p, Times[Plus[m, 1, Times[I, b, n, p]], Power[Times[2, I, b, n], -1]], Plus[1, Times[Plus[m, 1, Times[I, b, n, p]], Power[Times[2, I, b, n], -1]]], Times[-1, Power[E, Times[2, I, a]], Power[Times[c, Power[x, n]], Times[2, I, b]]]]], x], And[FreeQ[List[a, b, c, m, n, p], x], NonzeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], Power[Plus[m, 1], 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Csc[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[x, Plus[m, 1]], Power[Plus[2, Times[-1, 2, Power[E, Times[2, I, a]], Power[Times[c, Power[x, n]], Times[2, I, b]]]], p], Power[Plus[m, 1, Times[I, b, n, p]], -1], Power[Times[-1, I, Power[E, Times[I, a]], Power[Times[c, Power[x, n]], Times[I, b]], Power[Plus[1, Times[-1, Power[E, Times[2, I, a]], Power[Times[c, Power[x, n]], Times[2, I, b]]]], -1]], p], Hypergeometric2F1[p, Times[Plus[m, 1, Times[I, b, n, p]], Power[Times[2, I, b, n], -1]], Plus[1, Times[Plus[m, 1, Times[I, b, n, p]], Power[Times[2, I, b, n], -1]]], Times[Power[E, Times[2, I, a]], Power[Times[c, Power[x, n]], Times[2, I, b]]]]], x], And[FreeQ[List[a, b, c, m, n, p], x], NonzeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], Power[Plus[m, 1], 2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Sin[Times[Power[Log[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Cos[Times[a, x, Power[Log[Times[b, x]], p]]], Power[a, -1]], x]], Times[-1, Dist[p, Int[Times[Sin[Times[a, x, Power[Log[Times[b, x]], p]]], Power[Log[Times[b, x]], Plus[p, -1]]], x], x]]], And[FreeQ[List[a, b], x], RationalQ[p], Greater[p, 0]]]], RuleDelayed[HoldPattern[Int[Times[Cos[Times[Power[Log[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Power[Log[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Sin[Times[a, x, Power[Log[Times[b, x]], p]]], Power[a, -1]], x], Times[-1, Dist[p, Int[Times[Cos[Times[a, x, Power[Log[Times[b, x]], p]]], Power[Log[Times[b, x]], 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RuleDelayed[HoldPattern[Int[Times[Cos[Times[Power[Log[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Log[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Sin[Times[a, Power[x, n], Power[Log[Times[b, x]], p]]], Power[Times[a, n, Power[x, Plus[n, -1]]], -1]], x], Plus[Times[-1, Dist[Times[p, Power[n, -1]], Int[Times[Cos[Times[a, Power[x, n], Power[Log[Times[b, x]], p]]], Power[Log[Times[b, x]], Plus[p, -1]]], x], x]], Dist[Times[Plus[n, -1], Power[Times[a, n], -1]], Int[Times[Sin[Times[a, Power[x, n], Power[Log[Times[b, x]], p]]], Power[Power[x, n], -1]], x], x]]], And[FreeQ[List[a, b], x], RationalQ[n, p], Greater[p, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], 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PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Power[Cos[Times[Optional[Pattern[a, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Power[d, -1], Subst[Int[Times[Power[Cos[Times[a, x]], n], Power[Power[x, 2], -1]], x], x, Times[1, Power[Plus[c, Times[d, x]], -1]]], x]], And[FreeQ[List[a, c, d], x], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Power[Sin[Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Power[d, -1], Subst[Int[Times[Power[Sin[Plus[Times[b, e, Power[d, -1]], Times[-1, e, Plus[Times[b, c], Times[-1, a, d]], x, Power[d, -1]]]], n], 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Condition[Module[List[Set[lst, QuotientOfLinearsParts[u, x]]], Int[Power[Sin[Times[Plus[Part[lst, 1], Times[Part[lst, 2], x]], Power[Plus[Part[lst, 3], Times[Part[lst, 4], x]], -1]]], n], x]], And[PositiveIntegerQ[n], QuotientOfLinearsQ[u, x]]]], RuleDelayed[HoldPattern[Int[Power[Cos[Pattern[u, Blank[]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[lst, QuotientOfLinearsParts[u, x]]], Int[Power[Cos[Times[Plus[Part[lst, 1], Times[Part[lst, 2], x]], Power[Plus[Part[lst, 3], Times[Part[lst, 4], x]], -1]]], n], x]], And[PositiveIntegerQ[n], QuotientOfLinearsQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Optional[Pattern[u, Blank[]]], Power[Sin[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Sin[Pattern[w, Blank[]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[u, Power[Sin[v], Plus[p, q]]], x], ZeroQ[Plus[v, Times[-1, w]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cos[Pattern[v, Blank[]]], 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And[BinomialQ[List[v, w], x], IndependentQ[Cancel[Times[v, Power[w, -1]]], x]]], PositiveIntegerQ[p, q]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sin[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Sin[Pattern[w, Blank[]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Power[x, m], Times[Power[Sin[v], p], Power[Sin[w], q]], x], x], And[PositiveIntegerQ[m, p, q], Or[And[PolynomialQ[v, x], PolynomialQ[w, x]], And[BinomialQ[List[v, w], x], IndependentQ[Cancel[Times[v, Power[w, -1]]], x]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cos[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Cos[Pattern[w, Blank[]]], Optional[Pattern[q, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Power[x, m], Times[Power[Cos[v], p], Power[Cos[w], q]], x], x], And[PositiveIntegerQ[m, p, q], 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Optional[Pattern[q, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sin[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Power[x, m], Times[Power[Sin[v], p], Power[Cos[w], q]], x], x], And[PositiveIntegerQ[m, p, q], Or[And[PolynomialQ[v, x], PolynomialQ[w, x]], And[BinomialQ[List[v, w], x], IndependentQ[Cancel[Times[v, Power[w, -1]]], x]]]]]], RuleDelayed[HoldPattern[Int[Times[Sin[Pattern[v, Blank[]]], Power[Tan[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Int[Times[Cos[v], Power[Tan[w], Plus[n, -1]]], x]], Dist[Cos[Plus[v, Times[-1, w]]], Int[Times[Sec[w], Power[Tan[w], Plus[n, -1]]], x], x]], And[RationalQ[n], Greater[n, 0], FreeQ[Plus[v, Times[-1, w]], x], NonzeroQ[Plus[v, Times[-1, w]]]]]], RuleDelayed[HoldPattern[Int[Times[Cos[Pattern[v, Blank[]]], Power[Cot[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Int[Times[Sin[v], Power[Cot[w], Plus[n, -1]]], x]], Dist[Cos[Plus[v, Times[-1, w]]], Int[Times[Csc[w], Power[Cot[w], Plus[n, -1]]], x], x]], And[RationalQ[n], Greater[n, 0], FreeQ[Plus[v, Times[-1, w]], x], NonzeroQ[Plus[v, Times[-1, w]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cot[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]], Sin[Pattern[v, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Int[Times[Cos[v], Power[Cot[w], Plus[n, -1]]], x], Dist[Sin[Plus[v, Times[-1, w]]], Int[Times[Csc[w], Power[Cot[w], Plus[n, -1]]], x], x]], And[RationalQ[n], Greater[n, 0], FreeQ[Plus[v, Times[-1, w]], x], NonzeroQ[Plus[v, Times[-1, w]]]]]], RuleDelayed[HoldPattern[Int[Times[Cos[Pattern[v, Blank[]]], Power[Tan[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Int[Times[Sin[v], Power[Tan[w], Plus[n, -1]]], x], Times[-1, Dist[Sin[Plus[v, Times[-1, w]]], Int[Times[Sec[w], Power[Tan[w], Plus[n, -1]]], x], x]]], And[RationalQ[n], Greater[n, 0], FreeQ[Plus[v, Times[-1, w]], x], NonzeroQ[Plus[v, Times[-1, w]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Sec[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]], Sin[Pattern[v, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Cos[Plus[v, Times[-1, w]]], Int[Times[Tan[w], Power[Sec[w], Plus[n, -1]]], x], x], Dist[Sin[Plus[v, Times[-1, w]]], Int[Power[Sec[w], Plus[n, -1]], x], x]], And[RationalQ[n], Greater[n, 0], FreeQ[Plus[v, Times[-1, w]], x], NonzeroQ[Plus[v, Times[-1, w]]]]]], RuleDelayed[HoldPattern[Int[Times[Cos[Pattern[v, Blank[]]], Power[Csc[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Cos[Plus[v, Times[-1, w]]], Int[Times[Cot[w], Power[Csc[w], Plus[n, -1]]], x], x], Times[-1, Dist[Sin[Plus[v, Times[-1, w]]], Int[Power[Csc[w], Plus[n, -1]], x], x]]], And[RationalQ[n], Greater[n, 0], FreeQ[Plus[v, Times[-1, w]], x], NonzeroQ[Plus[v, Times[-1, w]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Csc[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]], Sin[Pattern[v, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Sin[Plus[v, Times[-1, w]]], Int[Times[Cot[w], Power[Csc[w], Plus[n, -1]]], x], x], Dist[Cos[Plus[v, Times[-1, w]]], Int[Power[Csc[w], Plus[n, -1]], x], x]], And[RationalQ[n], Greater[n, 0], FreeQ[Plus[v, Times[-1, w]], x], NonzeroQ[Plus[v, Times[-1, w]]]]]], RuleDelayed[HoldPattern[Int[Times[Cos[Pattern[v, Blank[]]], Power[Sec[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Dist[Sin[Plus[v, Times[-1, w]]], Int[Times[Tan[w], Power[Sec[w], Plus[n, -1]]], x], x]], Dist[Cos[Plus[v, Times[-1, w]]], Int[Power[Sec[w], Plus[n, -1]], x], x]], And[RationalQ[n], Greater[n, 0], FreeQ[Plus[v, Times[-1, w]], x], NonzeroQ[Plus[v, Times[-1, w]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[e, Times[f, x]], m], Power[Plus[a, Times[b, Sin[Plus[Times[2, c], Times[2, d, x]]], Power[2, -1]]], n]], x], FreeQ[List[a, b, c, d, e, f, m, n], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Power[2, n], -1]], Int[Times[Power[x, m], Power[Plus[Times[2, a], b, Times[-1, b, Cos[Plus[Times[2, c], Times[2, d, x]]]]], n]], x], x], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[a, b]], IntegersQ[m, n], Greater[m, 0], Less[n, 0], Or[Equal[n, -1], And[Equal[m, 1], Equal[n, -2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[Power[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Power[2, n], -1]], Int[Times[Power[x, m], Power[Plus[Times[2, a], b, Times[b, Cos[Plus[Times[2, c], Times[2, d, x]]]]], n]], x], x], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[a, b]], IntegersQ[m, n], Greater[m, 0], Less[n, 0], Or[Equal[n, -1], And[Equal[m, 1], Equal[n, -2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sin[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Power[d, Plus[m, 1]], -1]], Subst[Int[Times[Power[Plus[Times[d, e], Times[-1, c, f], Times[f, x]], m], Power[Sin[Plus[a, Times[b, Power[x, n]]]], p]], x], x, Plus[c, Times[d, x]]], x], And[FreeQ[List[a, b, c, d, e, f, n], x], PositiveIntegerQ[m], RationalQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cos[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Power[d, Plus[m, 1]], -1]], Subst[Int[Times[Power[Plus[Times[d, e], Times[-1, c, f], Times[f, x]], m], Power[Cos[Plus[a, Times[b, Power[x, n]]]], p]], x], x, Plus[c, Times[d, x]]], x], And[FreeQ[List[a, b, c, d, e, f, n], x], PositiveIntegerQ[m], RationalQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[Cos[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Sin[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[2, Int[Times[Power[Plus[f, Times[g, x]], m], Power[Plus[Times[2, a], b, c, Times[Plus[b, Times[-1, c]], Cos[Plus[Times[2, d], 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Optional[Pattern[m, Blank[]]]], Power[Sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Optional[Pattern[b, Blank[]]], Times[Optional[Pattern[a, Blank[]]], Power[Sec[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]], Times[Optional[Pattern[c, Blank[]]], Power[Tan[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[2, Int[Times[Power[Plus[f, Times[g, x]], m], Power[Plus[Times[2, a], b, c, Times[Plus[b, Times[-1, c]], Cos[Plus[Times[2, d], Times[2, e, x]]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, g], x], PositiveIntegerQ[m], NonzeroQ[Plus[a, b]], NonzeroQ[Plus[a, c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Csc[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Times[Power[Cot[Plus[Optional[Pattern[d, 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Blank[]]]], -1], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[2, Int[Times[Power[Plus[f, Times[g, x]], m], Power[Plus[Times[2, a], b, c, Times[Plus[b, Times[-1, c]], Cos[Plus[Times[2, d], Times[2, e, x]]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, g], x], PositiveIntegerQ[m], NonzeroQ[Plus[a, b]], NonzeroQ[Plus[a, c]]]]], RuleDelayed[HoldPattern[Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[I, Power[Plus[e, Times[f, x]], Plus[m, 1]], Power[Times[b, f, Plus[m, 1]], -1]], x]], Plus[Int[Times[Power[Plus[e, Times[f, x]], m], Power[E, Times[I, Plus[c, Times[d, x]]]], Power[Plus[a, Times[-1, Rt[Plus[Power[a, 2], Times[-1, Power[b, 2]]], 2]], Times[-1, I, b, Power[E, Times[I, Plus[c, Times[d, x]]]]]], -1]], x], Int[Times[Power[Plus[e, Times[f, x]], m], Power[E, Times[I, Plus[c, Times[d, x]]]], Power[Plus[a, Rt[Plus[Power[a, 2], Times[-1, Power[b, 2]]], 2], Times[-1, I, b, Power[E, Times[I, Plus[c, Times[d, x]]]]]], -1]], x]]], And[FreeQ[List[a, b, c, d, e, f], x], PositiveIntegerQ[m], PosQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[I, Power[Plus[e, Times[f, x]], Plus[m, 1]], Power[Times[b, f, Plus[m, 1]], -1]], x], Plus[Times[-1, Dist[I, Int[Times[Power[Plus[e, Times[f, x]], m], Power[E, Times[I, Plus[c, Times[d, x]]]], Power[Plus[a, Times[-1, Rt[Plus[Power[a, 2], Times[-1, Power[b, 2]]], 2]], Times[b, Power[E, Times[I, Plus[c, Times[d, x]]]]]], -1]], x], x]], Times[-1, Dist[I, Int[Times[Power[Plus[e, Times[f, x]], m], Power[E, Times[I, Plus[c, Times[d, x]]]], Power[Plus[a, Rt[Plus[Power[a, 2], Times[-1, Power[b, 2]]], 2], Times[b, Power[E, Times[I, Plus[c, Times[d, x]]]]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d, e, f], x], PositiveIntegerQ[m], PosQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[I, Power[Plus[e, Times[f, x]], Plus[m, 1]], Power[Times[b, f, Plus[m, 1]], -1]], x]], Plus[Dist[I, Int[Times[Power[Plus[e, Times[f, x]], m], Power[E, Times[I, Plus[c, Times[d, x]]]], Power[Plus[Times[I, a], Times[-1, Rt[Plus[Times[-1, Power[a, 2]], Power[b, 2]], 2]], Times[b, Power[E, Times[I, Plus[c, Times[d, x]]]]]], -1]], x], x], Dist[I, Int[Times[Power[Plus[e, Times[f, x]], m], Power[E, Times[I, Plus[c, Times[d, x]]]], Power[Plus[Times[I, a], Rt[Plus[Times[-1, Power[a, 2]], Power[b, 2]], 2], Times[b, Power[E, Times[I, Plus[c, Times[d, x]]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f], x], PositiveIntegerQ[m], NegQ[Plus[Power[a, 2], Times[-1, Power[b, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[Cos[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], -1], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Sin[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[I, Power[Plus[e, Times[f, x]], Plus[m, 1]], Power[Times[b, f, Plus[m, 1]], -1]], x], Plus[Int[Times[Power[Plus[e, Times[f, x]], m], Power[E, Times[I, Plus[c, Times[d, x]]]], Power[Plus[Times[I, a], Times[-1, Rt[Plus[Times[-1, Power[a, 2]], Power[b, 2]], 2]], Times[I, b, Power[E, Times[I, Plus[c, Times[d, x]]]]]], -1]], x], Int[Times[Power[Plus[e, Times[f, x]], m], Power[E, Times[I, Plus[c, Times[d, x]]]], 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Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Times[Power[c, Plus[m, 1]], Sqrt[d]], -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], n], Power[Sin[x], m]], x], x, ArcSin[Times[c, x]]], x], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], PositiveQ[d], PositiveIntegerQ[n], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, 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Simp[Times[b, c, Power[Times[f, x], Plus[m, 2]], HypergeometricPFQ[List[1, Plus[1, Times[m, Power[2, -1]]], Plus[1, Times[m, Power[2, -1]]]], List[Plus[Times[3, Power[2, -1]], Times[m, Power[2, -1]]], Plus[2, Times[m, Power[2, -1]]]], Times[Power[c, 2], Power[x, 2]]], Power[Times[Sqrt[d], Power[f, 2], Plus[m, 1], Plus[m, 2]], -1]], x]]], And[FreeQ[List[a, b, c, d, e, f, m], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], PositiveQ[d], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[f, x], Plus[m, 1]], Plus[a, Times[b, ArcCos[Times[c, x]]]], Hypergeometric2F1[Times[1, Power[2, -1]], Times[Plus[1, m], Power[2, -1]], Times[Plus[3, m], Power[2, -1]], Times[Power[c, 2], Power[x, 2]]], Power[Times[Sqrt[d], f, Plus[m, 1]], -1]], x], Simp[Times[b, c, Power[Times[f, x], Plus[m, 2]], HypergeometricPFQ[List[1, Plus[1, Times[m, Power[2, -1]]], Plus[1, Times[m, Power[2, -1]]]], List[Plus[Times[3, Power[2, -1]], Times[m, Power[2, -1]]], Plus[2, Times[m, Power[2, -1]]]], Times[Power[c, 2], Power[x, 2]]], Power[Times[Sqrt[d], Power[f, 2], Plus[m, 1], Plus[m, 2]], -1]], x]], And[FreeQ[List[a, b, c, d, e, f, m], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], PositiveQ[d], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]]], Power[Sqrt[Plus[d, Times[e, Power[x, 2]]]], -1]], Int[Times[Power[Times[f, x], m], Power[Plus[a, Times[b, ArcSin[Times[c, x]]]], n], Power[Sqrt[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, m], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[n], Greater[n, 0], Not[PositiveQ[d]], Or[IntegerQ[m], Equal[n, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]]], Power[Sqrt[Plus[d, Times[e, Power[x, 2]]]], -1]], Int[Times[Power[Times[f, x], m], Power[Plus[a, Times[b, ArcCos[Times[c, x]]]], n], Power[Sqrt[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, m], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[n], Greater[n, 0], Not[PositiveQ[d]], Or[IntegerQ[m], Equal[n, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[f, Power[Times[f, x], Plus[m, -1]], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcSin[Times[c, x]]]], n], Power[Times[e, Plus[m, Times[2, p], 1]], -1]], x], Plus[Dist[Times[Power[f, 2], Plus[m, -1], Power[Times[Power[c, 2], Plus[m, Times[2, p], 1]], -1]], Int[Times[Power[Times[f, x], Plus[m, -2]], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcSin[Times[c, x]]]], n]], x], x], Dist[Times[b, f, n, Power[d, IntPart[p]], Power[Plus[d, Times[e, Power[x, 2]]], FracPart[p]], Power[Times[c, Plus[m, Times[2, p], 1], Power[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]], FracPart[p]]], -1]], Int[Times[Power[Times[f, x], Plus[m, -1]], Power[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcSin[Times[c, x]]]], Plus[n, -1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, p], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[m, n], Greater[n, 0], Greater[m, 1], NonzeroQ[Plus[m, Times[2, p], 1]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[f, Power[Times[f, x], Plus[m, -1]], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCos[Times[c, x]]]], n], Power[Times[e, Plus[m, Times[2, p], 1]], -1]], x], Plus[Dist[Times[Power[f, 2], Plus[m, -1], Power[Times[Power[c, 2], Plus[m, Times[2, p], 1]], -1]], Int[Times[Power[Times[f, x], Plus[m, -2]], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcCos[Times[c, x]]]], n]], x], x], Times[-1, Dist[Times[b, f, n, Power[d, IntPart[p]], Power[Plus[d, Times[e, Power[x, 2]]], FracPart[p]], Power[Times[c, Plus[m, Times[2, p], 1], Power[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]], FracPart[p]]], -1]], Int[Times[Power[Times[f, x], Plus[m, -1]], Power[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCos[Times[c, x]]]], Plus[n, -1]]], x], x]]]], And[FreeQ[List[a, b, c, d, e, f, p], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[m, n], Greater[n, 0], Greater[m, 1], NonzeroQ[Plus[m, Times[2, p], 1]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[f, x], m], Sqrt[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]]], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcSin[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Plus[n, 1]], -1]], x], Times[-1, Dist[Times[f, m, Power[d, IntPart[p]], Power[Plus[d, Times[e, Power[x, 2]]], FracPart[p]], Power[Times[b, c, Plus[n, 1], Power[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]], FracPart[p]]], -1]], Int[Times[Power[Times[f, x], Plus[m, -1]], Power[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcSin[Times[c, x]]]], Plus[n, 1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, m, p], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[n], Less[n, -1], ZeroQ[Plus[m, Times[2, p], 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Times[f, x], m], Sqrt[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]]], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcCos[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Plus[n, 1]], -1]], x]], Dist[Times[f, m, Power[d, IntPart[p]], Power[Plus[d, Times[e, Power[x, 2]]], FracPart[p]], Power[Times[b, c, Plus[n, 1], Power[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]], FracPart[p]]], -1]], Int[Times[Power[Times[f, x], Plus[m, -1]], Power[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCos[Times[c, x]]]], Plus[n, 1]]], x], x]], And[FreeQ[List[a, b, c, d, e, f, m, p], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[n], Less[n, -1], ZeroQ[Plus[m, Times[2, p], 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[f, x], m], Power[Plus[a, Times[b, ArcSin[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Sqrt[d], Plus[n, 1]], -1]], x], Times[-1, Dist[Times[f, m, Power[Times[b, c, Sqrt[d], Plus[n, 1]], -1]], Int[Times[Power[Times[f, x], Plus[m, -1]], Power[Plus[a, Times[b, ArcSin[Times[c, x]]]], Plus[n, 1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, m], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[n], Less[n, -1], PositiveQ[d]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Times[f, x], m], Power[Plus[a, Times[b, ArcCos[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Sqrt[d], Plus[n, 1]], -1]], x]], Dist[Times[f, m, Power[Times[b, c, Sqrt[d], Plus[n, 1]], -1]], Int[Times[Power[Times[f, x], Plus[m, -1]], Power[Plus[a, Times[b, ArcCos[Times[c, x]]]], Plus[n, 1]]], x], x]], And[FreeQ[List[a, b, c, d, e, f, m], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[n], Less[n, -1], PositiveQ[d]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[f, x], m], Sqrt[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]]], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcSin[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Plus[n, 1]], -1]], x], Plus[Times[-1, Dist[Times[f, m, Power[d, IntPart[p]], Power[Plus[d, Times[e, Power[x, 2]]], FracPart[p]], Power[Times[b, c, Plus[n, 1], Power[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]], FracPart[p]]], -1]], Int[Times[Power[Times[f, x], Plus[m, -1]], Power[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[a, Times[b, 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Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Times[f, x], m], Sqrt[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]]], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcCos[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Plus[n, 1]], -1]], x]], Plus[Dist[Times[f, m, Power[d, IntPart[p]], Power[Plus[d, Times[e, Power[x, 2]]], FracPart[p]], Power[Times[b, c, Plus[n, 1], Power[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]], FracPart[p]]], -1]], Int[Times[Power[Times[f, x], Plus[m, -1]], Power[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCos[Times[c, x]]]], Plus[n, 1]]], x], x], Times[-1, Dist[Times[c, Plus[m, Times[2, p], 1], Power[d, IntPart[p]], Power[Plus[d, Times[e, Power[x, 2]]], FracPart[p]], Power[Times[b, f, Plus[n, 1], Power[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]], FracPart[p]]], -1]], Int[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCos[Times[c, x]]]], Plus[n, 1]]], x], x]]]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[n], Less[n, -1], IntegerQ[m], Greater[m, -3], PositiveIntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[d, p], Power[Power[c, Plus[m, 1]], -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], n], Power[Sin[x], m], Power[Cos[x], Plus[Times[2, p], 1]]], x], x, ArcSin[Times[c, x]]], x], And[FreeQ[List[a, b, c, d, e, n], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], IntegerQ[Times[2, p]], Greater[p, -1], PositiveIntegerQ[m], Or[IntegerQ[p], PositiveQ[d]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Times[Power[d, p], Power[Power[c, Plus[m, 1]], -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], n], Power[Cos[x], m], Power[Sin[x], Plus[Times[2, p], 1]]], x], x, ArcCos[Times[c, x]]], x]], And[FreeQ[List[a, b, c, d, e, n], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], IntegerQ[Times[2, p]], Greater[p, -1], PositiveIntegerQ[m], Or[IntegerQ[p], PositiveQ[d]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[d, IntPart[p]], Power[Plus[d, Times[e, Power[x, 2]]], FracPart[p]], Power[Power[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]], FracPart[p]], -1]], Int[Times[Power[x, m], Power[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]], p], Power[Plus[a, Times[b, ArcSin[Times[c, x]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, n], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], IntegerQ[Times[2, p]], Greater[p, -1], PositiveIntegerQ[m], Not[Or[IntegerQ[p], PositiveQ[d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], 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Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, IntHide[Times[Power[Plus[f, Times[g, x]], p], Power[Plus[d, Times[e, x]], m]], x]]], Plus[Dist[Power[Plus[a, Times[b, ArcSin[Times[c, x]]]], n], u, x], Times[-1, Dist[Times[b, c, n], Int[SimplifyIntegrand[Times[u, Power[Plus[a, Times[b, ArcSin[Times[c, x]]]], Plus[n, -1]], Power[Sqrt[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]]], -1]], x], x], x]]]], And[FreeQ[List[a, b, c, d, e, f, g], x], PositiveIntegerQ[n, p], NegativeIntegerQ[m], Less[Plus[m, p, 1], 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, IntHide[Times[Power[Plus[f, Times[g, x]], p], Power[Plus[d, Times[e, x]], m]], x]]], Plus[Dist[Power[Plus[a, Times[b, ArcCos[Times[c, x]]]], n], u, x], Dist[Times[b, c, n], Int[SimplifyIntegrand[Times[u, Power[Plus[a, Times[b, ArcCos[Times[c, x]]]], Plus[n, -1]], Power[Sqrt[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]]], -1]], x], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g], x], PositiveIntegerQ[n, p], NegativeIntegerQ[m], Less[Plus[m, p, 1], 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], 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x], x], And[FreeQ[List[a, b, c, d, e, f, g], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], IntegerQ[m], PositiveIntegerQ[Plus[p, Times[1, Power[2, -1]]]], PositiveQ[d], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[f, Times[g, x]], m], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcSin[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Sqrt[d], Plus[n, 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[b, c, Sqrt[d], Plus[n, 1]], -1]], 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Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Plus[f, Times[g, x]], m], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCos[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Sqrt[d], Plus[n, 1]], -1]], x]], Dist[Times[1, Power[Times[b, c, Sqrt[d], Plus[n, 1]], -1]], Int[ExpandIntegrand[Times[Power[Plus[f, Times[g, x]], Plus[m, -1]], Power[Plus[a, Times[b, ArcCos[Times[c, x]]]], Plus[n, 1]]], Times[Plus[Times[d, g, m], Times[e, f, Plus[Times[2, p], 1], x], Times[e, g, Plus[m, Times[2, p], 1], Power[x, 2]]], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, Times[-1, 1, Power[2, -1]]]]], x], x], x]], And[FreeQ[List[a, b, c, d, e, f, g], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], IntegerQ[m], PositiveIntegerQ[Plus[p, Times[-1, 1, Power[2, -1]]]], PositiveQ[d], PositiveIntegerQ[n], Less[m, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], 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Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Plus[f, Times[g, x]], m], Power[Plus[a, Times[b, ArcCos[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Sqrt[d], Plus[n, 1]], -1]], x]], Dist[Times[g, m, Power[Times[b, c, Sqrt[d], Plus[n, 1]], -1]], Int[Times[Power[Plus[f, Times[g, x]], Plus[m, -1]], Power[Plus[a, Times[b, ArcCos[Times[c, x]]]], Plus[n, 1]]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], IntegerQ[m], PositiveQ[d], Greater[m, 0], RationalQ[n], Less[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], 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Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Power[Times[Power[c, Plus[m, 1]], Sqrt[d]], -1], Subst[Int[Times[Power[Plus[a, Times[b, x]], n], Power[Plus[Times[c, f], Times[g, Cos[x]]], m]], x], x, ArcCos[Times[c, x]]], x]], And[FreeQ[List[a, b, c, d, e, f, g, n], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], IntegerQ[m], PositiveQ[d], Or[Greater[m, 0], PositiveIntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[a, Times[b, ArcSin[Times[c, x]]]], n], Power[Sqrt[Plus[d, Times[e, Power[x, 2]]]], -1]], Times[Power[Plus[f, Times[g, x]], m], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, Times[1, Power[2, -1]]]]], x], x], And[FreeQ[List[a, b, c, d, e, f, g], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], IntegerQ[m], NegativeIntegerQ[Plus[p, Times[1, Power[2, -1]]]], PositiveQ[d], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[a, Times[b, ArcCos[Times[c, x]]]], n], Power[Sqrt[Plus[d, Times[e, Power[x, 2]]]], -1]], Times[Power[Plus[f, Times[g, x]], m], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, Times[1, Power[2, -1]]]]], x], x], And[FreeQ[List[a, b, c, d, e, f, g], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], IntegerQ[m], NegativeIntegerQ[Plus[p, Times[1, Power[2, -1]]]], PositiveQ[d], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[d, IntPart[p]], Power[Plus[d, Times[e, 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2]]], FracPart[p]], Power[Power[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]], FracPart[p]], -1]], Int[Times[Power[Plus[f, Times[g, x]], m], Power[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]], p], Power[Plus[a, Times[b, ArcCos[Times[c, x]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, n], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], IntegerQ[m], IntegerQ[Plus[p, Times[-1, 1, Power[2, -1]]]], Not[PositiveQ[d]]]]], RuleDelayed[HoldPattern[Int[Times[Log[Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Log[Times[h, Power[Plus[f, Times[g, x]], m]]], Power[Plus[a, Times[b, ArcSin[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Sqrt[d], Plus[n, 1]], -1]], x], Times[-1, Dist[Times[g, m, Power[Times[b, c, Sqrt[d], Plus[n, 1]], -1]], Int[Times[Power[Plus[a, Times[b, ArcSin[Times[c, x]]]], Plus[n, 1]], Power[Plus[f, Times[g, x]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g, h, m], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], PositiveQ[d], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Log[Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Log[Times[h, Power[Plus[f, Times[g, x]], m]]], Power[Plus[a, Times[b, ArcCos[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Sqrt[d], Plus[n, 1]], -1]], x]], Dist[Times[g, m, Power[Times[b, c, Sqrt[d], Plus[n, 1]], -1]], Int[Times[Power[Plus[a, Times[b, ArcCos[Times[c, x]]]], Plus[n, 1]], Power[Plus[f, Times[g, x]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f, g, h, m], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], PositiveQ[d], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Log[Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[d, IntPart[p]], Power[Plus[d, Times[e, Power[x, 2]]], FracPart[p]], Power[Power[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]], FracPart[p]], -1]], Int[Times[Log[Times[h, Power[Plus[f, Times[g, x]], m]]], Power[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]], p], Power[Plus[a, Times[b, ArcSin[Times[c, x]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, h, m, n], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], IntegerQ[Plus[p, Times[-1, 1, Power[2, -1]]]], Not[PositiveQ[d]]]]], RuleDelayed[HoldPattern[Int[Times[Log[Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[d, IntPart[p]], Power[Plus[d, Times[e, Power[x, 2]]], FracPart[p]], Power[Power[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]], FracPart[p]], -1]], Int[Times[Log[Times[h, Power[Plus[f, Times[g, x]], m]]], Power[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]], p], Power[Plus[a, Times[b, ArcCos[Times[c, x]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, h, m, n], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], IntegerQ[Plus[p, Times[-1, 1, Power[2, -1]]]], Not[PositiveQ[d]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, IntHide[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], m]], x]]], Plus[Dist[Plus[a, Times[b, ArcSin[Times[c, x]]]], u, x], Times[-1, Dist[Times[b, c], Int[Dist[Times[1, Power[Sqrt[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]]], -1]], u, x], x], x]]]], And[FreeQ[List[a, b, c, d, e, f, g], x], NegativeIntegerQ[Plus[m, Times[1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, IntHide[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], m]], x]]], Plus[Dist[Plus[a, Times[b, ArcCos[Times[c, x]]]], u, x], Dist[Times[b, c], Int[Dist[Times[1, Power[Sqrt[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]]], -1]], u, x], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g], x], NegativeIntegerQ[Plus[m, Times[1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], m], Power[Plus[a, Times[b, ArcSin[Times[c, x]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, n], x], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], m], Power[Plus[a, Times[b, ArcCos[Times[c, x]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, n], x], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[v, IntHide[u, x]]], Condition[Plus[Dist[Plus[a, Times[b, ArcSin[Times[c, x]]]], v, x], Times[-1, Dist[Times[b, c], Int[SimplifyIntegrand[Times[v, Power[Sqrt[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]]], -1]], x], x], x]]], InverseFunctionFreeQ[v, x]]], FreeQ[List[a, b, c], x]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[v, IntHide[u, x]]], Condition[Plus[Dist[Plus[a, Times[b, ArcCos[Times[c, x]]]], v, x], Dist[Times[b, c], Int[SimplifyIntegrand[Times[v, Power[Sqrt[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]]], -1]], x], x], x]], InverseFunctionFreeQ[v, x]]], FreeQ[List[a, b, c], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, ExpandIntegrand[Times[Px, Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcSin[Times[c, x]]]], n]], x]]], Condition[Int[u, x], SumQ[u]]], And[FreeQ[List[a, b, c, d, e, n], x], PolynomialQ[Px, x], ZeroQ[Plus[Times[Power[c, 2], d], e]], IntegerQ[Plus[p, Times[-1, 1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, ExpandIntegrand[Times[Px, Power[Plus[d, Times[e, Power[x, 2]]], p], 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PolynomialQ[Px, x], ZeroQ[Plus[Times[Power[c, 2], d], e]], PositiveIntegerQ[Plus[p, Times[1, Power[2, -1]]]], IntegersQ[m, n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[Px, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, ExpandIntegrand[Times[Px, Power[Plus[f, Times[g, Power[Plus[d, Times[e, Power[x, 2]]], p]]], m], Power[Plus[a, Times[b, ArcCos[Times[c, x]]]], n]], x]]], Condition[Int[u, x], SumQ[u]]], And[FreeQ[List[a, b, c, d, e, f, g], x], PolynomialQ[Px, x], ZeroQ[Plus[Times[Power[c, 2], d], e]], PositiveIntegerQ[Plus[p, Times[1, Power[2, -1]]]], IntegersQ[m, n]]]], RuleDelayed[HoldPattern[Int[Times[Power[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[RFx, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, ExpandIntegrand[Power[ArcSin[Times[c, x]], n], RFx, x]]], Condition[Int[u, x], SumQ[u]]], And[FreeQ[c, x], RationalFunctionQ[RFx, x], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[RFx, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, ExpandIntegrand[Power[ArcCos[Times[c, x]], n], RFx, x]]], Condition[Int[u, x], SumQ[u]]], And[FreeQ[c, x], RationalFunctionQ[RFx, x], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]], Pattern[RFx, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[RFx, Power[Plus[a, Times[b, ArcSin[Times[c, x]]]], n]], x], x], And[FreeQ[List[a, b, c], x], RationalFunctionQ[RFx, x], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[ArcCos[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]], Pattern[RFx, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[RFx, Power[Plus[a, Times[b, ArcCos[Times[c, x]]]], n]], x], x], And[FreeQ[List[a, b, c], x], RationalFunctionQ[RFx, x], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[ArcSin[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[RFx, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, 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x]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSin[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[x, Power[Plus[a, Times[b, ArcSin[Plus[c, Times[d, Power[x, 2]]]]]], n]], x], Plus[Times[-1, Dist[Times[4, Power[b, 2], n, Plus[n, -1]], Int[Power[Plus[a, Times[b, ArcSin[Plus[c, Times[d, Power[x, 2]]]]]], Plus[n, -2]], x], x]], Simp[Times[2, b, n, Sqrt[Plus[Times[-2, c, d, Power[x, 2]], Times[-1, Power[d, 2], Power[x, 4]]]], Power[Plus[a, Times[b, ArcSin[Plus[c, Times[d, Power[x, 2]]]]]], Plus[n, -1]], Power[Times[d, x], -1]], x]]], And[FreeQ[List[a, b, c, d], x], ZeroQ[Plus[Power[c, 2], -1]], RationalQ[n], Greater[n, 1]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCos[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 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Power[Times[b, c, d, Plus[n, 1], Plus[d, Times[e, Power[x, 2]]]], -1]], x]], Plus[Times[-1, Dist[Times[4, Power[Times[Power[b, 2], Plus[n, 1], Plus[n, 2]], -1]], Int[Times[x, Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], Plus[n, 2]], Power[Power[Plus[d, Times[e, Power[x, 2]]], 2], -1]], x], x]], Times[-1, Simp[Times[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]], Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], Plus[n, 2]], Power[Times[Power[b, 2], e, Plus[n, 1], Plus[n, 2], Plus[d, Times[e, Power[x, 2]]]], -1]], x]]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[n], Less[n, -1], Unequal[n, -2]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, 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Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Times[4, Power[c, 3], d, Power[Plus[p, 1], 2]], -1]], x], Plus[Times[-1, Dist[Times[1, Power[Times[2, Power[c, 2], d, Plus[p, 1]], -1]], Int[Times[Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Plus[a, Times[b, ArcCot[Times[c, x]]]]], x], x]], Simp[Times[x, Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Plus[a, Times[b, ArcCot[Times[c, x]]]], Power[Times[2, Power[c, 2], d, Plus[p, 1]], -1]], x]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[p], Less[p, -1], Unequal[p, Times[-5, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -2]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], Plus[n, 1]], Power[Times[2, b, Power[c, 3], Power[d, 2], Plus[n, 1]], -1]], x], Plus[Dist[Times[b, n, Power[Times[2, c], -1]], Int[Times[x, Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], Plus[n, -1]], Power[Power[Plus[d, Times[e, Power[x, 2]]], 2], -1]], x], x], Times[-1, Simp[Times[x, Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], n], Power[Times[2, Power[c, 2], d, Plus[d, Times[e, Power[x, 2]]]], -1]], x]]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[n], Greater[n, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -2]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], Plus[n, 1]], Power[Times[2, b, Power[c, 3], Power[d, 2], Plus[n, 1]], -1]], x]], Plus[Times[-1, Dist[Times[b, n, Power[Times[2, c], -1]], Int[Times[x, Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], Plus[n, -1]], Power[Power[Plus[d, Times[e, Power[x, 2]]], 2], -1]], x], x]], Times[-1, Simp[Times[x, Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], n], Power[Times[2, Power[c, 2], d, Plus[d, Times[e, Power[x, 2]]]], -1]], x]]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[n], Greater[n, 0]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, Power[x, m], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Times[c, d, Power[m, 2]], -1]], x], Plus[Dist[Times[Plus[m, -1], Power[Times[Power[c, 2], d, m], -1]], Int[Times[Power[x, Plus[m, -2]], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Plus[a, Times[b, ArcTan[Times[c, x]]]]], x], x], Times[-1, Simp[Times[Power[x, Plus[m, -1]], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Plus[a, Times[b, ArcTan[Times[c, x]]]], Power[Times[Power[c, 2], d, m], -1]], x]]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], ZeroQ[Plus[m, Times[2, p], 2]], RationalQ[p], Less[p, -1]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[b, Power[x, m], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Times[c, d, Power[m, 2]], -1]], x]], Plus[Dist[Times[Plus[m, -1], Power[Times[Power[c, 2], d, m], -1]], Int[Times[Power[x, Plus[m, -2]], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Plus[a, Times[b, ArcCot[Times[c, x]]]]], x], x], Times[-1, Simp[Times[Power[x, Plus[m, -1]], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Plus[a, Times[b, ArcCot[Times[c, x]]]], Power[Times[Power[c, 2], d, m], -1]], x]]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], ZeroQ[Plus[m, Times[2, p], 2]], RationalQ[p], Less[p, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, n, Power[x, m], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], Plus[n, -1]], Power[Times[c, d, Power[m, 2]], -1]], x], Plus[Dist[Times[Plus[m, -1], Power[Times[Power[c, 2], d, m], -1]], Int[Times[Power[x, Plus[m, -2]], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], n]], x], x], Times[-1, Dist[Times[Power[b, 2], n, Plus[n, -1], Power[Power[m, 2], -1]], Int[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], Plus[n, -2]]], x], x]], Times[-1, Simp[Times[Power[x, Plus[m, -1]], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], n], Power[Times[Power[c, 2], d, m], -1]], x]]]], And[FreeQ[List[a, b, c, d, e, m], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], ZeroQ[Plus[m, Times[2, p], 2]], RationalQ[n, p], Less[p, -1], Greater[n, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[b, n, Power[x, m], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], Plus[n, -1]], Power[Times[c, d, Power[m, 2]], -1]], x]], Plus[Dist[Times[Plus[m, -1], Power[Times[Power[c, 2], d, m], -1]], Int[Times[Power[x, Plus[m, -2]], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], n]], x], x], Times[-1, Dist[Times[Power[b, 2], n, Plus[n, -1], Power[Power[m, 2], -1]], Int[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], Plus[n, -2]]], x], x]], Times[-1, Simp[Times[Power[x, Plus[m, -1]], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], n], Power[Times[Power[c, 2], d, m], -1]], x]]]], And[FreeQ[List[a, b, c, d, e, m], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], ZeroQ[Plus[m, Times[2, p], 2]], RationalQ[n, p], Less[p, -1], Greater[n, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, d, Plus[n, 1]], -1]], x], Times[-1, Dist[Times[m, Power[Times[b, c, Plus[n, 1]], -1]], Int[Times[Power[x, Plus[m, -1]], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], Plus[n, 1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, m, p], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], ZeroQ[Plus[m, Times[2, p], 2]], RationalQ[n], Less[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, d, Plus[n, 1]], -1]], x]], Dist[Times[m, Power[Times[b, c, Plus[n, 1]], -1]], Int[Times[Power[x, Plus[m, -1]], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], Plus[n, 1]]], x], x]], And[FreeQ[List[a, b, c, d, e, m, p], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], ZeroQ[Plus[m, Times[2, p], 2]], RationalQ[n], Less[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[m, 1]], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], n], Power[Times[d, Plus[m, 1]], -1]], x], Times[-1, Dist[Times[b, c, n, Power[Plus[m, 1], -1]], Int[Times[Power[x, Plus[m, 1]], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], Plus[n, -1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, m, p], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], ZeroQ[Plus[m, Times[2, p], 3]], RationalQ[n], Greater[n, 0], NonzeroQ[Plus[m, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[m, 1]], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], n], Power[Times[d, Plus[m, 1]], -1]], x], Dist[Times[b, c, n, Power[Plus[m, 1], -1]], Int[Times[Power[x, Plus[m, 1]], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], Plus[n, -1]]], x], x]], And[FreeQ[List[a, b, c, d, e, m, p], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], ZeroQ[Plus[m, Times[2, p], 3]], RationalQ[n], Greater[n, 0], NonzeroQ[Plus[m, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[m, 1]], Sqrt[Plus[d, Times[e, Power[x, 2]]]], Plus[a, Times[b, ArcTan[Times[c, x]]]], Power[Plus[m, 2], -1]], x], Plus[Dist[Times[d, Power[Plus[m, 2], -1]], Int[Times[Power[x, m], Plus[a, Times[b, ArcTan[Times[c, x]]]], Power[Sqrt[Plus[d, Times[e, Power[x, 2]]]], -1]], x], x], Times[-1, Dist[Times[b, c, d, Power[Plus[m, 2], -1]], Int[Times[Power[x, Plus[m, 1]], Power[Sqrt[Plus[d, Times[e, Power[x, 2]]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d, e, m], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], NonzeroQ[Plus[m, 2]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[m, 1]], Sqrt[Plus[d, Times[e, Power[x, 2]]]], Plus[a, Times[b, ArcCot[Times[c, x]]]], Power[Plus[m, 2], -1]], x], Plus[Dist[Times[d, Power[Plus[m, 2], -1]], Int[Times[Power[x, m], Plus[a, Times[b, ArcCot[Times[c, x]]]], Power[Sqrt[Plus[d, Times[e, Power[x, 2]]]], -1]], x], x], Dist[Times[b, c, d, Power[Plus[m, 2], -1]], Int[Times[Power[x, Plus[m, 1]], Power[Sqrt[Plus[d, Times[e, Power[x, 2]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, m], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], NonzeroQ[Plus[m, 2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, m], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], PositiveIntegerQ[n], IntegerQ[p], Greater[p, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, m], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], PositiveIntegerQ[n], IntegerQ[p], Greater[p, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[d, Int[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, -1]], Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], n]], x], x], Dist[Times[Power[c, 2], d], Int[Times[Power[x, Plus[m, 2]], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, -1]], Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], n]], x], x]], And[FreeQ[List[a, b, c, d, e, m], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[p], Greater[p, 0], PositiveIntegerQ[n], Or[RationalQ[m], And[EqQ[n, 1], IntegerQ[p]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[d, Int[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, -1]], Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], n]], x], x], Dist[Times[Power[c, 2], d], Int[Times[Power[x, Plus[m, 2]], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, -1]], Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], n]], x], x]], And[FreeQ[List[a, b, c, d, e, m], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[p], Greater[p, 0], PositiveIntegerQ[n], Or[RationalQ[m], And[EqQ[n, 1], IntegerQ[p]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[m, -1]], Sqrt[Plus[d, Times[e, Power[x, 2]]]], Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], n], Power[Times[Power[c, 2], d, m], -1]], x], Plus[Times[-1, Dist[Times[b, n, Power[Times[c, m], -1]], Int[Times[Power[x, Plus[m, -1]], Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], Plus[n, -1]], Power[Sqrt[Plus[d, Times[e, Power[x, 2]]]], -1]], x], x]], Times[-1, Dist[Times[Plus[m, -1], Power[Times[Power[c, 2], m], -1]], Int[Times[Power[x, Plus[m, -2]], Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], n], Power[Sqrt[Plus[d, Times[e, Power[x, 2]]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[m, n], Greater[n, 0], Greater[m, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[m, -1]], Sqrt[Plus[d, Times[e, Power[x, 2]]]], Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], n], Power[Times[Power[c, 2], d, m], -1]], x], Plus[Dist[Times[b, n, Power[Times[c, m], -1]], Int[Times[Power[x, Plus[m, -1]], Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], Plus[n, -1]], Power[Sqrt[Plus[d, Times[e, Power[x, 2]]]], -1]], x], x], Times[-1, Dist[Times[Plus[m, -1], Power[Times[Power[c, 2], m], -1]], Int[Times[Power[x, Plus[m, -2]], Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], n], Power[Sqrt[Plus[d, Times[e, Power[x, 2]]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[m, n], Greater[n, 0], Greater[m, 1]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[-2, Power[Sqrt[d], -1], Plus[a, Times[b, ArcTan[Times[c, x]]]], ArcTanh[Times[Sqrt[Plus[1, Times[I, c, x]]], Power[Sqrt[Plus[1, Times[-1, I, c, x]]], -1]]]], x], Plus[Simp[Times[I, b, Power[Sqrt[d], -1], PolyLog[2, Times[-1, Sqrt[Plus[1, Times[I, c, x]]], Power[Sqrt[Plus[1, Times[-1, I, c, x]]], -1]]]], x], Times[-1, Simp[Times[I, b, Power[Sqrt[d], -1], PolyLog[2, Times[Sqrt[Plus[1, Times[I, c, x]]], Power[Sqrt[Plus[1, Times[-1, I, c, x]]], -1]]]], x]]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], PositiveQ[d]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[-2, Power[Sqrt[d], -1], Plus[a, Times[b, ArcCot[Times[c, x]]]], ArcTanh[Times[Sqrt[Plus[1, Times[I, c, x]]], Power[Sqrt[Plus[1, Times[-1, I, c, x]]], -1]]]], x], Plus[Times[-1, Simp[Times[I, b, Power[Sqrt[d], -1], PolyLog[2, Times[-1, Sqrt[Plus[1, Times[I, c, x]]], Power[Sqrt[Plus[1, Times[-1, I, c, x]]], -1]]]], x]], Simp[Times[I, b, Power[Sqrt[d], -1], PolyLog[2, Times[Sqrt[Plus[1, Times[I, c, x]]], Power[Sqrt[Plus[1, Times[-1, I, c, x]]], -1]]]], x]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], PositiveQ[d]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Sqrt[d], -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], n], Csc[x]], x], x, ArcTan[Times[c, x]]], x], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], PositiveIntegerQ[n], PositiveQ[d]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Times[c, x, Sqrt[Plus[1, Times[1, Power[Times[Power[c, 2], Power[x, 2]], -1]]]], Power[Sqrt[Plus[d, Times[e, Power[x, 2]]]], -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], n], Sec[x]], x], x, ArcCot[Times[c, x]]], x]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], PositiveIntegerQ[n], PositiveQ[d]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Plus[1, Times[Power[c, 2], Power[x, 2]]]], Power[Sqrt[Plus[d, Times[e, Power[x, 2]]]], -1]], Int[Times[Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], n], Power[Times[x, Sqrt[Plus[1, Times[Power[c, 2], Power[x, 2]]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], PositiveIntegerQ[n], Not[PositiveQ[d]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Plus[1, Times[Power[c, 2], Power[x, 2]]]], Power[Sqrt[Plus[d, Times[e, Power[x, 2]]]], -1]], Int[Times[Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], n], Power[Times[x, Sqrt[Plus[1, Times[Power[c, 2], Power[x, 2]]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], PositiveIntegerQ[n], Not[PositiveQ[d]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], -2], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Sqrt[Plus[d, Times[e, Power[x, 2]]]], Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], n], Power[Times[d, x], -1]], x]], Dist[Times[b, c, n], Int[Times[Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], Plus[n, -1]], Power[Times[x, Sqrt[Plus[d, Times[e, Power[x, 2]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[n], Greater[n, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], -2], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Sqrt[Plus[d, Times[e, Power[x, 2]]]], Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], n], Power[Times[d, x], -1]], x]], Times[-1, Dist[Times[b, c, n], Int[Times[Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], Plus[n, -1]], Power[Times[x, Sqrt[Plus[d, Times[e, Power[x, 2]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[n], Greater[n, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[m, 1]], Sqrt[Plus[d, Times[e, Power[x, 2]]]], Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], n], Power[Times[d, Plus[m, 1]], -1]], x], Plus[Times[-1, Dist[Times[Power[c, 2], Plus[m, 2], Power[Plus[m, 1], -1]], Int[Times[Power[x, Plus[m, 2]], Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], n], Power[Sqrt[Plus[d, Times[e, Power[x, 2]]]], -1]], x], x]], Times[-1, Dist[Times[b, c, n, Power[Plus[m, 1], -1]], Int[Times[Power[x, Plus[m, 1]], Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], Plus[n, -1]], Power[Sqrt[Plus[d, Times[e, Power[x, 2]]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[m, n], Greater[n, 0], Less[m, -1], Unequal[m, -2]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[m, 1]], Sqrt[Plus[d, Times[e, Power[x, 2]]]], Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], n], Power[Times[d, Plus[m, 1]], -1]], x], Plus[Times[-1, Dist[Times[Power[c, 2], Plus[m, 2], Power[Plus[m, 1], -1]], Int[Times[Power[x, Plus[m, 2]], Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], n], Power[Sqrt[Plus[d, Times[e, Power[x, 2]]]], -1]], x], x]], Dist[Times[b, c, n, Power[Plus[m, 1], -1]], Int[Times[Power[x, Plus[m, 1]], Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], Plus[n, -1]], Power[Sqrt[Plus[d, Times[e, Power[x, 2]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[m, n], Greater[n, 0], Less[m, -1], Unequal[m, -2]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[e, -1]], Int[Times[Power[x, Plus[m, -2]], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], n]], x], x], Times[-1, Dist[Times[d, Power[e, -1]], Int[Times[Power[x, Plus[m, -2]], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], n]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], IntegersQ[m, n, Times[2, p]], Less[p, -1], Greater[m, 1], Unequal[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[e, -1]], Int[Times[Power[x, Plus[m, -2]], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], n]], x], x], Times[-1, Dist[Times[d, Power[e, -1]], Int[Times[Power[x, Plus[m, -2]], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], n]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], IntegersQ[m, n, Times[2, p]], Less[p, -1], Greater[m, 1], Unequal[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[d, -1]], Int[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], n]], x], x], Times[-1, Dist[Times[e, Power[d, -1]], Int[Times[Power[x, Plus[m, 2]], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], n]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], IntegersQ[m, n, Times[2, p]], Less[p, -1], Less[m, 0], Unequal[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[d, -1]], Int[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], n]], x], x], Times[-1, Dist[Times[e, Power[d, -1]], Int[Times[Power[x, Plus[m, 2]], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], n]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], IntegersQ[m, n, Times[2, p]], Less[p, -1], Less[m, 0], Unequal[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, d, Plus[n, 1]], -1]], x], Plus[Times[-1, Dist[Times[c, Plus[m, Times[2, p], 2], Power[Times[b, Plus[n, 1]], -1]], Int[Times[Power[x, Plus[m, 1]], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], Plus[n, 1]]], x], x]], Times[-1, Dist[Times[m, Power[Times[b, c, Plus[n, 1]], -1]], Int[Times[Power[x, Plus[m, -1]], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], Plus[n, 1]]], x], x]]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[m, n, p], Less[p, -1], Less[n, -1], NonzeroQ[Plus[m, Times[2, p], 2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, d, Plus[n, 1]], -1]], x]], Plus[Dist[Times[c, Plus[m, Times[2, p], 2], Power[Times[b, Plus[n, 1]], -1]], Int[Times[Power[x, Plus[m, 1]], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], Plus[n, 1]]], x], x], Dist[Times[m, Power[Times[b, c, Plus[n, 1]], -1]], Int[Times[Power[x, Plus[m, -1]], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], Plus[n, 1]]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[m, n, p], Less[p, -1], Less[n, -1], NonzeroQ[Plus[m, Times[2, p], 2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[d, p], Power[Power[c, Plus[m, 1]], -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], n], Power[Sin[x], m], Power[Power[Cos[x], Plus[m, Times[2, Plus[p, 1]]]], -1]], x], x, ArcTan[Times[c, x]]], x], And[FreeQ[List[a, b, c, d, e, n], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], PositiveIntegerQ[m], NegativeIntegerQ[Plus[m, Times[2, p], 1]], Or[IntegerQ[p], PositiveQ[d]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[d, Plus[p, Times[1, Power[2, -1]]]], Sqrt[Plus[1, Times[Power[c, 2], Power[x, 2]]]], Power[Sqrt[Plus[d, Times[e, Power[x, 2]]]], -1]], Int[Times[Power[x, m], Power[Plus[1, Times[Power[c, 2], Power[x, 2]]], p], Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, n], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], PositiveIntegerQ[m], NegativeIntegerQ[Plus[m, Times[2, p], 1]], Not[Or[IntegerQ[p], PositiveQ[d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], 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Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[x, m], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], n]], x], FreeQ[List[a, b, c, d, e, m, n, p], x]]], RuleDelayed[HoldPattern[Int[Times[ArcTanh[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[2, -1]], Int[Times[Log[Plus[1, u]], Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], n], Power[Plus[d, Times[e, Power[x, 2]]], -1]], x], x], Times[-1, Dist[Times[1, Power[2, -1]], Int[Times[Log[Plus[1, Times[-1, u]]], Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], n], Power[Plus[d, Times[e, Power[x, 2]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[n], Greater[n, 0], ZeroQ[Plus[Power[u, 2], Times[-1, Power[Plus[1, Times[-1, 2, I, Power[Plus[I, Times[c, x]], -1]]], 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[ArcCoth[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[2, -1]], Int[Times[Log[SimplifyIntegrand[Plus[1, Times[1, Power[u, -1]]], x]], Power[Plus[a, Times[b, ArcCot[Times[c, x]]]], n], Power[Plus[d, Times[e, Power[x, 2]]], -1]], x], x], Times[-1, Dist[Times[1, Power[2, -1]], Int[Times[Log[SimplifyIntegrand[Plus[1, Times[-1, 1, 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u]]]], Power[Plus[d, Times[e, Power[x, 2]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[n], Greater[n, 0], ZeroQ[Plus[Power[Plus[1, Times[-1, u]], 2], Times[-1, Power[Plus[1, Times[-1, 2, I, Power[Plus[I, Times[c, x]], -1]]], 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Log[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[I, Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], n], PolyLog[2, Together[Plus[1, Times[-1, u]]]], Power[Times[2, c, d], -1]], x]], Dist[Times[b, n, I, Power[2, -1]], Int[Times[Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], Plus[n, -1]], PolyLog[2, Together[Plus[1, Times[-1, u]]]], 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Power[Plus[d, Times[e, Power[x, 2]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[n], Greater[n, 0], ZeroQ[Plus[Power[Plus[1, Times[-1, u]], 2], Times[-1, Power[Plus[1, Times[-1, 2, I, Power[Plus[I, Times[-1, c, x]], -1]]], 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], PolyLog[Pattern[p, Blank[]], Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[I, Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], n], PolyLog[Plus[p, 1], u], Power[Times[2, c, d], -1]], x]], Dist[Times[b, n, I, Power[2, -1]], Int[Times[Power[Plus[a, Times[b, ArcTan[Times[c, x]]]], Plus[n, -1]], PolyLog[Plus[p, 1], u], Power[Plus[d, Times[e, 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Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[1, Times[-1, I, a, x]], Times[Plus[Times[I, n], 1], Power[2, -1]]], Power[Times[Power[Plus[1, Times[I, a, x]], Times[Plus[Times[I, n], -1], Power[2, -1]]], Sqrt[Plus[1, Times[Power[a, 2], Power[x, 2]]]]], -1]], x], And[FreeQ[a, x], OddQ[Times[I, n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[x, m], Times[Power[Plus[1, Times[-1, I, a, x]], Times[Plus[Times[I, n], 1], Power[2, -1]]], Power[Times[Power[Plus[1, Times[I, a, x]], Times[Plus[Times[I, n], -1], Power[2, -1]]], Sqrt[Plus[1, Times[Power[a, 2], Power[x, 2]]]]], -1]]], x], And[FreeQ[List[a, m], x], OddQ[Times[I, n]]]]], RuleDelayed[HoldPattern[Int[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[1, Times[-1, I, a, x]], Times[I, n, Power[2, -1]]], Power[Power[Plus[1, Times[I, a, x]], Times[I, n, Power[2, -1]]], -1]], x], And[FreeQ[List[a, n], x], Not[OddQ[Times[I, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[x, m], Power[Plus[1, Times[-1, I, a, x]], Times[I, n, Power[2, -1]]], Power[Power[Plus[1, Times[I, a, x]], Times[I, n, Power[2, -1]]], -1]], x], And[FreeQ[List[a, m, n], x], Not[OddQ[Times[I, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, p], Int[Times[u, Power[Plus[1, Times[d, x, Power[c, -1]]], p], Power[Plus[1, Times[-1, I, a, x]], Times[I, n, Power[2, -1]]], Power[Power[Plus[1, Times[I, a, x]], Times[I, n, Power[2, -1]]], -1]], x], x], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[Times[Power[a, 2], Power[c, 2]], Power[d, 2]]], Or[IntegerQ[p], PositiveQ[c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[u, Power[Plus[c, Times[d, x]], p], Power[Plus[1, Times[-1, I, a, x]], Times[I, n, Power[2, -1]]], Power[Power[Plus[1, Times[I, a, x]], Times[I, n, Power[2, -1]]], -1]], x], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[Times[Power[a, 2], Power[c, 2]], Power[d, 2]]], Not[Or[IntegerQ[p], PositiveQ[c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[d, p], Int[Times[u, Power[Power[x, p], -1], Power[Plus[1, Times[c, x, Power[d, -1]]], p], Power[E, Times[n, ArcTan[Times[a, x]]]]], x], x], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[Power[c, 2], Times[Power[a, 2], Power[d, 2]]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[-1, Times[n, Power[2, -1]]], Power[c, p]], Int[Times[u, Power[Plus[1, Times[d, Power[Times[c, x], -1]]], p], Power[Plus[1, Times[-1, 1, Power[Times[I, a, x], -1]]], Times[I, n, Power[2, -1]]], Power[Power[Plus[1, Times[1, Power[Times[I, a, x], -1]]], Times[I, n, Power[2, -1]]], -1]], x], x], And[FreeQ[List[a, c, d, p], x], ZeroQ[Plus[Power[c, 2], Times[Power[a, 2], Power[d, 2]]]], Not[IntegerQ[p]], IntegerQ[Times[I, n, Power[2, -1]]], PositiveQ[c]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[u, Power[Plus[c, Times[d, Power[x, -1]]], p], Power[Plus[1, Times[-1, I, a, x]], Times[I, n, Power[2, -1]]], Power[Power[Plus[1, Times[I, a, x]], Times[I, n, Power[2, -1]]], -1]], x], And[FreeQ[List[a, c, d, p], x], ZeroQ[Plus[Power[c, 2], Times[Power[a, 2], Power[d, 2]]]], Not[IntegerQ[p]], IntegerQ[Times[I, n, Power[2, -1]]], Not[PositiveQ[c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[x, p], Power[Plus[c, Times[d, Power[x, -1]]], p], Power[Power[Plus[1, Times[c, x, Power[d, -1]]], p], -1]], Int[Times[u, Power[Power[x, p], -1], Power[Plus[1, Times[c, x, Power[d, -1]]], p], Power[E, Times[n, ArcTan[Times[a, x]]]]], x], x], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[Power[c, 2], Times[Power[a, 2], Power[d, 2]]]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Plus[n, Times[a, x]], Power[E, Times[n, ArcTan[Times[a, x]]]], Power[Times[a, c, Plus[Power[n, 2], 1], Sqrt[Plus[c, Times[d, Power[x, 2]]]]], -1]], x], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[d, Times[-1, Power[a, 2], c]]], Not[IntegerQ[Times[I, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[n, Times[-1, 2, a, Plus[p, 1], x]], Power[Plus[c, Times[d, Power[x, 2]]], Plus[p, 1]], Power[E, Times[n, ArcTan[Times[a, x]]]], Power[Times[a, c, Plus[Power[n, 2], Times[4, Power[Plus[p, 1], 2]]]], -1]], x], Dist[Times[2, Plus[p, 1], Plus[Times[2, p], 3], Power[Times[c, Plus[Power[n, 2], Times[4, Power[Plus[p, 1], 2]]]], -1]], Int[Times[Power[Plus[c, Times[d, Power[x, 2]]], Plus[p, 1]], Power[E, Times[n, ArcTan[Times[a, x]]]]], x], x]], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[d, Times[-1, Power[a, 2], c]]], RationalQ[p], Less[p, -1], Not[IntegerQ[Times[I, n]]], NonzeroQ[Plus[Power[n, 2], Times[4, Power[Plus[p, 1], 2]]]], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[E, Times[n, ArcTan[Times[a, x]]]], Power[Times[a, c, n], -1]], x], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[d, Times[-1, Power[a, 2], c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, p], Int[Times[Power[Plus[1, Times[Power[a, 2], Power[x, 2]]], Plus[p, Times[-1, I, n, Power[2, -1]]]], Power[Plus[1, Times[-1, I, a, x]], Times[I, n]]], x], x], And[FreeQ[List[a, c, d, p], x], ZeroQ[Plus[d, Times[-1, Power[a, 2], c]]], IntegerQ[p], IntegerQ[Times[Plus[Times[I, n], 1], Power[2, -1]]], Not[IntegerQ[Plus[p, Times[-1, I, n, Power[2, -1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, p], Int[Times[Power[Plus[1, Times[-1, I, a, x]], Plus[p, Times[I, n, Power[2, -1]]]], Power[Plus[1, Times[I, a, x]], Plus[p, Times[-1, I, n, Power[2, -1]]]]], x], x], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[d, Times[-1, Power[a, 2], c]]], Or[IntegerQ[p], PositiveQ[c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, Times[I, n, Power[2, -1]]], Int[Times[Power[Plus[c, Times[d, Power[x, 2]]], Plus[p, Times[-1, I, n, Power[2, -1]]]], Power[Plus[1, Times[-1, I, a, x]], Times[I, n]]], x], x], And[FreeQ[List[a, c, d, p], x], ZeroQ[Plus[d, Times[-1, Power[a, 2], c]]], Not[Or[IntegerQ[p], PositiveQ[c]]], PositiveIntegerQ[Times[I, n, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Power[c, Times[I, n, Power[2, -1]]], -1]], Int[Times[Power[Plus[c, Times[d, Power[x, 2]]], Plus[p, Times[I, n, Power[2, -1]]]], Power[Power[Plus[1, Times[I, a, x]], Times[I, n]], -1]], x], x], And[FreeQ[List[a, c, d, p], x], ZeroQ[Plus[d, Times[-1, Power[a, 2], c]]], Not[Or[IntegerQ[p], PositiveQ[c]]], NegativeIntegerQ[Times[I, n, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[c, IntPart[p]], Power[Plus[c, Times[d, Power[x, 2]]], FracPart[p]], Power[Power[Plus[1, Times[Power[a, 2], Power[x, 2]]], FracPart[p]], -1]], Int[Times[Power[Plus[1, Times[Power[a, 2], Power[x, 2]]], p], Power[E, Times[n, ArcTan[Times[a, x]]]]], x], x], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[d, Times[-1, Power[a, 2], c]]], Not[Or[IntegerQ[p], PositiveQ[c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Simp[Times[Plus[1, Times[-1, a, n, x]], Power[E, Times[n, ArcTan[Times[a, x]]]], Power[Times[d, Plus[Power[n, 2], 1], Sqrt[Plus[c, Times[d, Power[x, 2]]]]], -1]], x]], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[d, Times[-1, Power[a, 2], c]]], Not[IntegerQ[Times[I, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[c, Times[d, Power[x, 2]]], Plus[p, 1]], Power[E, Times[n, ArcTan[Times[a, x]]]], Power[Times[2, d, Plus[p, 1]], -1]], x], Times[-1, Dist[Times[a, c, n, Power[Times[2, d, Plus[p, 1]], -1]], Int[Times[Power[Plus[c, Times[d, Power[x, 2]]], p], Power[E, Times[n, ArcTan[Times[a, x]]]]], x], x]]], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[d, Times[-1, Power[a, 2], c]]], RationalQ[p], Less[p, -1], Not[IntegerQ[Times[I, n]]], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Simp[Times[Plus[1, Times[-1, a, n, x]], Power[Plus[c, Times[d, Power[x, 2]]], Plus[p, 1]], Power[E, Times[n, ArcTan[Times[a, x]]]], Power[Times[a, d, n, Plus[Power[n, 2], 1]], -1]], x]], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[d, Times[-1, Power[a, 2], c]]], ZeroQ[Plus[Power[n, 2], Times[-1, 2, Plus[p, 1]]]], Not[IntegerQ[Times[I, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[n, Times[-1, 2, Plus[p, 1], a, x]], Power[Plus[c, Times[d, Power[x, 2]]], Plus[p, 1]], Power[E, Times[n, ArcTan[Times[a, x]]]], Power[Times[a, d, Plus[Power[n, 2], Times[4, Power[Plus[p, 1], 2]]]], -1]], x]], Dist[Times[Plus[Power[n, 2], Times[-1, 2, Plus[p, 1]]], Power[Times[d, Plus[Power[n, 2], Times[4, Power[Plus[p, 1], 2]]]], -1]], Int[Times[Power[Plus[c, Times[d, Power[x, 2]]], Plus[p, 1]], Power[E, Times[n, ArcTan[Times[a, x]]]]], x], x]], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[d, Times[-1, Power[a, 2], c]]], RationalQ[p], Less[p, -1], Not[IntegerQ[Times[I, n]]], NonzeroQ[Plus[Power[n, 2], Times[4, Power[Plus[p, 1], 2]]]], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, p], Int[Times[Power[x, m], Power[Plus[1, Times[Power[a, 2], Power[x, 2]]], Plus[p, Times[-1, I, n, Power[2, -1]]]], Power[Plus[1, Times[-1, I, a, x]], Times[I, n]]], x], x], And[FreeQ[List[a, c, d, m, p], x], ZeroQ[Plus[d, Times[-1, Power[a, 2], c]]], Or[IntegerQ[p], PositiveQ[c]], IntegerQ[Times[Plus[Times[I, n], 1], Power[2, -1]]], Not[IntegerQ[Plus[p, Times[-1, I, n, Power[2, -1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, p], Int[Times[Power[x, m], Power[Plus[1, Times[-1, I, a, x]], Plus[p, Times[I, n, Power[2, -1]]]], Power[Plus[1, Times[I, a, x]], Plus[p, Times[-1, I, n, Power[2, -1]]]]], x], x], And[FreeQ[List[a, c, d, m, n, p], x], ZeroQ[Plus[d, Times[-1, Power[a, 2], c]]], Or[IntegerQ[p], PositiveQ[c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, Times[I, n, Power[2, -1]]], Int[Times[Power[x, m], Power[Plus[c, Times[d, Power[x, 2]]], Plus[p, Times[-1, I, n, Power[2, -1]]]], Power[Plus[1, Times[-1, I, a, x]], Times[I, n]]], x], x], And[FreeQ[List[a, c, d, m, p], x], ZeroQ[Plus[d, Times[-1, Power[a, 2], c]]], Not[Or[IntegerQ[p], PositiveQ[c]]], PositiveIntegerQ[Times[I, n, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Power[c, Times[I, n, Power[2, -1]]], -1]], Int[Times[Power[x, m], Power[Plus[c, Times[d, Power[x, 2]]], Plus[p, Times[I, n, Power[2, -1]]]], Power[Power[Plus[1, Times[I, a, x]], Times[I, n]], -1]], x], x], And[FreeQ[List[a, c, d, m, p], x], ZeroQ[Plus[d, Times[-1, Power[a, 2], c]]], Not[Or[IntegerQ[p], PositiveQ[c]]], NegativeIntegerQ[Times[I, n, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[c, IntPart[p]], Power[Plus[c, Times[d, Power[x, 2]]], FracPart[p]], Power[Power[Plus[1, Times[Power[a, 2], Power[x, 2]]], FracPart[p]], -1]], Int[Times[Power[x, m], Power[Plus[1, Times[Power[a, 2], Power[x, 2]]], p], Power[E, Times[n, ArcTan[Times[a, x]]]]], x], x], And[FreeQ[List[a, c, d, m, n, p], x], ZeroQ[Plus[d, Times[-1, Power[a, 2], c]]], Not[Or[IntegerQ[p], PositiveQ[c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[u, Blank[]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, p], Int[Times[u, Power[Plus[1, Times[-1, I, a, x]], Plus[p, Times[I, n, Power[2, -1]]]], Power[Plus[1, Times[I, a, x]], Plus[p, Times[-1, I, n, Power[2, -1]]]]], x], x], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[d, Times[-1, Power[a, 2], c]]], Or[IntegerQ[p], PositiveQ[c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[u, Blank[]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[c, IntPart[p]], Power[Plus[c, Times[d, Power[x, 2]]], FracPart[p]], Power[Times[Power[Plus[1, Times[-1, I, a, x]], FracPart[p]], Power[Plus[1, Times[I, a, x]], FracPart[p]]], -1]], Int[Times[u, Power[Plus[1, Times[-1, I, a, x]], Plus[p, Times[I, n, Power[2, -1]]]], Power[Plus[1, Times[I, a, x]], Plus[p, Times[-1, I, n, Power[2, -1]]]]], x], x], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[d, Times[-1, Power[a, 2], c]]], Or[IntegerQ[p], PositiveQ[c]], IntegerQ[Times[I, n, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[u, Blank[]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[c, IntPart[p]], Power[Plus[c, Times[d, Power[x, 2]]], FracPart[p]], Power[Power[Plus[1, Times[Power[a, 2], Power[x, 2]]], FracPart[p]], -1]], Int[Times[u, Power[Plus[1, Times[Power[a, 2], Power[x, 2]]], p], Power[E, Times[n, ArcTan[Times[a, x]]]]], x], x], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[d, Times[-1, Power[a, 2], c]]], Not[Or[IntegerQ[p], PositiveQ[c]]], Not[IntegerQ[Times[I, n, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[d, p], Int[Times[u, Power[Power[x, Times[2, p]], -1], Power[Plus[1, Times[Power[a, 2], Power[x, 2]]], p], Power[E, Times[n, ArcTan[Times[a, x]]]]], x], x], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[c, Times[-1, Power[a, 2], d]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, p], Int[Times[u, Power[Plus[1, Times[-1, I, Power[Times[a, x], -1]]], p], Power[Plus[1, Times[I, Power[Times[a, x], -1]]], p], Power[E, Times[n, ArcTan[Times[a, x]]]]], x], x], And[FreeQ[List[a, c, d, p], x], ZeroQ[Plus[c, Times[-1, Power[a, 2], d]]], Not[IntegerQ[p]], IntegerQ[Times[I, n, Power[2, -1]]], PositiveQ[c]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[x, Times[2, p]], Power[Plus[c, Times[d, Power[Power[x, 2], -1]]], p], Power[Times[Power[Plus[1, Times[-1, I, a, x]], p], Power[Plus[1, Times[I, a, x]], p]], -1]], Int[Times[u, Power[Power[x, Times[2, p]], -1], Power[Plus[1, Times[-1, I, a, x]], p], Power[Plus[1, Times[I, a, x]], p], Power[E, Times[n, ArcTan[Times[a, x]]]]], x], x], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[c, Times[-1, Power[a, 2], d]]], Not[IntegerQ[p]], IntegerQ[Times[I, n, Power[2, -1]]], Not[PositiveQ[c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[x, Times[2, p]], Power[Plus[c, Times[d, Power[Power[x, 2], -1]]], p], Power[Power[Plus[1, Times[Power[a, 2], Power[x, 2]]], p], -1]], Int[Times[u, Power[Power[x, Times[2, p]], -1], Power[Plus[1, Times[Power[a, 2], Power[x, 2]]], p], Power[E, Times[n, ArcTan[Times[a, x]]]]], x], x], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[c, Times[-1, Power[a, 2], d]]], Not[IntegerQ[p]], Not[IntegerQ[Times[I, n, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Power[E, Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[1, Times[-1, I, a, c], Times[-1, I, b, c, x]], Times[I, n, Power[2, -1]]], Power[Power[Plus[1, Times[I, a, c], Times[I, b, c, x]], Times[I, n, Power[2, -1]]], -1]], x], FreeQ[List[a, b, c, n], x]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[4, Power[Times[Power[I, m], n, Power[b, Plus[m, 1]], Power[c, Plus[m, 1]]], -1]], Subst[Int[Times[Power[x, Times[2, Power[Times[I, n], -1]]], Power[Plus[1, Times[-1, I, a, c], Times[-1, Plus[1, Times[I, a, c]], Power[x, Times[2, Power[Times[I, n], -1]]]]], m], Power[Power[Plus[1, Power[x, Times[2, Power[Times[I, n], -1]]]], Plus[m, 2]], -1]], x], x, Times[Power[Plus[1, Times[-1, I, c, Plus[a, Times[b, x]]]], Times[I, n, Power[2, -1]]], Power[Power[Plus[1, Times[I, c, Plus[a, Times[b, x]]]], Times[I, n, Power[2, -1]]], -1]]], x], And[FreeQ[List[a, b, c], x], NegativeIntegerQ[m], RationalQ[Times[I, n]], Less[-1, Times[I, n], 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[1, Times[-1, I, a, c], Times[-1, I, b, c, x]], Times[I, n, Power[2, -1]]], Power[Power[Plus[1, Times[I, a, c], Times[I, b, c, x]], Times[I, n, Power[2, -1]]], -1]], x], FreeQ[List[a, b, c, d, e, m, n], x]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[Times[c, Power[Plus[1, Power[a, 2]], -1]], p], Int[Times[u, Power[Plus[1, Times[-1, I, a], Times[-1, I, b, x]], Plus[p, Times[I, n, Power[2, -1]]]], Power[Plus[1, Times[I, a], Times[I, b, x]], Plus[p, Times[-1, I, n, Power[2, -1]]]]], x], x], And[FreeQ[List[a, b, c, d, e, n, p], x], ZeroQ[Plus[Times[b, d], Times[-1, 2, a, e]]], ZeroQ[Plus[Times[Power[b, 2], c], Times[-1, e, Plus[1, Power[a, 2]]]]], Or[IntegerQ[p], PositiveQ[Times[c, Power[Plus[1, Power[a, 2]], -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[c, Times[d, x], Times[e, Power[x, 2]]], p], Power[Power[Plus[1, Power[a, 2], Times[2, a, b, x], Times[Power[b, 2], Power[x, 2]]], p], -1]], Int[Times[u, Power[Plus[1, Power[a, 2], Times[2, a, b, x], Times[Power[b, 2], Power[x, 2]]], p], Power[E, Times[n, ArcTan[Times[a, x]]]]], x], x], And[FreeQ[List[a, b, c, d, e, n, p], x], ZeroQ[Plus[Times[b, d], Times[-1, 2, a, e]]], ZeroQ[Plus[Times[Power[b, 2], c], Times[-1, e, Plus[1, Power[a, 2]]]]], Not[Or[IntegerQ[p], PositiveQ[Times[c, Power[Plus[1, Power[a, 2]], -1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTan[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[u, Power[E, Times[n, ArcCot[Plus[Times[a, Power[c, -1]], Times[b, x, Power[c, -1]]]]]]], x], FreeQ[List[a, b, c, n], x]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[-1, Times[I, n, Power[2, -1]]], Int[Times[u, Power[E, Times[-1, n, ArcTan[Times[a, x]]]]], x], x], And[FreeQ[a, x], IntegerQ[Times[I, n, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Subst[Int[Times[Power[Plus[1, Times[-1, I, x, Power[a, -1]]], Times[Plus[Times[I, n], 1], Power[2, -1]]], Power[Times[Power[x, 2], Power[Plus[1, Times[I, x, Power[a, -1]]], Times[Plus[Times[I, n], -1], Power[2, -1]]], Sqrt[Plus[1, Times[Power[x, 2], Power[Power[a, 2], -1]]]]], -1]], x], x, Times[1, Power[x, -1]]]], And[FreeQ[a, x], OddQ[Times[I, n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Subst[Int[Times[Power[Plus[1, Times[-1, I, x, Power[a, -1]]], Times[Plus[Times[I, n], 1], Power[2, -1]]], Power[Times[Power[x, Plus[m, 2]], Power[Plus[1, Times[I, x, Power[a, -1]]], Times[Plus[Times[I, n], -1], Power[2, -1]]], Sqrt[Plus[1, Times[Power[x, 2], Power[Power[a, 2], -1]]]]], -1]], x], x, Times[1, Power[x, -1]]]], And[FreeQ[a, x], OddQ[Times[I, n]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Subst[Int[Times[Power[Plus[1, Times[-1, I, x, Power[a, -1]]], Times[I, n, Power[2, -1]]], Power[Times[Power[x, 2], Power[Plus[1, Times[I, x, Power[a, -1]]], Times[I, n, Power[2, -1]]]], -1]], x], x, Times[1, Power[x, -1]]]], And[FreeQ[List[a, n], x], Not[IntegerQ[Times[I, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Subst[Int[Times[Power[Plus[1, Times[-1, I, x, Power[a, -1]]], Times[n, Power[2, -1]]], Power[Times[Power[x, Plus[m, 2]], Power[Plus[1, Times[I, x, Power[a, -1]]], Times[n, Power[2, -1]]]], -1]], x], x, Times[1, Power[x, -1]]]], And[FreeQ[List[a, n], x], Not[IntegerQ[Times[I, n]]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Times[Power[x, m], Power[Times[1, Power[x, -1]], m]], Subst[Int[Times[Power[Plus[1, Times[-1, I, x, Power[a, -1]]], Times[Plus[Times[I, n], 1], Power[2, -1]]], Power[Times[Power[x, Plus[m, 2]], Power[Plus[1, Times[I, x, Power[a, -1]]], Times[Plus[Times[I, n], -1], Power[2, -1]]], Sqrt[Plus[1, Times[Power[x, 2], Power[Power[a, 2], -1]]]]], -1]], x], x, Times[1, Power[x, -1]]], x]], And[FreeQ[List[a, m], x], OddQ[Times[I, n]], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Subst[Int[Times[Power[Plus[1, Times[-1, I, x, Power[a, -1]]], Times[n, Power[2, -1]]], Power[Times[Power[x, Plus[m, 2]], Power[Plus[1, Times[I, x, Power[a, -1]]], Times[n, Power[2, -1]]]], -1]], x], x, Times[1, Power[x, -1]]]], And[FreeQ[List[a, m, n], x], Not[IntegerQ[Times[I, n, Power[2, -1]]]], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[d, p], Int[Times[u, Power[x, p], Power[Plus[1, Times[c, Power[Times[d, x], -1]]], p], Power[E, Times[n, ArcCot[Times[a, x]]]]], x], x], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[Times[Power[a, 2], Power[c, 2]], Power[d, 2]]], Not[IntegerQ[Times[I, n, Power[2, -1]]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[c, Times[d, x]], p], Power[Times[Power[x, p], Power[Plus[1, Times[c, Power[Times[d, x], -1]]], p]], -1]], Int[Times[u, Power[x, p], Power[Plus[1, Times[c, Power[Times[d, x], -1]]], p], Power[E, Times[n, ArcCot[Times[a, x]]]]], x], x], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[Times[Power[a, 2], Power[c, 2]], Power[d, 2]]], Not[IntegerQ[Times[I, n, Power[2, -1]]]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Power[c, p], Subst[Int[Times[Power[Plus[1, Times[d, x, Power[c, -1]]], p], Power[Plus[1, Times[-1, I, x, Power[a, -1]]], Times[I, n, Power[2, -1]]], Power[Times[Power[x, 2], Power[Plus[1, Times[I, x, Power[a, -1]]], Times[I, n, Power[2, -1]]]], -1]], x], x, Times[1, Power[x, -1]]], x]], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[Power[c, 2], Times[Power[a, 2], Power[d, 2]]]], Not[IntegerQ[Times[I, n, Power[2, -1]]]], Or[IntegerQ[p], PositiveQ[c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Power[c, p], Subst[Int[Times[Power[Plus[1, Times[d, x, Power[c, -1]]], p], Power[Plus[1, Times[-1, I, x, Power[a, -1]]], Times[I, n, Power[2, -1]]], Power[Times[Power[x, Plus[m, 2]], Power[Plus[1, Times[I, x, Power[a, -1]]], Times[I, n, Power[2, -1]]]], -1]], x], x, Times[1, Power[x, -1]]], x]], And[FreeQ[List[a, c, d, m, n, p], x], ZeroQ[Plus[Power[c, 2], Times[Power[a, 2], Power[d, 2]]]], Not[IntegerQ[Times[I, n, Power[2, -1]]]], Or[IntegerQ[p], PositiveQ[c]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[c, Times[d, Power[x, -1]]], p], Power[Power[Plus[1, Times[d, Power[Times[c, x], -1]]], p], -1]], Int[Times[Power[Plus[1, Times[d, Power[Times[c, x], -1]]], p], Power[E, Times[n, ArcCot[Times[a, x]]]]], x], x], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[Power[c, 2], Times[Power[a, 2], Power[d, 2]]]], Not[IntegerQ[Times[I, n, Power[2, -1]]]], Not[Or[IntegerQ[p], PositiveQ[c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Times[Power[c, p], Power[x, m], Power[Times[1, Power[x, -1]], m]], Subst[Int[Times[Power[Plus[1, Times[d, x, Power[c, -1]]], p], Power[Plus[1, Times[-1, I, x, Power[a, -1]]], Times[I, n, Power[2, -1]]], Power[Times[Power[x, Plus[m, 2]], Power[Plus[1, Times[I, x, Power[a, -1]]], Times[I, n, Power[2, -1]]]], -1]], x], x, Times[1, Power[x, -1]]], x]], And[FreeQ[List[a, c, d, m, n, p], x], ZeroQ[Plus[Power[c, 2], Times[Power[a, 2], Power[d, 2]]]], Not[IntegerQ[Times[I, n, Power[2, -1]]]], Or[IntegerQ[p], PositiveQ[c]], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[c, Times[d, Power[x, -1]]], p], Power[Power[Plus[1, Times[d, Power[Times[c, x], -1]]], p], -1]], Int[Times[u, Power[Plus[1, Times[d, Power[Times[c, x], -1]]], p], Power[E, Times[n, ArcCot[Times[a, x]]]]], x], x], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[Power[c, 2], Times[Power[a, 2], Power[d, 2]]]], Not[IntegerQ[Times[I, n, Power[2, -1]]]], Not[Or[IntegerQ[p], PositiveQ[c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Simp[Times[Power[E, Times[n, ArcCot[Times[a, x]]]], Power[Times[a, c, n], -1]], x]], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[d, Times[-1, Power[a, 2], c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Simp[Times[Plus[n, Times[-1, a, x]], Power[E, Times[n, ArcCot[Times[a, x]]]], Power[Times[a, c, Plus[Power[n, 2], 1], Sqrt[Plus[c, Times[d, Power[x, 2]]]]], -1]], x]], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[d, Times[-1, Power[a, 2], c]]], Not[OddQ[Times[I, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[n, Times[2, a, Plus[p, 1], x]], Power[Plus[c, Times[d, Power[x, 2]]], Plus[p, 1]], Power[E, Times[n, ArcCot[Times[a, x]]]], Power[Times[a, c, Plus[Power[n, 2], Times[4, Power[Plus[p, 1], 2]]]], -1]], x]], Dist[Times[2, Plus[p, 1], Plus[Times[2, p], 3], Power[Times[c, Plus[Power[n, 2], Times[4, Power[Plus[p, 1], 2]]]], -1]], Int[Times[Power[Plus[c, Times[d, Power[x, 2]]], Plus[p, 1]], Power[E, Times[n, ArcCot[Times[a, x]]]]], x], x]], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[d, Times[-1, Power[a, 2], c]]], RationalQ[p], Less[p, -1], Unequal[p, Times[-3, Power[2, -1]]], NonzeroQ[Plus[Power[n, 2], Times[4, Power[Plus[p, 1], 2]]]], Not[And[IntegerQ[p], EvenQ[Times[I, n]]]], Not[And[Not[IntegerQ[p]], OddQ[Times[I, n]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Simp[Times[Plus[1, Times[a, n, x]], Power[E, Times[n, ArcCot[Times[a, x]]]], Power[Times[Power[a, 2], c, Plus[Power[n, 2], 1], Sqrt[Plus[c, Times[d, Power[x, 2]]]]], -1]], x]], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[d, Times[-1, Power[a, 2], c]]], Not[OddQ[Times[I, n]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[Times[2, Plus[p, 1]], Times[-1, a, n, x]], Power[Plus[c, Times[d, Power[x, 2]]], Plus[p, 1]], Power[E, Times[n, ArcCot[Times[a, x]]]], Power[Times[Power[a, 2], c, Plus[Power[n, 2], Times[4, Power[Plus[p, 1], 2]]]], -1]], x], Dist[Times[n, Plus[Times[2, p], 3], Power[Times[a, c, Plus[Power[n, 2], Times[4, Power[Plus[p, 1], 2]]]], -1]], Int[Times[Power[Plus[c, Times[d, Power[x, 2]]], Plus[p, 1]], Power[E, Times[n, ArcCot[Times[a, x]]]]], x], x]], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[d, Times[-1, Power[a, 2], c]]], RationalQ[p], LessEqual[p, -1], Unequal[p, Times[-3, Power[2, -1]]], NonzeroQ[Plus[Power[n, 2], Times[4, Power[Plus[p, 1], 2]]]], Not[And[IntegerQ[p], EvenQ[Times[I, n]]]], Not[And[Not[IntegerQ[p]], OddQ[Times[I, n]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Plus[n, Times[2, Plus[p, 1], a, x]], Power[Plus[c, Times[d, Power[x, 2]]], Plus[p, 1]], Power[E, Times[n, ArcCot[Times[a, x]]]], Power[Times[Power[a, 3], c, Power[n, 2], Plus[Power[n, 2], 1]], -1]], x], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[d, Times[-1, Power[a, 2], c]]], ZeroQ[Plus[Power[n, 2], Times[-1, 2, Plus[p, 1]]]], NonzeroQ[Plus[Power[n, 2], 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[n, Times[2, Plus[p, 1], a, x]], Power[Plus[c, Times[d, Power[x, 2]]], Plus[p, 1]], Power[E, Times[n, ArcCot[Times[a, x]]]], Power[Times[Power[a, 3], c, Plus[Power[n, 2], Times[4, Power[Plus[p, 1], 2]]]], -1]], x], Dist[Times[Plus[Power[n, 2], Times[-1, 2, Plus[p, 1]]], Power[Times[Power[a, 2], c, Plus[Power[n, 2], Times[4, Power[Plus[p, 1], 2]]]], -1]], Int[Times[Power[Plus[c, Times[d, Power[x, 2]]], Plus[p, 1]], Power[E, Times[n, ArcCot[Times[a, x]]]]], x], x]], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[d, Times[-1, Power[a, 2], c]]], RationalQ[p], LessEqual[p, -1], NonzeroQ[Plus[Power[n, 2], Times[-1, 2, Plus[p, 1]]]], NonzeroQ[Plus[Power[n, 2], Times[4, Power[Plus[p, 1], 2]]]], Not[And[IntegerQ[p], EvenQ[Times[I, n]]]], Not[And[Not[IntegerQ[p]], OddQ[Times[I, n]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Times[Power[c, p], Power[Power[a, Plus[m, 1]], -1]], Subst[Int[Times[Power[E, Times[n, x]], Power[Cot[x], Plus[m, Times[2, Plus[p, 1]]]], Power[Power[Cos[x], Times[2, Plus[p, 1]]], -1]], x], x, ArcCot[Times[a, x]]], x]], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[d, Times[-1, Power[a, 2], c]]], IntegerQ[m], LessEqual[3, m, Times[-2, Plus[p, 1]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[d, p], Int[Times[u, Power[x, Times[2, p]], Power[Plus[1, Times[1, Power[Times[Power[a, 2], Power[x, 2]], -1]]], p], Power[E, Times[n, ArcCot[Times[a, x]]]]], x], x], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[d, Times[-1, Power[a, 2], c]]], Not[IntegerQ[Times[I, n, Power[2, -1]]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[c, Times[d, Power[x, 2]]], p], Power[Times[Power[x, Times[2, p]], Power[Plus[1, Times[1, Power[Times[Power[a, 2], Power[x, 2]], -1]]], p]], -1]], Int[Times[u, Power[x, Times[2, p]], Power[Plus[1, Times[1, Power[Times[Power[a, 2], Power[x, 2]], -1]]], p], Power[E, Times[n, ArcCot[Times[a, x]]]]], x], x], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[d, Times[-1, Power[a, 2], c]]], Not[IntegerQ[Times[I, n, Power[2, -1]]]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[c, p], Power[Power[Times[I, a], Times[2, p]], -1]], Int[Times[u, Power[Power[x, Times[2, p]], -1], Power[Plus[-1, Times[I, a, x]], Plus[p, Times[-1, I, n, Power[2, -1]]]], Power[Plus[1, Times[I, a, x]], Plus[p, Times[I, n, Power[2, -1]]]]], x], x], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[c, Times[-1, Power[a, 2], d]]], Not[IntegerQ[Times[I, n, Power[2, -1]]]], Or[IntegerQ[p], PositiveQ[c]], IntegersQ[Times[2, p], Plus[p, Times[I, n, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Power[c, p], Subst[Int[Times[Power[Plus[1, Times[-1, I, x, Power[a, -1]]], Plus[p, Times[I, n, Power[2, -1]]]], Power[Plus[1, Times[I, x, Power[a, -1]]], Plus[p, Times[-1, I, n, Power[2, -1]]]], Power[Power[x, 2], -1]], x], x, Times[1, Power[x, -1]]], x]], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[c, Times[-1, Power[a, 2], d]]], Not[IntegerQ[Times[I, n, Power[2, -1]]]], Or[IntegerQ[p], PositiveQ[c]], Not[IntegersQ[Times[2, p], Plus[p, Times[I, n, Power[2, -1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Power[c, p], Subst[Int[Times[Power[Plus[1, Times[-1, I, x, Power[a, -1]]], Plus[p, Times[I, n, Power[2, -1]]]], Power[Plus[1, Times[I, x, Power[a, -1]]], Plus[p, Times[-1, I, n, Power[2, -1]]]], Power[Power[x, Plus[m, 2]], -1]], x], x, Times[1, Power[x, -1]]], x]], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[c, Times[-1, Power[a, 2], d]]], Not[IntegerQ[Times[I, n, Power[2, -1]]]], Or[IntegerQ[p], PositiveQ[c]], Not[IntegersQ[Times[2, p], Plus[p, Times[I, n, Power[2, -1]]]]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Times[Power[c, p], Power[x, m], Power[Times[1, Power[x, -1]], m]], Subst[Int[Times[Power[Plus[1, Times[-1, I, x, Power[a, -1]]], Plus[p, Times[I, n, Power[2, -1]]]], Power[Plus[1, Times[I, x, Power[a, -1]]], Plus[p, Times[-1, I, n, Power[2, -1]]]], Power[Power[x, Plus[m, 2]], -1]], x], x, Times[1, Power[x, -1]]], x]], And[FreeQ[List[a, c, d, m, n, p], x], ZeroQ[Plus[c, Times[-1, Power[a, 2], d]]], Not[IntegerQ[Times[I, n, Power[2, -1]]]], Or[IntegerQ[p], PositiveQ[c]], Not[IntegersQ[Times[2, p], Plus[p, Times[I, n, Power[2, -1]]]]], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[c, Times[d, Power[Power[x, 2], -1]]], p], Power[Power[Plus[1, Times[1, Power[Times[Power[a, 2], Power[x, 2]], -1]]], p], -1]], Int[Times[u, Power[Plus[1, Times[1, Power[Times[Power[a, 2], Power[x, 2]], -1]]], p], Power[E, Times[n, ArcCot[Times[a, x]]]]], x], x], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[c, Times[-1, Power[a, 2], d]]], Not[IntegerQ[Times[I, n, Power[2, -1]]]], Not[Or[IntegerQ[p], PositiveQ[c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[-1, Times[I, n, Power[2, -1]]], Int[Times[u, Power[E, Times[-1, n, ArcTan[Times[c, Plus[a, Times[b, x]]]]]]], x], x], And[FreeQ[List[a, b, c], x], IntegerQ[Times[I, n, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Power[E, Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[I, c, Plus[a, Times[b, x]]], Times[I, n, Power[2, -1]]], Power[Plus[1, Times[1, Power[Times[I, c, Plus[a, Times[b, x]]], -1]]], Times[I, n, Power[2, -1]]], Power[Power[Plus[1, Times[I, a, c], Times[I, b, c, x]], Times[I, n, Power[2, -1]]], -1]], Int[Times[Power[Plus[1, Times[I, a, c], Times[I, b, c, x]], Times[I, n, Power[2, -1]]], Power[Power[Plus[-1, Times[I, a, c], Times[I, b, c, x]], Times[I, n, Power[2, -1]]], -1]], x], x], And[FreeQ[List[a, b, c, n], x], Not[IntegerQ[Times[I, n, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[4, Power[Times[Power[I, m], n, Power[b, Plus[m, 1]], Power[c, Plus[m, 1]]], -1]], Subst[Int[Times[Power[x, Times[2, Power[Times[I, n], -1]]], Power[Plus[1, Times[I, a, c], Times[Plus[1, Times[-1, I, a, c]], Power[x, Times[2, Power[Times[I, n], -1]]]]], m], Power[Power[Plus[-1, Power[x, Times[2, Power[Times[I, n], -1]]]], Plus[m, 2]], -1]], x], x, Times[Power[Plus[1, Times[1, Power[Times[I, c, Plus[a, Times[b, x]]], -1]]], Times[I, n, Power[2, -1]]], Power[Power[Plus[1, Times[-1, 1, Power[Times[I, c, Plus[a, Times[b, x]]], -1]]], Times[I, n, Power[2, -1]]], -1]]], x], And[FreeQ[List[a, b, c], x], NegativeIntegerQ[m], RationalQ[Times[I, n]], Less[-1, Times[I, n], 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[I, c, Plus[a, Times[b, x]]], Times[I, n, Power[2, -1]]], Power[Plus[1, Times[1, Power[Times[I, c, Plus[a, Times[b, x]]], -1]]], Times[I, n, Power[2, -1]]], Power[Power[Plus[1, Times[I, a, c], Times[I, b, c, x]], Times[I, n, Power[2, -1]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[1, Times[I, a, c], Times[I, b, c, x]], Times[I, n, Power[2, -1]]], Power[Power[Plus[-1, Times[I, a, c], Times[I, b, c, x]], Times[I, n, Power[2, -1]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, m, n], x], Not[IntegerQ[Times[I, n, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[c, Power[Plus[1, Power[a, 2]], -1]], p], Power[Times[Plus[Times[I, a], Times[I, b, x]], Power[Plus[1, Times[I, a], Times[I, b, x]], -1]], Times[I, n, Power[2, -1]]], Power[Times[Plus[1, Times[I, a], Times[I, b, x]], Power[Plus[Times[I, a], Times[I, b, x]], -1]], Times[I, n, Power[2, -1]]], Times[Power[Plus[1, Times[-1, I, a], Times[-1, I, b, x]], Times[I, n, Power[2, -1]]], Power[Power[Plus[-1, Times[I, a], Times[I, b, x]], Times[I, n, Power[2, -1]]], -1]]], Int[Times[u, Power[Plus[1, Times[-1, I, a], Times[-1, I, b, x]], Plus[p, Times[-1, I, n, Power[2, -1]]]], Power[Plus[1, Times[I, a], Times[I, b, x]], Plus[p, Times[I, n, Power[2, -1]]]]], x], x], And[FreeQ[List[a, b, c, d, e, n, p], x], Not[IntegerQ[Times[I, n, Power[2, -1]]]], ZeroQ[Plus[Times[b, d], Times[-1, 2, a, e]]], ZeroQ[Plus[Times[Power[b, 2], c], Times[-1, e, Plus[1, Power[a, 2]]]]], Or[IntegerQ[p], PositiveQ[Times[c, Power[Plus[1, Power[a, 2]], -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[c, Times[d, x], Times[e, Power[x, 2]]], p], Power[Power[Plus[1, Power[a, 2], Times[2, a, b, x], Times[Power[b, 2], Power[x, 2]]], p], -1]], Int[Times[u, Power[Plus[1, Power[a, 2], Times[2, a, b, x], Times[Power[b, 2], Power[x, 2]]], p], Power[E, Times[n, ArcCot[Times[a, x]]]]], x], x], And[FreeQ[List[a, b, c, d, e, n, p], x], Not[IntegerQ[Times[I, n, Power[2, -1]]]], ZeroQ[Plus[Times[b, d], Times[-1, 2, a, e]]], ZeroQ[Plus[Times[Power[b, 2], c], Times[-1, e, Plus[1, Power[a, 2]]]]], Not[Or[IntegerQ[p], PositiveQ[Times[c, Power[Plus[1, Power[a, 2]], -1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCot[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[u, Power[E, Times[n, ArcTan[Plus[Times[a, Power[c, -1]], Times[b, x, Power[c, -1]]]]]]], x], FreeQ[List[a, b, c, n], x]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTan[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, 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Condition[Plus[Simp[Times[Power[e, m], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Cosh[Plus[c, Times[d, x]]], Power[Times[b, n, Plus[p, 1]], -1]], x], Times[-1, Dist[Times[d, Power[e, m], Power[Times[b, n, Plus[p, 1]], -1]], Int[Times[Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Sinh[Plus[c, Times[d, x]]]], x], x]]], And[FreeQ[List[a, b, c, d, e, m, n], x], IntegerQ[p], ZeroQ[Plus[m, Times[-1, n], 1]], RationalQ[p], Less[p, -1], Or[IntegerQ[n], PositiveQ[e]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[m, Times[-1, n], 1]], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Sinh[Plus[c, Times[d, x]]], Power[Times[b, n, Plus[p, 1]], -1]], x], Plus[Times[-1, Dist[Times[Plus[m, Times[-1, n], 1], Power[Times[b, n, Plus[p, 1]], -1]], Int[Times[Power[x, Plus[m, Times[-1, n]]], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Sinh[Plus[c, Times[d, x]]]], x], x]], Times[-1, Dist[Times[d, Power[Times[b, n, Plus[p, 1]], -1]], Int[Times[Power[x, Plus[m, Times[-1, n], 1]], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Cosh[Plus[c, Times[d, x]]]], x], x]]]], And[FreeQ[List[a, b, c, d], x], IntegerQ[p], PositiveIntegerQ[n], RationalQ[m], Less[p, -1], Or[Greater[Plus[m, Times[-1, n], 1], 0], Greater[n, 2]]]]], RuleDelayed[HoldPattern[Int[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[m, Times[-1, n], 1]], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Cosh[Plus[c, Times[d, x]]], Power[Times[b, n, Plus[p, 1]], -1]], x], Plus[Times[-1, Dist[Times[Plus[m, Times[-1, n], 1], Power[Times[b, n, Plus[p, 1]], -1]], Int[Times[Power[x, Plus[m, Times[-1, n]]], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Cosh[Plus[c, Times[d, x]]]], x], x]], Times[-1, Dist[Times[d, Power[Times[b, n, Plus[p, 1]], -1]], Int[Times[Power[x, Plus[m, Times[-1, n], 1]], Power[Plus[a, Times[b, Power[x, n]]], Plus[p, 1]], Sinh[Plus[c, Times[d, x]]]], x], x]]]], And[FreeQ[List[a, b, c, d], x], IntegerQ[p], PositiveIntegerQ[n], RationalQ[m], Less[p, -1], Or[Greater[Plus[m, Times[-1, n], 1], 0], Greater[n, 2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Sinh[Plus[c, Times[d, x]]], Times[Power[x, m], Power[Plus[a, Times[b, Power[x, n]]], p]], x], x], And[FreeQ[List[a, b, c, d], x], NegativeIntegerQ[p], IntegerQ[m], PositiveIntegerQ[n], Or[Equal[n, 2], Equal[p, -1]]]]], RuleDelayed[HoldPattern[Int[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Cosh[Plus[c, Times[d, x]]], Times[Power[x, m], Power[Plus[a, Times[b, Power[x, n]]], p]], x], x], And[FreeQ[List[a, b, c, d], x], NegativeIntegerQ[p], IntegerQ[m], PositiveIntegerQ[n], Or[Equal[n, 2], Equal[p, -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[x, Plus[m, Times[n, p]]], Power[Plus[b, Times[a, Power[x, Times[-1, n]]]], p], Sinh[Plus[c, Times[d, x]]]], x], And[FreeQ[List[a, b, c, d, m], x], NegativeIntegerQ[p], NegativeIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[x, Plus[m, Times[n, p]]], Power[Plus[b, Times[a, Power[x, Times[-1, n]]]], p], Cosh[Plus[c, Times[d, x]]]], x], And[FreeQ[List[a, b, c, d, m], x], NegativeIntegerQ[p], NegativeIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Times[e, x], m], Power[Plus[a, Times[b, Power[x, n]]], p], Sinh[Plus[c, Times[d, x]]]], x], FreeQ[List[a, b, c, d, e, m, n, p], x]]], RuleDelayed[HoldPattern[Int[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Times[e, x], m], Power[Plus[a, Times[b, Power[x, n]]], p], Cosh[Plus[c, Times[d, x]]]], x], FreeQ[List[a, b, c, d, e, m, n, p], x]]], RuleDelayed[HoldPattern[Int[Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[2, -1]], Int[Power[E, Plus[c, Times[d, Power[x, n]]]], x], x], Times[-1, Dist[Times[1, Power[2, -1]], Int[Power[E, Plus[Times[-1, c], Times[-1, d, Power[x, n]]]], x], x]]], And[FreeQ[List[c, d], x], IntegerQ[n], Greater[n, 1]]]], RuleDelayed[HoldPattern[Int[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[2, -1]], Int[Power[E, Plus[c, Times[d, Power[x, n]]]], x], x], Dist[Times[1, Power[2, -1]], Int[Power[E, Plus[Times[-1, c], Times[-1, d, Power[x, n]]]], x], x]], And[FreeQ[List[c, d], x], IntegerQ[n], Greater[n, 1]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Power[Plus[a, Times[b, Sinh[Plus[c, Times[d, Power[x, n]]]]]], p], x], x], And[FreeQ[List[a, b, c, d], x], IntegersQ[n, p], Greater[n, 1], Greater[p, 1]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Power[Plus[a, Times[b, Cosh[Plus[c, Times[d, Power[x, n]]]]]], p], x], x], And[FreeQ[List[a, b, c, d], x], IntegersQ[n, p], Greater[n, 1], Greater[p, 1]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Subst[Int[Times[Power[Plus[a, Times[b, Sinh[Plus[c, Times[d, Power[x, Times[-1, n]]]]]]], p], Power[Power[x, 2], -1]], x], x, Times[1, Power[x, -1]]]], And[FreeQ[List[a, b, c, d], x], IntegerQ[p], NegativeIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Subst[Int[Times[Power[Plus[a, Times[b, Cosh[Plus[c, Times[d, Power[x, Times[-1, n]]]]]]], p], Power[Power[x, 2], -1]], x], x, Times[1, Power[x, -1]]]], And[FreeQ[List[a, b, c, d], x], IntegerQ[p], NegativeIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[k, Denominator[n]]], Dist[k, Subst[Int[Times[Power[x, Plus[k, -1]], Power[Plus[a, Times[b, Sinh[Plus[c, Times[d, Power[x, Times[k, n]]]]]]], p]], x], x, Power[x, Times[1, Power[k, -1]]]], x]], And[FreeQ[List[a, b, c, d], x], IntegerQ[p], FractionQ[n]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[k, Denominator[n]]], Dist[k, Subst[Int[Times[Power[x, Plus[k, -1]], Power[Plus[a, Times[b, Cosh[Plus[c, Times[d, Power[x, Times[k, n]]]]]]], p]], x], x, Power[x, Times[1, Power[k, -1]]]], x]], And[FreeQ[List[a, b, c, d], x], IntegerQ[p], FractionQ[n]]]], RuleDelayed[HoldPattern[Int[Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[2, -1]], Int[Power[E, Plus[c, Times[d, Power[x, n]]]], x], x], Times[-1, Dist[Times[1, Power[2, -1]], Int[Power[E, Plus[Times[-1, c], Times[-1, d, Power[x, n]]]], x], x]]], FreeQ[List[c, d, n], x]]], RuleDelayed[HoldPattern[Int[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[2, -1]], Int[Power[E, Plus[c, Times[d, Power[x, n]]]], x], x], Dist[Times[1, Power[2, -1]], Int[Power[E, Plus[Times[-1, c], Times[-1, d, Power[x, n]]]], x], x]], FreeQ[List[c, d, n], x]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Power[Plus[a, Times[b, Sinh[Plus[c, Times[d, Power[x, n]]]]]], p], x], x], And[FreeQ[List[a, b, c, d, n], x], PositiveIntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Power[Plus[a, Times[b, Cosh[Plus[c, Times[d, Power[x, n]]]]]], p], x], x], And[FreeQ[List[a, b, c, d, n], x], PositiveIntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Coefficient[u, x, 1], -1]], Subst[Int[Power[Plus[a, Times[b, Sinh[Plus[c, Times[d, Power[x, n]]]]]], p], x], x, u], x], And[FreeQ[List[a, b, c, d, n], x], IntegerQ[p], LinearQ[u, x], NonzeroQ[Plus[u, Times[-1, x]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Coefficient[u, x, 1], -1]], Subst[Int[Power[Plus[a, Times[b, Cosh[Plus[c, Times[d, Power[x, n]]]]]], p], x], x, u], x], And[FreeQ[List[a, b, c, d, n], x], IntegerQ[p], LinearQ[u, x], NonzeroQ[Plus[u, Times[-1, x]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Power[Plus[a, Times[b, Sinh[Plus[c, Times[d, Power[u, n]]]]]], p], x], And[FreeQ[List[a, b, c, d, n, p], x], LinearQ[u, x]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Power[Plus[a, Times[b, Cosh[Plus[c, Times[d, Power[u, n]]]]]], p], x], And[FreeQ[List[a, b, c, d, n, p], x], LinearQ[u, x]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Pattern[u, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Power[Plus[a, Times[b, Sinh[ExpandToSum[u, x]]]], p], x], And[FreeQ[List[a, b, p], x], BinomialQ[u, x], Not[BinomialMatchQ[u, x]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Pattern[u, Blank[]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Power[Plus[a, Times[b, Cosh[ExpandToSum[u, x]]]], p], x], And[FreeQ[List[a, b, p], x], BinomialQ[u, x], Not[BinomialMatchQ[u, x]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], -1], Sinh[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[SinhIntegral[Times[d, Power[x, n]]], Power[n, -1]], x], FreeQ[List[d, n], x]]], RuleDelayed[HoldPattern[Int[Times[Cosh[Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[CoshIntegral[Times[d, Power[x, n]]], Power[n, -1]], x], FreeQ[List[d, n], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], -1], Sinh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Sinh[c], Int[Times[Cosh[Times[d, Power[x, n]]], Power[x, -1]], x], x], Dist[Cosh[c], Int[Times[Sinh[Times[d, Power[x, n]]], Power[x, -1]], x], x]], FreeQ[List[c, d, n], x]]], RuleDelayed[HoldPattern[Int[Times[Cosh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Power[Pattern[x, Blank[]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Cosh[c], Int[Times[Cosh[Times[d, Power[x, n]]], Power[x, -1]], x], x], Dist[Sinh[c], Int[Times[Sinh[Times[d, Power[x, n]]], Power[x, -1]], x], x]], FreeQ[List[c, d, n], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[mn, Simplify[Times[Plus[m, 1], Power[n, -1]]]]], Condition[Dist[Times[1, Power[n, -1]], Subst[Int[Times[Power[x, Plus[mn, -1]], Power[Plus[a, Times[b, Sinh[Plus[c, Times[d, x]]]]], p]], x], x, Power[x, n]], x], And[IntegerQ[mn], Or[Equal[p, 1], Greater[mn, 0]]]]], And[FreeQ[List[a, b, c, d, m, n], x], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[mn, Simplify[Times[Plus[m, 1], Power[n, -1]]]]], Condition[Dist[Times[1, Power[n, -1]], Subst[Int[Times[Power[x, Plus[mn, -1]], Power[Plus[a, Times[b, Cosh[Plus[c, Times[d, x]]]]], p]], x], x, Power[x, n]], x], And[IntegerQ[mn], Or[Equal[p, 1], Greater[mn, 0]]]]], And[FreeQ[List[a, b, c, d, m, n], x], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[mn, Simplify[Times[Plus[m, 1], Power[n, -1]]]]], Condition[Dist[Times[Power[e, IntPart[m]], Power[Times[e, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Plus[a, Times[b, Sinh[Plus[c, Times[d, Power[x, n]]]]]], p]], x], x], And[IntegerQ[mn], Or[Equal[p, 1], Greater[mn, 0]]]]], And[FreeQ[List[a, b, c, d, e, m, n], x], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Times[Pattern[e, Blank[]], Pattern[x, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[mn, Simplify[Times[Plus[m, 1], Power[n, -1]]]]], Condition[Dist[Times[Power[e, IntPart[m]], Power[Times[e, x], FracPart[m]], Power[Power[x, FracPart[m]], -1]], Int[Times[Power[x, m], Power[Plus[a, Times[b, Cosh[Plus[c, Times[d, Power[x, n]]]]]], p]], x], x], And[IntegerQ[mn], Or[Equal[p, 1], Greater[mn, 0]]]]], And[FreeQ[List[a, b, c, d, e, m, n], x], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[e, Plus[n, -1]], Power[Times[e, x], Plus[m, Times[-1, n], 1]], Cosh[Plus[c, Times[d, Power[x, n]]]], Power[Times[d, n], -1]], x], Times[-1, Dist[Times[Power[e, n], Plus[m, Times[-1, n], 1], Power[Times[d, n], -1]], Int[Times[Power[Times[e, x], Plus[m, Times[-1, n]]], Cosh[Plus[c, Times[d, Power[x, n]]]]], x], x]]], And[FreeQ[List[c, d, e], x], PositiveIntegerQ[n], RationalQ[m], Less[0, n, Plus[m, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[e, Plus[n, -1]], Power[Times[e, x], Plus[m, Times[-1, n], 1]], Sinh[Plus[c, Times[d, Power[x, n]]]], Power[Times[d, n], -1]], x], Times[-1, Dist[Times[Power[e, n], Plus[m, Times[-1, n], 1], Power[Times[d, n], -1]], Int[Times[Power[Times[e, x], Plus[m, Times[-1, n]]], Sinh[Plus[c, Times[d, Power[x, n]]]]], x], x]]], And[FreeQ[List[c, d, e], x], PositiveIntegerQ[n], RationalQ[m], Less[0, n, Plus[m, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[e, x], Plus[m, 1]], Sinh[Plus[c, Times[d, Power[x, n]]]], Power[Times[e, Plus[m, 1]], -1]], x], Times[-1, Dist[Times[d, n, Power[Times[Power[e, n], Plus[m, 1]], -1]], Int[Times[Power[Times[e, x], Plus[m, n]], Cosh[Plus[c, Times[d, Power[x, n]]]]], x], x]]], And[FreeQ[List[c, d, e], x], PositiveIntegerQ[n], RationalQ[m], Less[m, -1]]]], RuleDelayed[HoldPattern[Int[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[e, x], Plus[m, 1]], Cosh[Plus[c, Times[d, Power[x, n]]]], Power[Times[e, Plus[m, 1]], -1]], x], Times[-1, Dist[Times[d, n, Power[Times[Power[e, n], Plus[m, 1]], -1]], Int[Times[Power[Times[e, x], Plus[m, n]], Sinh[Plus[c, Times[d, Power[x, n]]]]], x], x]]], And[FreeQ[List[c, d, e], x], PositiveIntegerQ[n], RationalQ[m], Less[m, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[2, -1]], Int[Times[Power[Times[e, x], m], Power[E, Plus[c, Times[d, Power[x, n]]]]], x], x], Times[-1, Dist[Times[1, Power[2, -1]], Int[Times[Power[Times[e, x], m], Power[E, Plus[Times[-1, c], Times[-1, d, Power[x, n]]]]], x], x]]], And[FreeQ[List[c, d, e, m], x], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[2, -1]], Int[Times[Power[Times[e, x], m], Power[E, Plus[c, Times[d, Power[x, n]]]]], x], x], Dist[Times[1, Power[2, -1]], Int[Times[Power[Times[e, x], m], Power[E, Plus[Times[-1, c], Times[-1, d, Power[x, n]]]]], x], x]], And[FreeQ[List[c, d, e, m], x], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Sinh[Plus[a, Times[b, Power[x, n]]]], p], Power[Times[Plus[n, -1], Power[x, Plus[n, -1]]], -1]], x]], Dist[Times[b, n, p, Power[Plus[n, -1], -1]], Int[Times[Power[Sinh[Plus[a, Times[b, Power[x, n]]]], Plus[p, -1]], Cosh[Plus[a, Times[b, Power[x, n]]]]], x], x]], And[FreeQ[List[a, b], x], IntegersQ[n, p], ZeroQ[Plus[m, n]], Greater[p, 1], NonzeroQ[Plus[n, -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Cosh[Plus[a, Times[b, Power[x, n]]]], p], Power[Times[Plus[n, -1], Power[x, Plus[n, -1]]], -1]], x]], Dist[Times[b, n, p, Power[Plus[n, -1], -1]], Int[Times[Power[Cosh[Plus[a, Times[b, Power[x, n]]]], Plus[p, -1]], Sinh[Plus[a, Times[b, Power[x, n]]]]], x], x]], And[FreeQ[List[a, b], x], IntegersQ[n, p], ZeroQ[Plus[m, n]], Greater[p, 1], NonzeroQ[Plus[n, -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[n, Power[Sinh[Plus[a, Times[b, Power[x, n]]]], p], Power[Times[Power[b, 2], Power[n, 2], Power[p, 2]], -1]], x]], Plus[Times[-1, Dist[Times[Plus[p, -1], Power[p, -1]], Int[Times[Power[x, m], Power[Sinh[Plus[a, Times[b, Power[x, n]]]], Plus[p, -2]]], x], x]], Simp[Times[Power[x, n], Cosh[Plus[a, Times[b, Power[x, n]]]], Power[Sinh[Plus[a, Times[b, Power[x, n]]]], Plus[p, -1]], Power[Times[b, n, p], -1]], x]]], And[FreeQ[List[a, b, m, n], x], ZeroQ[Plus[m, Times[-1, 2, n], 1]], RationalQ[p], Greater[p, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[n, Power[Cosh[Plus[a, Times[b, Power[x, n]]]], p], Power[Times[Power[b, 2], Power[n, 2], Power[p, 2]], -1]], x]], Plus[Dist[Times[Plus[p, -1], Power[p, -1]], Int[Times[Power[x, m], Power[Cosh[Plus[a, Times[b, Power[x, n]]]], Plus[p, -2]]], x], x], Simp[Times[Power[x, n], Sinh[Plus[a, Times[b, Power[x, n]]]], Power[Cosh[Plus[a, Times[b, Power[x, n]]]], Plus[p, -1]], Power[Times[b, n, p], -1]], x]]], And[FreeQ[List[a, b, m, n], x], ZeroQ[Plus[m, Times[-1, 2, n], 1]], RationalQ[p], Greater[p, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[m, Times[-1, n], 1], Power[x, Plus[m, Times[-1, 2, n], 1]], Power[Sinh[Plus[a, Times[b, Power[x, n]]]], p], Power[Times[Power[b, 2], Power[n, 2], Power[p, 2]], -1]], x]], Plus[Times[-1, Dist[Times[Plus[p, -1], Power[p, -1]], Int[Times[Power[x, m], Power[Sinh[Plus[a, Times[b, Power[x, n]]]], Plus[p, -2]]], x], x]], Dist[Times[Plus[m, Times[-1, n], 1], Plus[m, Times[-1, 2, n], 1], Power[Times[Power[b, 2], Power[n, 2], Power[p, 2]], -1]], Int[Times[Power[x, Plus[m, Times[-1, 2, n]]], Power[Sinh[Plus[a, Times[b, Power[x, n]]]], p]], x], x], Simp[Times[Power[x, Plus[m, Times[-1, n], 1]], Cosh[Plus[a, Times[b, Power[x, n]]]], Power[Sinh[Plus[a, Times[b, Power[x, n]]]], Plus[p, -1]], Power[Times[b, n, p], -1]], x]]], And[FreeQ[List[a, b], x], IntegersQ[m, n], RationalQ[p], Greater[p, 1], Less[0, Times[2, n], Plus[m, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[m, Times[-1, n], 1], Power[x, Plus[m, Times[-1, 2, n], 1]], Power[Cosh[Plus[a, Times[b, Power[x, n]]]], p], Power[Times[Power[b, 2], Power[n, 2], Power[p, 2]], -1]], x]], Plus[Dist[Times[Plus[p, -1], Power[p, -1]], Int[Times[Power[x, m], Power[Cosh[Plus[a, Times[b, Power[x, n]]]], Plus[p, -2]]], x], x], Dist[Times[Plus[m, Times[-1, n], 1], Plus[m, Times[-1, 2, n], 1], Power[Times[Power[b, 2], Power[n, 2], Power[p, 2]], -1]], Int[Times[Power[x, Plus[m, Times[-1, 2, n]]], Power[Cosh[Plus[a, Times[b, Power[x, n]]]], p]], x], x], Simp[Times[Power[x, Plus[m, Times[-1, n], 1]], Sinh[Plus[a, Times[b, Power[x, n]]]], Power[Cosh[Plus[a, Times[b, Power[x, n]]]], Plus[p, -1]], Power[Times[b, n, p], -1]], x]]], And[FreeQ[List[a, b], x], IntegersQ[m, n], RationalQ[p], Greater[p, 1], Less[0, Times[2, n], Plus[m, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[m, 1]], Power[Sinh[Plus[a, Times[b, Power[x, n]]]], p], Power[Plus[m, 1], -1]], x], Plus[Dist[Times[Power[b, 2], Power[n, 2], p, Plus[p, -1], Power[Times[Plus[m, 1], Plus[m, n, 1]], -1]], Int[Times[Power[x, Plus[m, Times[2, n]]], Power[Sinh[Plus[a, Times[b, Power[x, n]]]], Plus[p, -2]]], x], x], Dist[Times[Power[b, 2], Power[n, 2], Power[p, 2], Power[Times[Plus[m, 1], Plus[m, n, 1]], -1]], Int[Times[Power[x, Plus[m, Times[2, n]]], Power[Sinh[Plus[a, Times[b, Power[x, n]]]], p]], x], x], Times[-1, Simp[Times[b, n, p, Power[x, Plus[m, n, 1]], Cosh[Plus[a, Times[b, Power[x, n]]]], Power[Sinh[Plus[a, Times[b, Power[x, n]]]], Plus[p, -1]], Power[Times[Plus[m, 1], Plus[m, n, 1]], -1]], x]]]], And[FreeQ[List[a, b], x], IntegersQ[m, n], RationalQ[p], Greater[p, 1], Less[0, Times[2, n], Plus[1, Times[-1, m]]], NonzeroQ[Plus[m, n, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 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NonzeroQ[Plus[m, n, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[k, Denominator[m]]], Dist[Times[k, Power[e, -1]], Subst[Int[Times[Power[x, Plus[Times[k, Plus[m, 1]], -1]], Power[Plus[a, Times[b, Sinh[Plus[c, Times[d, Power[x, Times[k, n]], Power[Power[e, n], -1]]]]]], p]], x], x, Power[Times[e, x], Times[1, Power[k, -1]]]], x]], And[FreeQ[List[a, b, c, d, e], x], IntegerQ[p], PositiveIntegerQ[n], FractionQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, 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Condition[Int[ExpandTrigReduce[Power[Times[e, x], m], Power[Plus[a, Times[b, Sinh[Plus[c, Times[d, Power[x, n]]]]]], p], x], x], And[FreeQ[List[a, b, c, d, e, m], x], IntegerQ[p], PositiveIntegerQ[n], Greater[p, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Power[Times[e, x], m], Power[Plus[a, Times[b, Cosh[Plus[c, Times[d, Power[x, n]]]]]], p], x], x], And[FreeQ[List[a, b, c, d, e, m], x], IntegerQ[p], PositiveIntegerQ[n], Greater[p, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, n], Cosh[Plus[a, Times[b, Power[x, n]]]], Power[Sinh[Plus[a, Times[b, Power[x, n]]]], Plus[p, 1]], Power[Times[b, n, Plus[p, 1]], -1]], x], Plus[Times[-1, Dist[Times[Plus[p, 2], Power[Plus[p, 1], -1]], Int[Times[Power[x, m], Power[Sinh[Plus[a, Times[b, Power[x, n]]]], Plus[p, 2]]], x], x]], Times[-1, Simp[Times[n, Power[Sinh[Plus[a, Times[b, Power[x, n]]]], Plus[p, 2]], Power[Times[Power[b, 2], Power[n, 2], Plus[p, 1], Plus[p, 2]], -1]], x]]]], And[FreeQ[List[a, b, m, n], x], ZeroQ[Plus[m, Times[-1, 2, n], 1]], RationalQ[p], Less[p, -1], Unequal[p, -2]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[x, n], Sinh[Plus[a, Times[b, Power[x, n]]]], Power[Cosh[Plus[a, Times[b, Power[x, n]]]], Plus[p, 1]], Power[Times[b, n, Plus[p, 1]], -1]], x]], Plus[Dist[Times[Plus[p, 2], Power[Plus[p, 1], -1]], Int[Times[Power[x, m], Power[Cosh[Plus[a, Times[b, Power[x, n]]]], Plus[p, 2]]], x], x], Simp[Times[n, Power[Cosh[Plus[a, Times[b, Power[x, n]]]], Plus[p, 2]], Power[Times[Power[b, 2], Power[n, 2], Plus[p, 1], Plus[p, 2]], -1]], x]]], And[FreeQ[List[a, b, m, n], x], ZeroQ[Plus[m, Times[-1, 2, n], 1]], RationalQ[p], Less[p, -1], Unequal[p, -2]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[m, Times[-1, n], 1]], Cosh[Plus[a, Times[b, Power[x, n]]]], Power[Sinh[Plus[a, Times[b, 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Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[x, Plus[m, Times[-1, n], 1]], Sinh[Plus[a, Times[b, Power[x, n]]]], Power[Cosh[Plus[a, Times[b, Power[x, n]]]], Plus[p, 1]], Power[Times[b, n, Plus[p, 1]], -1]], x]], Plus[Dist[Times[Plus[p, 2], Power[Plus[p, 1], -1]], Int[Times[Power[x, m], Power[Cosh[Plus[a, Times[b, Power[x, n]]]], Plus[p, 2]]], x], x], Times[-1, Dist[Times[Plus[m, Times[-1, n], 1], Plus[m, Times[-1, 2, n], 1], Power[Times[Power[b, 2], Power[n, 2], Plus[p, 1], Plus[p, 2]], -1]], Int[Times[Power[x, Plus[m, Times[-1, 2, n]]], Power[Cosh[Plus[a, Times[b, Power[x, n]]]], Plus[p, 2]]], x], x]], Simp[Times[Plus[m, Times[-1, n], 1], Power[x, Plus[m, Times[-1, 2, n], 1]], Power[Cosh[Plus[a, Times[b, Power[x, n]]]], Plus[p, 2]], Power[Times[Power[b, 2], Power[n, 2], Plus[p, 1], Plus[p, 2]], -1]], x]]], And[FreeQ[List[a, b], x], IntegersQ[m, n], RationalQ[p], Less[p, -1], Unequal[p, 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RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Times[Power[Times[e, x], m], Power[Power[x, -1], m]], Subst[Int[Times[Power[Plus[a, Times[b, Sinh[Plus[c, Times[d, Power[x, Times[-1, n]]]]]]], p], Power[Power[x, Plus[m, 2]], -1]], x], x, Times[1, Power[x, -1]]], x]], And[FreeQ[List[a, b, c, d, e, m], x], IntegerQ[p], NegativeIntegerQ[n], Not[RationalQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, 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x], IntegerQ[p], NonzeroQ[Plus[m, 1]], PositiveIntegerQ[Simplify[Times[n, Power[Plus[m, 1], -1]]]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[2, -1]], Int[Times[Power[Times[e, x], m], Power[E, Plus[c, Times[d, Power[x, n]]]]], x], x], Times[-1, Dist[Times[1, Power[2, -1]], Int[Times[Power[Times[e, x], m], Power[E, Plus[Times[-1, c], Times[-1, d, Power[x, n]]]]], x], x]]], FreeQ[List[c, d, e, m, n], x]]], RuleDelayed[HoldPattern[Int[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[2, -1]], Int[Times[Power[Times[e, x], m], Power[E, Plus[c, Times[d, Power[x, n]]]]], x], x], Dist[Times[1, Power[2, -1]], Int[Times[Power[Times[e, x], m], Power[E, Plus[Times[-1, c], Times[-1, d, Power[x, n]]]]], x], x]], FreeQ[List[c, d, e, m, n], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Power[Times[e, x], m], Power[Plus[a, Times[b, Sinh[Plus[c, Times[d, Power[x, n]]]]]], p], x], x], And[FreeQ[List[a, b, c, d, e, m, n], x], PositiveIntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Pattern[p, Blank[]]], Power[Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Power[Times[e, x], m], Power[Plus[a, Times[b, Cosh[Plus[c, Times[d, Power[x, n]]]]]], p], x], x], And[FreeQ[List[a, b, c, d, e, m, n], x], PositiveIntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[u, Blank[]], Pattern[n, Blank[]]]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Power[Coefficient[u, x, 1], Plus[m, 1]], -1]], Subst[Int[Times[Power[Plus[x, Times[-1, 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RuleDelayed[HoldPattern[Int[Times[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[e, Sinh[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], Power[Times[2, c], -1]], x], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[Times[b, e], Times[-1, 2, c, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[e, Cosh[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], Power[Times[2, c], -1]], x], Times[-1, Dist[Times[Plus[Times[b, e], 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x]], Plus[m, 1]], Cosh[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]], x], x]]]], And[FreeQ[List[a, b, c, d, e], x], RationalQ[m], Less[m, -1], NonzeroQ[Plus[Times[b, e], Times[-1, 2, c, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Cosh[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], Power[Times[e, Plus[m, 1]], -1]], x], Plus[Times[-1, Dist[Times[2, c, Power[Times[Power[e, 2], Plus[m, 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 2]], Sinh[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]], x], x]], Times[-1, Dist[Times[Plus[Times[b, e], Times[-1, 2, c, d]], Power[Times[Power[e, 2], Plus[m, 1]], -1]], Int[Times[Power[Plus[d, Times[e, x]], Plus[m, 1]], Sinh[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]], x], x]]]], And[FreeQ[List[a, b, c, d, e], x], RationalQ[m], Less[m, -1], NonzeroQ[Plus[Times[b, e], Times[-1, 2, c, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Plus[d, Times[e, x]], m], Sinh[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]], x], FreeQ[List[a, b, c, d, e, m], x]]], RuleDelayed[HoldPattern[Int[Times[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Plus[d, Times[e, x]], m], Cosh[Plus[a, Times[b, x], Times[c, Power[x, 2]]]]], x], FreeQ[List[a, b, c, d, e, m], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Power[Plus[d, Times[e, x]], m], Power[Sinh[Plus[a, Times[b, x], Times[c, Power[x, 2]]]], n], x], x], And[FreeQ[List[a, b, c, d, e, m], x], IntegerQ[n], Greater[n, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, 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Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Power[Plus[c, Times[d, x]], m], Times[Power[Sinh[Plus[a, Times[b, x]]], n], Power[Cosh[Plus[a, Times[b, x]]], p]], x], x], And[FreeQ[List[a, b, c, d, m], x], PositiveIntegerQ[n, p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Int[Times[Power[Plus[c, Times[d, x]], m], Power[Sinh[Plus[a, Times[b, x]]], n], Power[Tanh[Plus[a, Times[b, x]]], Plus[p, 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Condition[Plus[Simp[Times[Power[Plus[e, Times[f, x]], m], Power[Plus[a, Times[b, Sinh[Plus[c, Times[d, x]]]]], Plus[n, 1]], Power[Times[b, d, Plus[n, 1]], -1]], x], Times[-1, Dist[Times[f, m, Power[Times[b, d, Plus[n, 1]], -1]], Int[Times[Power[Plus[e, Times[f, x]], Plus[m, -1]], Power[Plus[a, Times[b, Sinh[Plus[c, Times[d, x]]]]], Plus[n, 1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, n], x], PositiveIntegerQ[m], NonzeroQ[Plus[n, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[e, Times[f, x]], m], Power[Plus[a, Times[b, Cosh[Plus[c, Times[d, x]]]]], Plus[n, 1]], Power[Times[b, d, Plus[n, 1]], -1]], x], Times[-1, Dist[Times[f, m, Power[Times[b, d, Plus[n, 1]], -1]], Int[Times[Power[Plus[e, Times[f, x]], Plus[m, -1]], Power[Plus[a, Times[b, Cosh[Plus[c, Times[d, x]]]]], Plus[n, 1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, n], x], PositiveIntegerQ[m], NonzeroQ[Plus[n, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sech[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Tanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[e, Times[f, x]], m], Power[Plus[a, Times[b, Tanh[Plus[c, Times[d, x]]]]], Plus[n, 1]], Power[Times[b, d, Plus[n, 1]], -1]], x], Times[-1, Dist[Times[f, m, Power[Times[b, d, Plus[n, 1]], -1]], Int[Times[Power[Plus[e, Times[f, x]], Plus[m, -1]], Power[Plus[a, Times[b, Tanh[Plus[c, Times[d, x]]]]], Plus[n, 1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, n], x], PositiveIntegerQ[m], NonzeroQ[Plus[n, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Csch[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Times[Coth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Plus[e, Times[f, x]], m], Power[Plus[a, Times[b, Coth[Plus[c, Times[d, x]]]]], Plus[n, 1]], Power[Times[b, d, Plus[n, 1]], -1]], x]], Dist[Times[f, m, Power[Times[b, d, Plus[n, 1]], -1]], Int[Times[Power[Plus[e, Times[f, x]], Plus[m, -1]], Power[Plus[a, Times[b, Coth[Plus[c, Times[d, x]]]]], Plus[n, 1]]], x], x]], And[FreeQ[List[a, b, c, d, e, f, n], x], PositiveIntegerQ[m], NonzeroQ[Plus[n, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Sech[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Sech[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]], Tanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Plus[e, Times[f, x]], m], Power[Plus[a, Times[b, Sech[Plus[c, Times[d, x]]]]], Plus[n, 1]], Power[Times[b, d, Plus[n, 1]], -1]], x]], Dist[Times[f, m, Power[Times[b, d, Plus[n, 1]], -1]], Int[Times[Power[Plus[e, Times[f, x]], Plus[m, -1]], Power[Plus[a, Times[b, Sech[Plus[c, Times[d, x]]]]], Plus[n, 1]]], x], x]], And[FreeQ[List[a, b, c, d, e, f, n], x], PositiveIntegerQ[m], NonzeroQ[Plus[n, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Coth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Csch[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Power[Plus[Times[Csch[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Plus[e, Times[f, x]], m], Power[Plus[a, Times[b, Csch[Plus[c, Times[d, x]]]]], Plus[n, 1]], Power[Times[b, d, Plus[n, 1]], -1]], x]], Dist[Times[f, m, Power[Times[b, d, Plus[n, 1]], -1]], Int[Times[Power[Plus[e, Times[f, x]], Plus[m, -1]], Power[Plus[a, Times[b, Csch[Plus[c, Times[d, x]]]]], Plus[n, 1]]], x], x]], And[FreeQ[List[a, b, c, d, e, f, n], x], PositiveIntegerQ[m], NonzeroQ[Plus[n, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Power[Plus[e, Times[f, x]], m], Times[Power[Sinh[Plus[a, Times[b, x]]], p], Power[Sinh[Plus[c, Times[d, x]]], q]], x], x], And[FreeQ[List[a, b, c, d, e, f], x], PositiveIntegerQ[p, q], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Power[Plus[e, Times[f, x]], m], Times[Power[Cosh[Plus[a, Times[b, x]]], p], Power[Cosh[Plus[c, Times[d, x]]], q]], x], x], And[FreeQ[List[a, b, c, d, e, f], x], PositiveIntegerQ[p, q], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Power[Plus[e, Times[f, x]], m], Times[Power[Sinh[Plus[a, Times[b, x]]], p], Power[Cosh[Plus[c, Times[d, x]]], q]], x], x], And[FreeQ[List[a, b, c, d, e, f, m], x], PositiveIntegerQ[p, q]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Pattern[F, Blank[]][Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[G, Blank[]][Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigExpand[Times[Power[Plus[e, Times[f, x]], m], Power[G[Plus[c, Times[d, x]]], q]], F, Plus[c, Times[d, x]], p, Times[b, Power[d, -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, m], x], MemberQ[List[Sinh, Cosh], F], MemberQ[List[Sech, Csch], G], PositiveIntegerQ[p, q], ZeroQ[Plus[Times[b, c], Times[-1, a, d]]], PositiveIntegerQ[Plus[Times[b, Power[d, -1]], -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Sinh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[b, c, Log[F], Power[F, Times[c, Plus[a, Times[b, x]]]], Sinh[Plus[d, Times[e, x]]], Power[Plus[Power[e, 2], Times[-1, Power[b, 2], Power[c, 2], Power[Log[F], 2]]], -1]], x]], Simp[Times[e, Power[F, Times[c, Plus[a, Times[b, x]]]], Cosh[Plus[d, Times[e, x]]], Power[Plus[Power[e, 2], Times[-1, Power[b, 2], Power[c, 2], Power[Log[F], 2]]], -1]], x]], And[FreeQ[List[F, a, b, c, d, e], x], NonzeroQ[Plus[Power[e, 2], Times[-1, Power[b, 2], Power[c, 2], Power[Log[F], 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Cosh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[b, c, Log[F], Power[F, Times[c, Plus[a, Times[b, x]]]], Cosh[Plus[d, Times[e, x]]], Power[Plus[Power[e, 2], Times[-1, Power[b, 2], Power[c, 2], Power[Log[F], 2]]], -1]], x]], Simp[Times[e, Power[F, Times[c, Plus[a, Times[b, x]]]], Sinh[Plus[d, Times[e, x]]], Power[Plus[Power[e, 2], Times[-1, Power[b, 2], Power[c, 2], Power[Log[F], 2]]], -1]], x]], And[FreeQ[List[F, a, b, c, d, e], x], NonzeroQ[Plus[Power[e, 2], Times[-1, Power[b, 2], Power[c, 2], Power[Log[F], 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sinh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[b, c, Log[F], Power[F, Times[c, Plus[a, Times[b, x]]]], Power[Sinh[Plus[d, Times[e, x]]], n], Power[Plus[Times[Power[e, 2], Power[n, 2]], Times[-1, Power[b, 2], Power[c, 2], Power[Log[F], 2]]], -1]], x]], Plus[Times[-1, Dist[Times[n, Plus[n, -1], Power[e, 2], Power[Plus[Times[Power[e, 2], Power[n, 2]], Times[-1, Power[b, 2], Power[c, 2], Power[Log[F], 2]]], -1]], Int[Times[Power[F, Times[c, Plus[a, Times[b, x]]]], Power[Sinh[Plus[d, Times[e, x]]], Plus[n, -2]]], x], x]], Simp[Times[e, n, Power[F, Times[c, Plus[a, Times[b, x]]]], Cosh[Plus[d, Times[e, x]]], Power[Sinh[Plus[d, Times[e, x]]], Plus[n, -1]], Power[Plus[Times[Power[e, 2], Power[n, 2]], Times[-1, Power[b, 2], Power[c, 2], Power[Log[F], 2]]], -1]], x]]], And[FreeQ[List[F, a, b, c, d, e], x], NonzeroQ[Plus[Times[Power[e, 2], Power[n, 2]], Times[-1, Power[b, 2], Power[c, 2], Power[Log[F], 2]]]], RationalQ[n], Greater[n, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cosh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[b, c, Log[F], Power[F, Times[c, Plus[a, Times[b, x]]]], Power[Cosh[Plus[d, Times[e, x]]], n], Power[Plus[Times[Power[e, 2], Power[n, 2]], Times[-1, Power[b, 2], Power[c, 2], Power[Log[F], 2]]], -1]], x]], Plus[Dist[Times[n, Plus[n, -1], Power[e, 2], Power[Plus[Times[Power[e, 2], Power[n, 2]], Times[-1, Power[b, 2], Power[c, 2], Power[Log[F], 2]]], -1]], Int[Times[Power[F, Times[c, Plus[a, Times[b, x]]]], Power[Cosh[Plus[d, Times[e, x]]], Plus[n, -2]]], x], x], Simp[Times[e, n, Power[F, Times[c, Plus[a, Times[b, x]]]], Sinh[Plus[d, Times[e, x]]], Power[Cosh[Plus[d, Times[e, x]]], Plus[n, -1]], Power[Plus[Times[Power[e, 2], Power[n, 2]], Times[-1, Power[b, 2], Power[c, 2], Power[Log[F], 2]]], -1]], x]]], And[FreeQ[List[F, a, b, c, d, e], x], NonzeroQ[Plus[Times[Power[e, 2], Power[n, 2]], Times[-1, Power[b, 2], Power[c, 2], Power[Log[F], 2]]]], RationalQ[n], Greater[n, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sinh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[b, c, Log[F], Power[F, Times[c, Plus[a, Times[b, x]]]], Power[Sinh[Plus[d, Times[e, x]]], Plus[n, 2]], Power[Times[Power[e, 2], Plus[n, 1], Plus[n, 2]], -1]], x]], Simp[Times[Power[F, Times[c, Plus[a, Times[b, x]]]], Cosh[Plus[d, Times[e, x]]], Power[Sinh[Plus[d, Times[e, x]]], Plus[n, 1]], Power[Times[e, Plus[n, 1]], -1]], x]], And[FreeQ[List[F, a, b, c, d, e, n], x], ZeroQ[Plus[Times[Power[e, 2], Power[Plus[n, 2], 2]], Times[-1, Power[b, 2], Power[c, 2], Power[Log[F], 2]]]], NonzeroQ[Plus[n, 1]], NonzeroQ[Plus[n, 2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cosh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, c, Log[F], Power[F, Times[c, Plus[a, Times[b, x]]]], Power[Cosh[Plus[d, Times[e, x]]], Plus[n, 2]], Power[Times[Power[e, 2], Plus[n, 1], Plus[n, 2]], -1]], x], Times[-1, Simp[Times[Power[F, Times[c, Plus[a, Times[b, x]]]], Sinh[Plus[d, Times[e, x]]], Power[Cosh[Plus[d, Times[e, x]]], Plus[n, 1]], Power[Times[e, Plus[n, 1]], -1]], x]]], And[FreeQ[List[F, a, b, c, d, e, n], x], ZeroQ[Plus[Times[Power[e, 2], Power[Plus[n, 2], 2]], Times[-1, Power[b, 2], Power[c, 2], Power[Log[F], 2]]]], NonzeroQ[Plus[n, 1]], NonzeroQ[Plus[n, 2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sinh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[b, c, Log[F], Power[F, Times[c, Plus[a, Times[b, x]]]], Power[Sinh[Plus[d, Times[e, x]]], Plus[n, 2]], Power[Times[Power[e, 2], Plus[n, 1], Plus[n, 2]], -1]], x]], Plus[Times[-1, Dist[Times[Plus[Times[Power[e, 2], Power[Plus[n, 2], 2]], Times[-1, Power[b, 2], Power[c, 2], Power[Log[F], 2]]], Power[Times[Power[e, 2], Plus[n, 1], Plus[n, 2]], -1]], Int[Times[Power[F, Times[c, Plus[a, Times[b, x]]]], Power[Sinh[Plus[d, Times[e, x]]], Plus[n, 2]]], x], x]], Simp[Times[Power[F, Times[c, Plus[a, Times[b, x]]]], Cosh[Plus[d, Times[e, x]]], Power[Sinh[Plus[d, Times[e, x]]], Plus[n, 1]], Power[Times[e, Plus[n, 1]], -1]], x]]], And[FreeQ[List[F, a, b, c, d, e], x], NonzeroQ[Plus[Times[Power[e, 2], Power[Plus[n, 2], 2]], Times[-1, Power[b, 2], Power[c, 2], Power[Log[F], 2]]]], RationalQ[n], Less[n, -1], Unequal[n, -2]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cosh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[b, c, Log[F], Power[F, Times[c, Plus[a, Times[b, x]]]], Power[Cosh[Plus[d, Times[e, x]]], Plus[n, 2]], Power[Times[Power[e, 2], Plus[n, 1], Plus[n, 2]], -1]], x], Plus[Dist[Times[Plus[Times[Power[e, 2], Power[Plus[n, 2], 2]], Times[-1, Power[b, 2], Power[c, 2], Power[Log[F], 2]]], Power[Times[Power[e, 2], Plus[n, 1], Plus[n, 2]], -1]], Int[Times[Power[F, Times[c, Plus[a, Times[b, x]]]], Power[Cosh[Plus[d, Times[e, x]]], Plus[n, 2]]], x], x], Times[-1, Simp[Times[Power[F, Times[c, Plus[a, Times[b, x]]]], Sinh[Plus[d, Times[e, x]]], Power[Cosh[Plus[d, Times[e, x]]], Plus[n, 1]], Power[Times[e, Plus[n, 1]], -1]], x]]]], And[FreeQ[List[F, a, b, c, d, e], x], NonzeroQ[Plus[Times[Power[e, 2], Power[Plus[n, 2], 2]], Times[-1, Power[b, 2], Power[c, 2], Power[Log[F], 2]]]], RationalQ[n], Less[n, -1], Unequal[n, -2]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sinh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[E, Times[n, Plus[d, Times[e, x]]]], Power[Sinh[Plus[d, Times[e, x]]], n], Power[Power[Plus[-1, Power[E, Times[2, Plus[d, Times[e, x]]]]], n], -1]], Int[Times[Power[F, Times[c, Plus[a, Times[b, x]]]], Power[Plus[-1, Power[E, Times[2, Plus[d, Times[e, x]]]]], n], Power[Power[E, Times[n, Plus[d, Times[e, x]]]], -1]], x], x], And[FreeQ[List[F, a, b, c, d, e, n], x], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cosh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Pattern[n, Blank[]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[E, Times[n, Plus[d, Times[e, x]]]], Power[Cosh[Plus[d, Times[e, x]]], n], Power[Power[Plus[1, Power[E, Times[2, Plus[d, Times[e, x]]]]], n], -1]], Int[Times[Power[F, Times[c, Plus[a, Times[b, x]]]], Power[Plus[1, Power[E, Times[2, Plus[d, Times[e, x]]]]], n], Power[Power[E, Times[n, Plus[d, Times[e, x]]]], -1]], x], x], And[FreeQ[List[F, a, b, c, d, e, n], x], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Tanh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[F, 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Condition[With[List[Set[u, IntHide[Times[Power[F, Times[c, Plus[a, Times[b, x]]]], Power[Sinh[Plus[d, Times[e, x]]], n]], x]]], Plus[Simp[Times[Power[x, m], u], x], Times[-1, Dist[m, Int[Times[Power[x, Plus[m, -1]], u], x], x]]]], And[FreeQ[List[F, a, b, c, d, e], x], RationalQ[m], Greater[m, 0], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cosh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, IntHide[Times[Power[F, Times[c, Plus[a, Times[b, x]]]], Power[Cosh[Plus[d, Times[e, x]]], n]], x]]], Plus[Simp[Times[Power[x, m], u], x], Times[-1, Dist[m, Int[Times[Power[x, Plus[m, -1]], u], x], x]]]], And[FreeQ[List[F, a, b, c, d, e], x], RationalQ[m], Greater[m, 0], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cosh[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Sinh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Power[F, Times[c, Plus[a, Times[b, x]]]], Times[Power[Sinh[Plus[d, Times[e, x]]], m], Power[Cosh[Plus[f, Times[g, x]]], n]], x], x], And[FreeQ[List[F, a, b, c, d, e, f, g], x], PositiveIntegerQ[m, n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cosh[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Times[Power[x, p], Power[F, Times[c, Plus[a, Times[b, x]]]]], Times[Power[Sinh[Plus[d, Times[e, x]]], m], Power[Cosh[Plus[f, Times[g, x]]], n]], x], x], And[FreeQ[List[F, a, b, c, d, e, f, g], x], PositiveIntegerQ[m, n, p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[F, Blank[]], Times[Optional[Pattern[c, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Power[Pattern[G, Blank[]][Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Pattern[H, Blank[]][Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigToExp[Power[F, Times[c, Plus[a, Times[b, x]]]], Times[Power[G[Plus[d, Times[e, x]]], m], Power[H[Plus[d, Times[e, x]]], n]], x], x], And[FreeQ[List[F, a, b, c, d, e], x], PositiveIntegerQ[m, n], HyperbolicQ[G], HyperbolicQ[H]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[F, Blank[]], Pattern[u, Blank[]]], Power[Sinh[Pattern[v, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigToExp[Power[F, u], Power[Sinh[v], n], x], x], And[FreeQ[F, x], Or[LinearQ[u, x], PolyQ[u, x, 2]], Or[LinearQ[v, x], PolyQ[v, x, 2]], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cosh[Pattern[v, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigToExp[Power[F, u], Power[Cosh[v], n], x], x], And[FreeQ[F, x], Or[LinearQ[u, x], PolyQ[u, x, 2]], Or[LinearQ[v, x], PolyQ[v, x, 2]], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cosh[Pattern[v, Blank[]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[F, Blank[]], Pattern[u, Blank[]]], Power[Sinh[Pattern[v, Blank[]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigToExp[Power[F, u], Times[Power[Sinh[v], m], Power[Cosh[v], n]], x], x], And[FreeQ[F, x], Or[LinearQ[u, x], PolyQ[u, x, 2]], Or[LinearQ[v, x], PolyQ[v, x, 2]], PositiveIntegerQ[m, n]]]], RuleDelayed[HoldPattern[Int[Power[Sinh[Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Power[Plus[Times[Power[Times[c, Power[x, n]], b], Power[2, -1]], Times[-1, 1, Power[Times[2, Power[Times[c, 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Power[Sinh[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 2]], Power[Times[b, n, Plus[p, 1]], -1]], x]], And[FreeQ[List[a, b, c, n, p], x], ZeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[Plus[p, 2], 2]], -1]], NonzeroQ[Plus[p, 1]]]]], RuleDelayed[HoldPattern[Int[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[x, Plus[p, 2], Power[Cosh[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 2]], Power[Plus[p, 1], -1]], x], Times[-1, Simp[Times[x, Tanh[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Power[Cosh[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 2]], Power[Times[b, n, Plus[p, 1]], -1]], x]]], And[FreeQ[List[a, b, c, n, p], x], ZeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[Plus[p, 2], 2]], -1]], NonzeroQ[Plus[p, 1]]]]], RuleDelayed[HoldPattern[Int[Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Rational[1, 2]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[x, Sqrt[Sinh[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]]], Power[Sqrt[Plus[-1, Times[Power[E, Times[2, a]], Power[Times[c, Power[x, n]], Times[4, Power[n, -1]]]]]], -1]], Int[Times[Sqrt[Plus[-1, Times[Power[E, Times[2, a]], Power[Times[c, Power[x, n]], Times[4, Power[n, -1]]]]]], Power[x, -1]], x], x], And[FreeQ[List[a, b, c, n], x], ZeroQ[Plus[Times[b, n], -2]]]]], RuleDelayed[HoldPattern[Int[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Rational[1, 2]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[x, Sqrt[Cosh[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]]], Power[Sqrt[Plus[1, Times[Power[E, Times[2, a]], Power[Times[c, Power[x, n]], Times[4, Power[n, -1]]]]]], -1]], Int[Times[Sqrt[Plus[1, Times[Power[E, Times[2, a]], Power[Times[c, Power[x, n]], Times[4, Power[n, -1]]]]]], Power[x, -1]], x], x], And[FreeQ[List[a, b, c, n], x], ZeroQ[Plus[Times[b, n], -2]]]]], RuleDelayed[HoldPattern[Int[Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Power[Plus[Times[-1, Power[E, Times[-1, a, b, n, p]], Power[Times[2, b, n, p], -1], Power[Times[c, Power[x, n]], Times[-1, Power[Times[n, p], -1]]]], Times[Power[E, Times[a, b, n, p]], Power[Times[2, b, n, p], -1], Power[Times[c, Power[x, n]], Times[1, Power[Times[n, p], -1]]]]], p], x], x], And[FreeQ[List[a, b, c, n], x], PositiveIntegerQ[p], ZeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], -1]]]]], RuleDelayed[HoldPattern[Int[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Power[Plus[Times[Power[E, Times[-1, a, b, n, p]], Power[2, -1], Power[Times[c, Power[x, n]], Times[-1, Power[Times[n, p], -1]]]], Times[Power[E, Times[a, b, n, p]], Power[2, -1], Power[Times[c, Power[x, n]], Times[1, Power[Times[n, p], -1]]]]], p], x], x], And[FreeQ[List[a, b, c, n], x], PositiveIntegerQ[p], ZeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], -1]]]]], RuleDelayed[HoldPattern[Int[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[x, Sinh[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Power[Plus[Times[Power[b, 2], Power[n, 2]], -1], -1]], x]], Simp[Times[b, n, x, Cosh[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Power[Plus[Times[Power[b, 2], Power[n, 2]], -1], -1]], x]], And[FreeQ[List[a, b, c, n], x], NonzeroQ[Plus[Times[Power[b, 2], Power[n, 2]], -1]]]]], RuleDelayed[HoldPattern[Int[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[x, Cosh[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Power[Plus[Times[Power[b, 2], Power[n, 2]], -1], -1]], x]], Simp[Times[b, n, x, Sinh[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Power[Plus[Times[Power[b, 2], Power[n, 2]], -1], -1]], x]], And[FreeQ[List[a, b, c, n], x], NonzeroQ[Plus[Times[Power[b, 2], Power[n, 2]], -1]]]]], RuleDelayed[HoldPattern[Int[Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[x, Power[Sinh[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], p], Power[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], -1], -1]], x]], Plus[Times[-1, Dist[Times[Power[b, 2], Power[n, 2], p, Plus[p, -1], Power[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], -1], -1]], Int[Power[Sinh[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, -2]], x], x]], Simp[Times[b, n, p, x, Cosh[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Power[Sinh[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, -1]], Power[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], -1], -1]], x]]], And[FreeQ[List[a, b, c, n], x], RationalQ[p], Greater[p, 1], NonzeroQ[Plus[Times[Power[b, 2], Power[n, 2], 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Power[n, 2], Power[p, 2]], -1]]]]], RuleDelayed[HoldPattern[Int[Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[x, Coth[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Power[Sinh[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 2]], Power[Times[b, n, Plus[p, 1]], -1]], x], Plus[Times[-1, Dist[Times[Plus[Times[Power[b, 2], Power[n, 2], Power[Plus[p, 2], 2]], -1], Power[Times[Power[b, 2], Power[n, 2], Plus[p, 1], Plus[p, 2]], -1]], Int[Power[Sinh[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 2]], x], x]], Times[-1, Simp[Times[x, Power[Sinh[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 2]], Power[Times[Power[b, 2], Power[n, 2], Plus[p, 1], Plus[p, 2]], -1]], x]]]], And[FreeQ[List[a, b, c, n], x], RationalQ[p], Less[p, -1], Unequal[p, -2], NonzeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[Plus[p, 2], 2]], -1]]]]], RuleDelayed[HoldPattern[Int[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[x, Tanh[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Power[Cosh[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 2]], Power[Times[b, n, Plus[p, 1]], -1]], x]], Plus[Dist[Times[Plus[Times[Power[b, 2], Power[n, 2], Power[Plus[p, 2], 2]], -1], Power[Times[Power[b, 2], Power[n, 2], Plus[p, 1], Plus[p, 2]], -1]], Int[Power[Cosh[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 2]], x], x], Simp[Times[x, Power[Cosh[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 2]], Power[Times[Power[b, 2], Power[n, 2], Plus[p, 1], Plus[p, 2]], -1]], x]]], And[FreeQ[List[a, b, c, n], x], RationalQ[p], Less[p, -1], Unequal[p, -2], NonzeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[Plus[p, 2], 2]], -1]]]]], RuleDelayed[HoldPattern[Int[Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[x, Power[Plus[Times[-1, Power[E, Times[-1, a]], Power[Times[c, Power[x, n]], Times[-1, b]]], Times[Power[E, a], Power[Times[c, Power[x, n]], b]]], p], Power[Times[Plus[Times[b, n, p], 1], Power[Plus[2, Times[-1, 2, Power[E, Times[-2, a]], Power[Times[c, Power[x, n]], Times[-2, b]]]], p]], -1], Hypergeometric2F1[Times[-1, p], Times[-1, Plus[1, Times[b, n, p]], Power[Times[2, b, n], -1]], Plus[1, Times[-1, Plus[1, Times[b, n, p]], Power[Times[2, b, n], -1]]], Times[Power[E, Times[-2, a]], Power[Times[c, Power[x, n]], Times[-2, b]]]]], x], And[FreeQ[List[a, b, c, n, p], x], NonzeroQ[Plus[Times[Power[b, 2], 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RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[p, 2], Power[x, Plus[m, 1]], Power[Sinh[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 2]], Power[Times[Plus[m, 1], Plus[p, 1]], -1]], x]], Simp[Times[Power[x, Plus[m, 1]], Coth[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Power[Sinh[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 2]], Power[Times[b, n, Plus[p, 1]], -1]], x]], And[FreeQ[List[a, b, c, m, n, p], x], ZeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[Plus[p, 2], 2]], Times[-1, Power[Plus[m, 1], 2]]]], NonzeroQ[Plus[p, 1]], NonzeroQ[Plus[m, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], 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Times[b, Log[Times[c, Power[x, n]]]]]], p], Power[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], -1], -1]], x]]]], And[FreeQ[List[a, b, c, n], x], RationalQ[p], Less[p, -1], NonzeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], -1]]]]], RuleDelayed[HoldPattern[Int[Power[Csch[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[p, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[b, n, p, x, Cosh[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Power[Csch[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 1]], Power[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], -1], -1]], x]], Plus[Times[-1, Dist[Times[Power[b, 2], Power[n, 2], p, Plus[p, 1], Power[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], -1], -1]], Int[Power[Csch[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 2]], x], x]], Times[-1, Simp[Times[x, Power[Csch[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], p], Power[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], -1], -1]], x]]]], And[FreeQ[List[a, b, c, n], x], RationalQ[p], Less[p, -1], NonzeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], -1]]]]], RuleDelayed[HoldPattern[Int[Power[Sech[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[2, p], x, Power[Plus[Times[Power[E, Times[2, a]], Power[Times[c, Power[x, n]], Times[2, b]]], 1], p], Power[Plus[Times[b, n, p], 1], -1], Power[Times[Power[E, a], Power[Times[c, Power[x, n]], b], Power[Plus[Times[Power[E, Times[2, a]], Power[Times[c, Power[x, n]], Times[2, b]]], 1], -1]], p], Hypergeometric2F1[p, Times[Plus[Times[b, n, p], 1], Power[Times[2, b, n], -1]], Plus[1, Times[Plus[Times[b, n, p], 1], Power[Times[2, b, n], -1]]], Times[-1, Power[E, Times[2, a]], Power[Times[c, Power[x, n]], Times[2, b]]]]], x], And[FreeQ[List[a, b, c, n, p], x], NonzeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], -1]]]]], RuleDelayed[HoldPattern[Int[Power[Csch[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[x, Power[Plus[2, Times[-1, 2, Power[E, Times[2, a]], Power[Times[c, Power[x, n]], Times[2, b]]]], p], Power[Plus[Times[b, n, p], 1], -1], Power[Times[Power[E, a], Power[Times[c, Power[x, n]], b], Power[Plus[Times[Power[E, Times[2, a]], Power[Times[c, Power[x, n]], Times[2, b]]], -1], -1]], p], Hypergeometric2F1[p, Times[Plus[Times[b, n, p], 1], Power[Times[2, b, n], -1]], Plus[1, Times[Plus[Times[b, n, p], 1], Power[Times[2, b, n], -1]]], Times[Power[E, Times[2, a]], Power[Times[c, Power[x, n]], Times[2, b]]]]], x], And[FreeQ[List[a, b, c, n, p], x], NonzeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sech[Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[2, p], Int[Times[Power[x, m], Power[Times[Power[Times[c, Power[x, n]], b], Power[Plus[1, Power[Times[c, Power[x, n]], Times[2, b]]], -1]], p]], x], x], And[FreeQ[c, x], RationalQ[b, m, n, p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Csch[Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[2, p], Int[Times[Power[x, m], Power[Times[Power[Times[c, Power[x, n]], b], Power[Plus[-1, Power[Times[c, Power[x, n]], Times[2, b]]], -1]], p]], x], x], And[FreeQ[c, x], RationalQ[b, m, n, p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Sec[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[2, Power[E, Times[-1, a, b, n, Power[Plus[m, 1], -1]]]], Int[Times[Power[x, m], Power[Times[c, Power[x, n]], Times[Plus[m, 1], Power[n, -1]]], Power[Plus[Power[E, Times[-2, a, b, n, Power[Plus[m, 1], -1]]], Power[Times[c, Power[x, n]], Times[2, Plus[m, 1], Power[n, -1]]]], -1]], x], x], And[FreeQ[List[a, b, c, m, n], x], ZeroQ[Plus[Times[Power[b, 2], Power[n, 2]], Times[-1, Power[Plus[m, 1], 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Csc[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[-2, b, n, Power[Plus[m, 1], -1], Power[E, Times[-1, a, b, n, Power[Plus[m, 1], -1]]]], Int[Times[Power[x, m], Power[Times[c, Power[x, n]], Times[Plus[m, 1], Power[n, -1]]], Power[Plus[Power[E, Times[-2, a, b, n, Power[Plus[m, 1], -1]]], Times[-1, Power[Times[c, Power[x, n]], Times[2, Plus[m, 1], Power[n, -1]]]]], -1]], x], x], And[FreeQ[List[a, b, c, m, n], x], ZeroQ[Plus[Times[Power[b, 2], Power[n, 2]], Times[-1, Power[Plus[m, 1], 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sech[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[p, -2], Power[x, Plus[m, 1]], Power[Sech[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, -2]], Power[Times[Plus[m, 1], Plus[p, -1]], -1]], x], Simp[Times[Power[x, Plus[m, 1]], Tanh[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Power[Sech[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, -2]], Power[Times[b, n, Plus[p, -1]], -1]], x]], And[FreeQ[List[a, b, c, m, n, p], x], ZeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[Plus[p, -2], 2]], Power[Plus[m, 1], 2]]], NonzeroQ[Plus[m, 1]], NonzeroQ[Plus[p, -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Csch[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[p, -2], Power[x, Plus[m, 1]], Power[Csch[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, -2]], Power[Times[Plus[m, 1], Plus[p, -1]], -1]], x]], Times[-1, Simp[Times[Power[x, Plus[m, 1]], Coth[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Power[Csch[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, -2]], Power[Times[b, n, Plus[p, -1]], -1]], x]]], And[FreeQ[List[a, b, c, m, n, p], x], ZeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[Plus[p, -2], 2]], Power[Plus[m, 1], 2]]], NonzeroQ[Plus[m, 1]], NonzeroQ[Plus[p, -1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sech[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[m, 1]], Tanh[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Power[Sech[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, -2]], Power[Times[b, n, Plus[p, -1]], -1]], x], Plus[Dist[Times[Plus[Times[Power[b, 2], Power[n, 2], Power[Plus[p, -2], 2]], Times[-1, Power[Plus[m, 1], 2]]], Power[Times[Power[b, 2], Power[n, 2], Plus[p, -1], Plus[p, -2]], -1]], Int[Times[Power[x, m], Power[Sech[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, -2]]], x], x], Simp[Times[Plus[m, 1], Power[x, Plus[m, 1]], Power[Sech[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, -2]], Power[Times[Power[b, 2], Power[n, 2], Plus[p, -1], Plus[p, -2]], -1]], x]]], And[FreeQ[List[a, b, c, m, n], x], RationalQ[p], Greater[p, 1], Unequal[p, 2], NonzeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[Plus[p, -2], 2]], Times[-1, Power[Plus[m, 1], 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Csch[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[x, Plus[m, 1]], Coth[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Power[Csch[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, -2]], Power[Times[b, n, Plus[p, -1]], -1]], x]], Plus[Times[-1, Dist[Times[Plus[Times[Power[b, 2], Power[n, 2], Power[Plus[p, -2], 2]], Times[-1, Power[Plus[m, 1], 2]]], Power[Times[Power[b, 2], Power[n, 2], Plus[p, -1], Plus[p, -2]], -1]], Int[Times[Power[x, m], Power[Csch[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, -2]]], x], x]], Times[-1, Simp[Times[Plus[m, 1], Power[x, Plus[m, 1]], Power[Csch[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, -2]], Power[Times[Power[b, 2], Power[n, 2], Plus[p, -1], Plus[p, -2]], -1]], x]]]], And[FreeQ[List[a, b, c, m, n], x], RationalQ[p], Greater[p, 1], Unequal[p, 2], NonzeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[Plus[p, -2], 2]], Times[-1, Power[Plus[m, 1], 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sech[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[m, 1], Power[x, Plus[m, 1]], Power[Sech[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], p], Power[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], Times[-1, Power[Plus[m, 1], 2]]], -1]], x]], Plus[Dist[Times[Power[b, 2], Power[n, 2], p, Plus[p, 1], Power[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], Times[-1, Power[Plus[m, 1], 2]]], -1]], Int[Times[Power[x, m], Power[Sech[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 2]]], x], x], Times[-1, Simp[Times[b, n, p, Power[x, Plus[m, 1]], Power[Sech[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 1]], Sinh[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Power[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], Times[-1, Power[Plus[m, 1], 2]]], -1]], x]]]], And[FreeQ[List[a, b, c, m, n], x], RationalQ[p], Less[p, -1], NonzeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], Times[-1, Power[Plus[m, 1], 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Csch[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Pattern[p, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[m, 1], Power[x, Plus[m, 1]], Power[Csch[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], p], Power[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], Times[-1, Power[Plus[m, 1], 2]]], -1]], x]], Plus[Times[-1, Dist[Times[Power[b, 2], Power[n, 2], p, Plus[p, 1], Power[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], Times[-1, Power[Plus[m, 1], 2]]], -1]], Int[Times[Power[x, m], Power[Csch[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 2]]], x], x]], Times[-1, Simp[Times[b, n, p, Power[x, Plus[m, 1]], Cosh[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Power[Csch[Plus[a, Times[b, Log[Times[c, Power[x, n]]]]]], Plus[p, 1]], Power[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], Times[-1, Power[Plus[m, 1], 2]]], -1]], x]]]], And[FreeQ[List[a, b, c, m, n], x], RationalQ[p], Less[p, -1], NonzeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], Times[-1, Power[Plus[m, 1], 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sech[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[2, p], Power[x, Plus[m, 1]], Power[Plus[Times[Power[E, Times[2, a]], Power[Times[c, Power[x, n]], Times[2, b]]], 1], p], Power[Plus[m, Times[b, n, p], 1], -1], Power[Times[Power[E, a], Power[Times[c, Power[x, n]], b], Power[Plus[Times[Power[E, Times[2, a]], Power[Times[c, Power[x, n]], Times[2, b]]], 1], -1]], p], Hypergeometric2F1[p, Times[Plus[m, Times[b, n, p], 1], Power[Times[2, b, n], -1]], Plus[1, Times[Plus[m, Times[b, n, p], 1], Power[Times[2, b, n], -1]]], Times[-1, Power[E, Times[2, a]], Power[Times[c, Power[x, n]], Times[2, b]]]]], x], And[FreeQ[List[a, b, c, m, n, p], x], NonzeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], Times[-1, Power[Plus[m, 1], 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Csch[Plus[Optional[Pattern[a, Blank[]]], Times[Log[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Optional[Pattern[b, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[2, p], Power[x, Plus[m, 1]], Power[Plus[Times[-1, Power[E, Times[2, a]], Power[Times[c, Power[x, n]], Times[2, b]]], 1], p], Power[Plus[m, Times[b, n, p], 1], -1], Power[Times[Power[E, a], Power[Times[c, Power[x, n]], b], Power[Plus[Times[Power[E, Times[2, a]], Power[Times[c, Power[x, n]], Times[2, b]]], -1], -1]], p], Hypergeometric2F1[p, Times[Plus[m, Times[b, n, p], 1], Power[Times[2, b, n], -1]], Plus[1, Times[Plus[m, Times[b, n, p], 1], Power[Times[2, b, n], -1]]], Times[Power[E, Times[2, a]], Power[Times[c, Power[x, n]], Times[2, b]]]]], x], And[FreeQ[List[a, b, c, m, n, p], x], NonzeroQ[Plus[Times[Power[b, 2], Power[n, 2], Power[p, 2]], Times[-1, Power[Plus[m, 1], 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Sinh[Times[Power[Log[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Cosh[Times[a, x, Power[Log[Times[b, x]], p]]], Power[a, -1]], x], Times[-1, Dist[p, Int[Times[Sinh[Times[a, x, Power[Log[Times[b, x]], p]]], Power[Log[Times[b, x]], Plus[p, -1]]], x], x]]], And[FreeQ[List[a, b], x], RationalQ[p], Greater[p, 0]]]], RuleDelayed[HoldPattern[Int[Times[Cosh[Times[Power[Log[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Power[Log[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Sinh[Times[a, x, Power[Log[Times[b, x]], p]]], Power[a, -1]], x], Times[-1, Dist[p, Int[Times[Cosh[Times[a, x, Power[Log[Times[b, x]], p]]], Power[Log[Times[b, x]], Plus[p, -1]]], x], x]]], And[FreeQ[List[a, b], x], RationalQ[p], Greater[p, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Log[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Sinh[Times[Power[Log[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Optional[Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Cosh[Times[a, Power[x, n], Power[Log[Times[b, x]], p]]], Power[Times[a, n, Power[x, Plus[n, -1]]], -1]], x], Plus[Times[-1, Dist[Times[p, Power[n, -1]], Int[Times[Sinh[Times[a, Power[x, n], Power[Log[Times[b, x]], p]]], Power[Log[Times[b, x]], Plus[p, -1]]], x], x]], Dist[Times[Plus[n, -1], Power[Times[a, n], -1]], Int[Times[Cosh[Times[a, Power[x, n], Power[Log[Times[b, x]], p]]], Power[Power[x, n], -1]], x], x]]], And[FreeQ[List[a, b], x], RationalQ[n, p], Greater[p, 0]]]], RuleDelayed[HoldPattern[Int[Times[Cosh[Times[Power[Log[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], 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-1]], x], Plus[Times[-1, Dist[Times[p, Power[n, -1]], Int[Times[Power[x, m], Cosh[Times[a, Power[x, n], Power[Log[Times[b, x]], p]]], Power[Log[Times[b, x]], Plus[p, -1]]], x], x]], Times[-1, Dist[Times[Plus[m, Times[-1, n], 1], Power[Times[a, n], -1]], Int[Times[Power[x, Plus[m, Times[-1, n]]], Sinh[Times[a, Power[x, n], Power[Log[Times[b, x]], p]]]], x], x]]]], And[FreeQ[List[a, b], x], RationalQ[m, n, p], Greater[p, 0], NonzeroQ[Plus[m, Times[-1, n], 1]]]]], RuleDelayed[HoldPattern[Int[Power[Sinh[Times[Optional[Pattern[a, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Power[d, -1], Subst[Int[Times[Power[Sinh[Times[a, x]], n], Power[Power[x, 2], -1]], x], x, Times[1, Power[Plus[c, Times[d, x]], -1]]], x]], And[FreeQ[List[a, c, d], x], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Power[Cosh[Times[Optional[Pattern[a, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Power[d, -1], Subst[Int[Times[Power[Cosh[Times[a, x]], n], Power[Power[x, 2], -1]], x], x, Times[1, Power[Plus[c, Times[d, x]], -1]]], x]], And[FreeQ[List[a, c, d], x], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Power[Sinh[Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Power[d, -1], Subst[Int[Times[Power[Sinh[Plus[Times[b, e, Power[d, -1]], Times[-1, e, Plus[Times[b, c], Times[-1, a, d]], x, Power[d, -1]]]], n], Power[Power[x, 2], -1]], x], x, Times[1, Power[Plus[c, Times[d, x]], -1]]], x]], And[FreeQ[List[a, b, c, d], x], PositiveIntegerQ[n], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]]]]], RuleDelayed[HoldPattern[Int[Power[Cosh[Times[Optional[Pattern[e, Blank[]]], Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], -1]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Power[d, -1], Subst[Int[Times[Power[Cosh[Plus[Times[b, e, Power[d, -1]], Times[-1, e, Plus[Times[b, c], Times[-1, a, d]], x, Power[d, -1]]]], n], Power[Power[x, 2], -1]], x], x, Times[1, Power[Plus[c, Times[d, x]], -1]]], x]], And[FreeQ[List[a, b, c, d], x], PositiveIntegerQ[n], NonzeroQ[Plus[Times[b, c], Times[-1, a, d]]]]]], RuleDelayed[HoldPattern[Int[Power[Sinh[Pattern[u, Blank[]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[lst, QuotientOfLinearsParts[u, x]]], Int[Power[Sinh[Times[Plus[Part[lst, 1], Times[Part[lst, 2], x]], Power[Plus[Part[lst, 3], Times[Part[lst, 4], x]], -1]]], n], x]], And[PositiveIntegerQ[n], QuotientOfLinearsQ[u, x]]]], RuleDelayed[HoldPattern[Int[Power[Cosh[Pattern[u, Blank[]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[lst, QuotientOfLinearsParts[u, x]]], Int[Power[Cosh[Times[Plus[Part[lst, 1], Times[Part[lst, 2], x]], Power[Plus[Part[lst, 3], Times[Part[lst, 4], x]], -1]]], n], x]], And[PositiveIntegerQ[n], QuotientOfLinearsQ[u, x]]]], RuleDelayed[HoldPattern[Int[Times[Optional[Pattern[u, Blank[]]], Power[Sinh[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Sinh[Pattern[w, Blank[]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[u, Power[Sinh[v], Plus[p, q]]], x], ZeroQ[Plus[v, Times[-1, w]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cosh[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Cosh[Pattern[w, Blank[]]], Optional[Pattern[q, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[u, Power[Cosh[v], Plus[p, q]]], x], ZeroQ[Plus[v, Times[-1, w]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Sinh[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Sinh[Pattern[w, Blank[]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Times[Power[Sinh[v], p], Power[Sinh[w], q]], x], x], And[PositiveIntegerQ[p, q], Or[And[PolynomialQ[v, x], PolynomialQ[w, x]], And[BinomialQ[List[v, w], x], IndependentQ[Cancel[Times[v, Power[w, -1]]], x]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cosh[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Cosh[Pattern[w, Blank[]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Times[Power[Cosh[v], p], Power[Cosh[w], q]], x], x], And[PositiveIntegerQ[p, q], Or[And[PolynomialQ[v, x], PolynomialQ[w, x]], And[BinomialQ[List[v, w], x], IndependentQ[Cancel[Times[v, Power[w, -1]]], x]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Sinh[Pattern[w, Blank[]]], Optional[Pattern[q, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Power[x, m], Times[Power[Sinh[v], p], Power[Sinh[w], q]], x], x], And[PositiveIntegerQ[m, p, q], Or[And[PolynomialQ[v, x], PolynomialQ[w, x]], And[BinomialQ[List[v, w], x], IndependentQ[Cancel[Times[v, Power[w, -1]]], x]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cosh[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]], Power[Cosh[Pattern[w, Blank[]]], Optional[Pattern[q, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Power[x, m], Times[Power[Cosh[v], p], Power[Cosh[w], q]], x], x], And[PositiveIntegerQ[m, p, q], Or[And[PolynomialQ[v, x], PolynomialQ[w, x]], And[BinomialQ[List[v, w], x], IndependentQ[Cancel[Times[v, Power[w, -1]]], x]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cosh[Pattern[w, Blank[]]], Optional[Pattern[p, Blank[]]]], Optional[Pattern[u, Blank[]]], Power[Sinh[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Power[2, p], -1]], Int[Times[u, Power[Sinh[Times[2, v]], p]], x], x], And[ZeroQ[Plus[v, Times[-1, w]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cosh[Pattern[w, Blank[]]], Optional[Pattern[q, Blank[]]]], Power[Sinh[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Times[Power[Sinh[v], p], Power[Cosh[w], q]], x], x], And[PositiveIntegerQ[p, q], Or[And[PolynomialQ[v, x], PolynomialQ[w, x]], And[BinomialQ[List[v, w], x], IndependentQ[Cancel[Times[v, Power[w, -1]]], x]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cosh[Pattern[w, Blank[]]], Optional[Pattern[q, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Pattern[v, Blank[]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandTrigReduce[Power[x, m], Times[Power[Sinh[v], p], Power[Cosh[w], q]], x], x], And[PositiveIntegerQ[m, p, q], Or[And[PolynomialQ[v, x], PolynomialQ[w, x]], And[BinomialQ[List[v, w], x], IndependentQ[Cancel[Times[v, Power[w, -1]]], x]]]]]], RuleDelayed[HoldPattern[Int[Times[Sinh[Pattern[v, Blank[]]], Power[Tanh[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Int[Times[Cosh[v], Power[Tanh[w], Plus[n, -1]]], x], Times[-1, Dist[Cosh[Plus[v, Times[-1, w]]], Int[Times[Sech[w], Power[Tanh[w], Plus[n, -1]]], x], x]]], And[RationalQ[n], Greater[n, 0], FreeQ[Plus[v, Times[-1, w]], x], NonzeroQ[Plus[v, Times[-1, w]]]]]], RuleDelayed[HoldPattern[Int[Times[Cosh[Pattern[v, Blank[]]], Power[Coth[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Int[Times[Sinh[v], Power[Coth[w], Plus[n, -1]]], x], Dist[Cosh[Plus[v, Times[-1, w]]], Int[Times[Csch[w], Power[Coth[w], Plus[n, -1]]], x], x]], And[RationalQ[n], Greater[n, 0], FreeQ[Plus[v, Times[-1, w]], x], NonzeroQ[Plus[v, Times[-1, w]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Coth[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]], Sinh[Pattern[v, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Int[Times[Cosh[v], Power[Coth[w], Plus[n, -1]]], x], Dist[Sinh[Plus[v, Times[-1, w]]], Int[Times[Csch[w], Power[Coth[w], Plus[n, -1]]], x], x]], And[RationalQ[n], Greater[n, 0], FreeQ[Plus[v, Times[-1, w]], x], NonzeroQ[Plus[v, Times[-1, w]]]]]], RuleDelayed[HoldPattern[Int[Times[Cosh[Pattern[v, Blank[]]], Power[Tanh[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Int[Times[Sinh[v], Power[Tanh[w], Plus[n, -1]]], x], Times[-1, Dist[Sinh[Plus[v, Times[-1, w]]], Int[Times[Sech[w], Power[Tanh[w], Plus[n, -1]]], x], x]]], And[RationalQ[n], Greater[n, 0], FreeQ[Plus[v, Times[-1, w]], x], NonzeroQ[Plus[v, Times[-1, w]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Sech[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]], Sinh[Pattern[v, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Cosh[Plus[v, Times[-1, w]]], Int[Times[Tanh[w], Power[Sech[w], Plus[n, -1]]], x], x], Dist[Sinh[Plus[v, Times[-1, w]]], Int[Power[Sech[w], Plus[n, -1]], x], x]], And[RationalQ[n], Greater[n, 0], FreeQ[Plus[v, Times[-1, w]], x], NonzeroQ[Plus[v, Times[-1, w]]]]]], RuleDelayed[HoldPattern[Int[Times[Cosh[Pattern[v, Blank[]]], Power[Csch[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Cosh[Plus[v, Times[-1, w]]], Int[Times[Coth[w], Power[Csch[w], Plus[n, -1]]], x], x], Dist[Sinh[Plus[v, Times[-1, w]]], Int[Power[Csch[w], Plus[n, -1]], x], x]], And[RationalQ[n], Greater[n, 0], FreeQ[Plus[v, Times[-1, w]], x], NonzeroQ[Plus[v, Times[-1, w]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Csch[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]], Sinh[Pattern[v, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Sinh[Plus[v, Times[-1, w]]], Int[Times[Coth[w], Power[Csch[w], Plus[n, -1]]], x], x], Dist[Cosh[Plus[v, Times[-1, w]]], Int[Power[Csch[w], Plus[n, -1]], x], x]], And[RationalQ[n], Greater[n, 0], FreeQ[Plus[v, Times[-1, w]], x], NonzeroQ[Plus[v, Times[-1, w]]]]]], RuleDelayed[HoldPattern[Int[Times[Cosh[Pattern[v, Blank[]]], Power[Sech[Pattern[w, Blank[]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Sinh[Plus[v, Times[-1, w]]], Int[Times[Tanh[w], Power[Sech[w], Plus[n, -1]]], x], x], Dist[Cosh[Plus[v, Times[-1, w]]], Int[Power[Sech[w], Plus[n, -1]], x], x]], And[RationalQ[n], Greater[n, 0], FreeQ[Plus[v, Times[-1, w]], x], NonzeroQ[Plus[v, Times[-1, w]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]], Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[e, Times[f, x]], m], Power[Plus[a, Times[b, Sinh[Plus[Times[2, c], Times[2, d, x]]], Power[2, -1]]], n]], x], FreeQ[List[a, b, c, d, e, f, m, n], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Sinh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], 2]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Power[2, n], -1]], Int[Times[Power[x, m], Power[Plus[Times[2, a], Times[-1, b], Times[b, Cosh[Plus[Times[2, c], Times[2, d, x]]]]], n]], x], x], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[a, Times[-1, b]]], IntegersQ[m, n], Greater[m, 0], Less[n, 0], Or[Equal[n, -1], And[Equal[m, 1], Equal[n, -2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[Power[Cosh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Pattern[n, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Power[2, n], -1]], Int[Times[Power[x, m], Power[Plus[Times[2, a], b, Times[b, Cosh[Plus[Times[2, c], Times[2, d, x]]]]], n]], x], x], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[a, b]], IntegersQ[m, n], Greater[m, 0], Less[n, 0], Or[Equal[n, -1], And[Equal[m, 1], Equal[n, -2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sinh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Power[d, Plus[m, 1]], -1]], Subst[Int[Times[Power[Plus[Times[d, e], Times[-1, c, f], Times[f, x]], m], Power[Sinh[Plus[a, Times[b, Power[x, n]]]], p]], x], x, Plus[c, Times[d, x]]], x], And[FreeQ[List[a, b, c, d, e, f, n], x], PositiveIntegerQ[m], RationalQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Cosh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Power[d, Plus[m, 1]], -1]], Subst[Int[Times[Power[Plus[Times[d, e], Times[-1, c, f], Times[f, x]], m], Power[Cosh[Plus[a, Times[b, Power[x, n]]]], p]], x], x, Plus[c, Times[d, x]]], x], And[FreeQ[List[a, b, c, d, e, f, n], x], PositiveIntegerQ[m], RationalQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Power[Cosh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Optional[Pattern[b, Blank[]]]], Times[Optional[Pattern[c, Blank[]]], Power[Sinh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[2, Int[Times[Power[Plus[f, Times[g, x]], m], Power[Plus[Times[2, a], b, Times[-1, c], Times[Plus[b, c], Cosh[Plus[Times[2, d], Times[2, e, x]]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, g], x], PositiveIntegerQ[m], NonzeroQ[Plus[a, b]], NonzeroQ[Plus[a, c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Sech[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2], Power[Plus[Pattern[b, Blank[]], Times[Optional[Pattern[c, Blank[]]], Power[Tanh[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[2, Int[Times[Power[Plus[f, Times[g, x]], m], Power[Plus[b, Times[-1, c], Times[Plus[b, c], Cosh[Plus[Times[2, d], Times[2, e, x]]]]], -1]], x], x], And[FreeQ[List[b, c, d, e, f, g], x], PositiveIntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[f, Blank[]]], 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Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[x, Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Plus[Times[2, p], 1], -1]], x], Plus[Times[-1, Dist[Times[b, c, n, Power[Times[-1, d], p], Power[Plus[Times[2, p], 1], -1]], Int[Times[x, Power[Plus[-1, Times[c, x]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[1, Times[c, x]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, -1]]], x], x]], Dist[Times[2, d, p, Power[Plus[Times[2, p], 1], -1]], Int[Times[Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, -1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[n, p], Greater[n, 0], Greater[p, 0], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, 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RationalQ[n], Greater[n, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[x, Sqrt[Plus[d1, Times[e1, x]]], Sqrt[Plus[d2, Times[e2, x]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[2, -1]], x], Plus[Times[-1, Dist[Times[Sqrt[Plus[d1, Times[e1, x]]], Sqrt[Plus[d2, Times[e2, x]]], Power[Times[2, Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]]], -1]], Int[Times[Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]]], -1]], x], x]], Times[-1, Dist[Times[b, c, n, Sqrt[Plus[d1, Times[e1, x]]], Sqrt[Plus[d2, Times[e2, x]]], Power[Times[2, Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]]], -1]], Int[Times[x, Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, -1]]], x], x]]]], And[FreeQ[List[a, b, c, d1, e1, d2, e2], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], RationalQ[n], Greater[n, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[x, Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n], Power[Plus[Times[2, p], 1], -1]], x], Plus[Dist[Times[2, d, p, Power[Plus[Times[2, p], 1], -1]], Int[Times[Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, -1]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n]], x], x], Times[-1, Dist[Times[b, c, n, Power[d, IntPart[p]], Power[Plus[d, Times[e, Power[x, 2]]], FracPart[p]], Power[Times[Plus[Times[2, p], 1], Power[Plus[1, Times[Power[c, 2], Power[x, 2]]], FracPart[p]]], -1]], Int[Times[x, Power[Plus[1, Times[Power[c, 2], Power[x, 2]]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], Plus[n, -1]]], x], x]]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[n, p], Greater[n, 0], Greater[p, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[x, Power[Plus[d1, Times[e1, x]], p], Power[Plus[d2, Times[e2, x]], p], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Plus[Times[2, p], 1], -1]], x], Plus[Dist[Times[2, d1, d2, p, Power[Plus[Times[2, p], 1], -1]], Int[Times[Power[Plus[d1, Times[e1, x]], Plus[p, -1]], Power[Plus[d2, Times[e2, x]], Plus[p, -1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], x], Times[-1, Dist[Times[b, c, n, Power[Times[-1, d1, d2], Plus[p, Times[-1, 1, Power[2, -1]]]], Sqrt[Plus[d1, Times[e1, x]]], Sqrt[Plus[d2, Times[e2, x]]], Power[Times[Plus[Times[2, p], 1], Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]]], -1]], Int[Times[x, Power[Plus[-1, Times[Power[c, 2], Power[x, 2]]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, -1]]], x], x]]]], And[FreeQ[List[a, b, c, d1, e1, d2, e2], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], RationalQ[n, p], Greater[n, 0], Greater[p, 0], IntegerQ[Plus[p, Times[-1, 1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[x, Power[Plus[d1, Times[e1, x]], p], Power[Plus[d2, Times[e2, x]], p], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Plus[Times[2, p], 1], -1]], x], Plus[Dist[Times[2, d1, d2, p, Power[Plus[Times[2, p], 1], -1]], Int[Times[Power[Plus[d1, Times[e1, x]], Plus[p, -1]], Power[Plus[d2, Times[e2, x]], Plus[p, -1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], x], Times[-1, Dist[Times[b, c, n, Power[Times[-1, d1, d2], IntPart[p]], Power[Plus[d1, Times[e1, x]], FracPart[p]], Power[Plus[d2, Times[e2, x]], FracPart[p]], Power[Times[Plus[Times[2, p], 1], Power[Plus[1, Times[c, x]], FracPart[p]], Power[Plus[-1, Times[c, x]], FracPart[p]]], -1]], Int[Times[x, Power[Plus[-1, Times[c, x]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[1, Times[c, x]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, -1]]], x], x]]]], And[FreeQ[List[a, b, c, d1, e1, d2, e2], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], RationalQ[n, p], Greater[n, 0], Greater[p, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[x, Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n], Power[Times[d, Sqrt[Plus[d, Times[e, Power[x, 2]]]]], -1]], x], Times[-1, Dist[Times[b, c, n, Sqrt[Plus[1, Times[Power[c, 2], Power[x, 2]]]], Power[Times[d, Sqrt[Plus[d, Times[e, Power[x, 2]]]]], -1]], Int[Times[x, Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], Plus[n, -1]], Power[Plus[1, Times[Power[c, 2], Power[x, 2]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[n], Greater[n, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Rational[-3, 2]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[x, Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[d1, d2, Sqrt[Plus[d1, Times[e1, x]]], Sqrt[Plus[d2, Times[e2, x]]]], -1]], x], Dist[Times[b, c, n, Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]], Power[Times[d1, d2, Sqrt[Plus[d1, Times[e1, x]]], Sqrt[Plus[d2, Times[e2, x]]]], -1]], Int[Times[x, Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, -1]], Power[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]], -1]], x], x]], And[FreeQ[List[a, b, c, d1, e1, d2, e2], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], RationalQ[n], Greater[n, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[x, Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[2, d, Plus[p, 1]], -1]], x]], Plus[Times[-1, Dist[Times[b, c, n, Power[Times[-1, d], p], Power[Times[2, Plus[p, 1]], -1]], Int[Times[x, Power[Plus[1, Times[c, x]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[-1, Times[c, x]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, -1]]], x], x]], Dist[Times[Plus[Times[2, p], 3], Power[Times[2, d, Plus[p, 1]], -1]], Int[Times[Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[n, p], Greater[n, 0], Less[p, -1], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[x, Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n], Power[Times[2, d, Plus[p, 1]], -1]], x]], Plus[Dist[Times[Plus[Times[2, p], 3], Power[Times[2, d, Plus[p, 1]], -1]], Int[Times[Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n]], x], x], Dist[Times[b, c, n, Power[d, IntPart[p]], Power[Plus[d, Times[e, Power[x, 2]]], FracPart[p]], Power[Times[2, Plus[p, 1], Power[Plus[1, Times[Power[c, 2], Power[x, 2]]], FracPart[p]]], -1]], Int[Times[x, Power[Plus[1, Times[Power[c, 2], Power[x, 2]]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], Plus[n, -1]]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[n, p], Greater[n, 0], Less[p, -1], Unequal[p, Times[-3, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[x, Power[Plus[d1, Times[e1, x]], Plus[p, 1]], Power[Plus[d2, Times[e2, x]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[2, d1, d2, Plus[p, 1]], -1]], x]], Plus[Dist[Times[Plus[Times[2, p], 3], Power[Times[2, d1, d2, Plus[p, 1]], -1]], Int[Times[Power[Plus[d1, Times[e1, x]], Plus[p, 1]], Power[Plus[d2, Times[e2, x]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], x], Times[-1, Dist[Times[b, c, n, Power[Times[-1, d1, d2], Plus[p, Times[1, Power[2, -1]]]], Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]], Power[Times[2, Plus[p, 1], Sqrt[Plus[d1, Times[e1, x]]], Sqrt[Plus[d2, Times[e2, x]]]], -1]], Int[Times[x, Power[Plus[-1, Times[Power[c, 2], Power[x, 2]]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, -1]]], x], x]]]], And[FreeQ[List[a, b, c, d1, e1, d2, e2], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], RationalQ[n, p], Greater[n, 0], Less[p, -1], Unequal[p, Times[-3, Power[2, -1]]], IntegerQ[Plus[p, Times[1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[x, Power[Plus[d1, Times[e1, x]], Plus[p, 1]], Power[Plus[d2, Times[e2, x]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[2, d1, d2, Plus[p, 1]], -1]], x]], Plus[Dist[Times[Plus[Times[2, p], 3], Power[Times[2, d1, d2, Plus[p, 1]], -1]], Int[Times[Power[Plus[d1, Times[e1, x]], Plus[p, 1]], Power[Plus[d2, Times[e2, x]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], x], Times[-1, Dist[Times[b, c, n, Power[Times[-1, d1, d2], IntPart[p]], Power[Plus[d1, Times[e1, x]], FracPart[p]], Power[Plus[d2, Times[e2, x]], FracPart[p]], Power[Times[2, Plus[p, 1], Power[Plus[1, Times[c, x]], FracPart[p]], Power[Plus[-1, Times[c, x]], FracPart[p]]], -1]], Int[Times[x, Power[Plus[1, Times[c, x]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[-1, Times[c, x]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, -1]]], x], x]]]], And[FreeQ[List[a, b, c, d1, e1, d2, e2], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], RationalQ[n, p], Greater[n, 0], Less[p, -1], Unequal[p, Times[-3, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Times[c, d], -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], n], Sech[x]], x], x, ArcSinh[Times[c, x]]], x], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Power[Times[c, d], -1], Subst[Int[Times[Power[Plus[a, Times[b, x]], n], Csch[x]], x], x, ArcCosh[Times[c, x]]], x]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[-1, d], p], Power[Plus[-1, Times[c, x]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[1, Times[c, x]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Plus[n, 1]], -1]], x], Times[-1, Dist[Times[c, Power[Times[-1, d], p], Plus[Times[2, p], 1], Power[Times[b, Plus[n, 1]], -1]], Int[Times[x, Power[Plus[-1, Times[c, x]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[1, Times[c, x]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, 1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, p], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[n], Less[n, -1], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Sqrt[Plus[1, Times[Power[c, 2], Power[x, 2]]]], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Plus[n, 1]], -1]], x], Times[-1, Dist[Times[c, Plus[Times[2, p], 1], Power[d, IntPart[p]], Power[Plus[d, Times[e, Power[x, 2]]], FracPart[p]], Power[Times[b, Plus[n, 1], Power[Plus[1, Times[Power[c, 2], Power[x, 2]]], FracPart[p]]], -1]], Int[Times[x, Power[Plus[1, Times[Power[c, 2], Power[x, 2]]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], Plus[n, 1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, p], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[n], Less[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]], Power[Plus[d1, Times[e1, x]], p], Power[Plus[d2, Times[e2, x]], p], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Plus[n, 1]], -1]], x], Times[-1, Dist[Times[c, Plus[Times[2, p], 1], Power[Times[-1, d1, d2], Plus[p, Times[-1, 1, Power[2, -1]]]], Sqrt[Plus[d1, Times[e1, x]]], Sqrt[Plus[d2, Times[e2, x]]], Power[Times[b, Plus[n, 1], Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]]], -1]], Int[Times[x, Power[Plus[-1, Times[Power[c, 2], Power[x, 2]]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, 1]]], x], x]]], And[FreeQ[List[a, b, c, d1, e1, d2, e2, p], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], RationalQ[n], Less[n, -1], IntegerQ[Plus[p, Times[-1, 1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]], Power[Plus[d1, Times[e1, x]], p], Power[Plus[d2, Times[e2, x]], p], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Plus[n, 1]], -1]], x], Times[-1, Dist[Times[c, Plus[Times[2, p], 1], Power[Times[-1, d1, d2], IntPart[p]], Power[Plus[d1, Times[e1, x]], FracPart[p]], Power[Plus[d2, Times[e2, x]], FracPart[p]], Power[Times[b, Plus[n, 1], Power[Plus[1, Times[c, x]], FracPart[p]], Power[Plus[-1, Times[c, x]], FracPart[p]]], -1]], Int[Times[x, Power[Plus[1, Times[c, x]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[-1, Times[c, x]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, 1]]], x], x]]], And[FreeQ[List[a, b, c, d1, e1, d2, e2, p], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], RationalQ[n], Less[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[d, p], Power[c, -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], n], Power[Cosh[x], Plus[Times[2, p], 1]]], x], x, ArcSinh[Times[c, x]]], x], And[FreeQ[List[a, b, c, d, e, n], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], PositiveIntegerQ[Times[2, p]], Or[IntegerQ[p], PositiveQ[d]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[-1, d], p], Power[c, -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], n], Power[Sinh[x], Plus[Times[2, p], 1]]], x], x, ArcCosh[Times[c, x]]], x], And[FreeQ[List[a, b, c, d, e, n], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], PositiveIntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[-1, d1, d2], p], Power[c, -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], n], Power[Sinh[x], Plus[Times[2, p], 1]]], x], x, ArcCosh[Times[c, x]]], x], And[FreeQ[List[a, b, c, d1, e1, d2, e2, n], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], PositiveIntegerQ[Plus[p, Times[1, Power[2, -1]]]], And[PositiveQ[d1], NegativeQ[d2]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[d, Plus[p, Times[-1, 1, Power[2, -1]]]], Sqrt[Plus[d, Times[e, Power[x, 2]]]], Power[Sqrt[Plus[1, Times[Power[c, 2], Power[x, 2]]]], -1]], Int[Times[Power[Plus[1, Times[Power[c, 2], Power[x, 2]]], p], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, n], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], PositiveIntegerQ[Times[2, p]], Not[Or[IntegerQ[p], PositiveQ[d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[-1, d1, d2], Plus[p, Times[-1, 1, Power[2, -1]]]], Sqrt[Plus[d1, Times[e1, x]]], Sqrt[Plus[d2, Times[e2, x]]], Power[Times[Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]]], -1]], Int[Times[Power[Plus[1, Times[c, x]], p], Power[Plus[-1, Times[c, x]], p], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], x], And[FreeQ[List[a, b, c, d1, e1, d2, e2, n], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], PositiveIntegerQ[Times[2, p]], Not[And[PositiveQ[d1], NegativeQ[d2]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, IntHide[Power[Plus[d, Times[e, Power[x, 2]]], p], x]]], Plus[Dist[Plus[a, Times[b, ArcSinh[Times[c, x]]]], u, x], Times[-1, Dist[Times[b, c], Int[SimplifyIntegrand[Times[u, Power[Sqrt[Plus[1, Times[Power[c, 2], Power[x, 2]]]], -1]], x], x], x]]]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[e, Times[-1, Power[c, 2], d]]], Or[PositiveIntegerQ[p], NegativeIntegerQ[Plus[p, Times[1, Power[2, -1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, IntHide[Power[Plus[d, Times[e, Power[x, 2]]], p], x]]], Plus[Dist[Plus[a, Times[b, ArcCosh[Times[c, x]]]], u, x], Times[-1, Dist[Times[b, c], Int[SimplifyIntegrand[Times[u, Power[Times[Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]]], -1]], x], x], x]]]], And[FreeQ[List[a, b, c, d, e], x], NonzeroQ[Plus[Times[Power[c, 2], d], e]], Or[PositiveIntegerQ[p], NegativeIntegerQ[Plus[p, Times[1, Power[2, -1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n], Power[Plus[d, Times[e, Power[x, 2]]], p], x], x], And[FreeQ[List[a, b, c, d, e, n], x], NonzeroQ[Plus[e, Times[-1, Power[c, 2], d]]], IntegerQ[p], Or[Greater[p, 0], PositiveIntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Plus[d, Times[e, Power[x, 2]]], p], x], x], And[FreeQ[List[a, b, c, d, e, n], x], NonzeroQ[Plus[Times[Power[c, 2], d], e]], IntegerQ[p], Or[Greater[p, 0], PositiveIntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n]], x], FreeQ[List[a, b, c, d, e, n, p], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], And[FreeQ[List[a, b, c, d, e, n, p], x], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[Plus[d1, Times[e1, x]], p], Power[Plus[d2, Times[e2, x]], p], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], FreeQ[List[a, b, c, d1, e1, d2, e2, n, p], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[d, Times[e, x]], FracPart[p]], Power[Plus[f, Times[g, x]], FracPart[p]], Power[Power[Plus[Times[d, f], Times[e, g, Power[x, 2]]], FracPart[p]], -1]], Int[Times[Power[Plus[Times[d, f], Times[e, g, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, n, p], x], ZeroQ[Plus[Times[e, f], Times[d, g]]], ZeroQ[Plus[Times[Power[c, 2], Power[f, 2]], Power[g, 2]]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[-1, d], IntPart[p]], Power[Plus[d, Times[e, Power[x, 2]]], FracPart[p]], Power[Times[Power[Plus[1, Times[c, x]], FracPart[p]], Power[Plus[-1, Times[c, x]], FracPart[p]]], -1]], Int[Times[Power[Plus[1, Times[c, x]], p], Power[Plus[-1, Times[c, x]], p], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, n, p], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[e, -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], n], Tanh[x]], x], x, ArcSinh[Times[c, x]]], x], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[e, -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], n], Coth[x]], x], x, ArcCosh[Times[c, x]]], x], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[2, e, Plus[p, 1]], -1]], x], Times[-1, Dist[Times[b, n, Power[Times[-1, d], p], Power[Times[2, c, Plus[p, 1]], -1]], Int[Times[Power[Plus[1, Times[c, x]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[-1, Times[c, x]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, -1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, p], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[n], Greater[n, 0], NonzeroQ[Plus[p, 1]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n], Power[Times[2, e, Plus[p, 1]], -1]], x], Times[-1, Dist[Times[b, n, Power[d, IntPart[p]], Power[Plus[d, Times[e, Power[x, 2]]], FracPart[p]], Power[Times[2, c, Plus[p, 1], Power[Plus[1, Times[Power[c, 2], Power[x, 2]]], FracPart[p]]], -1]], Int[Times[Power[Plus[1, Times[Power[c, 2], Power[x, 2]]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], Plus[n, -1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, p], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[n], Greater[n, 0], NonzeroQ[Plus[p, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d1, Times[e1, x]], Plus[p, 1]], Power[Plus[d2, Times[e2, x]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[2, e1, e2, Plus[p, 1]], -1]], x], Times[-1, Dist[Times[b, n, Power[Times[-1, d1, d2], IntPart[p]], Power[Plus[d1, Times[e1, x]], FracPart[p]], Power[Plus[d2, Times[e2, x]], FracPart[p]], Power[Times[2, c, Plus[p, 1], Power[Plus[1, Times[c, x]], FracPart[p]], Power[Plus[-1, Times[c, x]], FracPart[p]]], -1]], Int[Times[Power[Plus[-1, Times[Power[c, 2], Power[x, 2]]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, -1]]], x], x]]], And[FreeQ[List[a, b, c, d1, e1, d2, e2, p], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], RationalQ[n], Greater[n, 0], NonzeroQ[Plus[p, 1]], IntegerQ[Plus[p, Times[1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[d1, Times[e1, x]], Plus[p, 1]], Power[Plus[d2, Times[e2, x]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[2, e1, e2, Plus[p, 1]], -1]], x], Times[-1, Dist[Times[b, n, Power[Times[-1, d1, d2], IntPart[p]], Power[Plus[d1, Times[e1, x]], FracPart[p]], Power[Plus[d2, Times[e2, x]], FracPart[p]], Power[Times[2, c, Plus[p, 1], Power[Plus[1, Times[c, x]], FracPart[p]], Power[Plus[-1, Times[c, x]], FracPart[p]]], 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IntHide[Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, 2]]], p]], x]]], Plus[Dist[Plus[a, Times[b, ArcSinh[Times[c, x]]]], u, x], Times[-1, Dist[Times[b, c], Int[SimplifyIntegrand[Times[u, Power[Sqrt[Plus[1, Times[Power[c, 2], Power[x, 2]]]], -1]], x], x], x]]]], And[FreeQ[List[a, b, c, d, e, f, m], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], PositiveIntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, IntHide[Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, 2]]], p]], x]]], Plus[Dist[Plus[a, Times[b, ArcCosh[Times[c, x]]]], u, x], Times[-1, 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Power[Times[-1, d1, d2], p]], Int[SimplifyIntegrand[Times[u, Power[Times[Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]]], -1]], x], x], x]]]], And[FreeQ[List[a, b, c, d1, e1, d2, e2], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], IntegerQ[Plus[p, Times[-1, 1, Power[2, -1]]]], Or[PositiveIntegerQ[Times[Plus[m, 1], Power[2, -1]]], NegativeIntegerQ[Times[Plus[m, Times[2, p], 3], Power[2, -1]]]], Unequal[p, Times[-1, Power[2, -1]]], PositiveQ[d1], NegativeQ[d2]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, IntHide[Times[Power[x, m], Power[Plus[1, Times[Power[c, 2], Power[x, 2]]], p]], x]]], 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2]]]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n], Power[Times[f, Plus[m, 1]], -1]], x], Plus[Times[-1, Dist[Times[b, c, n, Sqrt[Plus[d, Times[e, Power[x, 2]]]], Power[Times[f, Plus[m, 1], Sqrt[Plus[1, Times[Power[c, 2], Power[x, 2]]]]], -1]], Int[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], Plus[n, -1]]], x], x]], Times[-1, Dist[Times[Power[c, 2], Sqrt[Plus[d, Times[e, Power[x, 2]]]], Power[Times[Power[f, 2], Plus[m, 1], Sqrt[Plus[1, Times[Power[c, 2], Power[x, 2]]]]], -1]], Int[Times[Power[Times[f, x], Plus[m, 2]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n], Power[Sqrt[Plus[1, Times[Power[c, 2], Power[x, 2]]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[m, n], Greater[n, 0], Less[m, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Rational[1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[f, x], Plus[m, 1]], Sqrt[Plus[d1, Times[e1, x]]], Sqrt[Plus[d2, Times[e2, x]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[f, Plus[m, 1]], -1]], x], Plus[Times[-1, Dist[Times[b, c, n, Sqrt[Plus[d1, Times[e1, x]]], Sqrt[Plus[d2, Times[e2, x]]], Power[Times[f, Plus[m, 1], Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]]], -1]], Int[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, -1]]], x], x]], Times[-1, Dist[Times[Power[c, 2], Sqrt[Plus[d1, Times[e1, x]]], Sqrt[Plus[d2, Times[e2, x]]], Power[Times[Power[f, 2], Plus[m, 1], Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]]], -1]], Int[Times[Times[Power[Times[f, x], Plus[m, 2]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], Power[Times[Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d1, e1, d2, e2, f], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], RationalQ[m, n], Greater[n, 0], Less[m, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[d, Times[e, 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Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[d1, Times[e1, x]], p], Power[Plus[d2, Times[e2, x]], p], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[f, Plus[m, 1]], -1]], x], Plus[Times[-1, Dist[Times[2, e1, e2, p, Power[Times[Power[f, 2], Plus[m, 1]], -1]], Int[Times[Power[Times[f, x], Plus[m, 2]], Power[Plus[d1, Times[e1, x]], Plus[p, -1]], Power[Plus[d2, Times[e2, x]], Plus[p, -1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], x]], Times[-1, 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Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[f, Plus[m, Times[2, p], 1]], -1]], x], Plus[Dist[Times[2, d, p, Power[Plus[m, Times[2, p], 1], -1]], Int[Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, -1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], x], Times[-1, Dist[Times[b, c, n, Power[Times[-1, d], p], Power[Times[f, Plus[m, Times[2, p], 1]], -1]], Int[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[1, Times[c, x]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[-1, Times[c, x]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, -1]]], x], x]]]], And[FreeQ[List[a, b, c, d, e, f, m], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[n, p], Greater[n, 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Rational[1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[f, x], Plus[m, 1]], Sqrt[Plus[d1, Times[e1, x]]], Sqrt[Plus[d2, Times[e2, x]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[f, Plus[m, 2]], -1]], x], Plus[Times[-1, Dist[Times[Sqrt[Plus[d1, Times[e1, x]]], Sqrt[Plus[d2, Times[e2, x]]], Power[Times[Plus[m, 2], Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]]], -1]], Int[Times[Power[Times[f, x], m], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]]], -1]], x], x]], Times[-1, Dist[Times[b, c, n, Sqrt[Plus[d1, Times[e1, x]]], Sqrt[Plus[d2, Times[e2, x]]], Power[Times[f, Plus[m, 2], Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]]], -1]], Int[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, -1]]], x], x]]]], And[FreeQ[List[a, b, c, d1, e1, d2, e2, f, m], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, 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Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[d1, Times[e1, x]], p], Power[Plus[d2, Times[e2, x]], p], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[f, Plus[m, Times[2, p], 1]], -1]], x], Plus[Dist[Times[2, d1, d2, p, Power[Plus[m, Times[2, p], 1], -1]], Int[Times[Power[Times[f, x], m], Power[Plus[d1, Times[e1, x]], Plus[p, -1]], Power[Plus[d2, Times[e2, x]], Plus[p, -1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], x], Times[-1, Dist[Times[b, c, n, Power[Times[-1, d1, d2], Plus[p, Times[-1, 1, Power[2, -1]]]], Sqrt[Plus[d1, Times[e1, x]]], Sqrt[Plus[d2, Times[e2, x]]], Power[Times[f, Plus[m, Times[2, p], 1], Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]]], -1]], Int[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[-1, Times[Power[c, 2], 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Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[d, f, Plus[m, 1]], -1]], x], Plus[Dist[Times[b, c, n, Power[Times[-1, d], p], Power[Times[f, Plus[m, 1]], -1]], Int[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[1, Times[c, x]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[-1, Times[c, x]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, -1]]], x], x], Dist[Times[Power[c, 2], Plus[m, Times[2, p], 3], Power[Times[Power[f, 2], Plus[m, 1]], -1]], Int[Times[Power[Times[f, x], Plus[m, 2]], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, p], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[m, n], Greater[n, 0], Less[m, -1], IntegerQ[m], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n], Power[Times[d, f, Plus[m, 1]], -1]], x], Plus[Times[-1, Dist[Times[Power[c, 2], Plus[m, Times[2, p], 3], Power[Times[Power[f, 2], Plus[m, 1]], -1]], Int[Times[Power[Times[f, x], Plus[m, 2]], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n]], x], x]], Times[-1, Dist[Times[b, c, n, Power[d, IntPart[p]], Power[Plus[d, Times[e, Power[x, 2]]], FracPart[p]], Power[Times[f, Plus[m, 1], Power[Plus[1, Times[Power[c, 2], Power[x, 2]]], FracPart[p]]], -1]], Int[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[1, Times[Power[c, 2], Power[x, 2]]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], Plus[n, -1]]], x], x]]]], And[FreeQ[List[a, b, c, d, e, f, p], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[m, n], Greater[n, 0], Less[m, -1], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[d1, Times[e1, x]], Plus[p, 1]], Power[Plus[d2, Times[e2, x]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[d1, d2, f, Plus[m, 1]], -1]], x], Plus[Dist[Times[Power[c, 2], Plus[m, Times[2, p], 3], Power[Times[Power[f, 2], Plus[m, 1]], -1]], Int[Times[Power[Times[f, x], Plus[m, 2]], Power[Plus[d1, Times[e1, x]], p], Power[Plus[d2, Times[e2, x]], p], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], x], Dist[Times[b, c, n, Power[Times[-1, d1, d2], IntPart[p]], Power[Plus[d1, Times[e1, x]], FracPart[p]], Power[Plus[d2, Times[e2, x]], FracPart[p]], Power[Times[f, Plus[m, 1], Power[Plus[1, Times[c, x]], FracPart[p]], Power[Plus[-1, Times[c, x]], FracPart[p]]], -1]], Int[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[-1, Times[Power[c, 2], Power[x, 2]]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, -1]]], x], x]]], And[FreeQ[List[a, b, c, d1, e1, d2, e2, f, p], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], RationalQ[m, n], Greater[n, 0], Less[m, -1], IntegerQ[m], IntegerQ[Plus[p, Times[1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[d1, Times[e1, x]], Plus[p, 1]], Power[Plus[d2, Times[e2, x]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[d1, d2, f, Plus[m, 1]], -1]], x], Plus[Dist[Times[Power[c, 2], Plus[m, Times[2, p], 3], Power[Times[Power[f, 2], Plus[m, 1]], -1]], Int[Times[Power[Times[f, x], Plus[m, 2]], Power[Plus[d1, Times[e1, x]], p], Power[Plus[d2, Times[e2, x]], p], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], x], Dist[Times[b, c, n, Power[Times[-1, d1, d2], IntPart[p]], Power[Plus[d1, Times[e1, x]], FracPart[p]], Power[Plus[d2, Times[e2, x]], FracPart[p]], Power[Times[f, Plus[m, 1], Power[Plus[1, Times[c, x]], FracPart[p]], Power[Plus[-1, Times[c, x]], FracPart[p]]], -1]], Int[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[1, Times[c, x]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[-1, Times[c, x]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, -1]]], x], x]]], And[FreeQ[List[a, b, c, d1, e1, d2, e2, f, p], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], RationalQ[m, n], Greater[n, 0], Less[m, -1], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[f, Power[Times[f, x], Plus[m, -1]], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[2, e, Plus[p, 1]], -1]], x], Plus[Times[-1, Dist[Times[b, f, n, Power[Times[-1, d], p], Power[Times[2, c, Plus[p, 1]], -1]], Int[Times[Power[Times[f, x], Plus[m, -1]], Power[Plus[1, Times[c, x]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[-1, Times[c, x]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, -1]]], x], x]], Times[-1, Dist[Times[Power[f, 2], Plus[m, -1], Power[Times[2, e, Plus[p, 1]], -1]], Int[Times[Power[Times[f, x], Plus[m, -2]], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], x]]]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[m, n], Greater[n, 0], Less[p, -1], Greater[m, 1], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[f, Power[Times[f, x], Plus[m, -1]], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n], Power[Times[2, e, Plus[p, 1]], -1]], x], Plus[Times[-1, Dist[Times[Power[f, 2], Plus[m, -1], Power[Times[2, e, Plus[p, 1]], -1]], Int[Times[Power[Times[f, x], Plus[m, -2]], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n]], x], x]], Times[-1, Dist[Times[b, f, n, Power[d, IntPart[p]], Power[Plus[d, Times[e, Power[x, 2]]], FracPart[p]], Power[Times[2, c, Plus[p, 1], Power[Plus[1, Times[Power[c, 2], Power[x, 2]]], FracPart[p]]], -1]], Int[Times[Power[Times[f, x], Plus[m, -1]], Power[Plus[1, Times[Power[c, 2], Power[x, 2]]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], Plus[n, -1]]], x], x]]]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[m, n, p], Greater[n, 0], Less[p, -1], Greater[m, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[f, Power[Times[f, x], Plus[m, -1]], Power[Plus[d1, Times[e1, x]], Plus[p, 1]], Power[Plus[d2, Times[e2, x]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[2, e1, e2, Plus[p, 1]], -1]], x], Plus[Times[-1, Dist[Times[Power[f, 2], Plus[m, -1], Power[Times[2, e1, e2, Plus[p, 1]], -1]], Int[Times[Power[Times[f, x], Plus[m, -2]], Power[Plus[d1, Times[e1, x]], Plus[p, 1]], Power[Plus[d2, Times[e2, x]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], x]], Times[-1, Dist[Times[b, f, n, Power[Times[-1, d1, d2], IntPart[p]], Power[Plus[d1, Times[e1, x]], FracPart[p]], Power[Plus[d2, Times[e2, x]], FracPart[p]], Power[Times[2, c, Plus[p, 1], Power[Plus[1, Times[c, x]], FracPart[p]], Power[Plus[-1, Times[c, x]], FracPart[p]]], -1]], Int[Times[Power[Times[f, x], Plus[m, -1]], Power[Plus[-1, Times[Power[c, 2], Power[x, 2]]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, -1]]], x], x]]]], And[FreeQ[List[a, b, c, d1, e1, d2, e2, f], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], RationalQ[m, n, p], Greater[n, 0], Less[p, -1], Greater[m, 1], IntegerQ[Plus[p, Times[1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[f, Power[Times[f, x], Plus[m, -1]], Power[Plus[d1, Times[e1, x]], Plus[p, 1]], Power[Plus[d2, Times[e2, x]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[2, e1, e2, Plus[p, 1]], -1]], x], Plus[Times[-1, Dist[Times[Power[f, 2], Plus[m, -1], Power[Times[2, e1, e2, Plus[p, 1]], -1]], Int[Times[Power[Times[f, x], Plus[m, -2]], Power[Plus[d1, Times[e1, x]], Plus[p, 1]], Power[Plus[d2, Times[e2, x]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], x]], Times[-1, Dist[Times[b, f, n, Power[Times[-1, d1, d2], IntPart[p]], Power[Plus[d1, Times[e1, x]], FracPart[p]], Power[Plus[d2, Times[e2, x]], FracPart[p]], Power[Times[2, c, Plus[p, 1], Power[Plus[1, Times[c, x]], FracPart[p]], Power[Plus[-1, Times[c, x]], FracPart[p]]], -1]], Int[Times[Power[Times[f, x], Plus[m, -1]], Power[Plus[1, Times[c, x]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[-1, Times[c, x]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, -1]]], x], x]]]], And[FreeQ[List[a, b, c, d1, e1, d2, e2, f], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], RationalQ[m, n, p], Greater[n, 0], Less[p, -1], Not[IntegerQ[p]], Greater[m, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[2, d, f, Plus[p, 1]], -1]], x]], Plus[Dist[Times[Plus[m, Times[2, p], 3], Power[Times[2, d, Plus[p, 1]], -1]], Int[Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], x], Times[-1, Dist[Times[b, c, n, Power[Times[-1, d], p], Power[Times[2, f, Plus[p, 1]], -1]], Int[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[1, Times[c, x]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[-1, Times[c, x]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, -1]]], x], x]]]], And[FreeQ[List[a, b, c, d, e, f, m], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[n, p], Greater[n, 0], Less[p, -1], Not[And[RationalQ[m], Greater[m, 1]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n], Power[Times[2, d, f, Plus[p, 1]], -1]], x]], Plus[Dist[Times[Plus[m, Times[2, p], 3], Power[Times[2, d, Plus[p, 1]], -1]], Int[Times[Power[Times[f, x], m], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n]], x], x], Dist[Times[b, c, n, Power[d, IntPart[p]], Power[Plus[d, Times[e, Power[x, 2]]], FracPart[p]], Power[Times[2, f, Plus[p, 1], Power[Plus[1, Times[Power[c, 2], Power[x, 2]]], FracPart[p]]], -1]], Int[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[1, Times[Power[c, 2], Power[x, 2]]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], Plus[n, -1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, m], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[n, p], Greater[n, 0], Less[p, -1], Not[And[RationalQ[m], Greater[m, 1]]], Or[IntegerQ[m], IntegerQ[p], Equal[n, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[d1, Times[e1, x]], Plus[p, 1]], Power[Plus[d2, Times[e2, x]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[2, d1, d2, f, Plus[p, 1]], -1]], x]], Plus[Dist[Times[Plus[m, Times[2, p], 3], Power[Times[2, d1, d2, Plus[p, 1]], -1]], Int[Times[Power[Times[f, x], m], Power[Plus[d1, Times[e1, x]], Plus[p, 1]], Power[Plus[d2, Times[e2, x]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], x], Times[-1, Dist[Times[b, c, n, Power[Times[-1, d1, d2], IntPart[p]], Power[Plus[d1, Times[e1, x]], FracPart[p]], Power[Plus[d2, Times[e2, x]], FracPart[p]], Power[Times[2, f, Plus[p, 1], Power[Plus[1, Times[c, x]], FracPart[p]], Power[Plus[-1, Times[c, x]], FracPart[p]]], -1]], Int[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[-1, Times[Power[c, 2], Power[x, 2]]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, -1]]], x], x]]]], And[FreeQ[List[a, b, c, d1, e1, d2, e2, f, m], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], RationalQ[n, p], Greater[n, 0], Less[p, -1], Not[And[RationalQ[m], Greater[m, 1]]], Or[IntegerQ[m], Equal[n, 1]], IntegerQ[Plus[p, Times[1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[d1, Times[e1, x]], Plus[p, 1]], Power[Plus[d2, Times[e2, x]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[2, d1, d2, f, Plus[p, 1]], -1]], x]], Plus[Dist[Times[Plus[m, Times[2, p], 3], Power[Times[2, d1, d2, Plus[p, 1]], -1]], Int[Times[Power[Times[f, x], m], Power[Plus[d1, Times[e1, x]], Plus[p, 1]], Power[Plus[d2, Times[e2, x]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], x], Times[-1, Dist[Times[b, c, n, Power[Times[-1, d1, d2], IntPart[p]], Power[Plus[d1, Times[e1, x]], FracPart[p]], Power[Plus[d2, Times[e2, x]], FracPart[p]], Power[Times[2, f, Plus[p, 1], Power[Plus[1, Times[c, x]], FracPart[p]], Power[Plus[-1, Times[c, x]], FracPart[p]]], -1]], Int[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[1, Times[c, x]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[-1, Times[c, x]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, -1]]], x], x]]]], And[FreeQ[List[a, b, c, d1, e1, d2, e2, f, m], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], RationalQ[n, p], Greater[n, 0], Less[p, -1], Not[And[RationalQ[m], Greater[m, 1]]], Or[IntegerQ[m], IntegerQ[p], Equal[n, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[f, Power[Times[f, x], Plus[m, -1]], Sqrt[Plus[d, Times[e, Power[x, 2]]]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n], Power[Times[e, m], -1]], x], Plus[Times[-1, Dist[Times[Power[f, 2], Plus[m, -1], Power[Times[Power[c, 2], m], -1]], Int[Times[Times[Power[Times[f, x], Plus[m, -2]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n]], Power[Sqrt[Plus[d, Times[e, Power[x, 2]]]], -1]], x], x]], Times[-1, Dist[Times[b, f, n, Sqrt[Plus[1, Times[Power[c, 2], Power[x, 2]]]], Power[Times[c, m, Sqrt[Plus[d, Times[e, Power[x, 2]]]]], -1]], Int[Times[Power[Times[f, x], Plus[m, -1]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], Plus[n, -1]]], x], x]]]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[m, n], Greater[n, 0], Greater[m, 1], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[f, Power[Times[f, x], Plus[m, -1]], Sqrt[Plus[d1, Times[e1, x]]], Sqrt[Plus[d2, Times[e2, x]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[e1, e2, m], -1]], x], Plus[Dist[Times[Power[f, 2], Plus[m, -1], Power[Times[Power[c, 2], m], -1]], Int[Times[Power[Times[f, x], Plus[m, -2]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[Sqrt[Plus[d1, Times[e1, x]]], Sqrt[Plus[d2, Times[e2, x]]]], -1]], x], x], Dist[Times[b, f, n, Sqrt[Plus[d1, Times[e1, x]]], Sqrt[Plus[d2, Times[e2, x]]], Power[Times[c, d1, d2, m, Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]]], -1]], Int[Times[Power[Times[f, x], Plus[m, -1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, -1]]], x], x]]], And[FreeQ[List[a, b, c, d1, e1, d2, e2, f], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], RationalQ[m, n], Greater[n, 0], Greater[m, 1], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Times[Power[c, Plus[m, 1]], Sqrt[d]], -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], n], Power[Sinh[x], m]], x], x, ArcSinh[Times[c, x]]], x], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], PositiveQ[d], PositiveIntegerQ[n], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Times[Power[c, Plus[m, 1]], Sqrt[Times[-1, d1, d2]]], -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], n], Power[Cosh[x], m]], x], x, ArcCosh[Times[c, x]]], x], And[FreeQ[List[a, b, c, d1, e1, d2, e2], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], PositiveIntegerQ[n], PositiveQ[d1], NegativeQ[d2], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[f, x], Plus[m, 1]], Plus[a, Times[b, ArcSinh[Times[c, x]]]], Hypergeometric2F1[Times[1, Power[2, -1]], Times[Plus[1, m], Power[2, -1]], Times[Plus[3, m], Power[2, -1]], Times[-1, Power[c, 2], Power[x, 2]]], Power[Times[Sqrt[d], f, Plus[m, 1]], -1]], x], Times[-1, Simp[Times[b, c, Power[Times[f, x], Plus[m, 2]], HypergeometricPFQ[List[1, Plus[1, Times[m, Power[2, -1]]], Plus[1, Times[m, Power[2, -1]]]], List[Plus[Times[3, Power[2, -1]], Times[m, Power[2, -1]]], Plus[2, Times[m, Power[2, -1]]]], Times[-1, Power[c, 2], Power[x, 2]]], Power[Times[Sqrt[d], Power[f, 2], Plus[m, 1], Plus[m, 2]], -1]], x]]], And[FreeQ[List[a, b, c, d, e, f, m], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], PositiveQ[d], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[f, x], Plus[m, 1]], Sqrt[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]]], Plus[a, Times[b, ArcCosh[Times[c, x]]]], Hypergeometric2F1[Times[1, Power[2, -1]], Times[Plus[1, m], Power[2, -1]], Times[Plus[3, m], Power[2, -1]], Times[Power[c, 2], Power[x, 2]]], Power[Times[f, Plus[m, 1], Sqrt[Plus[d1, Times[e1, x]]], Sqrt[Plus[d2, Times[e2, x]]]], -1]], x], Simp[Times[b, c, Power[Times[f, x], Plus[m, 2]], HypergeometricPFQ[List[1, Plus[1, Times[m, Power[2, -1]]], Plus[1, Times[m, Power[2, -1]]]], List[Plus[Times[3, Power[2, -1]], Times[m, Power[2, -1]]], Plus[2, Times[m, Power[2, -1]]]], Times[Power[c, 2], Power[x, 2]]], Power[Times[Sqrt[Times[-1, d1, d2]], Power[f, 2], Plus[m, 1], Plus[m, 2]], -1]], x]], And[FreeQ[List[a, b, c, d1, e1, d2, e2, f, m], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], PositiveQ[d1], NegativeQ[d2], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Plus[1, Times[Power[c, 2], Power[x, 2]]]], Power[Sqrt[Plus[d, Times[e, Power[x, 2]]]], -1]], Int[Times[Power[Times[f, x], m], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n], Power[Sqrt[Plus[1, Times[Power[c, 2], Power[x, 2]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, m], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[n], Greater[n, 0], Not[PositiveQ[d]], Or[IntegerQ[m], Equal[n, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]], Power[Times[Sqrt[Plus[d1, Times[e1, x]]], Sqrt[Plus[d2, Times[e2, x]]]], -1]], Int[Times[Power[Times[f, x], m], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]]], -1]], x], x], And[FreeQ[List[a, b, c, d1, e1, d2, e2, f, m], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], RationalQ[n], Greater[n, 0], Not[And[PositiveQ[d1], NegativeQ[d2]]], Or[IntegerQ[m], Equal[n, 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[f, Power[Times[f, x], Plus[m, -1]], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[e, Plus[m, Times[2, p], 1]], -1]], x], Plus[Times[-1, Dist[Times[b, f, n, Power[Times[-1, d], p], Power[Times[c, Plus[m, Times[2, p], 1]], -1]], Int[Times[Power[Times[f, x], Plus[m, -1]], Power[Plus[1, Times[c, x]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[-1, Times[c, x]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, -1]]], x], x]], Dist[Times[Power[f, 2], Plus[m, -1], Power[Times[Power[c, 2], Plus[m, Times[2, p], 1]], -1]], Int[Times[Power[Times[f, x], Plus[m, -2]], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, p], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[m, n], Greater[n, 0], Greater[m, 1], NonzeroQ[Plus[m, Times[2, p], 1]], IntegerQ[p], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[f, Power[Times[f, x], Plus[m, -1]], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, 1]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n], Power[Times[e, Plus[m, Times[2, p], 1]], -1]], x], Plus[Times[-1, Dist[Times[Power[f, 2], Plus[m, -1], Power[Times[Power[c, 2], Plus[m, Times[2, p], 1]], -1]], Int[Times[Power[Times[f, x], Plus[m, -2]], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n]], x], x]], Times[-1, Dist[Times[b, f, n, Power[d, IntPart[p]], Power[Plus[d, Times[e, Power[x, 2]]], FracPart[p]], Power[Times[c, Plus[m, Times[2, p], 1], Power[Plus[1, Times[Power[c, 2], Power[x, 2]]], FracPart[p]]], -1]], Int[Times[Power[Times[f, x], Plus[m, -1]], Power[Plus[1, Times[Power[c, 2], Power[x, 2]]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], Plus[n, -1]]], x], x]]]], And[FreeQ[List[a, b, c, d, e, f, p], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[m, n], Greater[n, 0], Greater[m, 1], NonzeroQ[Plus[m, Times[2, p], 1]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[f, Power[Times[f, x], Plus[m, -1]], Power[Plus[d1, Times[e1, x]], Plus[p, 1]], Power[Plus[d2, Times[e2, x]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[e1, e2, Plus[m, Times[2, p], 1]], -1]], x], Plus[Dist[Times[Power[f, 2], Plus[m, -1], Power[Times[Power[c, 2], Plus[m, Times[2, p], 1]], -1]], Int[Times[Power[Times[f, x], Plus[m, -2]], Power[Plus[d1, Times[e1, x]], p], Power[Plus[d2, Times[e2, x]], p], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], x], Times[-1, Dist[Times[b, f, n, Power[Times[-1, d1, d2], IntPart[p]], Power[Plus[d1, Times[e1, x]], FracPart[p]], Power[Plus[d2, Times[e2, x]], FracPart[p]], Power[Times[c, Plus[m, Times[2, p], 1], Power[Plus[1, Times[c, x]], FracPart[p]], Power[Plus[-1, Times[c, x]], FracPart[p]]], -1]], Int[Times[Power[Times[f, x], Plus[m, -1]], Power[Plus[-1, Times[Power[c, 2], Power[x, 2]]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, -1]]], x], x]]]], And[FreeQ[List[a, b, c, d1, e1, d2, e2, f, p], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], RationalQ[m, n], Greater[n, 0], Greater[m, 1], NonzeroQ[Plus[m, Times[2, p], 1]], IntegerQ[m], IntegerQ[Plus[p, Times[1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Pattern[m, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[f, Power[Times[f, x], Plus[m, -1]], Power[Plus[d1, Times[e1, x]], Plus[p, 1]], Power[Plus[d2, Times[e2, x]], Plus[p, 1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[e1, e2, Plus[m, Times[2, p], 1]], -1]], x], Plus[Dist[Times[Power[f, 2], Plus[m, -1], Power[Times[Power[c, 2], Plus[m, Times[2, p], 1]], -1]], Int[Times[Power[Times[f, x], Plus[m, -2]], Power[Plus[d1, Times[e1, x]], p], Power[Plus[d2, Times[e2, x]], p], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], x], Times[-1, Dist[Times[b, f, n, Power[Times[-1, d1, d2], IntPart[p]], Power[Plus[d1, Times[e1, x]], FracPart[p]], Power[Plus[d2, Times[e2, x]], FracPart[p]], Power[Times[c, Plus[m, Times[2, p], 1], Power[Plus[1, Times[c, x]], FracPart[p]], Power[Plus[-1, Times[c, x]], FracPart[p]]], -1]], Int[Times[Power[Times[f, x], Plus[m, -1]], Power[Plus[1, Times[c, x]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[-1, Times[c, x]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, -1]]], x], x]]]], And[FreeQ[List[a, b, c, d1, e1, d2, e2, f, p], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], RationalQ[m, n], Greater[n, 0], Greater[m, 1], NonzeroQ[Plus[m, Times[2, p], 1]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[f, x], m], Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Plus[n, 1]], -1]], x], Dist[Times[f, m, Power[Times[-1, d], p], Power[Times[b, c, Plus[n, 1]], -1]], Int[Times[Power[Times[f, x], Plus[m, -1]], Power[Plus[1, Times[c, x]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[-1, Times[c, x]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, 1]]], x], x]], And[FreeQ[List[a, b, c, d, e, f, m, p], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[n], Less[n, -1], ZeroQ[Plus[m, Times[2, p], 1]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[f, x], m], Sqrt[Plus[1, Times[Power[c, 2], Power[x, 2]]]], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Plus[n, 1]], -1]], x], Times[-1, Dist[Times[f, m, Power[d, IntPart[p]], Power[Plus[d, Times[e, Power[x, 2]]], FracPart[p]], Power[Times[b, c, Plus[n, 1], Power[Plus[1, Times[Power[c, 2], Power[x, 2]]], FracPart[p]]], -1]], Int[Times[Power[Times[f, x], Plus[m, -1]], Power[Plus[1, Times[Power[c, 2], Power[x, 2]]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], Plus[n, 1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, m, p], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[n], Less[n, -1], ZeroQ[Plus[m, Times[2, p], 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[f, x], m], Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]], Power[Plus[d1, Times[e1, x]], p], Power[Plus[d2, Times[e2, x]], p], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Plus[n, 1]], -1]], x], Dist[Times[f, m, Power[Times[-1, d1, d2], IntPart[p]], Power[Plus[d1, Times[e1, x]], FracPart[p]], Power[Plus[d2, Times[e2, x]], FracPart[p]], Power[Times[b, c, Plus[n, 1], Power[Plus[1, Times[c, x]], FracPart[p]], Power[Plus[-1, Times[c, x]], FracPart[p]]], -1]], Int[Times[Power[Times[f, x], Plus[m, -1]], Power[Plus[-1, Times[Power[c, 2], Power[x, 2]]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, 1]]], x], x]], And[FreeQ[List[a, b, c, d1, e1, d2, e2, f, m, p], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], RationalQ[n], Less[n, -1], ZeroQ[Plus[m, Times[2, p], 1]], IntegerQ[Plus[p, Times[-1, 1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[f, x], m], Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]], Power[Plus[d1, Times[e1, x]], p], Power[Plus[d2, Times[e2, x]], p], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Plus[n, 1]], -1]], x], Dist[Times[f, m, Power[Times[-1, d1, d2], IntPart[p]], Power[Plus[d1, Times[e1, x]], FracPart[p]], Power[Plus[d2, Times[e2, x]], FracPart[p]], Power[Times[b, c, Plus[n, 1], Power[Plus[1, Times[c, x]], FracPart[p]], Power[Plus[-1, Times[c, x]], FracPart[p]]], -1]], Int[Times[Power[Times[f, x], Plus[m, -1]], Power[Plus[1, Times[c, x]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[-1, Times[c, x]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, 1]]], x], x]], And[FreeQ[List[a, b, c, d1, e1, d2, e2, f, m, p], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], RationalQ[n], Less[n, -1], ZeroQ[Plus[m, Times[2, p], 1]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[f, x], m], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Sqrt[d], Plus[n, 1]], -1]], x], Times[-1, Dist[Times[f, m, Power[Times[b, c, Sqrt[d], Plus[n, 1]], -1]], Int[Times[Power[Times[f, x], Plus[m, -1]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], Plus[n, 1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, m], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[n], Less[n, -1], PositiveQ[d]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[f, x], m], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Sqrt[Times[-1, d1, d2]], Plus[n, 1]], -1]], x], Times[-1, Dist[Times[Times[f, m], Power[Times[b, c, Sqrt[Times[-1, d1, d2]], Plus[n, 1]], -1]], Int[Times[Power[Times[f, x], Plus[m, -1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, 1]]], x], x]]], And[FreeQ[List[a, b, c, d1, e1, d2, e2, f, m], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], RationalQ[n], Less[n, -1], PositiveQ[d1], NegativeQ[d2]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Plus[1, Times[Power[c, 2], Power[x, 2]]]], Power[Sqrt[Plus[d, Times[e, Power[x, 2]]]], -1]], Int[Times[Power[Times[f, x], m], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n], Power[Sqrt[Plus[1, Times[Power[c, 2], Power[x, 2]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, f, m], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[n], Less[n, -1], Not[PositiveQ[d]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]], Power[Times[Sqrt[Plus[d1, Times[e1, x]]], Sqrt[Plus[d2, Times[e2, x]]]], -1]], Int[Times[Power[Times[f, x], m], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]]], -1]], x], x], And[FreeQ[List[a, b, c, d1, e1, d2, e2, f, m], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], RationalQ[n], Less[n, -1], Not[And[PositiveQ[d1], NegativeQ[d2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[f, x], m], Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Plus[n, 1]], -1]], x], Plus[Dist[Times[f, m, Power[Times[-1, d], p], Power[Times[b, c, Plus[n, 1]], -1]], Int[Times[Power[Times[f, x], Plus[m, -1]], Power[Plus[1, Times[c, x]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[-1, Times[c, x]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, 1]]], x], x], Times[-1, Dist[Times[c, Power[Times[-1, d], p], Plus[m, Times[2, p], 1], Power[Times[b, f, Plus[n, 1]], -1]], Int[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[1, Times[c, x]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[-1, Times[c, x]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, 1]]], x], x]]]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[n], Less[n, -1], IntegerQ[m], Greater[m, -3], PositiveIntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[f, x], m], Sqrt[Plus[1, Times[Power[c, 2], Power[x, 2]]]], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Plus[n, 1]], -1]], x], Plus[Times[-1, Dist[Times[f, m, Power[d, IntPart[p]], Power[Plus[d, Times[e, Power[x, 2]]], FracPart[p]], Power[Times[b, c, Plus[n, 1], Power[Plus[1, Times[Power[c, 2], Power[x, 2]]], FracPart[p]]], -1]], Int[Times[Power[Times[f, x], Plus[m, -1]], Power[Plus[1, Times[Power[c, 2], Power[x, 2]]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], Plus[n, 1]]], x], x]], Times[-1, Dist[Times[c, Plus[m, Times[2, p], 1], Power[d, IntPart[p]], Power[Plus[d, Times[e, Power[x, 2]]], FracPart[p]], Power[Times[b, f, Plus[n, 1], Power[Plus[1, Times[Power[c, 2], Power[x, 2]]], FracPart[p]]], -1]], Int[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[1, Times[Power[c, 2], Power[x, 2]]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], Plus[n, 1]]], x], x]]]], And[FreeQ[List[a, b, c, d, e, f], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], RationalQ[n], Less[n, -1], IntegerQ[m], Greater[m, -3], PositiveIntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Times[f, x], m], Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]], Power[Plus[d1, Times[e1, x]], p], Power[Plus[d2, Times[e2, x]], p], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Plus[n, 1]], -1]], x], Plus[Dist[Times[f, m, Power[Times[-1, d1, d2], IntPart[p]], Power[Plus[d1, Times[e1, x]], FracPart[p]], Power[Plus[d2, Times[e2, x]], FracPart[p]], Power[Times[b, c, Plus[n, 1], Power[Plus[1, Times[c, x]], FracPart[p]], Power[Plus[-1, Times[c, x]], FracPart[p]]], -1]], Int[Times[Power[Times[f, x], Plus[m, -1]], Power[Plus[-1, Times[Power[c, 2], Power[x, 2]]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, 1]]], x], x], Times[-1, Dist[Times[c, Plus[m, Times[2, p], 1], Power[Times[-1, d1, d2], IntPart[p]], Power[Plus[d1, Times[e1, x]], FracPart[p]], Power[Plus[d2, Times[e2, x]], FracPart[p]], Power[Times[b, f, Plus[n, 1], Power[Plus[1, Times[c, x]], FracPart[p]], Power[Plus[-1, Times[c, x]], FracPart[p]]], -1]], Int[Times[Power[Times[f, x], Plus[m, 1]], Power[Plus[-1, Times[Power[c, 2], Power[x, 2]]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, 1]]], x], x]]]], And[FreeQ[List[a, b, c, d1, e1, d2, e2, f], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], RationalQ[n], Less[n, -1], IntegerQ[m], Greater[m, -3], PositiveIntegerQ[Plus[p, Times[1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[d, p], Power[Power[c, Plus[m, 1]], -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], n], Power[Sinh[x], m], Power[Cosh[x], Plus[Times[2, p], 1]]], x], x, ArcSinh[Times[c, x]]], x], And[FreeQ[List[a, b, c, d, e, n], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], IntegerQ[Times[2, p]], Greater[p, -1], PositiveIntegerQ[m], Or[IntegerQ[p], PositiveQ[d]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[-1, d], p], Power[Power[c, Plus[m, 1]], -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], n], Power[Cosh[x], m], Power[Sinh[x], Plus[Times[2, p], 1]]], x], x, ArcCosh[Times[c, x]]], x], And[FreeQ[List[a, b, c, d, e, n], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], PositiveIntegerQ[p], PositiveIntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[-1, d1, d2], p], Power[Power[c, Plus[m, 1]], -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], n], Power[Cosh[x], m], Power[Sinh[x], Plus[Times[2, p], 1]]], x], x, ArcCosh[Times[c, x]]], x], And[FreeQ[List[a, b, c, d1, e1, d2, e2, n], x], 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Subst[Int[Times[Power[Plus[a, Times[b, x]], n], Power[Plus[Times[c, d], Times[e, Cosh[x]]], m], Sinh[x]], x], x, ArcCosh[Times[c, x]]], x], And[FreeQ[List[a, b, c, d, e, n], x], PositiveIntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[Px, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, IntHide[Px, x]]], Plus[Dist[Plus[a, Times[b, ArcSinh[Times[c, x]]]], u, x], Times[-1, Dist[Times[b, c], Int[SimplifyIntegrand[Times[u, Power[Sqrt[Plus[1, Times[Power[c, 2], Power[x, 2]]]], -1]], x], x], x]]]], And[FreeQ[List[a, b, c], x], PolynomialQ[Px, x]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[Px, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, IntHide[Px, x]]], Plus[Dist[Plus[a, Times[b, ArcCosh[Times[c, x]]]], u, x], Times[-1, Dist[Times[b, c, Sqrt[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]]], Power[Times[Sqrt[Plus[-1, Times[c, x]]], Sqrt[Plus[1, Times[c, x]]]], -1]], Int[SimplifyIntegrand[Times[u, Power[Sqrt[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]]], -1]], x], x], x]]]], And[FreeQ[List[a, b, c], x], PolynomialQ[Px, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[Px, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Px, Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n]], x], x], And[FreeQ[List[a, b, c, n], x], PolynomialQ[Px, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[Px, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Px, Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], x], And[FreeQ[List[a, b, c, n], x], PolynomialQ[Px, x]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[Px, Blank[]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, IntHide[Times[Px, Power[Plus[d, Times[e, x]], m]], x]]], Plus[Dist[Plus[a, Times[b, ArcSinh[Times[c, x]]]], u, x], Times[-1, Dist[Times[b, c], Int[SimplifyIntegrand[Times[u, Power[Sqrt[Plus[1, Times[Power[c, 2], Power[x, 2]]]], -1]], x], x], x]]]], And[FreeQ[List[a, b, c, d, e, m], x], PolynomialQ[Px, x]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[Px, Blank[]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, IntHide[Times[Px, Power[Plus[d, Times[e, x]], m]], x]]], Plus[Dist[Plus[a, Times[b, ArcCosh[Times[c, x]]]], u, x], Times[-1, Dist[Times[b, c, Sqrt[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]]], Power[Times[Sqrt[Plus[-1, Times[c, x]]], Sqrt[Plus[1, Times[c, x]]]], -1]], Int[SimplifyIntegrand[Times[u, Power[Sqrt[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]]], -1]], x], x], x]]]], And[FreeQ[List[a, b, c, d, e, m], x], PolynomialQ[Px, x]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, IntHide[Times[Power[Plus[f, Times[g, x]], p], Power[Plus[d, Times[e, x]], m]], x]]], Plus[Dist[Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n], u, x], Times[-1, Dist[Times[b, c, n], Int[SimplifyIntegrand[Times[u, Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], Plus[n, -1]], Power[Sqrt[Plus[1, Times[Power[c, 2], Power[x, 2]]]], -1]], x], x], x]]]], And[FreeQ[List[a, b, c, d, e, f, g], x], PositiveIntegerQ[n, p], NegativeIntegerQ[m], Less[Plus[m, p, 1], 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, IntHide[Times[Power[Plus[f, Times[g, x]], p], Power[Plus[d, Times[e, x]], m]], x]]], Plus[Dist[Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], u, x], Times[-1, Dist[Times[b, c, n], Int[SimplifyIntegrand[Times[u, Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, -1]], Power[Times[Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]]], -1]], x], x], x]]]], And[FreeQ[List[a, b, c, d, e, f, g], x], PositiveIntegerQ[n, p], NegativeIntegerQ[m], Less[Plus[m, p, 1], 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -2], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, IntHide[Times[Power[Plus[f, Times[g, x], Times[h, Power[x, 2]]], p], Power[Power[Plus[d, Times[e, x]], 2], -1]], x]]], Plus[Dist[Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n], u, x], Times[-1, Dist[Times[b, c, n], Int[SimplifyIntegrand[Times[u, Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], Plus[n, -1]], Power[Sqrt[Plus[1, Times[Power[c, 2], Power[x, 2]]]], -1]], x], x], x]]]], And[FreeQ[List[a, b, c, d, e, f, g, h], x], PositiveIntegerQ[n, p], ZeroQ[Plus[Times[e, g], Times[-1, 2, d, h]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], -2], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[h, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, IntHide[Times[Power[Plus[f, Times[g, x], Times[h, Power[x, 2]]], p], Power[Power[Plus[d, Times[e, x]], 2], -1]], x]]], Plus[Dist[Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], u, x], Times[-1, Dist[Times[b, c, n], Int[SimplifyIntegrand[Times[u, Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, -1]], Power[Times[Sqrt[Plus[1, Times[c, x]]], Sqrt[Plus[-1, Times[c, x]]]], -1]], x], x], x]]]], And[FreeQ[List[a, b, c, d, e, f, g, h], x], PositiveIntegerQ[n, p], ZeroQ[Plus[Times[e, g], Times[-1, 2, d, h]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, 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NegativeIntegerQ[Plus[p, Times[1, Power[2, -1]]]], PositiveQ[d], Greater[m, 0], Or[Less[m, Plus[Times[-2, p], -1]], Greater[m, 3]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, IntHide[Times[Power[Plus[f, Times[g, x]], m], Power[Plus[d1, Times[e1, x]], p], Power[Plus[d2, Times[e2, x]], p]], x]]], Plus[Dist[Plus[a, Times[b, ArcCosh[Times[c, x]]]], u, x], Times[-1, Dist[Times[b, c], Int[Dist[Times[1, Power[Times[Sqrt[Plus[1, Times[c, x]]], 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And[FreeQ[List[a, b, c, d, e, f, g], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], IntegerQ[m], IntegerQ[Plus[p, Times[1, Power[2, -1]]]], PositiveQ[d], PositiveIntegerQ[n], Greater[m, 0], Or[And[Equal[n, 1], Greater[p, -1]], Greater[p, 0], Equal[m, 1], And[Equal[m, 2], Less[p, -2]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[d1, Times[e1, x]], p], Power[Plus[d2, Times[e2, x]], p], 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Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[f, Times[g, x]], m], Plus[Times[d1, d2], Times[e1, e2, Power[x, 2]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Sqrt[Times[-1, d1, d2]], Plus[n, 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[b, c, Sqrt[Times[-1, d1, d2]], Plus[n, 1]], -1]], Int[Times[Plus[Times[d1, d2, g, m], Times[2, e1, e2, f, x], Times[e1, e2, g, Plus[m, 2], Power[x, 2]]], Power[Plus[f, Times[g, x]], Plus[m, -1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, 1]]], x], x]]], And[FreeQ[List[a, b, c, d1, e1, d2, e2, f, g], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], IntegerQ[m], PositiveQ[d1], NegativeQ[d2], PositiveIntegerQ[n], Less[m, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, 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Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Sqrt[Plus[d1, Times[e1, x]]], Sqrt[Plus[d2, Times[e2, x]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], Times[Power[Plus[f, Times[g, x]], m], Power[Plus[d1, Times[e1, x]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[d2, Times[e2, x]], Plus[p, Times[-1, 1, Power[2, -1]]]]], x], x], And[FreeQ[List[a, b, c, d1, e1, d2, e2, f, g], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], IntegerQ[m], PositiveIntegerQ[Plus[p, Times[1, Power[2, -1]]]], PositiveQ[d1], NegativeQ[d2], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[f, Times[g, x]], m], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Sqrt[d], Plus[n, 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[b, c, Sqrt[d], Plus[n, 1]], -1]], Int[ExpandIntegrand[Times[Power[Plus[f, Times[g, x]], Plus[m, -1]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], Plus[n, 1]]], Times[Plus[Times[d, g, m], Times[e, f, Plus[Times[2, p], 1], x], Times[e, g, Plus[m, Times[2, p], 1], Power[x, 2]]], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, Times[-1, 1, Power[2, -1]]]]], x], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], IntegerQ[m], PositiveIntegerQ[Plus[p, Times[-1, 1, Power[2, -1]]]], PositiveQ[d], PositiveIntegerQ[n], Less[m, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[f, Times[g, x]], m], Power[Plus[d1, Times[e1, x]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[d2, Times[e2, x]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Sqrt[Times[-1, d1, d2]], Plus[n, 1]], -1]], x], Times[-1, Dist[Times[1, Power[Times[b, c, Sqrt[Times[-1, d1, d2]], Plus[n, 1]], -1]], Int[ExpandIntegrand[Times[Power[Plus[f, Times[g, x]], Plus[m, -1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, 1]]], Times[Plus[Times[d1, d2, g, m], Times[e1, e2, f, Plus[Times[2, p], 1], x], Times[e1, e2, g, Plus[m, Times[2, p], 1], Power[x, 2]]], Power[Plus[d1, Times[e1, x]], Plus[p, Times[-1, 1, Power[2, -1]]]], Power[Plus[d2, Times[e2, x]], Plus[p, Times[-1, 1, Power[2, -1]]]]], x], x], x]]], And[FreeQ[List[a, b, c, d1, e1, d2, e2, f, g], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], IntegerQ[m], PositiveIntegerQ[Plus[p, Times[-1, 1, Power[2, -1]]]], PositiveQ[d1], NegativeQ[d2], PositiveIntegerQ[n], Less[m, 0]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[f, Times[g, x]], m], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Sqrt[d], Plus[n, 1]], -1]], x], Times[-1, Dist[Times[g, m, Power[Times[b, c, Sqrt[d], Plus[n, 1]], -1]], Int[Times[Power[Plus[f, Times[g, x]], Plus[m, -1]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], Plus[n, 1]]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], IntegerQ[m], PositiveQ[d], Greater[m, 0], RationalQ[n], Less[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[f, Times[g, x]], m], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Sqrt[Times[-1, d1, d2]], Plus[n, 1]], -1]], x], Times[-1, Dist[Times[g, m, Power[Times[b, c, Sqrt[Times[-1, d1, d2]], Plus[n, 1]], -1]], Int[Times[Power[Plus[f, Times[g, x]], Plus[m, -1]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, 1]]], x], x]]], And[FreeQ[List[a, b, c, d1, e1, d2, e2, f, g], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], IntegerQ[m], PositiveQ[d1], NegativeQ[d2], Greater[m, 0], RationalQ[n], Less[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Times[Power[c, Plus[m, 1]], Sqrt[d]], -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], n], Power[Plus[Times[c, f], Times[g, Sinh[x]]], m]], x], x, ArcSinh[Times[c, x]]], x], And[FreeQ[List[a, b, c, d, e, f, g, n], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], IntegerQ[m], PositiveQ[d], Or[Greater[m, 0], PositiveIntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Times[Power[c, Plus[m, 1]], Sqrt[Times[-1, d1, d2]]], -1]], Subst[Int[Times[Power[Plus[a, Times[b, x]], n], Power[Plus[Times[c, f], Times[g, Cosh[x]]], m]], x], x, ArcCosh[Times[c, x]]], x], And[FreeQ[List[a, b, c, d1, e1, d2, e2, f, g, n], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], IntegerQ[m], PositiveQ[d1], NegativeQ[d2], Or[Greater[m, 0], PositiveIntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n], Power[Sqrt[Plus[d, Times[e, Power[x, 2]]]], -1]], Times[Power[Plus[f, Times[g, x]], m], Power[Plus[d, Times[e, Power[x, 2]]], Plus[p, Times[1, Power[2, -1]]]]], x], x], And[FreeQ[List[a, b, c, d, e, f, g], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], IntegerQ[m], NegativeIntegerQ[Plus[p, Times[1, Power[2, -1]]]], PositiveQ[d], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Power[Times[Sqrt[Plus[d1, Times[e1, x]]], Sqrt[Plus[d2, Times[e2, x]]]], -1]], Times[Power[Plus[f, Times[g, x]], m], Power[Plus[d1, Times[e1, x]], Plus[p, Times[1, Power[2, -1]]]], Power[Plus[d2, Times[e2, x]], Plus[p, Times[1, Power[2, -1]]]]], x], x], And[FreeQ[List[a, b, c, d1, e1, d2, e2, f, g], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], IntegerQ[m], NegativeIntegerQ[Plus[p, Times[1, Power[2, -1]]]], PositiveQ[d1], NegativeQ[d2], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[d, IntPart[p]], Power[Plus[d, Times[e, Power[x, 2]]], FracPart[p]], Power[Power[Plus[1, Times[Power[c, 2], Power[x, 2]]], FracPart[p]], -1]], Int[Times[Power[Plus[f, Times[g, x]], m], Power[Plus[1, Times[Power[c, 2], Power[x, 2]]], p], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, n], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], IntegerQ[m], IntegerQ[Plus[p, Times[-1, 1, Power[2, -1]]]], Not[PositiveQ[d]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[-1, d], IntPart[p]], Power[Plus[d, Times[e, Power[x, 2]]], FracPart[p]], Power[Times[Power[Plus[1, Times[c, x]], FracPart[p]], Power[Plus[-1, Times[c, x]], FracPart[p]]], -1]], Int[Times[Power[Plus[f, Times[g, x]], m], Power[Plus[1, Times[c, x]], p], Power[Plus[-1, Times[c, x]], p], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, n], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], IntegerQ[m], IntegerQ[Plus[p, Times[-1, 1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[-1, d1, d2], IntPart[p]], Power[Plus[d1, Times[e1, x]], FracPart[p]], Power[Plus[d2, Times[e2, x]], FracPart[p]], Power[Power[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]], FracPart[p]], -1]], Int[Times[Power[Plus[f, Times[g, x]], m], Power[Plus[1, Times[c, x]], p], Power[Plus[-1, Times[c, x]], p], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], x], And[FreeQ[List[a, b, c, d1, e1, d2, e2, f, g, n], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], IntegerQ[m], IntegerQ[Plus[p, Times[-1, 1, Power[2, -1]]]], Not[And[PositiveQ[d1], NegativeQ[d2]]]]]], RuleDelayed[HoldPattern[Int[Times[Log[Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Log[Times[h, Power[Plus[f, Times[g, x]], m]]], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Sqrt[d], Plus[n, 1]], -1]], x], Times[-1, Dist[Times[g, m, Power[Times[b, c, Sqrt[d], Plus[n, 1]], -1]], Int[Times[Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], Plus[n, 1]], Power[Plus[f, Times[g, x]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, f, g, h, m], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], PositiveQ[d], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Log[Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Log[Times[h, Power[Plus[f, Times[g, x]], m]]], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, 1]], Power[Times[b, c, Sqrt[Times[-1, d1, d2]], Plus[n, 1]], -1]], x], Times[-1, Dist[Times[g, m, Power[Times[b, c, Sqrt[Times[-1, d1, d2]], Plus[n, 1]], -1]], Int[Times[Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], Plus[n, 1]], Power[Plus[f, Times[g, x]], -1]], x], x]]], And[FreeQ[List[a, b, c, d1, e1, d2, e2, f, g, h, m], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], PositiveQ[d1], NegativeQ[d2], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Log[Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[d, IntPart[p]], Power[Plus[d, Times[e, Power[x, 2]]], FracPart[p]], Power[Power[Plus[1, Times[Power[c, 2], Power[x, 2]]], FracPart[p]], -1]], Int[Times[Log[Times[h, Power[Plus[f, Times[g, x]], m]]], Power[Plus[1, Times[Power[c, 2], Power[x, 2]]], p], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, h, m, n], x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], IntegerQ[Plus[p, Times[-1, 1, Power[2, -1]]]], Not[PositiveQ[d]]]]], RuleDelayed[HoldPattern[Int[Times[Log[Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[-1, d], IntPart[p]], Power[Plus[d, Times[e, Power[x, 2]]], FracPart[p]], Power[Times[Power[Plus[1, Times[c, x]], FracPart[p]], Power[Plus[-1, Times[c, x]], FracPart[p]]], -1]], Int[Times[Log[Times[h, Power[Plus[f, Times[g, x]], m]]], Power[Plus[1, Times[c, x]], p], Power[Plus[-1, Times[c, x]], p], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, h, m, n], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], IntegerQ[Plus[p, Times[-1, 1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Log[Times[Optional[Pattern[h, Blank[]]], Power[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[-1, d1, d2], IntPart[p]], Power[Plus[d1, Times[e1, x]], FracPart[p]], Power[Plus[d2, Times[e2, x]], FracPart[p]], Power[Times[Power[Plus[1, Times[c, x]], FracPart[p]], Power[Plus[-1, Times[c, x]], FracPart[p]]], -1]], Int[Times[Log[Times[h, Power[Plus[f, Times[g, x]], m]]], Power[Plus[1, Times[c, x]], p], Power[Plus[-1, Times[c, x]], p], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], x], And[FreeQ[List[a, b, c, d1, e1, d2, e2, f, g, h, m, n], x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], IntegerQ[Plus[p, Times[-1, 1, Power[2, -1]]]], Not[And[PositiveQ[d1], NegativeQ[d2]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, IntHide[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], m]], x]]], Plus[Dist[Plus[a, Times[b, ArcSinh[Times[c, x]]]], u, x], Times[-1, Dist[Times[b, c], Int[Dist[Times[1, Power[Sqrt[Plus[1, Times[Power[c, 2], Power[x, 2]]]], -1]], u, x], x], x]]]], And[FreeQ[List[a, b, c, d, e, f, g], x], NegativeIntegerQ[Plus[m, Times[1, Power[2, 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Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n], Times[Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], m]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, n], x], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n], Times[Power[Plus[d, Times[e, x]], m], Power[Plus[f, Times[g, x]], m]], x], x], And[FreeQ[List[a, b, c, d, e, f, g, n], x], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[v, IntHide[u, x]]], Condition[Plus[Dist[Plus[a, Times[b, ArcSinh[Times[c, x]]]], v, x], Times[-1, Dist[Times[b, c], Int[SimplifyIntegrand[Times[v, Power[Sqrt[Plus[1, Times[Power[c, 2], Power[x, 2]]]], -1]], x], x], x]]], InverseFunctionFreeQ[v, x]]], FreeQ[List[a, b, c], x]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[v, IntHide[u, x]]], Condition[Plus[Dist[Plus[a, Times[b, ArcCosh[Times[c, x]]]], v, x], Times[-1, Dist[Times[b, c, Sqrt[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]]], Power[Times[Sqrt[Plus[-1, Times[c, x]]], Sqrt[Plus[1, Times[c, x]]]], -1]], Int[SimplifyIntegrand[Times[v, Power[Sqrt[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]]], -1]], x], x], x]]], InverseFunctionFreeQ[v, x]]], FreeQ[List[a, b, c], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[Px, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, ExpandIntegrand[Times[Px, Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n]], x]]], Condition[Int[u, x], SumQ[u]]], And[FreeQ[List[a, b, c, d, e, n], x], PolynomialQ[Px, x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], IntegerQ[Plus[p, Times[-1, 1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Pattern[n, Blank[]]], Pattern[Px, Blank[]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, ExpandIntegrand[Times[Px, Power[Plus[d1, Times[e1, x]], p], Power[Plus[d2, Times[e2, x]], p], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x]]], Condition[Int[u, x], SumQ[u]]], And[FreeQ[List[a, b, c, d1, e1, d2, e2, n], x], PolynomialQ[Px, x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], IntegerQ[Plus[p, Times[-1, 1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[Px, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, ExpandIntegrand[Times[Px, Power[Plus[f, Times[g, Power[Plus[d, Times[e, Power[x, 2]]], p]]], m], Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n]], x]]], Condition[Int[u, x], SumQ[u]]], And[FreeQ[List[a, b, c, d, e, f, g], x], PolynomialQ[Px, x], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], PositiveIntegerQ[Plus[p, Times[1, Power[2, -1]]]], IntegersQ[m, n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[Px, Blank[]]], Power[Plus[Pattern[f, Blank[]], Times[Optional[Pattern[g, Blank[]]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, ExpandIntegrand[Times[Px, Power[Plus[f, Times[g, Power[Plus[d1, Times[e1, x]], p], Power[Plus[d2, Times[e2, x]], p]]], m], Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x]]], Condition[Int[u, x], SumQ[u]]], And[FreeQ[List[a, b, c, d1, e1, d2, e2, f, g], x], PolynomialQ[Px, x], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], PositiveIntegerQ[Plus[p, Times[1, Power[2, -1]]]], IntegersQ[m, n]]]], RuleDelayed[HoldPattern[Int[Times[Power[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[RFx, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, ExpandIntegrand[Power[ArcSinh[Times[c, x]], n], RFx, x]]], Condition[Int[u, x], SumQ[u]]], And[FreeQ[c, x], RationalFunctionQ[RFx, x], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[RFx, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, ExpandIntegrand[Power[ArcCosh[Times[c, x]], n], RFx, x]]], Condition[Int[u, x], SumQ[u]]], And[FreeQ[c, x], RationalFunctionQ[RFx, x], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]], Pattern[RFx, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[RFx, Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n]], x], x], And[FreeQ[List[a, b, c], x], RationalFunctionQ[RFx, x], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]], Pattern[RFx, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Times[RFx, Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], x], And[FreeQ[List[a, b, c], x], RationalFunctionQ[RFx, x], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[RFx, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, ExpandIntegrand[Times[Power[Plus[d, Times[e, Power[x, 2]]], p], Power[ArcSinh[Times[c, x]], n]], RFx, x]]], Condition[Int[u, x], SumQ[u]]], And[FreeQ[List[c, d, e], x], RationalFunctionQ[RFx, x], PositiveIntegerQ[n], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], IntegerQ[Plus[p, Times[-1, 1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]], Pattern[RFx, Blank[]], Power[Plus[Pattern[d1, Blank[]], Times[Optional[Pattern[e1, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]], Power[Plus[Pattern[d2, Blank[]], Times[Optional[Pattern[e2, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, ExpandIntegrand[Times[Power[Plus[d1, Times[e1, x]], p], Power[Plus[d2, Times[e2, x]], p], Power[ArcCosh[Times[c, x]], n]], RFx, x]]], Condition[Int[u, x], SumQ[u]]], And[FreeQ[List[c, d1, e1, d2, e2], x], RationalFunctionQ[RFx, x], PositiveIntegerQ[n], ZeroQ[Plus[e1, Times[-1, c, d1]]], ZeroQ[Plus[e2, Times[c, d2]]], IntegerQ[Plus[p, Times[-1, 1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[ArcSinh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Optional[Pattern[n, Blank[]]]], Pattern[RFx, Blank[]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Power[Plus[d, Times[e, Power[x, 2]]], p], Times[RFx, Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n]], x], x], And[FreeQ[List[a, b, c, d, e], x], RationalFunctionQ[RFx, x], PositiveIntegerQ[n], ZeroQ[Plus[e, Times[-1, Power[c, 2], d]]], IntegerQ[Plus[p, Times[-1, 1, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], 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Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[u, Power[Plus[a, Times[b, ArcSinh[Times[c, x]]]], n]], x], FreeQ[List[a, b, c, n], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[u, Power[Plus[a, Times[b, ArcCosh[Times[c, x]]]], n]], x], FreeQ[List[a, b, c, n], x]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[d, -1]], Subst[Int[Power[Plus[a, Times[b, ArcSinh[x]]], n], x], x, Plus[c, Times[d, x]]], x], FreeQ[List[a, b, c, d, n], x]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[d, -1]], Subst[Int[Power[Plus[a, Times[b, ArcCosh[x]]], n], x], x, Plus[c, Times[d, x]]], x], FreeQ[List[a, b, c, d, n], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[d, -1]], Subst[Int[Times[Power[Plus[Times[Plus[Times[d, e], Times[-1, c, f]], Power[d, -1]], Times[f, x, Power[d, -1]]], m], Power[Plus[a, Times[b, ArcSinh[x]]], n]], x], x, Plus[c, Times[d, x]]], x], FreeQ[List[a, b, c, d, e, f, m, n], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[d, -1]], Subst[Int[Times[Power[Plus[Times[Plus[Times[d, e], Times[-1, c, f]], Power[d, -1]], Times[f, x, Power[d, -1]]], m], Power[Plus[a, Times[b, ArcCosh[x]]], n]], x], x, Plus[c, Times[d, x]]], x], FreeQ[List[a, b, c, d, e, f, m, n], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[d, -1]], Subst[Int[Times[Power[Plus[Times[C, Power[Power[d, 2], -1]], Times[C, Power[Power[d, 2], -1], Power[x, 2]]], p], Power[Plus[a, Times[b, ArcSinh[x]]], n]], x], x, Plus[c, Times[d, x]]], x], And[FreeQ[List[a, b, c, d, A, B, C, n, p], x], ZeroQ[Plus[Times[B, Plus[1, Power[c, 2]]], Times[-1, 2, A, c, d]]], ZeroQ[Plus[Times[2, c, C], Times[-1, B, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[d, -1]], Subst[Int[Times[Power[Plus[Times[-1, C, Power[Power[d, 2], -1]], Times[C, Power[Power[d, 2], -1], Power[x, 2]]], p], Power[Plus[a, Times[b, ArcCosh[x]]], n]], x], x, Plus[c, Times[d, x]]], x], And[FreeQ[List[a, b, c, d, A, B, C, n, p], x], ZeroQ[Plus[Times[B, Plus[1, Times[-1, Power[c, 2]]]], Times[2, A, c, d]]], ZeroQ[Plus[Times[2, c, C], Times[-1, B, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[d, -1]], Subst[Int[Times[Power[Plus[Times[Plus[Times[d, e], Times[-1, c, f]], Power[d, -1]], Times[f, x, Power[d, -1]]], m], Power[Plus[Times[C, Power[Power[d, 2], -1]], Times[C, Power[Power[d, 2], -1], Power[x, 2]]], p], Power[Plus[a, Times[b, ArcSinh[x]]], n]], x], x, Plus[c, Times[d, x]]], x], And[FreeQ[List[a, b, c, d, e, f, A, B, C, m, n, p], x], ZeroQ[Plus[Times[B, Plus[1, Power[c, 2]]], Times[-1, 2, A, c, d]]], ZeroQ[Plus[Times[2, c, C], Times[-1, B, d]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCosh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[A, Blank[]]], Times[Optional[Pattern[B, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[C, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[d, -1]], Subst[Int[Times[Power[Plus[Times[Plus[Times[d, e], Times[-1, c, f]], Power[d, -1]], Times[f, x, Power[d, -1]]], m], Power[Plus[Times[-1, C, Power[Power[d, 2], -1]], Times[C, Power[Power[d, 2], -1], Power[x, 2]]], p], Power[Plus[a, Times[b, ArcCosh[x]]], n]], x], x, Plus[c, Times[d, x]]], x], And[FreeQ[List[a, b, c, d, e, f, A, B, C, m, n, p], x], ZeroQ[Plus[Times[B, Plus[1, Times[-1, Power[c, 2]]]], Times[2, A, c, d]]], ZeroQ[Plus[Times[2, c, C], Times[-1, B, d]]]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcSinh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[b, Blank[]]]]], 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Condition[With[List[Set[u, IntHide[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, 2]]], p]], x]]], Plus[Dist[Plus[a, Times[b, ArcCoth[Times[c, x]]]], u, x], Times[-1, Dist[Times[b, c], Int[SimplifyIntegrand[Times[u, Power[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]], -1]], x], x], x]]]], And[FreeQ[List[a, b, c, d, e, m, p], x], Or[And[PositiveIntegerQ[p], Not[And[NegativeIntegerQ[Times[Plus[m, -1], Power[2, -1]]], Greater[Plus[m, Times[2, p], 3], 0]]]], And[PositiveIntegerQ[Times[Plus[m, 1], Power[2, -1]]], Not[And[NegativeIntegerQ[p], Greater[Plus[m, Times[2, p], 3], 0]]]], And[NegativeIntegerQ[Times[Plus[m, Times[2, p], 1], Power[2, -1]]], Not[NegativeIntegerQ[Times[Plus[m, -1], Power[2, -1]]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Power[Plus[a, Times[b, ArcTanh[Times[c, x]]]], n], Times[Power[x, m], Power[Plus[d, Times[e, Power[x, 2]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, m], x], IntegerQ[p], PositiveIntegerQ[n], Or[Greater[p, 0], And[Less[p, -1], IntegerQ[m], Unequal[m, 1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[ExpandIntegrand[Power[Plus[a, Times[b, ArcCoth[Times[c, x]]]], n], Times[Power[x, m], Power[Plus[d, Times[e, Power[x, 2]]], p]], x], x], And[FreeQ[List[a, b, c, d, e, m], x], IntegerQ[p], PositiveIntegerQ[n], Or[Greater[p, 0], And[Less[p, -1], IntegerQ[m], Unequal[m, 1]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[a, Int[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, 2]]], p]], x], x], Dist[b, Int[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, 2]]], p], ArcTanh[Times[c, x]]], x], x]], FreeQ[List[a, b, c, d, e, m, p], x]]], RuleDelayed[HoldPattern[Int[Times[Plus[Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]], Pattern[a, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[a, Int[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, 2]]], p]], x], x], Dist[b, Int[Times[Power[x, m], Power[Plus[d, Times[e, Power[x, 2]]], p], ArcCoth[Times[c, x]]], x], x]], FreeQ[List[a, b, c, d, e, m, p], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[x, m], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcTanh[Times[c, x]]]], n]], x], FreeQ[List[a, b, c, d, e, m, n, p], x]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Defer[Int][Times[Power[x, m], Power[Plus[d, Times[e, Power[x, 2]]], p], Power[Plus[a, Times[b, ArcCoth[Times[c, x]]]], n]], x], FreeQ[List[a, b, c, d, e, m, n, p], x]]], RuleDelayed[HoldPattern[Int[Times[ArcTanh[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[2, -1]], Int[Times[Log[Plus[1, u]], Power[Plus[a, Times[b, ArcTanh[Times[c, x]]]], n], Power[Plus[d, Times[e, Power[x, 2]]], -1]], x], x], Times[-1, Dist[Times[1, Power[2, -1]], Int[Times[Log[Plus[1, Times[-1, u]]], Power[Plus[a, Times[b, ArcTanh[Times[c, x]]]], n], Power[Plus[d, Times[e, Power[x, 2]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[n], Greater[n, 0], ZeroQ[Plus[Power[u, 2], Times[-1, Power[Plus[1, Times[-1, 2, Power[Plus[1, Times[c, x]], -1]]], 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[ArcCoth[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[2, -1]], Int[Times[Log[SimplifyIntegrand[Plus[1, Times[1, Power[u, -1]]], x]], Power[Plus[a, Times[b, ArcCoth[Times[c, x]]]], n], Power[Plus[d, Times[e, Power[x, 2]]], -1]], x], x], Times[-1, Dist[Times[1, Power[2, -1]], Int[Times[Log[SimplifyIntegrand[Plus[1, Times[-1, 1, Power[u, -1]]], x]], Power[Plus[a, Times[b, ArcCoth[Times[c, x]]]], n], Power[Plus[d, Times[e, Power[x, 2]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[n], Greater[n, 0], ZeroQ[Plus[Power[u, 2], Times[-1, Power[Plus[1, Times[-1, 2, Power[Plus[1, Times[c, x]], -1]]], 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[ArcTanh[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[2, -1]], Int[Times[Log[Plus[1, u]], Power[Plus[a, Times[b, ArcTanh[Times[c, x]]]], n], Power[Plus[d, Times[e, Power[x, 2]]], -1]], x], x], Times[-1, Dist[Times[1, Power[2, -1]], Int[Times[Log[Plus[1, Times[-1, u]]], Power[Plus[a, Times[b, ArcTanh[Times[c, x]]]], n], Power[Plus[d, Times[e, Power[x, 2]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[n], Greater[n, 0], ZeroQ[Plus[Power[u, 2], Times[-1, Power[Plus[1, Times[-1, 2, Power[Plus[1, Times[-1, c, x]], -1]]], 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[ArcCoth[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Dist[Times[1, Power[2, -1]], Int[Times[Log[SimplifyIntegrand[Plus[1, Times[1, Power[u, -1]]], x]], Power[Plus[a, Times[b, ArcCoth[Times[c, x]]]], n], Power[Plus[d, Times[e, Power[x, 2]]], -1]], x], x], Times[-1, Dist[Times[1, Power[2, -1]], Int[Times[Log[SimplifyIntegrand[Plus[1, Times[-1, 1, Power[u, -1]]], x]], Power[Plus[a, Times[b, ArcCoth[Times[c, x]]]], n], Power[Plus[d, Times[e, Power[x, 2]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[n], Greater[n, 0], ZeroQ[Plus[Power[u, 2], Times[-1, Power[Plus[1, Times[-1, 2, Power[Plus[1, Times[-1, c, x]], -1]]], 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Log[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[a, Times[b, ArcTanh[Times[c, x]]]], n], PolyLog[2, Together[Plus[1, Times[-1, u]]]], Power[Times[2, c, d], -1]], x], Times[-1, Dist[Times[b, n, Power[2, -1]], Int[Times[Power[Plus[a, Times[b, ArcTanh[Times[c, x]]]], Plus[n, -1]], PolyLog[2, Together[Plus[1, Times[-1, u]]]], Power[Plus[d, Times[e, Power[x, 2]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[n], Greater[n, 0], ZeroQ[Plus[Power[Plus[1, Times[-1, u]], 2], Times[-1, Power[Plus[1, Times[-1, 2, Power[Plus[1, Times[c, x]], -1]]], 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Log[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[a, Times[b, ArcCoth[Times[c, x]]]], n], PolyLog[2, Together[Plus[1, Times[-1, u]]]], Power[Times[2, c, d], -1]], x], Times[-1, Dist[Times[b, n, Power[2, -1]], Int[Times[Power[Plus[a, Times[b, ArcCoth[Times[c, x]]]], Plus[n, -1]], PolyLog[2, Together[Plus[1, Times[-1, u]]]], Power[Plus[d, Times[e, Power[x, 2]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[n], Greater[n, 0], ZeroQ[Plus[Power[Plus[1, Times[-1, u]], 2], Times[-1, Power[Plus[1, Times[-1, 2, Power[Plus[1, Times[c, x]], -1]]], 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Log[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Plus[a, Times[b, ArcTanh[Times[c, x]]]], n], PolyLog[2, Together[Plus[1, Times[-1, u]]]], Power[Times[2, c, d], -1]], x]], Dist[Times[b, n, Power[2, -1]], Int[Times[Power[Plus[a, Times[b, ArcTanh[Times[c, x]]]], Plus[n, -1]], PolyLog[2, Together[Plus[1, Times[-1, u]]]], Power[Plus[d, Times[e, Power[x, 2]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[n], Greater[n, 0], ZeroQ[Plus[Power[Plus[1, Times[-1, u]], 2], Times[-1, Power[Plus[1, Times[-1, 2, Power[Plus[1, Times[-1, c, x]], -1]]], 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Log[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Plus[a, Times[b, ArcCoth[Times[c, x]]]], n], PolyLog[2, Together[Plus[1, Times[-1, u]]]], Power[Times[2, c, d], -1]], x]], Dist[Times[b, n, Power[2, -1]], Int[Times[Power[Plus[a, Times[b, ArcCoth[Times[c, x]]]], Plus[n, -1]], PolyLog[2, Together[Plus[1, Times[-1, u]]]], Power[Plus[d, Times[e, Power[x, 2]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[n], Greater[n, 0], ZeroQ[Plus[Power[Plus[1, Times[-1, u]], 2], Times[-1, Power[Plus[1, Times[-1, 2, Power[Plus[1, Times[-1, c, x]], -1]]], 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], PolyLog[Pattern[p, Blank[]], Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Plus[a, Times[b, ArcTanh[Times[c, x]]]], n], PolyLog[Plus[p, 1], u], Power[Times[2, c, d], -1]], x]], Dist[Times[b, n, Power[2, -1]], Int[Times[Power[Plus[a, Times[b, ArcTanh[Times[c, x]]]], Plus[n, -1]], PolyLog[Plus[p, 1], u], Power[Plus[d, Times[e, Power[x, 2]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, p], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[n], Greater[n, 0], ZeroQ[Plus[Power[u, 2], Times[-1, Power[Plus[1, Times[-1, 2, Power[Plus[1, Times[c, x]], -1]]], 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], PolyLog[Pattern[p, Blank[]], Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Power[Plus[a, Times[b, ArcCoth[Times[c, x]]]], n], PolyLog[Plus[p, 1], u], Power[Times[2, c, d], -1]], x]], Dist[Times[b, n, Power[2, -1]], Int[Times[Power[Plus[a, Times[b, ArcCoth[Times[c, x]]]], Plus[n, -1]], PolyLog[Plus[p, 1], u], Power[Plus[d, Times[e, Power[x, 2]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, p], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[n], Greater[n, 0], ZeroQ[Plus[Power[u, 2], Times[-1, Power[Plus[1, Times[-1, 2, Power[Plus[1, Times[c, x]], -1]]], 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1], PolyLog[Pattern[p, Blank[]], Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[a, Times[b, 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Power[Times[2, c, d], -1]], x], Times[-1, Dist[Times[b, n, Power[2, -1]], Int[Times[Power[Plus[a, Times[b, ArcCoth[Times[c, x]]]], Plus[n, -1]], PolyLog[Plus[p, 1], u], Power[Plus[d, Times[e, Power[x, 2]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e, p], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], RationalQ[n], Greater[n, 0], ZeroQ[Plus[Power[u, 2], Times[-1, Power[Plus[1, Times[-1, 2, Power[Plus[1, Times[-1, c, x]], -1]]], 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], -1], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], -1], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Plus[Times[-1, Log[Plus[a, Times[b, ArcCoth[Times[c, x]]]]]], Log[Plus[a, Times[b, ArcTanh[Times[c, x]]]]]], Power[Times[Power[b, 2], c, d, Plus[ArcCoth[Times[c, x]], Times[-1, ArcTanh[Times[c, x]]]]], -1]], x], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[Times[Power[c, 2], d], e]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[a, Times[b, ArcCoth[Times[c, x]]]], Plus[m, 1]], Power[Plus[a, Times[b, ArcTanh[Times[c, x]]]], n], Power[Times[b, c, d, Plus[m, 1]], -1]], x], Times[-1, Dist[Times[n, Power[Plus[m, 1], -1]], Int[Times[Power[Plus[a, Times[b, ArcCoth[Times[c, x]]]], Plus[m, 1]], Power[Plus[a, Times[b, ArcTanh[Times[c, x]]]], Plus[n, -1]], Power[Plus[d, Times[e, Power[x, 2]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d, e], x], ZeroQ[Plus[Times[Power[c, 2], d], e]], IntegersQ[m, n], Inequality[0, Less, n, LessEqual, m]]]], RuleDelayed[HoldPattern[Int[Times[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[d, Blank[]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[a, Times[b, ArcTanh[Times[c, x]]]], Plus[m, 1]], Power[Plus[a, Times[b, ArcCoth[Times[c, x]]]], n], 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Blank[Symbol]]]], Condition[Plus[Simp[Times[x, Plus[d, Times[e, Log[Plus[f, Times[g, Power[x, 2]]]]]], Plus[a, Times[b, ArcCoth[Times[c, x]]]]], x], Plus[Times[-1, Dist[Times[b, c], Int[Times[x, Plus[d, Times[e, Log[Plus[f, Times[g, Power[x, 2]]]]]], Power[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]], -1]], x], x]], Times[-1, Dist[Times[2, e, g], Int[Times[Power[x, 2], Plus[a, Times[b, ArcCoth[Times[c, x]]]], Power[Plus[f, Times[g, Power[x, 2]]], -1]], x], x]]]], FreeQ[List[a, b, c, d, e, f, g], x]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Log[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[e, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, 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Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[e, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, IntHide[Times[Power[x, m], Plus[d, Times[e, Log[Plus[f, Times[g, Power[x, 2]]]]]]], x]]], Plus[Dist[Plus[a, Times[b, ArcTanh[Times[c, x]]]], u, x], Times[-1, Dist[Times[b, c], Int[ExpandIntegrand[Times[u, Power[Plus[1, Times[-1, Power[c, 2], Power[x, 2]]], -1]], x], x], x]]]], And[FreeQ[List[a, b, c, d, e, f, g], x], PositiveIntegerQ[Times[Plus[m, 1], Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[b, Blank[]]]]], Plus[Optional[Pattern[d, Blank[]]], Times[Log[Plus[Optional[Pattern[f, Blank[]]], Times[Optional[Pattern[g, Blank[]]], Power[Pattern[x, Blank[]], 2]]]], Optional[Pattern[e, Blank[]]]]], Power[Pattern[x, Blank[]], 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RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[d, p], Int[Times[u, Power[Plus[1, Times[c, x, Power[d, -1]]], p], Power[E, Times[n, ArcTanh[Times[a, x]]]], Power[Power[x, p], -1]], x], x], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[Power[c, 2], Times[-1, Power[a, 2], Power[d, 2]]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[-1, Times[n, Power[2, -1]]], Power[c, p]], Int[Times[u, Power[Plus[1, Times[d, Power[Times[c, x], -1]]], p], Power[Plus[1, Times[1, Power[Times[a, x], -1]]], Times[n, Power[2, -1]]], Power[Power[Plus[1, Times[-1, 1, Power[Times[a, x], -1]]], Times[n, Power[2, -1]]], -1]], x], x], And[FreeQ[List[a, c, d, p], x], ZeroQ[Plus[Power[c, 2], Times[-1, Power[a, 2], Power[d, 2]]]], Not[IntegerQ[p]], IntegerQ[Times[n, Power[2, -1]]], PositiveQ[c]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[u, Power[Plus[c, Times[d, Power[x, -1]]], p], Power[Plus[1, Times[a, x]], Times[n, Power[2, -1]]], Power[Power[Plus[1, Times[-1, a, x]], Times[n, Power[2, -1]]], -1]], x], And[FreeQ[List[a, c, d, p], x], ZeroQ[Plus[Power[c, 2], Times[-1, 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Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Plus[n, Times[-1, a, x]], Power[E, Times[n, ArcTanh[Times[a, x]]]], Power[Times[a, c, Plus[Power[n, 2], -1], Sqrt[Plus[c, Times[d, Power[x, 2]]]]], -1]], x], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[Times[Power[a, 2], c], d]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Plus[n, Times[2, a, Plus[p, 1], x]], Power[Plus[c, Times[d, Power[x, 2]]], Plus[p, 1]], Power[E, Times[n, ArcTanh[Times[a, x]]]], Power[Times[a, c, Plus[Power[n, 2], Times[-1, 4, Power[Plus[p, 1], 2]]]], -1]], x], Times[-1, Dist[Times[2, Plus[p, 1], Plus[Times[2, p], 3], Power[Times[c, Plus[Power[n, 2], Times[-1, 4, Power[Plus[p, 1], 2]]]], -1]], Int[Times[Power[Plus[c, Times[d, Power[x, 2]]], Plus[p, 1]], Power[E, Times[n, ArcTanh[Times[a, x]]]]], x], x]]], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[Times[Power[a, 2], c], d]], RationalQ[p], Less[p, -1], Not[IntegerQ[n]], NonzeroQ[Plus[Power[n, 2], Times[-1, 4, Power[Plus[p, 1], 2]]]], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[E, Times[n, ArcTanh[Times[a, x]]]], Power[Times[a, c, n], -1]], x], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[Times[Power[a, 2], c], d]], Not[IntegerQ[Times[n, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, p], Int[Times[Power[Plus[1, Times[-1, Power[a, 2], Power[x, 2]]], Plus[p, Times[-1, n, Power[2, -1]]]], Power[Plus[1, Times[a, x]], n]], x], x], And[FreeQ[List[a, c, d, p], x], ZeroQ[Plus[Times[Power[a, 2], c], d]], IntegerQ[p], PositiveIntegerQ[Times[Plus[n, 1], Power[2, -1]]], Not[IntegerQ[Plus[p, Times[-1, n, Power[2, -1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, p], Int[Times[Power[Plus[1, Times[-1, Power[a, 2], Power[x, 2]]], Plus[p, Times[n, Power[2, -1]]]], Power[Power[Plus[1, Times[-1, a, x]], n], -1]], x], x], And[FreeQ[List[a, c, d, p], x], ZeroQ[Plus[Times[Power[a, 2], c], d]], IntegerQ[p], NegativeIntegerQ[Times[Plus[n, -1], Power[2, -1]]], Not[IntegerQ[Plus[p, Times[-1, n, Power[2, -1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, p], Int[Times[Power[Plus[1, Times[-1, a, x]], Plus[p, Times[-1, n, Power[2, -1]]]], Power[Plus[1, Times[a, x]], Plus[p, Times[n, Power[2, -1]]]]], x], x], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[Times[Power[a, 2], c], d]], Or[IntegerQ[p], PositiveQ[c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, Times[n, Power[2, -1]]], Int[Times[Power[Plus[c, Times[d, Power[x, 2]]], Plus[p, Times[-1, n, Power[2, -1]]]], Power[Plus[1, Times[a, x]], n]], x], x], And[FreeQ[List[a, c, d, p], x], ZeroQ[Plus[Times[Power[a, 2], c], d]], Not[Or[IntegerQ[p], PositiveQ[c]]], PositiveIntegerQ[Times[n, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Power[c, Times[n, Power[2, -1]]], -1]], Int[Times[Power[Plus[c, Times[d, Power[x, 2]]], Plus[p, Times[n, Power[2, -1]]]], Power[Power[Plus[1, Times[-1, a, x]], n], -1]], x], x], And[FreeQ[List[a, c, d, p], x], ZeroQ[Plus[Times[Power[a, 2], c], d]], Not[Or[IntegerQ[p], PositiveQ[c]]], NegativeIntegerQ[Times[n, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[c, IntPart[p]], Power[Plus[c, Times[d, Power[x, 2]]], FracPart[p]], Power[Power[Plus[1, Times[-1, Power[a, 2], Power[x, 2]]], FracPart[p]], -1]], Int[Times[Power[Plus[1, Times[-1, Power[a, 2], Power[x, 2]]], p], Power[E, Times[n, ArcTanh[Times[a, x]]]]], x], x], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[Times[Power[a, 2], c], d]], Not[Or[IntegerQ[p], PositiveQ[c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Plus[1, Times[-1, a, n, x]], Power[E, Times[n, ArcTanh[Times[a, x]]]], Power[Times[d, Plus[Power[n, 2], -1], Sqrt[Plus[c, Times[d, Power[x, 2]]]]], -1]], x], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[Times[Power[a, 2], c], d]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[c, Times[d, Power[x, 2]]], Plus[p, 1]], Power[E, Times[n, ArcTanh[Times[a, x]]]], Power[Times[2, d, Plus[p, 1]], -1]], x], Times[-1, Dist[Times[a, c, n, Power[Times[2, d, Plus[p, 1]], -1]], Int[Times[Power[Plus[c, Times[d, Power[x, 2]]], p], Power[E, Times[n, ArcTanh[Times[a, x]]]]], x], x]]], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[Times[Power[a, 2], c], d]], RationalQ[p], Less[p, -1], Not[IntegerQ[n]], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Plus[1, Times[-1, a, n, x]], Power[Plus[c, Times[d, Power[x, 2]]], Plus[p, 1]], Power[E, Times[n, ArcTanh[Times[a, x]]]], Power[Times[a, d, n, Plus[Power[n, 2], -1]], -1]], x], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[Times[Power[a, 2], c], d]], ZeroQ[Plus[Power[n, 2], Times[2, Plus[p, 1]]]], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], 2], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Times[-1, Simp[Times[Plus[n, Times[2, Plus[p, 1], a, x]], Power[Plus[c, Times[d, Power[x, 2]]], Plus[p, 1]], Power[E, Times[n, ArcTanh[Times[a, x]]]], Power[Times[a, d, Plus[Power[n, 2], Times[-1, 4, Power[Plus[p, 1], 2]]]], -1]], x]], Dist[Times[Plus[Power[n, 2], Times[2, Plus[p, 1]]], Power[Times[d, Plus[Power[n, 2], Times[-1, 4, Power[Plus[p, 1], 2]]]], -1]], Int[Times[Power[Plus[c, Times[d, Power[x, 2]]], Plus[p, 1]], Power[E, Times[n, ArcTanh[Times[a, x]]]]], x], x]], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[Times[Power[a, 2], c], d]], RationalQ[p], Less[p, -1], Not[IntegerQ[n]], NonzeroQ[Plus[Power[n, 2], Times[-1, 4, Power[Plus[p, 1], 2]]]], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, p], Int[Times[Power[x, m], Power[Plus[1, Times[-1, Power[a, 2], Power[x, 2]]], Plus[p, Times[-1, n, Power[2, -1]]]], Power[Plus[1, Times[a, x]], n]], x], x], And[FreeQ[List[a, c, d, m, p], x], ZeroQ[Plus[Times[Power[a, 2], c], d]], Or[IntegerQ[p], PositiveQ[c]], PositiveIntegerQ[Times[Plus[n, 1], Power[2, -1]]], Not[IntegerQ[Plus[p, Times[-1, n, Power[2, -1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, p], Int[Times[Power[x, m], Power[Plus[1, Times[-1, Power[a, 2], Power[x, 2]]], Plus[p, Times[n, Power[2, -1]]]], Power[Power[Plus[1, Times[-1, a, x]], n], -1]], x], x], And[FreeQ[List[a, c, d, m, p], x], ZeroQ[Plus[Times[Power[a, 2], c], d]], Or[IntegerQ[p], PositiveQ[c]], NegativeIntegerQ[Times[Plus[n, -1], Power[2, -1]]], Not[IntegerQ[Plus[p, Times[-1, n, Power[2, -1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, p], Int[Times[Power[x, m], Power[Plus[1, Times[-1, a, x]], Plus[p, Times[-1, n, Power[2, -1]]]], Power[Plus[1, Times[a, x]], Plus[p, Times[n, Power[2, -1]]]]], x], x], And[FreeQ[List[a, c, d, m, n, p], x], ZeroQ[Plus[Times[Power[a, 2], c], d]], Or[IntegerQ[p], PositiveQ[c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, Times[n, Power[2, -1]]], Int[Times[Power[x, m], Power[Plus[c, Times[d, Power[x, 2]]], Plus[p, Times[-1, n, Power[2, -1]]]], Power[Plus[1, Times[a, x]], n]], x], x], And[FreeQ[List[a, c, d, m, p], x], ZeroQ[Plus[Times[Power[a, 2], c], d]], Not[Or[IntegerQ[p], PositiveQ[c]]], PositiveIntegerQ[Times[n, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Power[c, Times[n, Power[2, -1]]], -1]], Int[Times[Power[x, m], Power[Plus[c, Times[d, Power[x, 2]]], Plus[p, Times[n, Power[2, -1]]]], Power[Power[Plus[1, Times[-1, a, x]], n], -1]], x], x], And[FreeQ[List[a, c, d, m, p], x], ZeroQ[Plus[Times[Power[a, 2], c], d]], Not[Or[IntegerQ[p], PositiveQ[c]]], NegativeIntegerQ[Times[n, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[c, IntPart[p]], Power[Plus[c, Times[d, Power[x, 2]]], FracPart[p]], Power[Power[Plus[1, Times[-1, Power[a, 2], Power[x, 2]]], FracPart[p]], -1]], Int[Times[Power[x, m], Power[Plus[1, Times[-1, Power[a, 2], Power[x, 2]]], p], Power[E, Times[n, ArcTanh[Times[a, x]]]]], x], x], And[FreeQ[List[a, c, d, m, n, p], x], ZeroQ[Plus[Times[Power[a, 2], c], d]], Not[Or[IntegerQ[p], PositiveQ[c]]], Not[IntegerQ[Times[n, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[u, Blank[]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, p], Int[Times[u, Power[Plus[1, Times[-1, a, x]], Plus[p, Times[-1, n, Power[2, -1]]]], Power[Plus[1, Times[a, x]], Plus[p, Times[n, Power[2, -1]]]]], x], x], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[Times[Power[a, 2], c], d]], Or[IntegerQ[p], PositiveQ[c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[u, Blank[]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[c, IntPart[p]], Power[Plus[c, Times[d, Power[x, 2]]], FracPart[p]], Power[Times[Power[Plus[1, Times[-1, a, x]], FracPart[p]], Power[Plus[1, Times[a, x]], FracPart[p]]], -1]], Int[Times[u, Power[Plus[1, Times[-1, a, x]], Plus[p, Times[-1, n, Power[2, -1]]]], Power[Plus[1, Times[a, x]], Plus[p, Times[n, Power[2, -1]]]]], x], x], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[Times[Power[a, 2], c], d]], Not[Or[IntegerQ[p], PositiveQ[c]]], IntegerQ[Times[n, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[u, Blank[]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[c, IntPart[p]], Power[Plus[c, Times[d, Power[x, 2]]], FracPart[p]], Power[Power[Plus[1, Times[-1, Power[a, 2], Power[x, 2]]], FracPart[p]], -1]], Int[Times[u, Power[Plus[1, Times[-1, Power[a, 2], Power[x, 2]]], p], Power[E, Times[n, ArcTanh[Times[a, x]]]]], x], x], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[Times[Power[a, 2], c], d]], Not[Or[IntegerQ[p], PositiveQ[c]]], Not[IntegerQ[Times[n, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[d, p], Int[Times[u, Power[Power[x, Times[2, p]], -1], Power[Plus[1, Times[-1, Power[a, 2], Power[x, 2]]], p], Power[E, Times[n, ArcTanh[Times[a, x]]]]], x], x], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[c, Times[Power[a, 2], d]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[c, p], Int[Times[u, Power[Plus[1, Times[-1, 1, Power[Times[a, x], -1]]], p], Power[Plus[1, Times[1, Power[Times[a, x], -1]]], p], Power[E, Times[n, ArcTanh[Times[a, x]]]]], x], x], And[FreeQ[List[a, c, d, p], x], ZeroQ[Plus[c, Times[Power[a, 2], d]]], Not[IntegerQ[p]], IntegerQ[Times[n, Power[2, -1]]], PositiveQ[c]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[x, Times[2, p]], Power[Plus[c, Times[d, Power[Power[x, 2], -1]]], p], Power[Times[Power[Plus[1, Times[-1, a, x]], p], Power[Plus[1, Times[a, x]], p]], -1]], Int[Times[u, Power[Power[x, Times[2, p]], -1], Power[Plus[1, Times[-1, a, x]], p], Power[Plus[1, Times[a, x]], p], Power[E, Times[n, ArcTanh[Times[a, x]]]]], x], x], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[c, Times[Power[a, 2], d]]], Not[IntegerQ[p]], IntegerQ[Times[n, Power[2, -1]]], Not[PositiveQ[c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[x, Times[2, p]], Power[Plus[c, Times[d, Power[Power[x, 2], -1]]], p], Power[Power[Plus[1, Times[c, Power[x, 2], Power[d, -1]]], p], -1]], Int[Times[u, Power[Power[x, Times[2, p]], -1], Power[Plus[1, Times[c, Power[x, 2], Power[d, -1]]], p], Power[E, Times[n, ArcTanh[Times[a, x]]]]], x], x], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[c, Times[Power[a, 2], d]]], Not[IntegerQ[p]], Not[IntegerQ[Times[n, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Power[E, Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[1, Times[a, c], Times[b, c, x]], Times[n, Power[2, -1]]], Power[Power[Plus[1, Times[-1, a, c], Times[-1, b, c, x]], Times[n, Power[2, -1]]], -1]], x], FreeQ[List[a, b, c, n], x]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[4, Power[Times[n, Power[b, Plus[m, 1]], Power[c, Plus[m, 1]]], -1]], Subst[Int[Times[Power[x, Times[2, Power[n, -1]]], Power[Plus[-1, Times[-1, a, c], Times[Plus[1, Times[-1, a, c]], Power[x, Times[2, Power[n, -1]]]]], m], Power[Power[Plus[1, Power[x, Times[2, Power[n, -1]]]], Plus[m, 2]], -1]], x], x, Times[Power[Plus[1, Times[c, Plus[a, Times[b, x]]]], Times[n, Power[2, -1]]], Power[Power[Plus[1, Times[-1, c, Plus[a, Times[b, x]]]], Times[n, Power[2, -1]]], -1]]], x], And[FreeQ[List[a, b, c], x], NegativeIntegerQ[m], RationalQ[n], Less[-1, n, 1]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[1, Times[a, c], Times[b, c, x]], Times[n, Power[2, -1]]], Power[Power[Plus[1, Times[-1, a, c], Times[-1, b, c, x]], Times[n, Power[2, -1]]], -1]], x], FreeQ[List[a, b, c, d, e, m, n], x]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[Times[c, Power[Plus[1, Times[-1, Power[a, 2]]], -1]], p], Int[Times[u, Power[Plus[1, Times[-1, a], Times[-1, b, x]], Plus[p, Times[-1, n, Power[2, -1]]]], Power[Plus[1, a, Times[b, x]], Plus[p, Times[n, Power[2, -1]]]]], x], x], And[FreeQ[List[a, b, c, d, e, n, p], x], ZeroQ[Plus[Times[b, d], Times[-1, 2, a, e]]], ZeroQ[Plus[Times[Power[b, 2], c], Times[e, Plus[1, Times[-1, Power[a, 2]]]]]], Or[IntegerQ[p], PositiveQ[Times[c, Power[Plus[1, Times[-1, Power[a, 2]]], -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[c, Times[d, x], Times[e, Power[x, 2]]], p], Power[Power[Plus[1, Times[-1, Power[a, 2]], Times[-1, 2, a, b, x], Times[-1, Power[b, 2], Power[x, 2]]], p], -1]], Int[Times[u, Power[Plus[1, Times[-1, Power[a, 2]], Times[-1, 2, a, b, x], Times[-1, Power[b, 2], Power[x, 2]]], p], Power[E, Times[n, ArcTanh[Times[a, x]]]]], x], x], And[FreeQ[List[a, b, c, d, e, n, p], x], ZeroQ[Plus[Times[b, d], Times[-1, 2, a, e]]], ZeroQ[Plus[Times[Power[b, 2], c], Times[e, Plus[1, Times[-1, Power[a, 2]]]]]], Not[Or[IntegerQ[p], PositiveQ[Times[c, Power[Plus[1, Times[-1, Power[a, 2]]], -1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcTanh[Times[Optional[Pattern[c, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], -1]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[u, Power[E, Times[n, ArcCoth[Plus[Times[a, Power[c, -1]], Times[b, x, Power[c, -1]]]]]]], x], FreeQ[List[a, b, c, n], x]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[-1, Times[n, Power[2, -1]]], Int[Times[u, Power[E, Times[n, ArcTanh[Times[a, x]]]]], x], x], And[FreeQ[a, x], IntegerQ[Times[n, Power[2, -1]]]]]], RuleDelayed[HoldPattern[Int[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Subst[Int[Times[Power[Plus[1, Times[x, Power[a, -1]]], Times[Plus[n, 1], Power[2, -1]]], Power[Times[Power[x, 2], Power[Plus[1, Times[-1, x, Power[a, -1]]], Times[Plus[n, -1], Power[2, -1]]], Sqrt[Plus[1, Times[-1, Power[x, 2], Power[Power[a, 2], -1]]]]], -1]], x], x, Times[1, Power[x, -1]]]], And[FreeQ[a, x], OddQ[n]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Subst[Int[Times[Power[Plus[1, Times[x, Power[a, -1]]], Times[Plus[n, 1], Power[2, -1]]], Power[Times[Power[x, Plus[m, 2]], Power[Plus[1, Times[-1, x, Power[a, -1]]], Times[Plus[n, -1], Power[2, -1]]], Sqrt[Plus[1, Times[-1, Power[x, 2], Power[Power[a, 2], -1]]]]], -1]], x], x, Times[1, Power[x, -1]]]], And[FreeQ[a, x], OddQ[n], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Subst[Int[Times[Power[Plus[1, Times[x, Power[a, -1]]], Times[n, Power[2, -1]]], Power[Times[Power[x, 2], Power[Plus[1, Times[-1, x, Power[a, -1]]], Times[n, Power[2, -1]]]], -1]], x], x, Times[1, Power[x, -1]]]], And[FreeQ[List[a, n], x], Not[IntegerQ[n]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Subst[Int[Times[Power[Plus[1, Times[x, Power[a, -1]]], Times[n, Power[2, -1]]], Power[Times[Power[x, Plus[m, 2]], Power[Plus[1, Times[-1, x, Power[a, -1]]], Times[n, Power[2, -1]]]], -1]], x], x, Times[1, Power[x, -1]]]], And[FreeQ[List[a, n], x], Not[IntegerQ[n]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Times[Power[x, m], Power[Times[1, Power[x, -1]], m]], Subst[Int[Times[Power[Plus[1, Times[x, Power[a, -1]]], Times[Plus[n, 1], Power[2, -1]]], Power[Times[Power[x, Plus[m, 2]], Power[Plus[1, Times[-1, x, Power[a, -1]]], Times[Plus[n, -1], Power[2, -1]]], Sqrt[Plus[1, Times[-1, Power[x, 2], Power[Power[a, 2], -1]]]]], -1]], x], x, Times[1, Power[x, -1]]], x]], And[FreeQ[List[a, m], x], OddQ[n], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Times[Power[x, m], Power[Times[1, Power[x, -1]], m]], Subst[Int[Times[Power[Plus[1, Times[x, Power[a, -1]]], Times[n, Power[2, -1]]], Power[Times[Power[x, Plus[m, 2]], Power[Plus[1, Times[-1, x, Power[a, -1]]], Times[n, Power[2, -1]]]], -1]], x], x, Times[1, Power[x, -1]]], x]], And[FreeQ[List[a, m, n], x], Not[IntegerQ[n]], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Plus[1, Times[a, x]], Power[Plus[c, Times[d, x]], p], Power[E, Times[n, ArcCoth[Times[a, x]]]], Power[Times[a, Plus[p, 1]], -1]], x], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[Times[a, c], d]], ZeroQ[Plus[p, Times[-1, n, Power[2, -1]]]], Not[IntegerQ[Times[n, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[d, p], Int[Times[u, Power[x, p], Power[Plus[1, Times[c, Power[Times[d, x], -1]]], p], Power[E, Times[n, ArcCoth[Times[a, x]]]]], x], x], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[Times[Power[a, 2], Power[c, 2]], Times[-1, Power[d, 2]]]], Not[IntegerQ[Times[n, Power[2, -1]]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[c, Times[d, x]], p], Power[Times[Power[x, p], Power[Plus[1, Times[c, Power[Times[d, x], -1]]], p]], -1]], Int[Times[u, Power[x, p], Power[Plus[1, Times[c, Power[Times[d, x], -1]]], p], Power[E, Times[n, ArcCoth[Times[a, x]]]]], x], x], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[Times[Power[a, 2], Power[c, 2]], Times[-1, Power[d, 2]]]], Not[IntegerQ[Times[n, Power[2, -1]]]], Not[IntegerQ[p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Power[c, n], Subst[Int[Times[Power[Plus[c, Times[d, x]], Plus[p, Times[-1, n]]], Power[Plus[1, Times[-1, Power[x, 2], Power[Power[a, 2], -1]]], Times[n, Power[2, -1]]], Power[Power[x, 2], -1]], x], x, Times[1, Power[x, -1]]], x]], And[FreeQ[List[a, c, d, p], x], ZeroQ[Plus[c, Times[a, d]]], IntegerQ[Times[Plus[n, -1], Power[2, -1]]], Or[IntegerQ[p], ZeroQ[Plus[p, Times[-1, n, Power[2, -1]]]], ZeroQ[Plus[p, Times[-1, n, Power[2, -1]], -1]]], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Power[c, n], Subst[Int[Times[Power[Plus[c, Times[d, x]], Plus[p, Times[-1, n]]], Power[Plus[1, Times[-1, Power[x, 2], Power[Power[a, 2], -1]]], Times[n, Power[2, -1]]], Power[Power[x, Plus[m, 2]], -1]], x], x, Times[1, Power[x, -1]]], x]], And[FreeQ[List[a, c, d, p], x], ZeroQ[Plus[c, Times[a, d]]], IntegerQ[Times[Plus[n, -1], Power[2, -1]]], IntegerQ[m], Or[IntegerQ[p], ZeroQ[Plus[p, Times[-1, n, Power[2, -1]]]], ZeroQ[Plus[p, Times[-1, n, Power[2, -1]], -1]], Less[-5, m, -1]], IntegerQ[Times[2, p]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Power[c, p], Subst[Int[Times[Power[Plus[1, Times[d, x, Power[c, -1]]], p], Power[Plus[1, Times[x, Power[a, -1]]], Times[n, Power[2, -1]]], Power[Times[Power[x, 2], Power[Plus[1, Times[-1, x, Power[a, -1]]], Times[n, Power[2, -1]]]], -1]], x], x, Times[1, Power[x, -1]]], x]], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[Power[c, 2], Times[-1, Power[a, 2], Power[d, 2]]]], Not[IntegerQ[Times[n, Power[2, -1]]]], Or[IntegerQ[p], PositiveQ[c]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Power[c, p], Subst[Int[Times[Power[Plus[1, Times[d, x, Power[c, -1]]], p], Power[Plus[1, Times[x, Power[a, -1]]], Times[n, Power[2, -1]]], Power[Times[Power[x, Plus[m, 2]], Power[Plus[1, Times[-1, x, Power[a, -1]]], Times[n, Power[2, -1]]]], -1]], x], x, Times[1, Power[x, -1]]], x]], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[Power[c, 2], Times[-1, Power[a, 2], Power[d, 2]]]], Not[IntegerQ[Times[n, Power[2, -1]]]], Or[IntegerQ[p], PositiveQ[c]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Times[Power[c, p], Power[x, m], Power[Times[1, Power[x, -1]], m]], Subst[Int[Times[Power[Plus[1, Times[d, x, Power[c, -1]]], p], Power[Plus[1, Times[x, Power[a, -1]]], Times[n, Power[2, -1]]], Power[Times[Power[x, Plus[m, 2]], Power[Plus[1, Times[-1, x, Power[a, -1]]], Times[n, Power[2, -1]]]], -1]], x], x, Times[1, Power[x, -1]]], x]], And[FreeQ[List[a, c, d, m, n, p], x], ZeroQ[Plus[Power[c, 2], Times[-1, Power[a, 2], Power[d, 2]]]], Not[IntegerQ[Times[n, Power[2, -1]]]], Or[IntegerQ[p], PositiveQ[c]], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -1]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[c, Times[d, Power[x, -1]]], p], Power[Power[Plus[1, Times[d, Power[Times[c, x], -1]]], p], -1]], Int[Times[u, Power[Plus[1, Times[d, Power[Times[c, x], -1]]], p], Power[E, Times[n, ArcCoth[Times[a, x]]]]], x], x], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[Power[c, 2], Times[-1, Power[a, 2], Power[d, 2]]]], Not[IntegerQ[Times[n, Power[2, -1]]]], Not[Or[IntegerQ[p], PositiveQ[c]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], -1]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Power[E, Times[n, ArcCoth[Times[a, x]]]], Power[Times[a, c, n], -1]], x], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[Times[Power[a, 2], c], d]], Not[IntegerQ[Times[n, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Pattern[n, Blank[]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-3, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Simp[Times[Plus[n, Times[-1, a, x]], Power[E, Times[n, ArcCoth[Times[a, x]]]], Power[Times[a, c, Plus[Power[n, 2], -1], Sqrt[Plus[c, Times[d, Power[x, 2]]]]], -1]], x], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[Times[Power[a, 2], c], d]], 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Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Times[Power[Times[-1, c], p], Power[Power[a, Plus[m, 1]], -1]], Subst[Int[Times[Power[E, Times[n, x]], Power[Coth[x], Plus[m, Times[2, Plus[p, 1]]]], Power[Power[Cosh[x], Times[2, Plus[p, 1]]], -1]], x], x, ArcCoth[Times[a, x]]], x]], And[FreeQ[List[a, c, d, n], x], ZeroQ[Plus[Times[Power[a, 2], c], d]], Not[IntegerQ[Times[n, Power[2, -1]]]], IntegerQ[m], RationalQ[p], LessEqual[3, m, Times[-2, Plus[p, 1]]], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[d, p], Int[Times[u, Power[x, Times[2, p]], Power[Plus[1, 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Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Power[c, p], Subst[Int[Times[Power[Plus[1, Times[-1, x, Power[a, -1]]], Plus[p, Times[-1, n, Power[2, -1]]]], Power[Plus[1, Times[x, Power[a, -1]]], Plus[p, Times[n, Power[2, -1]]]], Power[Power[x, 2], -1]], x], x, Times[1, Power[x, -1]]], x]], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[c, Times[Power[a, 2], d]]], Not[IntegerQ[Times[n, Power[2, -1]]]], Or[IntegerQ[p], PositiveQ[c]], Not[IntegersQ[Times[2, p], Plus[p, Times[n, Power[2, -1]]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Power[c, p], Subst[Int[Times[Power[Plus[1, Times[-1, x, Power[a, -1]]], Plus[p, Times[-1, n, Power[2, -1]]]], Power[Plus[1, Times[x, Power[a, -1]]], Plus[p, Times[n, Power[2, -1]]]], Power[Power[x, Plus[m, 2]], -1]], x], x, Times[1, Power[x, -1]]], x]], And[FreeQ[List[a, c, d, n, p], x], ZeroQ[Plus[c, Times[Power[a, 2], d]]], Not[IntegerQ[Times[n, Power[2, -1]]]], Or[IntegerQ[p], PositiveQ[c]], Not[IntegersQ[Times[2, p], Plus[p, Times[n, Power[2, -1]]]]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Optional[Pattern[p, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Times[-1, Dist[Times[Power[c, p], Power[x, m], Power[Times[1, Power[x, -1]], m]], Subst[Int[Times[Power[Plus[1, Times[-1, x, Power[a, -1]]], Plus[p, Times[-1, n, Power[2, -1]]]], Power[Plus[1, Times[x, Power[a, -1]]], Plus[p, Times[n, Power[2, -1]]]], Power[Power[x, Plus[m, 2]], -1]], x], x, Times[1, Power[x, -1]]], x]], And[FreeQ[List[a, c, d, m, n, p], x], ZeroQ[Plus[c, Times[Power[a, 2], d]]], Not[IntegerQ[Times[n, Power[2, -1]]]], Or[IntegerQ[p], PositiveQ[c]], Not[IntegersQ[Times[2, p], Plus[p, Times[n, Power[2, -1]]]]], Not[IntegerQ[m]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], -2]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[c, IntPart[p]], Power[Plus[c, Times[d, Power[Power[x, 2], -1]]], FracPart[p]], Power[Power[Plus[1, Times[-1, 1, Power[Times[Power[a, 2], Power[x, 2]], -1]]], 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Blank[Symbol]]]], Condition[Dist[Times[Power[Times[c, Plus[a, Times[b, x]]], Times[n, Power[2, -1]]], Power[Plus[1, Times[1, Power[Times[c, Plus[a, Times[b, x]]], -1]]], Times[n, Power[2, -1]]], Power[Power[Plus[1, Times[a, c], Times[b, c, x]], Times[n, Power[2, -1]]], -1]], Int[Times[Power[Plus[1, Times[a, c], Times[b, c, x]], Times[n, Power[2, -1]]], Power[Power[Plus[-1, Times[a, c], Times[b, c, x]], Times[n, Power[2, -1]]], -1]], x], x], And[FreeQ[List[a, b, c, n], x], Not[IntegerQ[Times[n, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]], Pattern[n, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[-4, Power[Times[n, Power[b, Plus[m, 1]], Power[c, Plus[m, 1]]], -1]], Subst[Int[Times[Power[x, Times[2, Power[n, -1]]], Power[Plus[1, Times[a, c], Times[Plus[1, Times[-1, a, 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Times[a, c], Times[b, c, x]], Times[n, Power[2, -1]]], -1]], Int[Times[Power[Plus[d, Times[e, x]], m], Power[Plus[1, Times[a, c], Times[b, c, x]], Times[n, Power[2, -1]]], Power[Power[Plus[-1, Times[a, c], Times[b, c, x]], Times[n, Power[2, -1]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, m, n], x], Not[IntegerQ[Times[n, Power[2, -1]]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[E, Times[ArcCoth[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[n, Blank[]]]]], Optional[Pattern[u, Blank[]]], Power[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Times[c, Power[Plus[1, Times[-1, Power[a, 2]]], -1]], p], Power[Times[Plus[a, Times[b, x]], Power[Plus[1, a, Times[b, x]], -1]], Times[n, Power[2, -1]]], Power[Times[Plus[1, a, Times[b, x]], 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Optional[Pattern[u, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Int[Times[u, Power[E, Times[n, ArcTanh[Plus[Times[a, Power[c, -1]], Times[b, x, Power[c, -1]]]]]]], x], FreeQ[List[a, b, c, n], x]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcTanh[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[d, -1]], Subst[Int[Power[Plus[a, Times[b, ArcTanh[x]]], n], x], x, Plus[c, Times[d, x]]], x], And[FreeQ[List[a, b, c, d], x], PositiveIntegerQ[n]]]], RuleDelayed[HoldPattern[Int[Power[Plus[Optional[Pattern[a, Blank[]]], Times[ArcCoth[Plus[Pattern[c, Blank[]], Times[Optional[Pattern[d, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[b, Blank[]]]]], Optional[Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[d, -1]], Subst[Int[Power[Plus[a, Times[b, ArcCoth[x]]], 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Times[-1, Power[c, 2]]]], ZeroQ[Plus[Times[b, d], Times[-1, a, e]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[ArcCoth[Times[Optional[Pattern[c, Blank[]]], Pattern[x, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[d, Blank[]]], Times[Optional[Pattern[e, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Rational[-1, 2]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Sqrt[Plus[a, Times[b, Power[x, 2]]]], Power[Sqrt[Plus[d, Times[e, Power[x, 2]]]], -1]], Int[Times[Power[ArcCoth[Times[c, x, Power[Sqrt[Plus[a, Times[b, Power[x, 2]]]], -1]]], m], Power[Sqrt[Plus[a, Times[b, Power[x, 2]]]], -1]], x], x], And[FreeQ[List[a, b, c, d, e, m], x], ZeroQ[Plus[b, Times[-1, Power[c, 2]]]], ZeroQ[Plus[Times[b, d], Times[-1, a, e]]]]]], RuleDelayed[HoldPattern[Int[Times[ArcTanh[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, IntHide[Power[Plus[c, Times[d, Power[x, 2]]], n], x]]], Plus[Dist[ArcTanh[Times[a, x]], u, x], Times[-1, Dist[a, Int[Dist[Times[1, Power[Plus[1, Times[-1, Power[a, 2], Power[x, 2]]], -1]], u, x], x], x]]]], And[FreeQ[List[a, c, d], x], IntegerQ[Times[2, n]], LessEqual[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[ArcCoth[Times[Optional[Pattern[a, Blank[]]], Pattern[x, Blank[]]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], 2]]], Pattern[n, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[u, IntHide[Power[Plus[c, Times[d, Power[x, 2]]], n], x]]], Plus[Dist[ArcCoth[Times[a, x]], u, x], Times[-1, Dist[a, Int[Dist[Times[1, Power[Plus[1, Times[-1, Power[a, 2], Power[x, 2]]], -1]], u, x], x], x]]]], And[FreeQ[List[a, c, d], x], IntegerQ[Times[2, n]], LessEqual[n, -1]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Power[Pattern[v, Blank[]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[tmp, InverseFunctionOfLinear[u, x]]], Condition[Dist[Times[Power[Times[-1, Discriminant[v, x], Power[Times[4, Coefficient[v, x, 2]], -1]], n], Power[Coefficient[Part[tmp, 1], x, 1], -1]], Subst[Int[SimplifyIntegrand[Times[SubstForInverseFunction[u, tmp, x], Power[Sech[x], Times[2, Plus[n, 1]]]], x], x], x, tmp], x], And[Not[FalseQ[tmp]], SameQ[Head[tmp], ArcTanh], ZeroQ[Plus[Times[Discriminant[v, x], Power[Part[tmp, 1], 2]], Times[-1, Power[D[v, x], 2]]]]]]], And[QuadraticQ[v, x], IntegerQ[n], Less[n, 0], PosQ[Discriminant[v, x]], MatchQ[u, Condition[Times[Optional[Pattern[r, Blank[]]], Power[Pattern[f, Blank[]], Pattern[w, Blank[]]]], FreeQ[f, x]]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Power[Pattern[v, Blank[]], Optional[Pattern[n, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[With[List[Set[tmp, InverseFunctionOfLinear[u, x]]], Condition[Dist[Times[Power[Times[-1, Discriminant[v, x], Power[Times[4, Coefficient[v, x, 2]], -1]], n], Power[Coefficient[Part[tmp, 1], x, 1], -1]], Subst[Int[SimplifyIntegrand[Times[SubstForInverseFunction[u, tmp, x], Power[Times[-1, Power[Csch[x], 2]], Plus[n, 1]]], x], x], x, tmp], x], And[Not[FalseQ[tmp]], SameQ[Head[tmp], ArcCoth], ZeroQ[Plus[Times[Discriminant[v, x], Power[Part[tmp, 1], 2]], Times[-1, Power[D[v, x], 2]]]]]]], And[QuadraticQ[v, x], IntegerQ[n], Less[n, 0], PosQ[Discriminant[v, x]], MatchQ[u, Condition[Times[Optional[Pattern[r, Blank[]]], Power[Pattern[f, Blank[]], Pattern[w, Blank[]]]], FreeQ[f, x]]]]]], RuleDelayed[HoldPattern[Int[ArcTanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[x, ArcTanh[Plus[c, Times[d, Tanh[Plus[a, Times[b, x]]]]]]], x], Dist[b, Int[Times[x, Power[Plus[c, Times[-1, d], Times[c, Power[E, Plus[Times[2, a], Times[2, b, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d], x], ZeroQ[Plus[Power[Plus[c, Times[-1, d]], 2], -1]]]]], RuleDelayed[HoldPattern[Int[ArcCoth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[x, ArcCoth[Plus[c, Times[d, Tanh[Plus[a, Times[b, x]]]]]]], x], Dist[b, Int[Times[x, Power[Plus[c, Times[-1, d], Times[c, Power[E, Plus[Times[2, a], Times[2, b, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d], x], ZeroQ[Plus[Power[Plus[c, Times[-1, d]], 2], -1]]]]], RuleDelayed[HoldPattern[Int[ArcTanh[Plus[Optional[Pattern[c, Blank[]]], Times[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[x, ArcTanh[Plus[c, Times[d, Coth[Plus[a, Times[b, x]]]]]]], x], Dist[b, Int[Times[x, Power[Plus[c, Times[-1, d], Times[-1, c, Power[E, Plus[Times[2, a], Times[2, b, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d], x], ZeroQ[Plus[Power[Plus[c, Times[-1, d]], 2], -1]]]]], RuleDelayed[HoldPattern[Int[ArcCoth[Plus[Optional[Pattern[c, Blank[]]], Times[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[x, ArcCoth[Plus[c, Times[d, Coth[Plus[a, Times[b, x]]]]]]], x], Dist[b, Int[Times[x, Power[Plus[c, Times[-1, d], Times[-1, c, Power[E, Plus[Times[2, a], Times[2, b, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d], x], ZeroQ[Plus[Power[Plus[c, Times[-1, d]], 2], -1]]]]], RuleDelayed[HoldPattern[Int[ArcTanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[x, ArcTanh[Plus[c, Times[d, Tanh[Plus[a, Times[b, x]]]]]]], x], Plus[Dist[Times[b, Plus[1, Times[-1, c], Times[-1, d]]], Int[Times[x, Power[E, Plus[Times[2, a], Times[2, b, x]]], Power[Plus[1, Times[-1, c], d, Times[Plus[1, Times[-1, c], Times[-1, d]], Power[E, Plus[Times[2, a], Times[2, b, x]]]]], -1]], x], x], Times[-1, Dist[Times[b, Plus[1, c, d]], Int[Times[x, Power[E, Plus[Times[2, a], Times[2, b, x]]], Power[Plus[1, c, Times[-1, d], Times[Plus[1, c, d], Power[E, Plus[Times[2, a], Times[2, b, x]]]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Power[Plus[c, Times[-1, d]], 2], -1]]]]], RuleDelayed[HoldPattern[Int[ArcCoth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[x, ArcCoth[Plus[c, Times[d, Tanh[Plus[a, Times[b, x]]]]]]], x], Plus[Dist[Times[b, Plus[1, Times[-1, c], Times[-1, d]]], Int[Times[x, Power[E, Plus[Times[2, a], Times[2, b, x]]], Power[Plus[1, Times[-1, c], d, Times[Plus[1, Times[-1, c], Times[-1, d]], Power[E, Plus[Times[2, a], Times[2, b, x]]]]], -1]], x], x], Times[-1, Dist[Times[b, Plus[1, c, d]], Int[Times[x, Power[E, Plus[Times[2, a], Times[2, b, x]]], Power[Plus[1, c, Times[-1, d], Times[Plus[1, c, d], Power[E, Plus[Times[2, a], Times[2, b, x]]]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Power[Plus[c, Times[-1, d]], 2], -1]]]]], RuleDelayed[HoldPattern[Int[ArcTanh[Plus[Optional[Pattern[c, Blank[]]], Times[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[x, ArcTanh[Plus[c, Times[d, Coth[Plus[a, Times[b, x]]]]]]], x], Plus[Times[-1, Dist[Times[b, Plus[1, Times[-1, c], Times[-1, d]]], Int[Times[x, Power[E, Plus[Times[2, a], Times[2, b, x]]], Power[Plus[1, Times[-1, c], d, Times[-1, Plus[1, Times[-1, c], Times[-1, d]], Power[E, Plus[Times[2, a], Times[2, b, x]]]]], -1]], x], x]], Dist[Times[b, Plus[1, c, d]], Int[Times[x, Power[E, Plus[Times[2, a], Times[2, b, x]]], Power[Plus[1, c, Times[-1, d], Times[-1, Plus[1, c, d], Power[E, Plus[Times[2, a], Times[2, b, x]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Power[Plus[c, Times[-1, d]], 2], -1]]]]], RuleDelayed[HoldPattern[Int[ArcCoth[Plus[Optional[Pattern[c, Blank[]]], Times[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[x, ArcCoth[Plus[c, Times[d, Coth[Plus[a, Times[b, x]]]]]]], x], Plus[Times[-1, Dist[Times[b, Plus[1, Times[-1, c], Times[-1, d]]], Int[Times[x, Power[E, Plus[Times[2, a], Times[2, b, x]]], Power[Plus[1, Times[-1, c], d, Times[-1, Plus[1, Times[-1, c], Times[-1, d]], Power[E, Plus[Times[2, a], Times[2, b, x]]]]], -1]], x], x]], Dist[Times[b, Plus[1, c, d]], Int[Times[x, Power[E, Plus[Times[2, a], Times[2, b, x]]], Power[Plus[1, c, Times[-1, d], Times[-1, Plus[1, c, d], Power[E, Plus[Times[2, a], Times[2, b, x]]]]], -1]], x], x]]], And[FreeQ[List[a, b, c, d], x], NonzeroQ[Plus[Power[Plus[c, Times[-1, d]], 2], -1]]]]], RuleDelayed[HoldPattern[Int[Times[ArcTanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[e, Times[f, x]], Plus[m, 1]], ArcTanh[Plus[c, Times[d, Tanh[Plus[a, Times[b, x]]]]]], Power[Times[f, Plus[m, 1]], -1]], x], Dist[Times[b, Power[Times[f, Plus[m, 1]], -1]], Int[Times[Power[Plus[e, Times[f, x]], Plus[m, 1]], Power[Plus[c, Times[-1, d], Times[c, Power[E, Plus[Times[2, a], Times[2, b, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], PositiveIntegerQ[m], ZeroQ[Plus[Power[Plus[c, Times[-1, d]], 2], -1]]]]], RuleDelayed[HoldPattern[Int[Times[ArcCoth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[e, Times[f, x]], Plus[m, 1]], ArcCoth[Plus[c, Times[d, Tanh[Plus[a, Times[b, x]]]]]], Power[Times[f, Plus[m, 1]], -1]], x], Dist[Times[b, Power[Times[f, Plus[m, 1]], -1]], Int[Times[Power[Plus[e, Times[f, x]], Plus[m, 1]], Power[Plus[c, Times[-1, d], Times[c, Power[E, Plus[Times[2, a], Times[2, b, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], PositiveIntegerQ[m], ZeroQ[Plus[Power[Plus[c, Times[-1, d]], 2], -1]]]]], RuleDelayed[HoldPattern[Int[Times[ArcTanh[Plus[Optional[Pattern[c, Blank[]]], Times[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[e, Times[f, x]], Plus[m, 1]], ArcTanh[Plus[c, Times[d, Coth[Plus[a, Times[b, x]]]]]], Power[Times[f, Plus[m, 1]], -1]], x], Dist[Times[b, Power[Times[f, Plus[m, 1]], -1]], Int[Times[Power[Plus[e, Times[f, x]], Plus[m, 1]], Power[Plus[c, Times[-1, d], Times[-1, c, Power[E, Plus[Times[2, a], Times[2, b, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], PositiveIntegerQ[m], ZeroQ[Plus[Power[Plus[c, Times[-1, d]], 2], -1]]]]], RuleDelayed[HoldPattern[Int[Times[ArcCoth[Plus[Optional[Pattern[c, Blank[]]], Times[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[e, Times[f, x]], Plus[m, 1]], ArcCoth[Plus[c, Times[d, Coth[Plus[a, Times[b, x]]]]]], Power[Times[f, Plus[m, 1]], -1]], x], Dist[Times[b, Power[Times[f, Plus[m, 1]], -1]], Int[Times[Power[Plus[e, Times[f, x]], Plus[m, 1]], Power[Plus[c, Times[-1, d], Times[-1, c, Power[E, Plus[Times[2, a], Times[2, b, x]]]]], -1]], x], x]], And[FreeQ[List[a, b, c, d, e, f], x], PositiveIntegerQ[m], ZeroQ[Plus[Power[Plus[c, Times[-1, d]], 2], -1]]]]], RuleDelayed[HoldPattern[Int[Times[ArcTanh[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[e, Times[f, x]], Plus[m, 1]], ArcTanh[Plus[c, Times[d, Tanh[Plus[a, Times[b, x]]]]]], Power[Times[f, Plus[m, 1]], -1]], x], Plus[Dist[Times[b, Plus[1, Times[-1, c], Times[-1, d]], Power[Times[f, Plus[m, 1]], -1]], Int[Times[Power[Plus[e, Times[f, x]], Plus[m, 1]], Power[E, Plus[Times[2, a], Times[2, b, x]]], Power[Plus[1, Times[-1, c], d, Times[Plus[1, Times[-1, c], Times[-1, d]], Power[E, Plus[Times[2, a], Times[2, b, x]]]]], -1]], x], x], Times[-1, Dist[Times[b, Plus[1, c, d], Power[Times[f, Plus[m, 1]], -1]], Int[Times[Power[Plus[e, Times[f, x]], Plus[m, 1]], Power[E, Plus[Times[2, a], Times[2, b, x]]], Power[Plus[1, c, Times[-1, d], Times[Plus[1, c, d], Power[E, Plus[Times[2, a], Times[2, b, x]]]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d, e, f], x], PositiveIntegerQ[m], NonzeroQ[Plus[Power[Plus[c, Times[-1, d]], 2], -1]]]]], RuleDelayed[HoldPattern[Int[Times[ArcCoth[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Tanh[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[e, Times[f, x]], Plus[m, 1]], ArcCoth[Plus[c, Times[d, Tanh[Plus[a, Times[b, x]]]]]], Power[Times[f, Plus[m, 1]], -1]], x], Plus[Dist[Times[b, Plus[1, Times[-1, c], Times[-1, d]], Power[Times[f, Plus[m, 1]], -1]], Int[Times[Power[Plus[e, Times[f, x]], Plus[m, 1]], Power[E, Plus[Times[2, a], Times[2, b, x]]], Power[Plus[1, Times[-1, c], d, Times[Plus[1, Times[-1, c], Times[-1, d]], Power[E, Plus[Times[2, a], Times[2, b, x]]]]], -1]], x], x], Times[-1, Dist[Times[b, Plus[1, c, d], Power[Times[f, Plus[m, 1]], -1]], Int[Times[Power[Plus[e, Times[f, x]], Plus[m, 1]], Power[E, Plus[Times[2, a], Times[2, b, x]]], Power[Plus[1, c, Times[-1, d], Times[Plus[1, c, d], Power[E, Plus[Times[2, a], Times[2, b, x]]]]], -1]], x], x]]]], And[FreeQ[List[a, b, c, d, e, f], x], PositiveIntegerQ[m], NonzeroQ[Plus[Power[Plus[c, Times[-1, d]], 2], -1]]]]], RuleDelayed[HoldPattern[Int[Times[ArcTanh[Plus[Optional[Pattern[c, Blank[]]], Times[Coth[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]]], Optional[Pattern[d, Blank[]]]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Pattern[x, Blank[]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[Plus[e, Times[f, x]], Plus[m, 1]], ArcTanh[Plus[c, Times[d, Coth[Plus[a, Times[b, x]]]]]], Power[Times[f, Plus[m, 1]], 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1]], -1]], Int[Times[Power[x, Plus[m, 1]], Cos[Times[Pi, Power[b, 2], Power[x, 2]]]], x], x], Times[-1, Simp[Times[b, Power[x, Plus[m, 2]], Power[Times[2, Plus[m, 1], Plus[m, 2]], -1]], x]]]], And[FreeQ[List[b, c], x], ZeroQ[Plus[c, Times[-1, Pi, Power[2, -1], Power[b, 2]]]], IntegerQ[m], Less[m, -2], Equal[Mod[m, 4], 0]]]], RuleDelayed[HoldPattern[Int[Times[Cos[Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], 2]]], FresnelC[Times[Optional[Pattern[b, Blank[]]], Pattern[x, Blank[]]]], Power[Pattern[x, Blank[]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Plus[Simp[Times[Power[x, Plus[m, 1]], Cos[Times[Pi, Power[2, -1], Power[b, 2], Power[x, 2]]], FresnelC[Times[b, x]], Power[Plus[m, 1], -1]], x], Plus[Dist[Times[Pi, Power[b, 2], Power[Plus[m, 1], -1]], Int[Times[Power[x, Plus[m, 2]], Sin[Times[Pi, Power[2, -1], Power[b, 2], Power[x, 2]]], FresnelC[Times[b, x]]], x], x], Times[-1, Dist[Times[b, Power[Times[2, Plus[m, 1]], -1]], Int[Times[Power[x, Plus[m, 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2]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[Plus[Times[a, Power[e, 2]], Times[-1, c, Power[f, 2]]], m], Int[ExpandIntegrand[Times[u, Power[Plus[Times[e, Sqrt[Plus[a, Times[b, Power[x, n]]]]], Times[-1, f, Sqrt[Plus[c, Times[d, Power[x, n]]]]]], Times[-1, m]]], x], x], x], And[FreeQ[List[a, b, c, d, e, f, n], x], NegativeIntegerQ[m], ZeroQ[Plus[Times[b, Power[e, 2]], Times[-1, d, Power[f, 2]]]]]]], RuleDelayed[HoldPattern[Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Times[Optional[Pattern[e, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Rational[1, 2]]], Times[Optional[Pattern[f, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]]], Rational[1, 2]]]], Pattern[m, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Power[Plus[Times[b, Power[e, 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Condition[Dist[r, Subst[Int[Times[Power[Plus[a, Times[b, x]], m], Power[Plus[c, Times[d, x]], n], Power[Plus[e, Times[f, x]], p], Power[Plus[g, Times[h, x]], q]], x], x, y], x], Not[FalseQ[r]]]], And[FreeQ[List[a, b, c, d, e, f, g, h, m, n, p, q], x], ZeroQ[Plus[y, Times[-1, v]]], ZeroQ[Plus[y, Times[-1, w]]], ZeroQ[Plus[y, Times[-1, z]]]]]], RuleDelayed[HoldPattern[Int[Times[Optional[Pattern[u, Blank[]]], Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[y, Blank[]], Pattern[n, Blank[]]]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[q, DerivativeDivides[y, u, x]]], Condition[Plus[Dist[a, Int[u, x], x], Dist[Times[b, q], Subst[Int[Power[x, n], x], x, y], x]], Not[FalseQ[q]]]], FreeQ[List[a, b, n], x]]], RuleDelayed[HoldPattern[Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[y, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[q, DerivativeDivides[y, u, x]]], Condition[Dist[q, Subst[Int[Power[Plus[a, Times[b, Power[x, n]]], p], x], x, y], x], Not[FalseQ[q]]]], FreeQ[List[a, b, n, p], x]]], RuleDelayed[HoldPattern[Int[Times[Optional[Pattern[u, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[y, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[q, r], Condition[Dist[Times[q, r], Subst[Int[Times[Power[x, m], Power[Plus[a, Times[b, Power[x, n]]], p]], x], x, y], x], And[Not[FalseQ[Set[r, Divides[Power[y, m], Power[v, m], x]]]], Not[FalseQ[Set[q, DerivativeDivides[y, u, x]]]]]]], FreeQ[List[a, b, m, n, p], x]]], RuleDelayed[HoldPattern[Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[n2, 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RuleDelayed[HoldPattern[Int[Times[Optional[Pattern[u, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[w, Blank[]], Optional[Pattern[n2, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[y, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[q, r], Condition[Dist[Times[q, r], Subst[Int[Times[Power[x, m], Power[Plus[a, Times[b, Power[x, n]], Times[c, Power[x, Times[2, n]]]], p]], x], x, y], x], And[Not[FalseQ[Set[r, Divides[Power[y, m], Power[v, m], x]]]], Not[FalseQ[Set[q, DerivativeDivides[y, u, x]]]]]]], And[FreeQ[List[a, b, c, m, n, p], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[y, Times[-1, w]]]]]], RuleDelayed[HoldPattern[Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, 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Power[Pattern[y, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[q, DerivativeDivides[y, u, x]]], Condition[Dist[q, Subst[Int[Times[Power[Plus[a, Times[b, Power[x, n]]], m], Power[Plus[c, Times[d, Power[x, n]]], p]], x], x, y], x], Not[FalseQ[q]]]], And[FreeQ[List[a, b, c, d, m, n, p], x], ZeroQ[Plus[y, Times[-1, v]]]]]], RuleDelayed[HoldPattern[Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[c, Blank[]]], Times[Optional[Pattern[d, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[p, Blank[]]]], Power[Plus[Optional[Pattern[e, Blank[]]], Times[Optional[Pattern[f, Blank[]]], Power[Pattern[w, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[q, Blank[]]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[y, Blank[]], Pattern[n, Blank[]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[r, DerivativeDivides[y, u, x]]], Condition[Dist[r, Subst[Int[Times[Power[Plus[a, Times[b, Power[x, n]]], m], Power[Plus[c, Times[d, Power[x, n]]], p], Power[Plus[e, Times[f, Power[x, n]]], q]], x], x, y], x], Not[FalseQ[r]]]], And[FreeQ[List[a, b, c, d, e, f, m, n, p, q], x], ZeroQ[Plus[y, Times[-1, v]]], ZeroQ[Plus[y, Times[-1, w]]]]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[F, Blank[]], Pattern[v, Blank[]]], Pattern[u, Blank[]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[q, DerivativeDivides[v, u, x]]], Condition[Simp[Times[q, Power[F, v], Power[Log[F], -1]], x], Not[FalseQ[q]]]], FreeQ[F, x]]], RuleDelayed[HoldPattern[Int[Times[Power[Pattern[F, Blank[]], Pattern[v, Blank[]]], Pattern[u, Blank[]], Power[Pattern[w, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[q, DerivativeDivides[v, u, x]]], Condition[Dist[q, Subst[Int[Times[Power[x, m], Power[F, x]], x], x, v], x], Not[FalseQ[q]]]], And[FreeQ[List[F, m], x], ZeroQ[Plus[w, Times[-1, v]]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Pattern[w, Blank[]], Optional[Pattern[p, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[c, Simplify[Times[u, Power[Plus[Times[w, D[v, x]], Times[v, D[w, x]]], -1]]]]], Condition[Dist[c, Subst[Int[Power[Plus[a, Times[b, Power[x, p]]], m], x], x, Times[v, w]], x], FreeQ[c, x]]], And[FreeQ[List[a, b, m, p], x], IntegerQ[p]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Power[Pattern[v, Blank[]], Optional[Pattern[r, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Pattern[w, Blank[]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[c, Simplify[Times[u, Power[Plus[Times[p, w, D[v, x]], Times[q, v, D[w, x]]], -1]]]]], Condition[Dist[Times[c, p, Power[Plus[r, 1], -1]], Subst[Int[Power[Plus[a, Times[b, Power[x, Times[p, Power[Plus[r, 1], -1]]]]], m], x], x, Times[Power[v, Plus[r, 1]], w]], x], FreeQ[c, x]]], And[FreeQ[List[a, b, m, p, q, r], x], ZeroQ[Plus[p, Times[-1, q, Plus[r, 1]]]], NonzeroQ[Plus[r, 1]], IntegerQ[Times[p, Power[Plus[r, 1], -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Power[Pattern[v, Blank[]], Optional[Pattern[r, Blank[]]]], Power[Pattern[w, Blank[]], Optional[Pattern[s, Blank[]]]], Power[Plus[Pattern[a, Blank[]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[p, Blank[]]]], Power[Pattern[w, Blank[]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[c, Simplify[Times[u, Power[Plus[Times[p, w, D[v, x]], Times[q, v, D[w, x]]], -1]]]]], Condition[Dist[Times[c, p, Power[Plus[r, 1], -1]], Subst[Int[Power[Plus[a, Times[b, Power[x, Times[p, Power[Plus[r, 1], -1]]]]], m], x], x, Times[Power[v, Plus[r, 1]], Power[w, Plus[s, 1]]]], x], FreeQ[c, x]]], And[FreeQ[List[a, b, m, p, q, r, s], x], ZeroQ[Plus[Times[p, Plus[s, 1]], Times[-1, q, Plus[r, 1]]]], NonzeroQ[Plus[r, 1]], IntegerQ[Times[p, Power[Plus[r, 1], -1]]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[p, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[w, Blank[]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[c, Simplify[Times[u, Power[Plus[Times[p, w, D[v, x]], Times[-1, q, v, D[w, x]]], -1]]]]], Condition[Dist[Times[c, p], Subst[Int[Power[Plus[b, Times[a, Power[x, p]]], m], x], x, Times[v, Power[w, Plus[Times[m, q], 1]]]], x], FreeQ[c, x]]], And[FreeQ[List[a, b, m, p, q], x], ZeroQ[Plus[p, Times[q, Plus[Times[m, p], 1]]]], IntegerQ[p], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Power[Pattern[v, Blank[]], Optional[Pattern[r, Blank[]]]], Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[p, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[w, Blank[]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[c, Simplify[Times[u, Power[Plus[Times[p, w, D[v, x]], Times[-1, q, v, D[w, x]]], -1]]]]], Condition[Times[-1, Dist[Times[c, q], Subst[Int[Power[Plus[a, Times[b, Power[x, q]]], m], x], x, Times[Power[v, Plus[Times[m, p], r, 1]], w]], x]], FreeQ[c, x]]], And[FreeQ[List[a, b, m, p, q, r], x], ZeroQ[Plus[p, Times[q, Plus[Times[m, p], r, 1]]]], IntegerQ[q], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Power[Pattern[w, Blank[]], Optional[Pattern[s, Blank[]]]], 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Power[Plus[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[p, Blank[]]]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[w, Blank[]], Optional[Pattern[q, Blank[]]]]]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Module[List[Set[c, Simplify[Times[u, Power[Plus[Times[p, w, D[v, x]], Times[-1, q, v, D[w, x]]], -1]]]]], Condition[Times[-1, Dist[Times[c, q, Power[Plus[s, 1], -1]], Subst[Int[Power[Plus[a, Times[b, Power[x, Times[q, Power[Plus[s, 1], -1]]]]], m], x], x, Times[Power[v, Plus[Times[m, p], r, 1]], Power[w, Plus[s, 1]]]], x]], FreeQ[c, x]]], And[FreeQ[List[a, b, m, p, q, r, s], x], ZeroQ[Plus[Times[p, Plus[s, 1]], Times[q, Plus[Times[m, p], r, 1]]]], NonzeroQ[Plus[s, 1]], IntegerQ[Times[q, Power[Plus[s, 1], -1]]], IntegerQ[m]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Power[Pattern[x, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Plus[m, 1], 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Blank[]]]], Power[Pattern[w, Blank[]], Optional[Pattern[n, Blank[]]]], Power[Pattern[z, Blank[]], Optional[Pattern[q, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[a, IntPart[p]], Power[Times[a, Power[v, m], Power[w, n], Power[z, q]], FracPart[p]], Power[Times[Power[v, Times[m, FracPart[p]]], Power[w, Times[n, FracPart[p]]], Power[z, Times[q, FracPart[p]]]], -1]], Int[Times[u, Power[v, Times[m, p]], Power[w, Times[n, p]], Power[z, Times[p, q]]], x], x], And[FreeQ[List[a, m, n, p, q], x], Not[IntegerQ[p]], Not[FreeQ[v, x]], Not[FreeQ[w, x]], Not[FreeQ[z, x]]]]], RuleDelayed[HoldPattern[Int[Times[Optional[Pattern[u, Blank[]]], Power[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[m, Blank[]]]], Power[Pattern[w, Blank[]], Optional[Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[a, IntPart[p]], Power[Times[a, Power[v, m], Power[w, n]], FracPart[p]], Power[Times[Power[v, Times[m, FracPart[p]]], Power[w, Times[n, FracPart[p]]]], -1]], Int[Times[u, Power[v, Times[m, p]], Power[w, Times[n, p]]], x], x], And[FreeQ[List[a, m, n, p], x], Not[IntegerQ[p]], Not[FreeQ[v, x]], Not[FreeQ[w, x]]]]], RuleDelayed[HoldPattern[Int[Times[Optional[Pattern[u, Blank[]]], Power[Times[Optional[Pattern[a, Blank[]]], Power[Pattern[v, Blank[]], Optional[Pattern[m, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[a, IntPart[p]], Power[Times[a, Power[v, m]], FracPart[p]], Power[Power[v, Times[m, FracPart[p]]], -1]], Int[Times[u, Power[v, Times[m, p]]], x], x], And[FreeQ[List[a, m, p], x], Not[IntegerQ[p]], Not[FreeQ[v, x]], Not[And[EqQ[a, 1], EqQ[m, 1]]], Not[And[EqQ[v, x], EqQ[m, 1]]]]]], RuleDelayed[HoldPattern[Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[FullSimplify[Times[Sqrt[Plus[a, Times[b, Power[x, n]]]], Power[Times[Power[x, Times[n, Power[2, -1]]], Sqrt[Plus[b, Times[a, Power[Power[x, n], -1]]]]], -1]]], Int[Times[u, Power[x, Times[n, p]], Power[Plus[b, Times[a, Power[x, Times[-1, n]]]], p]], x], x], And[FreeQ[List[a, b, p], x], IntegerQ[Plus[p, Times[1, Power[2, -1]]]], NegativeIntegerQ[n], Not[RationalFunctionQ[u, x]]]]], RuleDelayed[HoldPattern[Int[Times[Optional[Pattern[u, Blank[]]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[v, Blank[]], Pattern[n, Blank[]]]]], Pattern[p, Blank[]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[Power[Plus[a, Times[b, Power[v, n]]], FracPart[p]], Power[Times[Power[v, Times[n, FracPart[p]]], Power[Plus[b, Times[a, Power[v, Times[-1, n]]]], FracPart[p]]], -1]], Int[Times[u, Power[v, Times[n, p]], Power[Plus[b, Times[a, Power[v, Times[-1, n]]]], p]], x], x], And[FreeQ[List[a, b, p], x], Not[IntegerQ[p]], 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RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Optional[Pattern[p, Blank[]]]]], Pattern[x, Blank[Symbol]]]], Condition[Dist[Times[1, Power[Times[Power[4, p], Power[c, p]], -1]], Int[Times[u, Power[Plus[b, Times[2, c, Power[x, n]]], Times[2, p]]], x], x], And[FreeQ[List[a, b, c, n], x], ZeroQ[Plus[n2, Times[-1, 2, n]]], ZeroQ[Plus[Power[b, 2], Times[-1, 4, a, c]]], IntegerQ[p], Not[AlgebraicFunctionQ[u, x]]]]], RuleDelayed[HoldPattern[Int[Times[Pattern[u, Blank[]], Power[Plus[Optional[Pattern[a, Blank[]]], Times[Optional[Pattern[b, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n, Blank[]]]]], Times[Optional[Pattern[c, Blank[]]], Power[Pattern[x, Blank[]], Optional[Pattern[n2, Blank[]]]]]], Pattern[p, Blank[]]]], Pattern[x, 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