1"""
2This code is automatically generated. Never edit it manually.
3For details of generating the code see `rubi_parsing_guide.md` in `parsetools`.
4"""
5
6from sympy.external import import_module
7matchpy = import_module("matchpy")
8
9if matchpy:
10    from matchpy import Pattern, ReplacementRule, CustomConstraint, is_match
11    from sympy.integrals.rubi.utility_function import (
12        Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ,
13        ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ,
14        PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ,
15        ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ,
16        NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart,
17        FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest,
18        SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient,
19        Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart,
20        IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan,
21        ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec,
22        ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less,
23        Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ,
24        PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ,
25        ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ,
26        Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ,
27        SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator,
28        NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ,
29        InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ,
30        EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree,
31        PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts,
32        TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ,
33        NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll,
34        ExpandLinearProduct, GCD, ContentFactor, NumericFactor,
35        NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst,
36        ExpandExpression, Apart, SmartApart, MatchQ,
37        PolynomialQuotientRemainder, FreeFactors, NonfreeFactors,
38        RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms,
39        ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup,
40        AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor,
41        RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon,
42        MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ,
43        GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList,
44        PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ,
45        RationalFunctionFactors, NonrationalFunctionFactors, Reverse,
46        RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand,
47        SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree,
48        CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree,
49        GeneralizedBinomialParts, GeneralizedTrinomialDegree,
50        GeneralizedTrinomialParts, MonomialQ, MonomialSumQ,
51        MinimumMonomialExponent, MonomialExponent, LinearMatchQ,
52        PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ,
53        TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ,
54        QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms,
55        NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial,
56        PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD,
57        AlgebraicFunctionFactors, NonalgebraicFunctionFactors,
58        QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ,
59        Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors,
60        NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop,
61        CombineExponents, FactorInteger, FactorAbsurdNumber,
62        SubstForInverseFunction, SubstForFractionalPower,
63        SubstForFractionalPowerOfQuotientOfLinears,
64        FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ,
65        SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ,
66        FractionalPowerSubexpressionQ, Apply, FactorNumericGcd,
67        MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ,
68        TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest,
69        OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors,
70        PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn,
71        PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree,
72        FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify,
73        FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand,
74        NormalizeIntegrandAux, NormalizeIntegrandFactor,
75        NormalizeIntegrandFactorBase, NormalizeTogether,
76        NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors,
77        SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm,
78        TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum,
79        UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear,
80        PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ,
81        IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor,
82        FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ,
83        FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator,
84        SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand,
85        SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM,
86        SubstForFractionalPowerOfLinear, FractionalPowerOfLinear,
87        InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig,
88        FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ,
89        PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ,
90        KnownTangentIntegrandQ, KnownCotangentIntegrandQ,
91        KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst,
92        AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand,
93        ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp,
94        ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ,
95        FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ,
96        PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ,
97        FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ,
98        FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ,
99        FunctionOfLog, PowerVariableExpn, PowerVariableDegree,
100        PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic,
101        SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ,
102        Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ,
103        SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2,
104        ConstantFactor, SameQ, ReplacePart, CommonFactors,
105        MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential,
106        FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux,
107        FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev,
108        rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent,
109        RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct,
110        SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma,
111        FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ,
112        _SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify,
113        _SimplifyAntiderivativeSum, SimplifyAntiderivativeSum,
114        _SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux,
115        TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor,
116        PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral,
117        CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial,
118        Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ, Null, rubi_exp as exp, rubi_log as log, Discriminant,
119        Negative, Quotient
120    )
121    from sympy import (Integral, S, sqrt, And, Or, Integer, Float, Mod, I, Abs, simplify, Mul,
122    Add, Pow, sign, EulerGamma)
123    from sympy.integrals.rubi.symbol import WC
124    from sympy.core.symbol import symbols, Symbol
125    from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf)
126    from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch)
127    from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec, atan2)
128    from sympy import pi as Pi
129
130    A_, B_, C_, F_, G_, H_, a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_, m_, n_, p_, q_, r_, t_, u_, v_, s_, w_, x_, y_, z_ = [WC(i) for i in 'ABCFGHabcdefghijklmnpqrtuvswxyz']
131    a1_, a2_, b1_, b2_, c1_, c2_, d1_, d2_, n1_, n2_, e1_, e2_, f1_, f2_, g1_, g2_, n1_, n2_, n3_, Pq_, Pm_, Px_, Qm_, Qr_, Qx_, jn_, mn_, non2_, RFx_, RGx_ = [WC(i) for i in ['a1', 'a2', 'b1', 'b2', 'c1', 'c2', 'd1', 'd2', 'n1', 'n2', 'e1', 'e2', 'f1', 'f2', 'g1', 'g2', 'n1', 'n2', 'n3', 'Pq', 'Pm', 'Px', 'Qm', 'Qr', 'Qx', 'jn', 'mn', 'non2', 'RFx', 'RGx']]
132    i, ii, Pqq, Q, R, r, C, k, u = symbols('i ii Pqq Q R r C k u')
133    _UseGamma = False
134    ShowSteps = False
135    StepCounter = None
136
137
138def special_functions():
139    from sympy.integrals.rubi.constraints import cons69, cons2, cons3, cons68, cons19, cons1266, cons8, cons29, cons20, cons168, cons1959, cons1960, cons96, cons263, cons1961, cons1834, cons64, cons1962, cons1963, cons1964, cons249, cons1965, cons1966, cons1967, cons1833, cons4, cons1257, cons21, cons1361, cons1968, cons1969, cons170, cons1970, cons1971, cons33, cons1972, cons1973, cons1974, cons802, cons89, cons90, cons5, cons52, cons91, cons385, cons50, cons1975, cons1976, cons1977, cons54, cons1978, cons1101, cons127, cons1245, cons13, cons139, cons1381, cons1979, cons1980, cons198, cons1981, cons1982, cons1983, cons152, cons465, cons1767, cons165, cons950, cons951, cons1984, cons1985, cons805, cons1986, cons1987, cons1988, cons1989, cons340, cons1990, cons1991, cons1992, cons1993, cons1994, cons1995, cons40, cons1996, cons349, cons1997, cons1998, cons1999, cons2000, cons2001, cons2002, cons2003
140
141
142    pattern6742 = Pattern(Integral(Erf(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons69)
143    rule6742 = ReplacementRule(pattern6742, replacement6742)
144
145    pattern6743 = Pattern(Integral(Erfc(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons69)
146    rule6743 = ReplacementRule(pattern6743, replacement6743)
147
148    pattern6744 = Pattern(Integral(Erfi(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons69)
149    rule6744 = ReplacementRule(pattern6744, replacement6744)
150
151    pattern6745 = Pattern(Integral(Erf(x_*WC('b', S(1)))/x_, x_), cons3, cons3)
152    rule6745 = ReplacementRule(pattern6745, replacement6745)
153
154    pattern6746 = Pattern(Integral(Erfc(x_*WC('b', S(1)))/x_, x_), cons3, cons3)
155    rule6746 = ReplacementRule(pattern6746, replacement6746)
156
157    pattern6747 = Pattern(Integral(Erfi(x_*WC('b', S(1)))/x_, x_), cons3, cons3)
158    rule6747 = ReplacementRule(pattern6747, replacement6747)
159
160    pattern6748 = Pattern(Integral(x_**WC('m', S(1))*Erf(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons19, cons68)
161    rule6748 = ReplacementRule(pattern6748, replacement6748)
162
163    pattern6749 = Pattern(Integral(x_**WC('m', S(1))*Erfc(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons19, cons68)
164    rule6749 = ReplacementRule(pattern6749, replacement6749)
165
166    pattern6750 = Pattern(Integral(x_**WC('m', S(1))*Erfi(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons19, cons68)
167    rule6750 = ReplacementRule(pattern6750, replacement6750)
168
169    pattern6751 = Pattern(Integral(x_*Erf(x_*WC('b', S(1)) + WC('a', S(0)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons1266)
170    rule6751 = ReplacementRule(pattern6751, replacement6751)
171
172    pattern6752 = Pattern(Integral(x_*Erfc(x_*WC('b', S(1)) + WC('a', S(0)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons1266)
173    rule6752 = ReplacementRule(pattern6752, replacement6752)
174
175    pattern6753 = Pattern(Integral(x_*Erfi(x_*WC('b', S(1)) + WC('a', S(0)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons1266)
176    rule6753 = ReplacementRule(pattern6753, replacement6753)
177
178    pattern6754 = Pattern(Integral(x_**m_*Erf(x_*WC('b', S(1)) + WC('a', S(0)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons168)
179    rule6754 = ReplacementRule(pattern6754, replacement6754)
180
181    pattern6755 = Pattern(Integral(x_**m_*Erfc(x_*WC('b', S(1)) + WC('a', S(0)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons168)
182    rule6755 = ReplacementRule(pattern6755, replacement6755)
183
184    pattern6756 = Pattern(Integral(x_**m_*Erfi(x_*WC('b', S(1)) + WC('a', S(0)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons168)
185    rule6756 = ReplacementRule(pattern6756, replacement6756)
186
187    pattern6757 = Pattern(Integral(Erf(x_*WC('b', S(1)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0)))/x_, x_), cons3, cons1959)
188    rule6757 = ReplacementRule(pattern6757, replacement6757)
189
190    pattern6758 = Pattern(Integral(Erfc(x_*WC('b', S(1)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0)))/x_, x_), cons3, cons1959)
191    rule6758 = ReplacementRule(pattern6758, replacement6758)
192
193    pattern6759 = Pattern(Integral(Erfi(x_*WC('b', S(1)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0)))/x_, x_), cons3, cons1960)
194    rule6759 = ReplacementRule(pattern6759, replacement6759)
195
196    pattern6760 = Pattern(Integral(x_**m_*Erf(x_*WC('b', S(1)) + WC('a', S(0)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons96)
197    rule6760 = ReplacementRule(pattern6760, replacement6760)
198
199    pattern6761 = Pattern(Integral(x_**m_*Erfc(x_*WC('b', S(1)) + WC('a', S(0)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons96)
200    rule6761 = ReplacementRule(pattern6761, replacement6761)
201
202    pattern6762 = Pattern(Integral(x_**m_*Erfi(x_*WC('b', S(1)) + WC('a', S(0)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons96)
203    rule6762 = ReplacementRule(pattern6762, replacement6762)
204
205    pattern6763 = Pattern(Integral(Erf(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons69)
206    rule6763 = ReplacementRule(pattern6763, replacement6763)
207
208    pattern6764 = Pattern(Integral(Erfc(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons69)
209    rule6764 = ReplacementRule(pattern6764, replacement6764)
210
211    pattern6765 = Pattern(Integral(Erfi(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons69)
212    rule6765 = ReplacementRule(pattern6765, replacement6765)
213
214    pattern6766 = Pattern(Integral(x_**WC('m', S(1))*Erf(x_*WC('b', S(1)))**S(2), x_), cons3, cons20, cons263, cons1961)
215    rule6766 = ReplacementRule(pattern6766, replacement6766)
216
217    pattern6767 = Pattern(Integral(x_**WC('m', S(1))*Erfc(x_*WC('b', S(1)))**S(2), x_), cons3, cons20, cons1834, cons1961)
218    rule6767 = ReplacementRule(pattern6767, replacement6767)
219
220    pattern6768 = Pattern(Integral(x_**WC('m', S(1))*Erfi(x_*WC('b', S(1)))**S(2), x_), cons3, cons20, cons1834, cons1961)
221    rule6768 = ReplacementRule(pattern6768, replacement6768)
222
223    pattern6769 = Pattern(Integral(x_**WC('m', S(1))*Erf(a_ + x_*WC('b', S(1)))**S(2), x_), cons2, cons3, cons64)
224    rule6769 = ReplacementRule(pattern6769, replacement6769)
225
226    pattern6770 = Pattern(Integral(x_**WC('m', S(1))*Erfc(a_ + x_*WC('b', S(1)))**S(2), x_), cons2, cons3, cons64)
227    rule6770 = ReplacementRule(pattern6770, replacement6770)
228
229    pattern6771 = Pattern(Integral(x_**WC('m', S(1))*Erfi(a_ + x_*WC('b', S(1)))**S(2), x_), cons2, cons3, cons64)
230    rule6771 = ReplacementRule(pattern6771, replacement6771)
231
232    pattern6772 = Pattern(Integral(FresnelS(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons69)
233    rule6772 = ReplacementRule(pattern6772, replacement6772)
234
235    pattern6773 = Pattern(Integral(FresnelC(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons69)
236    rule6773 = ReplacementRule(pattern6773, replacement6773)
237
238    pattern6774 = Pattern(Integral(FresnelS(x_*WC('b', S(1)))/x_, x_), cons3, cons3)
239    rule6774 = ReplacementRule(pattern6774, replacement6774)
240
241    pattern6775 = Pattern(Integral(FresnelC(x_*WC('b', S(1)))/x_, x_), cons3, cons3)
242    rule6775 = ReplacementRule(pattern6775, replacement6775)
243
244    pattern6776 = Pattern(Integral(x_**WC('m', S(1))*FresnelS(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons19, cons68)
245    rule6776 = ReplacementRule(pattern6776, replacement6776)
246
247    pattern6777 = Pattern(Integral(x_**WC('m', S(1))*FresnelC(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons19, cons68)
248    rule6777 = ReplacementRule(pattern6777, replacement6777)
249
250    pattern6778 = Pattern(Integral(FresnelS(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons69)
251    rule6778 = ReplacementRule(pattern6778, replacement6778)
252
253    pattern6779 = Pattern(Integral(FresnelC(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons69)
254    rule6779 = ReplacementRule(pattern6779, replacement6779)
255
256    pattern6780 = Pattern(Integral(x_**m_*FresnelS(x_*WC('b', S(1)))**S(2), x_), cons3, cons20, cons1834, cons1962)
257    rule6780 = ReplacementRule(pattern6780, replacement6780)
258
259    pattern6781 = Pattern(Integral(x_**m_*FresnelC(x_*WC('b', S(1)))**S(2), x_), cons3, cons20, cons1834, cons1962)
260    rule6781 = ReplacementRule(pattern6781, replacement6781)
261
262    pattern6782 = Pattern(Integral(x_*FresnelS(x_*WC('b', S(1)))*sin(x_**S(2)*WC('c', S(1))), x_), cons3, cons8, cons1963)
263    rule6782 = ReplacementRule(pattern6782, replacement6782)
264
265    pattern6783 = Pattern(Integral(x_*FresnelC(x_*WC('b', S(1)))*cos(x_**S(2)*WC('c', S(1))), x_), cons3, cons8, cons1963)
266    rule6783 = ReplacementRule(pattern6783, replacement6783)
267
268    pattern6784 = Pattern(Integral(x_**m_*FresnelS(x_*WC('b', S(1)))*sin(x_**S(2)*WC('c', S(1))), x_), cons3, cons8, cons1963, cons20, cons168, cons1964)
269    rule6784 = ReplacementRule(pattern6784, replacement6784)
270
271    pattern6785 = Pattern(Integral(x_**m_*FresnelC(x_*WC('b', S(1)))*cos(x_**S(2)*WC('c', S(1))), x_), cons3, cons8, cons1963, cons20, cons168, cons1964)
272    rule6785 = ReplacementRule(pattern6785, replacement6785)
273
274    pattern6786 = Pattern(Integral(x_**m_*FresnelS(x_*WC('b', S(1)))*sin(x_**S(2)*WC('c', S(1))), x_), cons3, cons8, cons1963, cons20, cons249, cons1965)
275    rule6786 = ReplacementRule(pattern6786, replacement6786)
276
277    pattern6787 = Pattern(Integral(x_**m_*FresnelC(x_*WC('b', S(1)))*cos(x_**S(2)*WC('c', S(1))), x_), cons3, cons8, cons1963, cons20, cons249, cons1965)
278    rule6787 = ReplacementRule(pattern6787, replacement6787)
279
280    pattern6788 = Pattern(Integral(x_*FresnelS(x_*WC('b', S(1)))*cos(x_**S(2)*WC('c', S(1))), x_), cons3, cons8, cons1963)
281    rule6788 = ReplacementRule(pattern6788, replacement6788)
282
283    pattern6789 = Pattern(Integral(x_*FresnelC(x_*WC('b', S(1)))*sin(x_**S(2)*WC('c', S(1))), x_), cons3, cons8, cons1963)
284    rule6789 = ReplacementRule(pattern6789, replacement6789)
285
286    pattern6790 = Pattern(Integral(x_**m_*FresnelS(x_*WC('b', S(1)))*cos(x_**S(2)*WC('c', S(1))), x_), cons3, cons8, cons1963, cons20, cons168, cons1966)
287    rule6790 = ReplacementRule(pattern6790, replacement6790)
288
289    pattern6791 = Pattern(Integral(x_**m_*FresnelC(x_*WC('b', S(1)))*sin(x_**S(2)*WC('c', S(1))), x_), cons3, cons8, cons1963, cons20, cons168, cons1966)
290    rule6791 = ReplacementRule(pattern6791, replacement6791)
291
292    pattern6792 = Pattern(Integral(x_**m_*FresnelS(x_*WC('b', S(1)))*cos(x_**S(2)*WC('c', S(1))), x_), cons3, cons8, cons1963, cons20, cons96, cons1967)
293    rule6792 = ReplacementRule(pattern6792, replacement6792)
294
295    pattern6793 = Pattern(Integral(x_**m_*FresnelC(x_*WC('b', S(1)))*sin(x_**S(2)*WC('c', S(1))), x_), cons3, cons8, cons1963, cons20, cons96, cons1967)
296    rule6793 = ReplacementRule(pattern6793, replacement6793)
297
298    pattern6794 = Pattern(Integral(ExpIntegralE(n_, x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons4, cons1833)
299    rule6794 = ReplacementRule(pattern6794, replacement6794)
300
301    pattern6795 = Pattern(Integral(x_**WC('m', S(1))*ExpIntegralE(n_, x_*WC('b', S(1))), x_), cons3, cons1257, cons64)
302    rule6795 = ReplacementRule(pattern6795, replacement6795)
303
304    pattern6796 = Pattern(Integral(ExpIntegralE(S(1), x_*WC('b', S(1)))/x_, x_), cons3, cons3)
305    rule6796 = ReplacementRule(pattern6796, replacement6796)
306
307    pattern6797 = Pattern(Integral(x_**m_*ExpIntegralE(n_, x_*WC('b', S(1))), x_), cons3, cons1257, cons20, cons96)
308    rule6797 = ReplacementRule(pattern6797, replacement6797)
309
310    pattern6798 = Pattern(Integral(x_**m_*ExpIntegralE(n_, x_*WC('b', S(1))), x_), cons3, cons19, cons4, cons1257, cons21)
311    rule6798 = ReplacementRule(pattern6798, replacement6798)
312
313    pattern6799 = Pattern(Integral(x_**WC('m', S(1))*ExpIntegralE(n_, x_*WC('b', S(1))), x_), cons3, cons19, cons4, cons1361)
314    rule6799 = ReplacementRule(pattern6799, replacement6799)
315
316    pattern6800 = Pattern(Integral(x_**WC('m', S(1))*ExpIntegralE(n_, a_ + x_*WC('b', S(1))), x_), cons2, cons3, cons19, cons4, cons1968)
317    rule6800 = ReplacementRule(pattern6800, replacement6800)
318
319    pattern6801 = Pattern(Integral(x_**WC('m', S(1))*ExpIntegralE(n_, a_ + x_*WC('b', S(1))), x_), cons2, cons3, cons19, cons1969, cons68)
320    rule6801 = ReplacementRule(pattern6801, replacement6801)
321
322    pattern6802 = Pattern(Integral(ExpIntegralEi(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons69)
323    rule6802 = ReplacementRule(pattern6802, replacement6802)
324
325    pattern6803 = Pattern(Integral(x_**WC('m', S(1))*ExpIntegralEi(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons19, cons68)
326    rule6803 = ReplacementRule(pattern6803, replacement6803)
327
328    pattern6804 = Pattern(Integral(ExpIntegralEi(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons69)
329    rule6804 = ReplacementRule(pattern6804, replacement6804)
330
331    pattern6805 = Pattern(Integral(x_**WC('m', S(1))*ExpIntegralEi(x_*WC('b', S(1)))**S(2), x_), cons3, cons64)
332    rule6805 = ReplacementRule(pattern6805, replacement6805)
333
334    pattern6806 = Pattern(Integral(x_**WC('m', S(1))*ExpIntegralEi(a_ + x_*WC('b', S(1)))**S(2), x_), cons2, cons3, cons64)
335    rule6806 = ReplacementRule(pattern6806, replacement6806)
336
337    pattern6807 = Pattern(Integral(ExpIntegralEi(x_*WC('d', S(1)) + WC('c', S(0)))*exp(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons1266)
338    rule6807 = ReplacementRule(pattern6807, replacement6807)
339
340    pattern6808 = Pattern(Integral(x_**WC('m', S(1))*ExpIntegralEi(x_*WC('d', S(1)) + WC('c', S(0)))*exp(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons64)
341    rule6808 = ReplacementRule(pattern6808, replacement6808)
342
343    pattern6809 = Pattern(Integral(x_**m_*ExpIntegralEi(x_*WC('d', S(1)) + WC('c', S(0)))*exp(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons96)
344    rule6809 = ReplacementRule(pattern6809, replacement6809)
345
346    pattern6810 = Pattern(Integral(LogIntegral(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons69)
347    rule6810 = ReplacementRule(pattern6810, replacement6810)
348
349    pattern6811 = Pattern(Integral(LogIntegral(x_*WC('b', S(1)))/x_, x_), cons3, cons3)
350    rule6811 = ReplacementRule(pattern6811, replacement6811)
351
352    pattern6812 = Pattern(Integral(x_**WC('m', S(1))*LogIntegral(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons19, cons68)
353    rule6812 = ReplacementRule(pattern6812, replacement6812)
354
355    pattern6813 = Pattern(Integral(SinIntegral(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons69)
356    rule6813 = ReplacementRule(pattern6813, replacement6813)
357
358    pattern6814 = Pattern(Integral(CosIntegral(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons69)
359    rule6814 = ReplacementRule(pattern6814, replacement6814)
360
361    pattern6815 = Pattern(Integral(SinIntegral(x_*WC('b', S(1)))/x_, x_), cons3, cons3)
362    rule6815 = ReplacementRule(pattern6815, replacement6815)
363
364    pattern6816 = Pattern(Integral(CosIntegral(x_*WC('b', S(1)))/x_, x_), cons3, cons3)
365    rule6816 = ReplacementRule(pattern6816, replacement6816)
366
367    pattern6817 = Pattern(Integral(x_**WC('m', S(1))*SinIntegral(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons19, cons68)
368    rule6817 = ReplacementRule(pattern6817, replacement6817)
369
370    pattern6818 = Pattern(Integral(x_**WC('m', S(1))*CosIntegral(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons19, cons68)
371    rule6818 = ReplacementRule(pattern6818, replacement6818)
372
373    pattern6819 = Pattern(Integral(SinIntegral(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons69)
374    rule6819 = ReplacementRule(pattern6819, replacement6819)
375
376    pattern6820 = Pattern(Integral(CosIntegral(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons69)
377    rule6820 = ReplacementRule(pattern6820, replacement6820)
378
379    pattern6821 = Pattern(Integral(x_**WC('m', S(1))*SinIntegral(x_*WC('b', S(1)))**S(2), x_), cons3, cons64)
380    rule6821 = ReplacementRule(pattern6821, replacement6821)
381
382    pattern6822 = Pattern(Integral(x_**WC('m', S(1))*CosIntegral(x_*WC('b', S(1)))**S(2), x_), cons3, cons64)
383    rule6822 = ReplacementRule(pattern6822, replacement6822)
384
385    pattern6823 = Pattern(Integral(x_**WC('m', S(1))*SinIntegral(a_ + x_*WC('b', S(1)))**S(2), x_), cons2, cons3, cons64)
386    rule6823 = ReplacementRule(pattern6823, replacement6823)
387
388    pattern6824 = Pattern(Integral(x_**WC('m', S(1))*CosIntegral(a_ + x_*WC('b', S(1)))**S(2), x_), cons2, cons3, cons64)
389    rule6824 = ReplacementRule(pattern6824, replacement6824)
390
391    pattern6825 = Pattern(Integral(SinIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sin(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons1266)
392    rule6825 = ReplacementRule(pattern6825, replacement6825)
393
394    pattern6826 = Pattern(Integral(CosIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cos(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons1266)
395    rule6826 = ReplacementRule(pattern6826, replacement6826)
396
397    pattern6827 = Pattern(Integral(x_**WC('m', S(1))*SinIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sin(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons64)
398    rule6827 = ReplacementRule(pattern6827, replacement6827)
399
400    pattern6828 = Pattern(Integral(x_**WC('m', S(1))*CosIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cos(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons64)
401    rule6828 = ReplacementRule(pattern6828, replacement6828)
402
403    pattern6829 = Pattern(Integral(x_**m_*SinIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sin(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons96)
404    rule6829 = ReplacementRule(pattern6829, replacement6829)
405
406    pattern6830 = Pattern(Integral(x_**WC('m', S(1))*CosIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cos(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons96)
407    rule6830 = ReplacementRule(pattern6830, replacement6830)
408
409    pattern6831 = Pattern(Integral(SinIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cos(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons1266)
410    rule6831 = ReplacementRule(pattern6831, replacement6831)
411
412    pattern6832 = Pattern(Integral(CosIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sin(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons1266)
413    rule6832 = ReplacementRule(pattern6832, replacement6832)
414
415    pattern6833 = Pattern(Integral(x_**WC('m', S(1))*SinIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cos(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons64)
416    rule6833 = ReplacementRule(pattern6833, replacement6833)
417
418    pattern6834 = Pattern(Integral(x_**WC('m', S(1))*CosIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sin(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons64)
419    rule6834 = ReplacementRule(pattern6834, replacement6834)
420
421    pattern6835 = Pattern(Integral(x_**WC('m', S(1))*SinIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cos(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons96)
422    rule6835 = ReplacementRule(pattern6835, replacement6835)
423
424    pattern6836 = Pattern(Integral(x_**m_*CosIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sin(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons96)
425    rule6836 = ReplacementRule(pattern6836, replacement6836)
426
427    pattern6837 = Pattern(Integral(SinhIntegral(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons69)
428    rule6837 = ReplacementRule(pattern6837, replacement6837)
429
430    pattern6838 = Pattern(Integral(CoshIntegral(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons69)
431    rule6838 = ReplacementRule(pattern6838, replacement6838)
432
433    pattern6839 = Pattern(Integral(SinhIntegral(x_*WC('b', S(1)))/x_, x_), cons3, cons3)
434    rule6839 = ReplacementRule(pattern6839, replacement6839)
435
436    pattern6840 = Pattern(Integral(CoshIntegral(x_*WC('b', S(1)))/x_, x_), cons3, cons3)
437    rule6840 = ReplacementRule(pattern6840, replacement6840)
438
439    pattern6841 = Pattern(Integral(x_**WC('m', S(1))*SinhIntegral(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons19, cons68)
440    rule6841 = ReplacementRule(pattern6841, replacement6841)
441
442    pattern6842 = Pattern(Integral(x_**WC('m', S(1))*CoshIntegral(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons19, cons68)
443    rule6842 = ReplacementRule(pattern6842, replacement6842)
444
445    pattern6843 = Pattern(Integral(SinhIntegral(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons69)
446    rule6843 = ReplacementRule(pattern6843, replacement6843)
447
448    pattern6844 = Pattern(Integral(CoshIntegral(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons69)
449    rule6844 = ReplacementRule(pattern6844, replacement6844)
450
451    pattern6845 = Pattern(Integral(x_**WC('m', S(1))*SinhIntegral(x_*WC('b', S(1)))**S(2), x_), cons3, cons64)
452    rule6845 = ReplacementRule(pattern6845, replacement6845)
453
454    pattern6846 = Pattern(Integral(x_**WC('m', S(1))*CoshIntegral(x_*WC('b', S(1)))**S(2), x_), cons3, cons64)
455    rule6846 = ReplacementRule(pattern6846, replacement6846)
456
457    pattern6847 = Pattern(Integral(x_**WC('m', S(1))*SinhIntegral(a_ + x_*WC('b', S(1)))**S(2), x_), cons2, cons3, cons64)
458    rule6847 = ReplacementRule(pattern6847, replacement6847)
459
460    pattern6848 = Pattern(Integral(x_**WC('m', S(1))*CoshIntegral(a_ + x_*WC('b', S(1)))**S(2), x_), cons2, cons3, cons64)
461    rule6848 = ReplacementRule(pattern6848, replacement6848)
462
463    pattern6849 = Pattern(Integral(SinhIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sinh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons1266)
464    rule6849 = ReplacementRule(pattern6849, replacement6849)
465
466    pattern6850 = Pattern(Integral(CoshIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cosh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons1266)
467    rule6850 = ReplacementRule(pattern6850, replacement6850)
468
469    pattern6851 = Pattern(Integral(x_**WC('m', S(1))*SinhIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sinh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons170)
470    rule6851 = ReplacementRule(pattern6851, replacement6851)
471
472    pattern6852 = Pattern(Integral(x_**WC('m', S(1))*CoshIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cosh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons170)
473    rule6852 = ReplacementRule(pattern6852, replacement6852)
474
475    pattern6853 = Pattern(Integral(x_**m_*SinhIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sinh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons96)
476    rule6853 = ReplacementRule(pattern6853, replacement6853)
477
478    pattern6854 = Pattern(Integral(x_**WC('m', S(1))*CoshIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cosh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons96)
479    rule6854 = ReplacementRule(pattern6854, replacement6854)
480
481    pattern6855 = Pattern(Integral(SinhIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cosh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons1266)
482    rule6855 = ReplacementRule(pattern6855, replacement6855)
483
484    pattern6856 = Pattern(Integral(CoshIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sinh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons1266)
485    rule6856 = ReplacementRule(pattern6856, replacement6856)
486
487    pattern6857 = Pattern(Integral(x_**WC('m', S(1))*SinhIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cosh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons64)
488    rule6857 = ReplacementRule(pattern6857, replacement6857)
489
490    pattern6858 = Pattern(Integral(x_**WC('m', S(1))*CoshIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sinh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons64)
491    rule6858 = ReplacementRule(pattern6858, replacement6858)
492
493    pattern6859 = Pattern(Integral(x_**WC('m', S(1))*SinhIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cosh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons96)
494    rule6859 = ReplacementRule(pattern6859, replacement6859)
495
496    pattern6860 = Pattern(Integral(x_**m_*CoshIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sinh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons96)
497    rule6860 = ReplacementRule(pattern6860, replacement6860)
498
499    pattern6861 = Pattern(Integral(Gamma(n_, x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons69)
500    rule6861 = ReplacementRule(pattern6861, replacement6861)
501
502    pattern6862 = Pattern(Integral(Gamma(n_, b_*x_)/x_, x_), cons3, cons4, cons1970)
503    rule6862 = ReplacementRule(pattern6862, replacement6862)
504
505    pattern6863 = Pattern(Integral(x_**WC('m', S(1))*Gamma(n_, b_*x_), x_), cons3, cons19, cons4, cons68)
506    rule6863 = ReplacementRule(pattern6863, replacement6863)
507
508    pattern6864 = Pattern(Integral(x_**WC('m', S(1))*Gamma(n_, a_ + x_*WC('b', S(1))), x_), cons2, cons3, cons19, cons4, cons1971, cons68)
509    rule6864 = ReplacementRule(pattern6864, With6864)
510
511    pattern6865 = Pattern(Integral(LogGamma(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons69)
512    rule6865 = ReplacementRule(pattern6865, replacement6865)
513
514    pattern6866 = Pattern(Integral(x_**WC('m', S(1))*LogGamma(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons33, cons170)
515    rule6866 = ReplacementRule(pattern6866, replacement6866)
516
517    pattern6867 = Pattern(Integral(PolyGamma(n_, x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons4, cons1833)
518    rule6867 = ReplacementRule(pattern6867, replacement6867)
519
520    pattern6868 = Pattern(Integral(x_**WC('m', S(1))*PolyGamma(n_, x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons4, cons33, cons170)
521    rule6868 = ReplacementRule(pattern6868, replacement6868)
522
523    pattern6869 = Pattern(Integral(x_**WC('m', S(1))*PolyGamma(n_, x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons4, cons33, cons96)
524    rule6869 = ReplacementRule(pattern6869, replacement6869)
525
526    pattern6870 = Pattern(Integral(Gamma(x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))*PolyGamma(S(0), x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons4, cons1833)
527    rule6870 = ReplacementRule(pattern6870, replacement6870)
528
529    pattern6871 = Pattern(Integral(Factorial(x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))*PolyGamma(S(0), x_*WC('b', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons4, cons1972)
530    rule6871 = ReplacementRule(pattern6871, replacement6871)
531
532    pattern6872 = Pattern(Integral(Zeta(S(2), x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons69)
533    rule6872 = ReplacementRule(pattern6872, replacement6872)
534
535    pattern6873 = Pattern(Integral(Zeta(s_, x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons802, cons1973, cons1974)
536    rule6873 = ReplacementRule(pattern6873, replacement6873)
537
538    pattern6874 = Pattern(Integral(x_**WC('m', S(1))*Zeta(S(2), x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons33)
539    rule6874 = ReplacementRule(pattern6874, replacement6874)
540
541    pattern6875 = Pattern(Integral(x_**WC('m', S(1))*Zeta(s_, x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons802, cons1973, cons1974, cons33, cons170)
542    rule6875 = ReplacementRule(pattern6875, replacement6875)
543
544    pattern6876 = Pattern(Integral(x_**WC('m', S(1))*Zeta(s_, x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons802, cons1973, cons1974, cons33, cons96)
545    rule6876 = ReplacementRule(pattern6876, replacement6876)
546
547    pattern6877 = Pattern(Integral(PolyLog(n_, (x_**WC('p', S(1))*WC('b', S(1)))**WC('q', S(1))*WC('a', S(1))), x_), cons2, cons3, cons5, cons52, cons89, cons90)
548    rule6877 = ReplacementRule(pattern6877, replacement6877)
549
550    pattern6878 = Pattern(Integral(PolyLog(n_, (x_**WC('p', S(1))*WC('b', S(1)))**WC('q', S(1))*WC('a', S(1))), x_), cons2, cons3, cons5, cons52, cons89, cons91)
551    rule6878 = ReplacementRule(pattern6878, replacement6878)
552
553    pattern6879 = Pattern(Integral(PolyLog(n_, (x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*WC('c', S(1)))/(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons4, cons5, cons385)
554    rule6879 = ReplacementRule(pattern6879, replacement6879)
555
556    pattern6880 = Pattern(Integral(PolyLog(n_, (x_**WC('p', S(1))*WC('b', S(1)))**WC('q', S(1))*WC('a', S(1)))/x_, x_), cons2, cons3, cons4, cons5, cons52, cons1975)
557    rule6880 = ReplacementRule(pattern6880, replacement6880)
558
559    pattern6881 = Pattern(Integral(x_**WC('m', S(1))*PolyLog(n_, (x_**WC('p', S(1))*WC('b', S(1)))**WC('q', S(1))*WC('a', S(1))), x_), cons2, cons3, cons19, cons5, cons52, cons68, cons89, cons90)
560    rule6881 = ReplacementRule(pattern6881, replacement6881)
561
562    pattern6882 = Pattern(Integral(x_**WC('m', S(1))*PolyLog(n_, (x_**WC('p', S(1))*WC('b', S(1)))**WC('q', S(1))*WC('a', S(1))), x_), cons2, cons3, cons19, cons5, cons52, cons68, cons89, cons91)
563    rule6882 = ReplacementRule(pattern6882, replacement6882)
564
565    pattern6883 = Pattern(Integral(PolyLog(n_, (x_**WC('p', S(1))*WC('b', S(1)))**WC('q', S(1))*WC('a', S(1)))*log(x_**WC('m', S(1))*WC('c', S(1)))**WC('r', S(1))/x_, x_), cons2, cons3, cons8, cons19, cons4, cons52, cons54, cons1976, cons1977)
566    rule6883 = ReplacementRule(pattern6883, replacement6883)
567
568    pattern6884 = Pattern(Integral(PolyLog(n_, (x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*WC('c', S(1))), x_), cons2, cons3, cons8, cons5, cons89, cons90)
569    rule6884 = ReplacementRule(pattern6884, replacement6884)
570
571    pattern6885 = Pattern(Integral(x_**WC('m', S(1))*PolyLog(n_, (x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*WC('c', S(1))), x_), cons2, cons3, cons8, cons19, cons5, cons89, cons90, cons64)
572    rule6885 = ReplacementRule(pattern6885, replacement6885)
573
574    pattern6886 = Pattern(Integral(PolyLog(n_, (F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1))))**WC('p', S(1))*WC('d', S(1))), x_), cons1101, cons2, cons3, cons8, cons29, cons4, cons5, cons1978)
575    rule6886 = ReplacementRule(pattern6886, replacement6886)
576
577    pattern6887 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*PolyLog(n_, (F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1))))**WC('p', S(1))*WC('d', S(1))), x_), cons1101, cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons5, cons33, cons170)
578    rule6887 = ReplacementRule(pattern6887, replacement6887)
579
580    pattern6888 = Pattern(Integral(u_*PolyLog(n_, v_), x_), cons4, cons4, CustomConstraint(With6888))
581    rule6888 = ReplacementRule(pattern6888, replacement6888)
582
583    pattern6889 = Pattern(Integral(u_*PolyLog(n_, v_)*log(w_), x_), cons4, cons1245, CustomConstraint(With6889))
584    rule6889 = ReplacementRule(pattern6889, replacement6889)
585
586    pattern6890 = Pattern(Integral((ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))**p_, x_), cons2, cons3, cons8, cons13, cons139)
587    rule6890 = ReplacementRule(pattern6890, replacement6890)
588
589    pattern6891 = Pattern(Integral((ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons1381)
590    rule6891 = ReplacementRule(pattern6891, replacement6891)
591
592    pattern6892 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(a_ + x_*WC('b', S(1)))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons5, cons64)
593    rule6892 = ReplacementRule(pattern6892, replacement6892)
594
595    pattern6893 = Pattern(Integral((ProductLog(x_**n_*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons8, cons4, cons5, cons1979)
596    rule6893 = ReplacementRule(pattern6893, replacement6893)
597
598    pattern6894 = Pattern(Integral((ProductLog(x_**n_*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons8, cons4, cons1980)
599    rule6894 = ReplacementRule(pattern6894, replacement6894)
600
601    pattern6895 = Pattern(Integral((ProductLog(x_**n_*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons8, cons5, cons198)
602    rule6895 = ReplacementRule(pattern6895, replacement6895)
603
604    pattern6896 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons8, cons19, cons4, cons5, cons68, cons1981)
605    rule6896 = ReplacementRule(pattern6896, replacement6896)
606
607    pattern6897 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons8, cons19, cons4, cons5, cons1982)
608    rule6897 = ReplacementRule(pattern6897, replacement6897)
609
610    pattern6898 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons8, cons19, cons1983)
611    rule6898 = ReplacementRule(pattern6898, replacement6898)
612
613    pattern6899 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_**n_*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons8, cons5, cons152, cons465, cons68)
614    rule6899 = ReplacementRule(pattern6899, replacement6899)
615
616    pattern6900 = Pattern(Integral(S(1)/(d_ + ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('d', S(1))), x_), cons2, cons3, cons29, cons1767)
617    rule6900 = ReplacementRule(pattern6900, replacement6900)
618
619    pattern6901 = Pattern(Integral(ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))/(d_ + ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('d', S(1))), x_), cons2, cons3, cons29, cons1767)
620    rule6901 = ReplacementRule(pattern6901, replacement6901)
621
622    pattern6902 = Pattern(Integral((ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))**p_/(d_ + ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons13, cons165)
623    rule6902 = ReplacementRule(pattern6902, replacement6902)
624
625    pattern6903 = Pattern(Integral(S(1)/((d_ + ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('d', S(1)))*ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons29, cons1767)
626    rule6903 = ReplacementRule(pattern6903, replacement6903)
627
628    pattern6904 = Pattern(Integral(S(1)/(sqrt(ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(d_ + ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons950)
629    rule6904 = ReplacementRule(pattern6904, replacement6904)
630
631    pattern6905 = Pattern(Integral(S(1)/(sqrt(ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(d_ + ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons951)
632    rule6905 = ReplacementRule(pattern6905, replacement6905)
633
634    pattern6906 = Pattern(Integral((ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))**p_/(d_ + ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons13, cons139)
635    rule6906 = ReplacementRule(pattern6906, replacement6906)
636
637    pattern6907 = Pattern(Integral((ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons5, cons1984)
638    rule6907 = ReplacementRule(pattern6907, replacement6907)
639
640    pattern6908 = Pattern(Integral(x_**WC('m', S(1))/(d_ + ProductLog(a_ + x_*WC('b', S(1)))*WC('d', S(1))), x_), cons2, cons3, cons29, cons64)
641    rule6908 = ReplacementRule(pattern6908, replacement6908)
642
643    pattern6909 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(a_ + x_*WC('b', S(1)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(a_ + x_*WC('b', S(1)))*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons5, cons64)
644    rule6909 = ReplacementRule(pattern6909, replacement6909)
645
646    pattern6910 = Pattern(Integral(S(1)/(d_ + ProductLog(x_**n_*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons29, cons198)
647    rule6910 = ReplacementRule(pattern6910, replacement6910)
648
649    pattern6911 = Pattern(Integral((ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons8, cons29, cons4, cons5, cons1985)
650    rule6911 = ReplacementRule(pattern6911, replacement6911)
651
652    pattern6912 = Pattern(Integral(ProductLog(x_**WC('n', S(1))*WC('a', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons29, cons805, cons1986)
653    rule6912 = ReplacementRule(pattern6912, replacement6912)
654
655    pattern6913 = Pattern(Integral((ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**p_/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons8, cons29, cons805, cons1987, cons1988)
656    rule6913 = ReplacementRule(pattern6913, replacement6913)
657
658    pattern6914 = Pattern(Integral((ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**p_/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons8, cons29, cons805, cons1987, cons1989)
659    rule6914 = ReplacementRule(pattern6914, replacement6914)
660
661    pattern6915 = Pattern(Integral((ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons8, cons29, cons340, cons90, cons1990)
662    rule6915 = ReplacementRule(pattern6915, replacement6915)
663
664    pattern6916 = Pattern(Integral((ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons8, cons29, cons340, cons90, cons1991)
665    rule6916 = ReplacementRule(pattern6916, replacement6916)
666
667    pattern6917 = Pattern(Integral((ProductLog(x_**n_*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_**n_*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons8, cons29, cons5, cons198)
668    rule6917 = ReplacementRule(pattern6917, replacement6917)
669
670    pattern6918 = Pattern(Integral(x_**WC('m', S(1))/(d_ + ProductLog(x_*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons29, cons33, cons170)
671    rule6918 = ReplacementRule(pattern6918, replacement6918)
672
673    pattern6919 = Pattern(Integral(S(1)/(x_*(d_ + ProductLog(x_*WC('a', S(1)))*WC('d', S(1)))), x_), cons2, cons29, cons1992)
674    rule6919 = ReplacementRule(pattern6919, replacement6919)
675
676    pattern6920 = Pattern(Integral(x_**WC('m', S(1))/(d_ + ProductLog(x_*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons29, cons33, cons96)
677    rule6920 = ReplacementRule(pattern6920, replacement6920)
678
679    pattern6921 = Pattern(Integral(x_**WC('m', S(1))/(d_ + ProductLog(x_*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons29, cons19, cons21)
680    rule6921 = ReplacementRule(pattern6921, replacement6921)
681
682    pattern6922 = Pattern(Integral(S(1)/(x_*(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1)))), x_), cons2, cons29, cons4, cons1993)
683    rule6922 = ReplacementRule(pattern6922, replacement6922)
684
685    pattern6923 = Pattern(Integral(x_**WC('m', S(1))/(d_ + ProductLog(x_**n_*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons29, cons152, cons465, cons68)
686    rule6923 = ReplacementRule(pattern6923, replacement6923)
687
688    pattern6924 = Pattern(Integral((ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1))/(x_*(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1)))), x_), cons2, cons8, cons29, cons4, cons5, cons1994)
689    rule6924 = ReplacementRule(pattern6924, replacement6924)
690
691    pattern6925 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons8, cons29, cons19, cons4, cons5, cons68, cons1995)
692    rule6925 = ReplacementRule(pattern6925, replacement6925)
693
694    pattern6926 = Pattern(Integral(x_**WC('m', S(1))*ProductLog(x_**WC('n', S(1))*WC('a', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons29, cons19, cons4, cons40, cons1996)
695    rule6926 = ReplacementRule(pattern6926, replacement6926)
696
697    pattern6927 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**p_/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons8, cons29, cons19, cons4, cons68, cons349, cons1997, cons1998)
698    rule6927 = ReplacementRule(pattern6927, replacement6927)
699
700    pattern6928 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**p_/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons8, cons29, cons19, cons4, cons68, cons349, cons1997, cons1999)
701    rule6928 = ReplacementRule(pattern6928, replacement6928)
702
703    pattern6929 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons8, cons29, cons19, cons4, cons5, cons68, cons2000, cons2001)
704    rule6929 = ReplacementRule(pattern6929, replacement6929)
705
706    pattern6930 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons8, cons29, cons19, cons4, cons5, cons68, cons2000, cons2002)
707    rule6930 = ReplacementRule(pattern6930, replacement6930)
708
709    pattern6931 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons8, cons29, cons19, cons5, cons68)
710    rule6931 = ReplacementRule(pattern6931, replacement6931)
711
712    pattern6932 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons8, cons29, cons5, cons68, cons152, cons465)
713    rule6932 = ReplacementRule(pattern6932, replacement6932)
714
715    pattern6933 = Pattern(Integral(u_, x_), cons2003)
716    rule6933 = ReplacementRule(pattern6933, replacement6933)
717    return [rule6742, rule6743, rule6744, rule6745, rule6746, rule6747, rule6748, rule6749, rule6750, rule6751, rule6752, rule6753, rule6754, rule6755, rule6756, rule6757, rule6758, rule6759, rule6760, rule6761, rule6762, rule6763, rule6764, rule6765, rule6766, rule6767, rule6768, rule6769, rule6770, rule6771, rule6772, rule6773, rule6774, rule6775, rule6776, rule6777, rule6778, rule6779, rule6780, rule6781, rule6782, rule6783, rule6784, rule6785, rule6786, rule6787, rule6788, rule6789, rule6790, rule6791, rule6792, rule6793, rule6794, rule6795, rule6796, rule6797, rule6798, rule6799, rule6800, rule6801, rule6802, rule6803, rule6804, rule6805, rule6806, rule6807, rule6808, rule6809, rule6810, rule6811, rule6812, rule6813, rule6814, rule6815, rule6816, rule6817, rule6818, rule6819, rule6820, rule6821, rule6822, rule6823, rule6824, rule6825, rule6826, rule6827, rule6828, rule6829, rule6830, rule6831, rule6832, rule6833, rule6834, rule6835, rule6836, rule6837, rule6838, rule6839, rule6840, rule6841, rule6842, rule6843, rule6844, rule6845, rule6846, rule6847, rule6848, rule6849, rule6850, rule6851, rule6852, rule6853, rule6854, rule6855, rule6856, rule6857, rule6858, rule6859, rule6860, rule6861, rule6862, rule6863, rule6864, rule6865, rule6866, rule6867, rule6868, rule6869, rule6870, rule6871, rule6872, rule6873, rule6874, rule6875, rule6876, rule6877, rule6878, rule6879, rule6880, rule6881, rule6882, rule6883, rule6884, rule6885, rule6886, rule6887, rule6888, rule6889, rule6890, rule6891, rule6892, rule6893, rule6894, rule6895, rule6896, rule6897, rule6898, rule6899, rule6900, rule6901, rule6902, rule6903, rule6904, rule6905, rule6906, rule6907, rule6908, rule6909, rule6910, rule6911, rule6912, rule6913, rule6914, rule6915, rule6916, rule6917, rule6918, rule6919, rule6920, rule6921, rule6922, rule6923, rule6924, rule6925, rule6926, rule6927, rule6928, rule6929, rule6930, rule6931, rule6932, rule6933, ]
718
719
720
721
722
723def replacement6742(a, b, x):
724    return Simp(exp(-(a + b*x)**S(2))/(sqrt(Pi)*b), x) + Simp((a + b*x)*Erf(a + b*x)/b, x)
725
726
727def replacement6743(a, b, x):
728    return -Simp(exp(-(a + b*x)**S(2))/(sqrt(Pi)*b), x) + Simp((a + b*x)*Erfc(a + b*x)/b, x)
729
730
731def replacement6744(a, b, x):
732    return -Simp(exp((a + b*x)**S(2))/(sqrt(Pi)*b), x) + Simp((a + b*x)*Erfi(a + b*x)/b, x)
733
734
735def replacement6745(b, x):
736    return Simp(S(2)*b*x*HypergeometricPFQ(List(S(1)/2, S(1)/2), List(S(3)/2, S(3)/2), -b**S(2)*x**S(2))/sqrt(Pi), x)
737
738
739def replacement6746(b, x):
740    return -Int(Erf(b*x)/x, x) + Simp(log(x), x)
741
742
743def replacement6747(b, x):
744    return Simp(S(2)*b*x*HypergeometricPFQ(List(S(1)/2, S(1)/2), List(S(3)/2, S(3)/2), b**S(2)*x**S(2))/sqrt(Pi), x)
745
746
747def replacement6748(a, b, m, x):
748    return -Dist(S(2)*b/(sqrt(Pi)*(m + S(1))), Int(x**(m + S(1))*exp(-(a + b*x)**S(2)), x), x) + Simp(x**(m + S(1))*Erf(a + b*x)/(m + S(1)), x)
749
750
751def replacement6749(a, b, m, x):
752    return Dist(S(2)*b/(sqrt(Pi)*(m + S(1))), Int(x**(m + S(1))*exp(-(a + b*x)**S(2)), x), x) + Simp(x**(m + S(1))*Erfc(a + b*x)/(m + S(1)), x)
753
754
755def replacement6750(a, b, m, x):
756    return -Dist(S(2)*b/(sqrt(Pi)*(m + S(1))), Int(x**(m + S(1))*exp((a + b*x)**S(2)), x), x) + Simp(x**(m + S(1))*Erfi(a + b*x)/(m + S(1)), x)
757
758
759def replacement6751(a, b, c, d, x):
760    return -Dist(b/(sqrt(Pi)*d), Int(exp(-a**S(2) - S(2)*a*b*x + c - x**S(2)*(b**S(2) - d)), x), x) + Simp(Erf(a + b*x)*exp(c + d*x**S(2))/(S(2)*d), x)
761
762
763def replacement6752(a, b, c, d, x):
764    return Dist(b/(sqrt(Pi)*d), Int(exp(-a**S(2) - S(2)*a*b*x + c - x**S(2)*(b**S(2) - d)), x), x) + Simp(Erfc(a + b*x)*exp(c + d*x**S(2))/(S(2)*d), x)
765
766
767def replacement6753(a, b, c, d, x):
768    return -Dist(b/(sqrt(Pi)*d), Int(exp(a**S(2) + S(2)*a*b*x + c + x**S(2)*(b**S(2) + d)), x), x) + Simp(Erfi(a + b*x)*exp(c + d*x**S(2))/(S(2)*d), x)
769
770
771def replacement6754(a, b, c, d, m, x):
772    return -Dist((m + S(-1))/(S(2)*d), Int(x**(m + S(-2))*Erf(a + b*x)*exp(c + d*x**S(2)), x), x) - Dist(b/(sqrt(Pi)*d), Int(x**(m + S(-1))*exp(-a**S(2) - S(2)*a*b*x + c - x**S(2)*(b**S(2) - d)), x), x) + Simp(x**(m + S(-1))*Erf(a + b*x)*exp(c + d*x**S(2))/(S(2)*d), x)
773
774
775def replacement6755(a, b, c, d, m, x):
776    return -Dist((m + S(-1))/(S(2)*d), Int(x**(m + S(-2))*Erfc(a + b*x)*exp(c + d*x**S(2)), x), x) + Dist(b/(sqrt(Pi)*d), Int(x**(m + S(-1))*exp(-a**S(2) - S(2)*a*b*x + c - x**S(2)*(b**S(2) - d)), x), x) + Simp(x**(m + S(-1))*Erfc(a + b*x)*exp(c + d*x**S(2))/(S(2)*d), x)
777
778
779def replacement6756(a, b, c, d, m, x):
780    return -Dist((m + S(-1))/(S(2)*d), Int(x**(m + S(-2))*Erfi(a + b*x)*exp(c + d*x**S(2)), x), x) - Dist(b/(sqrt(Pi)*d), Int(x**(m + S(-1))*exp(a**S(2) + S(2)*a*b*x + c + x**S(2)*(b**S(2) + d)), x), x) + Simp(x**(m + S(-1))*Erfi(a + b*x)*exp(c + d*x**S(2))/(S(2)*d), x)
781
782
783def replacement6757(b, c, d, x):
784    return Simp(S(2)*b*x*HypergeometricPFQ(List(S(1)/2, S(1)), List(S(3)/2, S(3)/2), d*x**S(2))*exp(c)/sqrt(Pi), x)
785
786
787def replacement6758(b, c, d, x):
788    return Int(exp(c + d*x**S(2))/x, x) - Int(Erf(b*x)*exp(c + d*x**S(2))/x, x)
789
790
791def replacement6759(b, c, d, x):
792    return Simp(S(2)*b*x*HypergeometricPFQ(List(S(1)/2, S(1)), List(S(3)/2, S(3)/2), d*x**S(2))*exp(c)/sqrt(Pi), x)
793
794
795def replacement6760(a, b, c, d, m, x):
796    return -Dist(S(2)*d/(m + S(1)), Int(x**(m + S(2))*Erf(a + b*x)*exp(c + d*x**S(2)), x), x) - Dist(S(2)*b/(sqrt(Pi)*(m + S(1))), Int(x**(m + S(1))*exp(-a**S(2) - S(2)*a*b*x + c - x**S(2)*(b**S(2) - d)), x), x) + Simp(x**(m + S(1))*Erf(a + b*x)*exp(c + d*x**S(2))/(m + S(1)), x)
797
798
799def replacement6761(a, b, c, d, m, x):
800    return -Dist(S(2)*d/(m + S(1)), Int(x**(m + S(2))*Erfc(a + b*x)*exp(c + d*x**S(2)), x), x) + Dist(S(2)*b/(sqrt(Pi)*(m + S(1))), Int(x**(m + S(1))*exp(-a**S(2) - S(2)*a*b*x + c - x**S(2)*(b**S(2) - d)), x), x) + Simp(x**(m + S(1))*Erfc(a + b*x)*exp(c + d*x**S(2))/(m + S(1)), x)
801
802
803def replacement6762(a, b, c, d, m, x):
804    return -Dist(S(2)*d/(m + S(1)), Int(x**(m + S(2))*Erfi(a + b*x)*exp(c + d*x**S(2)), x), x) - Dist(S(2)*b/(sqrt(Pi)*(m + S(1))), Int(x**(m + S(1))*exp(a**S(2) + S(2)*a*b*x + c + x**S(2)*(b**S(2) + d)), x), x) + Simp(x**(m + S(1))*Erfi(a + b*x)*exp(c + d*x**S(2))/(m + S(1)), x)
805
806
807def replacement6763(a, b, x):
808    return -Dist(S(4)/sqrt(Pi), Int((a + b*x)*Erf(a + b*x)*exp(-(a + b*x)**S(2)), x), x) + Simp((a + b*x)*Erf(a + b*x)**S(2)/b, x)
809
810
811def replacement6764(a, b, x):
812    return Dist(S(4)/sqrt(Pi), Int((a + b*x)*Erfc(a + b*x)*exp(-(a + b*x)**S(2)), x), x) + Simp((a + b*x)*Erfc(a + b*x)**S(2)/b, x)
813
814
815def replacement6765(a, b, x):
816    return -Dist(S(4)/sqrt(Pi), Int((a + b*x)*Erfi(a + b*x)*exp((a + b*x)**S(2)), x), x) + Simp((a + b*x)*Erfi(a + b*x)**S(2)/b, x)
817
818
819def replacement6766(b, m, x):
820    return -Dist(S(4)*b/(sqrt(Pi)*(m + S(1))), Int(x**(m + S(1))*Erf(b*x)*exp(-b**S(2)*x**S(2)), x), x) + Simp(x**(m + S(1))*Erf(b*x)**S(2)/(m + S(1)), x)
821
822
823def replacement6767(b, m, x):
824    return Dist(S(4)*b/(sqrt(Pi)*(m + S(1))), Int(x**(m + S(1))*Erfc(b*x)*exp(-b**S(2)*x**S(2)), x), x) + Simp(x**(m + S(1))*Erfc(b*x)**S(2)/(m + S(1)), x)
825
826
827def replacement6768(b, m, x):
828    return -Dist(S(4)*b/(sqrt(Pi)*(m + S(1))), Int(x**(m + S(1))*Erfi(b*x)*exp(b**S(2)*x**S(2)), x), x) + Simp(x**(m + S(1))*Erfi(b*x)**S(2)/(m + S(1)), x)
829
830
831def replacement6769(a, b, m, x):
832    return Dist(S(1)/b, Subst(Int((-a/b + x/b)**m*Erf(x)**S(2), x), x, a + b*x), x)
833
834
835def replacement6770(a, b, m, x):
836    return Dist(S(1)/b, Subst(Int((-a/b + x/b)**m*Erfc(x)**S(2), x), x, a + b*x), x)
837
838
839def replacement6771(a, b, m, x):
840    return Dist(S(1)/b, Subst(Int((-a/b + x/b)**m*Erfi(x)**S(2), x), x, a + b*x), x)
841
842
843def replacement6772(a, b, x):
844    return Simp(cos(Pi*(a + b*x)**S(2)/S(2))/(Pi*b), x) + Simp((a + b*x)*FresnelS(a + b*x)/b, x)
845
846
847def replacement6773(a, b, x):
848    return -Simp(sin(Pi*(a + b*x)**S(2)/S(2))/(Pi*b), x) + Simp((a + b*x)*FresnelC(a + b*x)/b, x)
849
850
851def replacement6774(b, x):
852    return Simp(I*b*x*HypergeometricPFQ(List(S(1)/2, S(1)/2), List(S(3)/2, S(3)/2), -I*Pi*b**S(2)*x**S(2)/S(2))/S(2), x) - Simp(I*b*x*HypergeometricPFQ(List(S(1)/2, S(1)/2), List(S(3)/2, S(3)/2), I*Pi*b**S(2)*x**S(2)/S(2))/S(2), x)
853
854
855def replacement6775(b, x):
856    return Simp(b*x*HypergeometricPFQ(List(S(1)/2, S(1)/2), List(S(3)/2, S(3)/2), -I*Pi*b**S(2)*x**S(2)/S(2))/S(2), x) + Simp(b*x*HypergeometricPFQ(List(S(1)/2, S(1)/2), List(S(3)/2, S(3)/2), I*Pi*b**S(2)*x**S(2)/S(2))/S(2), x)
857
858
859def replacement6776(a, b, m, x):
860    return -Dist(b/(m + S(1)), Int(x**(m + S(1))*sin(Pi*(a + b*x)**S(2)/S(2)), x), x) + Simp(x**(m + S(1))*FresnelS(a + b*x)/(m + S(1)), x)
861
862
863def replacement6777(a, b, m, x):
864    return -Dist(b/(m + S(1)), Int(x**(m + S(1))*cos(Pi*(a + b*x)**S(2)/S(2)), x), x) + Simp(x**(m + S(1))*FresnelC(a + b*x)/(m + S(1)), x)
865
866
867def replacement6778(a, b, x):
868    return -Dist(S(2), Int((a + b*x)*FresnelS(a + b*x)*sin(Pi*(a + b*x)**S(2)/S(2)), x), x) + Simp((a + b*x)*FresnelS(a + b*x)**S(2)/b, x)
869
870
871def replacement6779(a, b, x):
872    return -Dist(S(2), Int((a + b*x)*FresnelC(a + b*x)*cos(Pi*(a + b*x)**S(2)/S(2)), x), x) + Simp((a + b*x)*FresnelC(a + b*x)**S(2)/b, x)
873
874
875def replacement6780(b, m, x):
876    return -Dist(S(2)*b/(m + S(1)), Int(x**(m + S(1))*FresnelS(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2)), x), x) + Simp(x**(m + S(1))*FresnelS(b*x)**S(2)/(m + S(1)), x)
877
878
879def replacement6781(b, m, x):
880    return -Dist(S(2)*b/(m + S(1)), Int(x**(m + S(1))*FresnelC(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2)), x), x) + Simp(x**(m + S(1))*FresnelC(b*x)**S(2)/(m + S(1)), x)
881
882
883def replacement6782(b, c, x):
884    return Dist(S(1)/(S(2)*Pi*b), Int(sin(Pi*b**S(2)*x**S(2)), x), x) - Simp(FresnelS(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2))/(Pi*b**S(2)), x)
885
886
887def replacement6783(b, c, x):
888    return -Dist(S(1)/(S(2)*Pi*b), Int(sin(Pi*b**S(2)*x**S(2)), x), x) + Simp(FresnelC(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2))/(Pi*b**S(2)), x)
889
890
891def replacement6784(b, c, m, x):
892    return Dist(S(1)/(S(2)*Pi*b), Int(x**(m + S(-1))*sin(Pi*b**S(2)*x**S(2)), x), x) + Dist((m + S(-1))/(Pi*b**S(2)), Int(x**(m + S(-2))*FresnelS(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2)), x), x) - Simp(x**(m + S(-1))*FresnelS(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2))/(Pi*b**S(2)), x)
893
894
895def replacement6785(b, c, m, x):
896    return -Dist(S(1)/(S(2)*Pi*b), Int(x**(m + S(-1))*sin(Pi*b**S(2)*x**S(2)), x), x) - Dist((m + S(-1))/(Pi*b**S(2)), Int(x**(m + S(-2))*FresnelC(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2)), x), x) + Simp(x**(m + S(-1))*FresnelC(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2))/(Pi*b**S(2)), x)
897
898
899def replacement6786(b, c, m, x):
900    return Dist(b/(S(2)*m + S(2)), Int(x**(m + S(1))*cos(Pi*b**S(2)*x**S(2)), x), x) - Dist(Pi*b**S(2)/(m + S(1)), Int(x**(m + S(2))*FresnelS(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2)), x), x) - Simp(b*x**(m + S(2))/(S(2)*(m + S(1))*(m + S(2))), x) + Simp(x**(m + S(1))*FresnelS(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2))/(m + S(1)), x)
901
902
903def replacement6787(b, c, m, x):
904    return -Dist(b/(S(2)*m + S(2)), Int(x**(m + S(1))*cos(Pi*b**S(2)*x**S(2)), x), x) + Dist(Pi*b**S(2)/(m + S(1)), Int(x**(m + S(2))*FresnelC(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2)), x), x) - Simp(b*x**(m + S(2))/(S(2)*(m + S(1))*(m + S(2))), x) + Simp(x**(m + S(1))*FresnelC(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2))/(m + S(1)), x)
905
906
907def replacement6788(b, c, x):
908    return Dist(S(1)/(S(2)*Pi*b), Int(cos(Pi*b**S(2)*x**S(2)), x), x) - Simp(x/(S(2)*Pi*b), x) + Simp(FresnelS(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2))/(Pi*b**S(2)), x)
909
910
911def replacement6789(b, c, x):
912    return Dist(S(1)/(S(2)*Pi*b), Int(cos(Pi*b**S(2)*x**S(2)), x), x) + Simp(x/(S(2)*Pi*b), x) - Simp(FresnelC(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2))/(Pi*b**S(2)), x)
913
914
915def replacement6790(b, c, m, x):
916    return Dist(S(1)/(S(2)*Pi*b), Int(x**(m + S(-1))*cos(Pi*b**S(2)*x**S(2)), x), x) - Dist((m + S(-1))/(Pi*b**S(2)), Int(x**(m + S(-2))*FresnelS(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2)), x), x) - Simp(x**m/(S(2)*Pi*b*m), x) + Simp(x**(m + S(-1))*FresnelS(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2))/(Pi*b**S(2)), x)
917
918
919def replacement6791(b, c, m, x):
920    return Dist(S(1)/(S(2)*Pi*b), Int(x**(m + S(-1))*cos(Pi*b**S(2)*x**S(2)), x), x) + Dist((m + S(-1))/(Pi*b**S(2)), Int(x**(m + S(-2))*FresnelC(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2)), x), x) + Simp(x**m/(S(2)*Pi*b*m), x) - Simp(x**(m + S(-1))*FresnelC(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2))/(Pi*b**S(2)), x)
921
922
923def replacement6792(b, c, m, x):
924    return -Dist(b/(S(2)*m + S(2)), Int(x**(m + S(1))*sin(Pi*b**S(2)*x**S(2)), x), x) + Dist(Pi*b**S(2)/(m + S(1)), Int(x**(m + S(2))*FresnelS(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2)), x), x) + Simp(x**(m + S(1))*FresnelS(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2))/(m + S(1)), x)
925
926
927def replacement6793(b, c, m, x):
928    return -Dist(b/(S(2)*m + S(2)), Int(x**(m + S(1))*sin(Pi*b**S(2)*x**S(2)), x), x) - Dist(Pi*b**S(2)/(m + S(1)), Int(x**(m + S(2))*FresnelC(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2)), x), x) + Simp(x**(m + S(1))*FresnelC(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2))/(m + S(1)), x)
929
930
931def replacement6794(a, b, n, x):
932    return -Simp(ExpIntegralE(n + S(1), a + b*x)/b, x)
933
934
935def replacement6795(b, m, n, x):
936    return Dist(m/b, Int(x**(m + S(-1))*ExpIntegralE(n + S(1), b*x), x), x) - Simp(x**m*ExpIntegralE(n + S(1), b*x)/b, x)
937
938
939def replacement6796(b, x):
940    return -Simp(EulerGamma*log(x), x) + Simp(b*x*HypergeometricPFQ(List(S(1), S(1), S(1)), List(S(2), S(2), S(2)), -b*x), x) - Simp(log(b*x)**S(2)/S(2), x)
941
942
943def replacement6797(b, m, n, x):
944    return Dist(b/(m + S(1)), Int(x**(m + S(1))*ExpIntegralE(n + S(-1), b*x), x), x) + Simp(x**(m + S(1))*ExpIntegralE(n, b*x)/(m + S(1)), x)
945
946
947def replacement6798(b, m, n, x):
948    return -Simp(x**(m + S(1))*HypergeometricPFQ(List(m + S(1), m + S(1)), List(m + S(2), m + S(2)), -b*x)/(m + S(1))**S(2), x) + Simp(x**m*(b*x)**(-m)*Gamma(m + S(1))*log(x)/b, x)
949
950
951def replacement6799(b, m, n, x):
952    return -Simp(x**(m + S(1))*ExpIntegralE(-m, b*x)/(m + n), x) + Simp(x**(m + S(1))*ExpIntegralE(n, b*x)/(m + n), x)
953
954
955def replacement6800(a, b, m, n, x):
956    return Dist(m/b, Int(x**(m + S(-1))*ExpIntegralE(n + S(1), a + b*x), x), x) - Simp(x**m*ExpIntegralE(n + S(1), a + b*x)/b, x)
957
958
959def replacement6801(a, b, m, n, x):
960    return Dist(b/(m + S(1)), Int(x**(m + S(1))*ExpIntegralE(n + S(-1), a + b*x), x), x) + Simp(x**(m + S(1))*ExpIntegralE(n, a + b*x)/(m + S(1)), x)
961
962
963def replacement6802(a, b, x):
964    return -Simp(exp(a + b*x)/b, x) + Simp((a + b*x)*ExpIntegralEi(a + b*x)/b, x)
965
966
967def replacement6803(a, b, m, x):
968    return -Dist(b/(m + S(1)), Int(x**(m + S(1))*exp(a + b*x)/(a + b*x), x), x) + Simp(x**(m + S(1))*ExpIntegralEi(a + b*x)/(m + S(1)), x)
969
970
971def replacement6804(a, b, x):
972    return -Dist(S(2), Int(ExpIntegralEi(a + b*x)*exp(a + b*x), x), x) + Simp((a + b*x)*ExpIntegralEi(a + b*x)**S(2)/b, x)
973
974
975def replacement6805(b, m, x):
976    return -Dist(S(2)/(m + S(1)), Int(x**m*ExpIntegralEi(b*x)*exp(b*x), x), x) + Simp(x**(m + S(1))*ExpIntegralEi(b*x)**S(2)/(m + S(1)), x)
977
978
979def replacement6806(a, b, m, x):
980    return -Dist(a*m/(b*(m + S(1))), Int(x**(m + S(-1))*ExpIntegralEi(a + b*x)**S(2), x), x) - Dist(S(2)/(m + S(1)), Int(x**m*ExpIntegralEi(a + b*x)*exp(a + b*x), x), x) + Simp(x**(m + S(1))*ExpIntegralEi(a + b*x)**S(2)/(m + S(1)), x) + Simp(a*x**m*ExpIntegralEi(a + b*x)**S(2)/(b*(m + S(1))), x)
981
982
983def replacement6807(a, b, c, d, x):
984    return -Dist(d/b, Int(exp(a + c + x*(b + d))/(c + d*x), x), x) + Simp(ExpIntegralEi(c + d*x)*exp(a + b*x)/b, x)
985
986
987def replacement6808(a, b, c, d, m, x):
988    return -Dist(d/b, Int(x**m*exp(a + c + x*(b + d))/(c + d*x), x), x) - Dist(m/b, Int(x**(m + S(-1))*ExpIntegralEi(c + d*x)*exp(a + b*x), x), x) + Simp(x**m*ExpIntegralEi(c + d*x)*exp(a + b*x)/b, x)
989
990
991def replacement6809(a, b, c, d, m, x):
992    return -Dist(b/(m + S(1)), Int(x**(m + S(1))*ExpIntegralEi(c + d*x)*exp(a + b*x), x), x) - Dist(d/(m + S(1)), Int(x**(m + S(1))*exp(a + c + x*(b + d))/(c + d*x), x), x) + Simp(x**(m + S(1))*ExpIntegralEi(c + d*x)*exp(a + b*x)/(m + S(1)), x)
993
994
995def replacement6810(a, b, x):
996    return -Simp(ExpIntegralEi(S(2)*log(a + b*x))/b, x) + Simp((a + b*x)*LogIntegral(a + b*x)/b, x)
997
998
999def replacement6811(b, x):
1000    return -Simp(b*x, x) + Simp(LogIntegral(b*x)*log(b*x), x)
1001
1002
1003def replacement6812(a, b, m, x):
1004    return -Dist(b/(m + S(1)), Int(x**(m + S(1))/log(a + b*x), x), x) + Simp(x**(m + S(1))*LogIntegral(a + b*x)/(m + S(1)), x)
1005
1006
1007def replacement6813(a, b, x):
1008    return Simp(cos(a + b*x)/b, x) + Simp((a + b*x)*SinIntegral(a + b*x)/b, x)
1009
1010
1011def replacement6814(a, b, x):
1012    return -Simp(sin(a + b*x)/b, x) + Simp((a + b*x)*CosIntegral(a + b*x)/b, x)
1013
1014
1015def replacement6815(b, x):
1016    return Simp(b*x*HypergeometricPFQ(List(S(1), S(1), S(1)), List(S(2), S(2), S(2)), -I*b*x)/S(2), x) + Simp(b*x*HypergeometricPFQ(List(S(1), S(1), S(1)), List(S(2), S(2), S(2)), I*b*x)/S(2), x)
1017
1018
1019def replacement6816(b, x):
1020    return Simp(EulerGamma*log(x), x) - Simp(I*b*x*HypergeometricPFQ(List(S(1), S(1), S(1)), List(S(2), S(2), S(2)), -I*b*x)/S(2), x) + Simp(I*b*x*HypergeometricPFQ(List(S(1), S(1), S(1)), List(S(2), S(2), S(2)), I*b*x)/S(2), x) + Simp(log(b*x)**S(2)/S(2), x)
1021
1022
1023def replacement6817(a, b, m, x):
1024    return -Dist(b/(m + S(1)), Int(x**(m + S(1))*sin(a + b*x)/(a + b*x), x), x) + Simp(x**(m + S(1))*SinIntegral(a + b*x)/(m + S(1)), x)
1025
1026
1027def replacement6818(a, b, m, x):
1028    return -Dist(b/(m + S(1)), Int(x**(m + S(1))*cos(a + b*x)/(a + b*x), x), x) + Simp(x**(m + S(1))*CosIntegral(a + b*x)/(m + S(1)), x)
1029
1030
1031def replacement6819(a, b, x):
1032    return -Dist(S(2), Int(SinIntegral(a + b*x)*sin(a + b*x), x), x) + Simp((a + b*x)*SinIntegral(a + b*x)**S(2)/b, x)
1033
1034
1035def replacement6820(a, b, x):
1036    return -Dist(S(2), Int(CosIntegral(a + b*x)*cos(a + b*x), x), x) + Simp((a + b*x)*CosIntegral(a + b*x)**S(2)/b, x)
1037
1038
1039def replacement6821(b, m, x):
1040    return -Dist(S(2)/(m + S(1)), Int(x**m*SinIntegral(b*x)*sin(b*x), x), x) + Simp(x**(m + S(1))*SinIntegral(b*x)**S(2)/(m + S(1)), x)
1041
1042
1043def replacement6822(b, m, x):
1044    return -Dist(S(2)/(m + S(1)), Int(x**m*CosIntegral(b*x)*cos(b*x), x), x) + Simp(x**(m + S(1))*CosIntegral(b*x)**S(2)/(m + S(1)), x)
1045
1046
1047def replacement6823(a, b, m, x):
1048    return -Dist(a*m/(b*(m + S(1))), Int(x**(m + S(-1))*SinIntegral(a + b*x)**S(2), x), x) - Dist(S(2)/(m + S(1)), Int(x**m*SinIntegral(a + b*x)*sin(a + b*x), x), x) + Simp(x**(m + S(1))*SinIntegral(a + b*x)**S(2)/(m + S(1)), x) + Simp(a*x**m*SinIntegral(a + b*x)**S(2)/(b*(m + S(1))), x)
1049
1050
1051def replacement6824(a, b, m, x):
1052    return -Dist(a*m/(b*(m + S(1))), Int(x**(m + S(-1))*CosIntegral(a + b*x)**S(2), x), x) - Dist(S(2)/(m + S(1)), Int(x**m*CosIntegral(a + b*x)*cos(a + b*x), x), x) + Simp(x**(m + S(1))*CosIntegral(a + b*x)**S(2)/(m + S(1)), x) + Simp(a*x**m*CosIntegral(a + b*x)**S(2)/(b*(m + S(1))), x)
1053
1054
1055def replacement6825(a, b, c, d, x):
1056    return Dist(d/b, Int(sin(c + d*x)*cos(a + b*x)/(c + d*x), x), x) - Simp(SinIntegral(c + d*x)*cos(a + b*x)/b, x)
1057
1058
1059def replacement6826(a, b, c, d, x):
1060    return -Dist(d/b, Int(sin(a + b*x)*cos(c + d*x)/(c + d*x), x), x) + Simp(CosIntegral(c + d*x)*sin(a + b*x)/b, x)
1061
1062
1063def replacement6827(a, b, c, d, m, x):
1064    return Dist(d/b, Int(x**m*sin(c + d*x)*cos(a + b*x)/(c + d*x), x), x) + Dist(m/b, Int(x**(m + S(-1))*SinIntegral(c + d*x)*cos(a + b*x), x), x) - Simp(x**m*SinIntegral(c + d*x)*cos(a + b*x)/b, x)
1065
1066
1067def replacement6828(a, b, c, d, m, x):
1068    return -Dist(d/b, Int(x**m*sin(a + b*x)*cos(c + d*x)/(c + d*x), x), x) - Dist(m/b, Int(x**(m + S(-1))*CosIntegral(c + d*x)*sin(a + b*x), x), x) + Simp(x**m*CosIntegral(c + d*x)*sin(a + b*x)/b, x)
1069
1070
1071def replacement6829(a, b, c, d, m, x):
1072    return -Dist(b/(m + S(1)), Int(x**(m + S(1))*SinIntegral(c + d*x)*cos(a + b*x), x), x) - Dist(d/(m + S(1)), Int(x**(m + S(1))*sin(a + b*x)*sin(c + d*x)/(c + d*x), x), x) + Simp(x**(m + S(1))*SinIntegral(c + d*x)*sin(a + b*x)/(m + S(1)), x)
1073
1074
1075def replacement6830(a, b, c, d, m, x):
1076    return Dist(b/(m + S(1)), Int(x**(m + S(1))*CosIntegral(c + d*x)*sin(a + b*x), x), x) - Dist(d/(m + S(1)), Int(x**(m + S(1))*cos(a + b*x)*cos(c + d*x)/(c + d*x), x), x) + Simp(x**(m + S(1))*CosIntegral(c + d*x)*cos(a + b*x)/(m + S(1)), x)
1077
1078
1079def replacement6831(a, b, c, d, x):
1080    return -Dist(d/b, Int(sin(a + b*x)*sin(c + d*x)/(c + d*x), x), x) + Simp(SinIntegral(c + d*x)*sin(a + b*x)/b, x)
1081
1082
1083def replacement6832(a, b, c, d, x):
1084    return Dist(d/b, Int(cos(a + b*x)*cos(c + d*x)/(c + d*x), x), x) - Simp(CosIntegral(c + d*x)*cos(a + b*x)/b, x)
1085
1086
1087def replacement6833(a, b, c, d, m, x):
1088    return -Dist(d/b, Int(x**m*sin(a + b*x)*sin(c + d*x)/(c + d*x), x), x) - Dist(m/b, Int(x**(m + S(-1))*SinIntegral(c + d*x)*sin(a + b*x), x), x) + Simp(x**m*SinIntegral(c + d*x)*sin(a + b*x)/b, x)
1089
1090
1091def replacement6834(a, b, c, d, m, x):
1092    return Dist(d/b, Int(x**m*cos(a + b*x)*cos(c + d*x)/(c + d*x), x), x) + Dist(m/b, Int(x**(m + S(-1))*CosIntegral(c + d*x)*cos(a + b*x), x), x) - Simp(x**m*CosIntegral(c + d*x)*cos(a + b*x)/b, x)
1093
1094
1095def replacement6835(a, b, c, d, m, x):
1096    return Dist(b/(m + S(1)), Int(x**(m + S(1))*SinIntegral(c + d*x)*sin(a + b*x), x), x) - Dist(d/(m + S(1)), Int(x**(m + S(1))*sin(c + d*x)*cos(a + b*x)/(c + d*x), x), x) + Simp(x**(m + S(1))*SinIntegral(c + d*x)*cos(a + b*x)/(m + S(1)), x)
1097
1098
1099def replacement6836(a, b, c, d, m, x):
1100    return -Dist(b/(m + S(1)), Int(x**(m + S(1))*CosIntegral(c + d*x)*cos(a + b*x), x), x) - Dist(d/(m + S(1)), Int(x**(m + S(1))*sin(a + b*x)*cos(c + d*x)/(c + d*x), x), x) + Simp(x**(m + S(1))*CosIntegral(c + d*x)*sin(a + b*x)/(m + S(1)), x)
1101
1102
1103def replacement6837(a, b, x):
1104    return -Simp(cosh(a + b*x)/b, x) + Simp((a + b*x)*SinhIntegral(a + b*x)/b, x)
1105
1106
1107def replacement6838(a, b, x):
1108    return -Simp(sinh(a + b*x)/b, x) + Simp((a + b*x)*CoshIntegral(a + b*x)/b, x)
1109
1110
1111def replacement6839(b, x):
1112    return Simp(b*x*HypergeometricPFQ(List(S(1), S(1), S(1)), List(S(2), S(2), S(2)), -b*x)/S(2), x) + Simp(b*x*HypergeometricPFQ(List(S(1), S(1), S(1)), List(S(2), S(2), S(2)), b*x)/S(2), x)
1113
1114
1115def replacement6840(b, x):
1116    return Simp(EulerGamma*log(x), x) - Simp(b*x*HypergeometricPFQ(List(S(1), S(1), S(1)), List(S(2), S(2), S(2)), -b*x)/S(2), x) + Simp(b*x*HypergeometricPFQ(List(S(1), S(1), S(1)), List(S(2), S(2), S(2)), b*x)/S(2), x) + Simp(log(b*x)**S(2)/S(2), x)
1117
1118
1119def replacement6841(a, b, m, x):
1120    return -Dist(b/(m + S(1)), Int(x**(m + S(1))*sinh(a + b*x)/(a + b*x), x), x) + Simp(x**(m + S(1))*SinhIntegral(a + b*x)/(m + S(1)), x)
1121
1122
1123def replacement6842(a, b, m, x):
1124    return -Dist(b/(m + S(1)), Int(x**(m + S(1))*cosh(a + b*x)/(a + b*x), x), x) + Simp(x**(m + S(1))*CoshIntegral(a + b*x)/(m + S(1)), x)
1125
1126
1127def replacement6843(a, b, x):
1128    return -Dist(S(2), Int(SinhIntegral(a + b*x)*sinh(a + b*x), x), x) + Simp((a + b*x)*SinhIntegral(a + b*x)**S(2)/b, x)
1129
1130
1131def replacement6844(a, b, x):
1132    return -Dist(S(2), Int(CoshIntegral(a + b*x)*cosh(a + b*x), x), x) + Simp((a + b*x)*CoshIntegral(a + b*x)**S(2)/b, x)
1133
1134
1135def replacement6845(b, m, x):
1136    return -Dist(S(2)/(m + S(1)), Int(x**m*SinhIntegral(b*x)*sinh(b*x), x), x) + Simp(x**(m + S(1))*SinhIntegral(b*x)**S(2)/(m + S(1)), x)
1137
1138
1139def replacement6846(b, m, x):
1140    return -Dist(S(2)/(m + S(1)), Int(x**m*CoshIntegral(b*x)*cosh(b*x), x), x) + Simp(x**(m + S(1))*CoshIntegral(b*x)**S(2)/(m + S(1)), x)
1141
1142
1143def replacement6847(a, b, m, x):
1144    return -Dist(a*m/(b*(m + S(1))), Int(x**(m + S(-1))*SinhIntegral(a + b*x)**S(2), x), x) - Dist(S(2)/(m + S(1)), Int(x**m*SinhIntegral(a + b*x)*sinh(a + b*x), x), x) + Simp(x**(m + S(1))*SinhIntegral(a + b*x)**S(2)/(m + S(1)), x) + Simp(a*x**m*SinhIntegral(a + b*x)**S(2)/(b*(m + S(1))), x)
1145
1146
1147def replacement6848(a, b, m, x):
1148    return -Dist(a*m/(b*(m + S(1))), Int(x**(m + S(-1))*CoshIntegral(a + b*x)**S(2), x), x) - Dist(S(2)/(m + S(1)), Int(x**m*CoshIntegral(a + b*x)*cosh(a + b*x), x), x) + Simp(x**(m + S(1))*CoshIntegral(a + b*x)**S(2)/(m + S(1)), x) + Simp(a*x**m*CoshIntegral(a + b*x)**S(2)/(b*(m + S(1))), x)
1149
1150
1151def replacement6849(a, b, c, d, x):
1152    return -Dist(d/b, Int(sinh(c + d*x)*cosh(a + b*x)/(c + d*x), x), x) + Simp(SinhIntegral(c + d*x)*cosh(a + b*x)/b, x)
1153
1154
1155def replacement6850(a, b, c, d, x):
1156    return -Dist(d/b, Int(sinh(a + b*x)*cosh(c + d*x)/(c + d*x), x), x) + Simp(CoshIntegral(c + d*x)*sinh(a + b*x)/b, x)
1157
1158
1159def replacement6851(a, b, c, d, m, x):
1160    return -Dist(d/b, Int(x**m*sinh(c + d*x)*cosh(a + b*x)/(c + d*x), x), x) - Dist(m/b, Int(x**(m + S(-1))*SinhIntegral(c + d*x)*cosh(a + b*x), x), x) + Simp(x**m*SinhIntegral(c + d*x)*cosh(a + b*x)/b, x)
1161
1162
1163def replacement6852(a, b, c, d, m, x):
1164    return -Dist(d/b, Int(x**m*sinh(a + b*x)*cosh(c + d*x)/(c + d*x), x), x) - Dist(m/b, Int(x**(m + S(-1))*CoshIntegral(c + d*x)*sinh(a + b*x), x), x) + Simp(x**m*CoshIntegral(c + d*x)*sinh(a + b*x)/b, x)
1165
1166
1167def replacement6853(a, b, c, d, m, x):
1168    return -Dist(b/(m + S(1)), Int(x**(m + S(1))*SinhIntegral(c + d*x)*cosh(a + b*x), x), x) - Dist(d/(m + S(1)), Int(x**(m + S(1))*sinh(a + b*x)*sinh(c + d*x)/(c + d*x), x), x) + Simp(x**(m + S(1))*SinhIntegral(c + d*x)*sinh(a + b*x)/(m + S(1)), x)
1169
1170
1171def replacement6854(a, b, c, d, m, x):
1172    return -Dist(b/(m + S(1)), Int(x**(m + S(1))*CoshIntegral(c + d*x)*sinh(a + b*x), x), x) - Dist(d/(m + S(1)), Int(x**(m + S(1))*cosh(a + b*x)*cosh(c + d*x)/(c + d*x), x), x) + Simp(x**(m + S(1))*CoshIntegral(c + d*x)*cosh(a + b*x)/(m + S(1)), x)
1173
1174
1175def replacement6855(a, b, c, d, x):
1176    return -Dist(d/b, Int(sinh(a + b*x)*sinh(c + d*x)/(c + d*x), x), x) + Simp(SinhIntegral(c + d*x)*sinh(a + b*x)/b, x)
1177
1178
1179def replacement6856(a, b, c, d, x):
1180    return -Dist(d/b, Int(cosh(a + b*x)*cosh(c + d*x)/(c + d*x), x), x) + Simp(CoshIntegral(c + d*x)*cosh(a + b*x)/b, x)
1181
1182
1183def replacement6857(a, b, c, d, m, x):
1184    return -Dist(d/b, Int(x**m*sinh(a + b*x)*sinh(c + d*x)/(c + d*x), x), x) - Dist(m/b, Int(x**(m + S(-1))*SinhIntegral(c + d*x)*sinh(a + b*x), x), x) + Simp(x**m*SinhIntegral(c + d*x)*sinh(a + b*x)/b, x)
1185
1186
1187def replacement6858(a, b, c, d, m, x):
1188    return -Dist(d/b, Int(x**m*cosh(a + b*x)*cosh(c + d*x)/(c + d*x), x), x) - Dist(m/b, Int(x**(m + S(-1))*CoshIntegral(c + d*x)*cosh(a + b*x), x), x) + Simp(x**m*CoshIntegral(c + d*x)*cosh(a + b*x)/b, x)
1189
1190
1191def replacement6859(a, b, c, d, m, x):
1192    return -Dist(b/(m + S(1)), Int(x**(m + S(1))*SinhIntegral(c + d*x)*sinh(a + b*x), x), x) - Dist(d/(m + S(1)), Int(x**(m + S(1))*sinh(c + d*x)*cosh(a + b*x)/(c + d*x), x), x) + Simp(x**(m + S(1))*SinhIntegral(c + d*x)*cosh(a + b*x)/(m + S(1)), x)
1193
1194
1195def replacement6860(a, b, c, d, m, x):
1196    return -Dist(b/(m + S(1)), Int(x**(m + S(1))*CoshIntegral(c + d*x)*cosh(a + b*x), x), x) - Dist(d/(m + S(1)), Int(x**(m + S(1))*sinh(a + b*x)*cosh(c + d*x)/(c + d*x), x), x) + Simp(x**(m + S(1))*CoshIntegral(c + d*x)*sinh(a + b*x)/(m + S(1)), x)
1197
1198
1199def replacement6861(a, b, n, x):
1200    return -Simp(Gamma(n + S(1), a + b*x)/b, x) + Simp((a + b*x)*Gamma(n, a + b*x)/b, x)
1201
1202
1203def replacement6862(b, n, x):
1204    return Simp(Gamma(n)*log(x), x) - Simp((b*x)**n*HypergeometricPFQ(List(n, n), List(n + S(1), n + S(1)), -b*x)/n**S(2), x)
1205
1206
1207def replacement6863(b, m, n, x):
1208    return Simp(x**(m + S(1))*Gamma(n, b*x)/(m + S(1)), x) - Simp(x**m*(b*x)**(-m)*Gamma(m + n + S(1), b*x)/(b*(m + S(1))), x)
1209
1210
1211def With6864(a, b, m, n, x):
1212    _UseGamma = True
1213    return Dist(b/(m + S(1)), Int(x**(m + S(1))*(a + b*x)**(n + S(-1))*exp(-a - b*x), x), x) + Simp(x**(m + S(1))*Gamma(n, a + b*x)/(m + S(1)), x)
1214
1215
1216def replacement6865(a, b, x):
1217    return Simp(PolyGamma(S(-2), a + b*x)/b, x)
1218
1219
1220def replacement6866(a, b, m, x):
1221    return -Dist(m/b, Int(x**(m + S(-1))*PolyGamma(S(-2), a + b*x), x), x) + Simp(x**m*PolyGamma(S(-2), a + b*x)/b, x)
1222
1223
1224def replacement6867(a, b, n, x):
1225    return Simp(PolyGamma(n + S(-1), a + b*x)/b, x)
1226
1227
1228def replacement6868(a, b, m, n, x):
1229    return -Dist(m/b, Int(x**(m + S(-1))*PolyGamma(n + S(-1), a + b*x), x), x) + Simp(x**m*PolyGamma(n + S(-1), a + b*x)/b, x)
1230
1231
1232def replacement6869(a, b, m, n, x):
1233    return -Dist(b/(m + S(1)), Int(x**(m + S(1))*PolyGamma(n + S(1), a + b*x), x), x) + Simp(x**(m + S(1))*PolyGamma(n, a + b*x)/(m + S(1)), x)
1234
1235
1236def replacement6870(a, b, n, x):
1237    return Simp(Gamma(a + b*x)**n/(b*n), x)
1238
1239
1240def replacement6871(a, b, c, n, x):
1241    return Simp(Factorial(a + b*x)**n/(b*n), x)
1242
1243
1244def replacement6872(a, b, x):
1245    return Int(PolyGamma(S(1), a + b*x), x)
1246
1247
1248def replacement6873(a, b, s, x):
1249    return -Simp(Zeta(s + S(-1), a + b*x)/(b*(s + S(-1))), x)
1250
1251
1252def replacement6874(a, b, m, x):
1253    return Int(x**m*PolyGamma(S(1), a + b*x), x)
1254
1255
1256def replacement6875(a, b, m, s, x):
1257    return Dist(m/(b*(s + S(-1))), Int(x**(m + S(-1))*Zeta(s + S(-1), a + b*x), x), x) - Simp(x**m*Zeta(s + S(-1), a + b*x)/(b*(s + S(-1))), x)
1258
1259
1260def replacement6876(a, b, m, s, x):
1261    return Dist(b*s/(m + S(1)), Int(x**(m + S(1))*Zeta(s + S(1), a + b*x), x), x) + Simp(x**(m + S(1))*Zeta(s, a + b*x)/(m + S(1)), x)
1262
1263
1264def replacement6877(a, b, n, p, q, x):
1265    return -Dist(p*q, Int(PolyLog(n + S(-1), a*(b*x**p)**q), x), x) + Simp(x*PolyLog(n, a*(b*x**p)**q), x)
1266
1267
1268def replacement6878(a, b, n, p, q, x):
1269    return -Dist(S(1)/(p*q), Int(PolyLog(n + S(1), a*(b*x**p)**q), x), x) + Simp(x*PolyLog(n + S(1), a*(b*x**p)**q)/(p*q), x)
1270
1271
1272def replacement6879(a, b, c, d, e, n, p, x):
1273    return Simp(PolyLog(n + S(1), c*(a + b*x)**p)/(e*p), x)
1274
1275
1276def replacement6880(a, b, n, p, q, x):
1277    return Simp(PolyLog(n + S(1), a*(b*x**p)**q)/(p*q), x)
1278
1279
1280def replacement6881(a, b, m, n, p, q, x):
1281    return -Dist(p*q/(m + S(1)), Int(x**m*PolyLog(n + S(-1), a*(b*x**p)**q), x), x) + Simp(x**(m + S(1))*PolyLog(n, a*(b*x**p)**q)/(m + S(1)), x)
1282
1283
1284def replacement6882(a, b, m, n, p, q, x):
1285    return -Dist((m + S(1))/(p*q), Int(x**m*PolyLog(n + S(1), a*(b*x**p)**q), x), x) + Simp(x**(m + S(1))*PolyLog(n + S(1), a*(b*x**p)**q)/(p*q), x)
1286
1287
1288def replacement6883(a, b, c, m, n, p, q, r, x):
1289    return -Dist(m*r/(p*q), Int(PolyLog(n + S(1), a*(b*x**p)**q)*log(c*x**m)**(r + S(-1))/x, x), x) + Simp(PolyLog(n + S(1), a*(b*x**p)**q)*log(c*x**m)**r/(p*q), x)
1290
1291
1292def replacement6884(a, b, c, n, p, x):
1293    return -Dist(p, Int(PolyLog(n + S(-1), c*(a + b*x)**p), x), x) + Dist(a*p, Int(PolyLog(n + S(-1), c*(a + b*x)**p)/(a + b*x), x), x) + Simp(x*PolyLog(n, c*(a + b*x)**p), x)
1294
1295
1296def replacement6885(a, b, c, m, n, p, x):
1297    return -Dist(b*p/(m + S(1)), Int(x**(m + S(1))*PolyLog(n + S(-1), c*(a + b*x)**p)/(a + b*x), x), x) + Simp(x**(m + S(1))*PolyLog(n, c*(a + b*x)**p)/(m + S(1)), x)
1298
1299
1300def replacement6886(F, a, b, c, d, n, p, x):
1301    return Simp(PolyLog(n + S(1), d*(F**(c*(a + b*x)))**p)/(b*c*p*log(F)), x)
1302
1303
1304def replacement6887(F, a, b, c, d, e, f, m, n, p, x):
1305    return -Dist(f*m/(b*c*p*log(F)), Int((e + f*x)**(m + S(-1))*PolyLog(n + S(1), d*(F**(c*(a + b*x)))**p), x), x) + Simp((e + f*x)**m*PolyLog(n + S(1), d*(F**(c*(a + b*x)))**p)/(b*c*p*log(F)), x)
1306
1307
1308def With6888(n, u, v, x):
1309    if isinstance(x, (int, Integer, float, Float)):
1310        return False
1311    try:
1312        w = DerivativeDivides(v, u*v, x)
1313        res = Not(FalseQ(w))
1314    except (TypeError, AttributeError):
1315        return False
1316    if res:
1317        return True
1318    return False
1319
1320
1321def replacement6888(n, u, v, x):
1322
1323    w = DerivativeDivides(v, u*v, x)
1324    return Simp(w*PolyLog(n + S(1), v), x)
1325
1326
1327def With6889(n, u, v, w, x):
1328    if isinstance(x, (int, Integer, float, Float)):
1329        return False
1330    try:
1331        z = DerivativeDivides(v, u*v, x)
1332        res = Not(FalseQ(z))
1333    except (TypeError, AttributeError):
1334        return False
1335    if res:
1336        return True
1337    return False
1338
1339
1340def replacement6889(n, u, v, w, x):
1341
1342    z = DerivativeDivides(v, u*v, x)
1343    return -Int(SimplifyIntegrand(z*D(w, x)*PolyLog(n + S(1), v)/w, x), x) + Simp(z*PolyLog(n + S(1), v)*log(w), x)
1344
1345
1346def replacement6890(a, b, c, p, x):
1347    return Dist(p/(c*(p + S(1))), Int((c*ProductLog(a + b*x))**(p + S(1))/(ProductLog(a + b*x) + S(1)), x), x) + Simp((c*ProductLog(a + b*x))**p*(a + b*x)/(b*(p + S(1))), x)
1348
1349
1350def replacement6891(a, b, c, p, x):
1351    return -Dist(p, Int((c*ProductLog(a + b*x))**p/(ProductLog(a + b*x) + S(1)), x), x) + Simp((c*ProductLog(a + b*x))**p*(a + b*x)/b, x)
1352
1353
1354def replacement6892(a, b, c, m, p, x):
1355    return Dist(S(1)/b, Subst(Int(ExpandIntegrand((c*ProductLog(x))**p, (-a/b + x/b)**m, x), x), x, a + b*x), x)
1356
1357
1358def replacement6893(a, c, n, p, x):
1359    return -Dist(n*p, Int((c*ProductLog(a*x**n))**p/(ProductLog(a*x**n) + S(1)), x), x) + Simp(x*(c*ProductLog(a*x**n))**p, x)
1360
1361
1362def replacement6894(a, c, n, p, x):
1363    return Dist(n*p/(c*(n*p + S(1))), Int((c*ProductLog(a*x**n))**(p + S(1))/(ProductLog(a*x**n) + S(1)), x), x) + Simp(x*(c*ProductLog(a*x**n))**p/(n*p + S(1)), x)
1364
1365
1366def replacement6895(a, c, n, p, x):
1367    return -Subst(Int((c*ProductLog(a*x**(-n)))**p/x**S(2), x), x, S(1)/x)
1368
1369
1370def replacement6896(a, c, m, n, p, x):
1371    return -Dist(n*p/(m + S(1)), Int(x**m*(c*ProductLog(a*x**n))**p/(ProductLog(a*x**n) + S(1)), x), x) + Simp(x**(m + S(1))*(c*ProductLog(a*x**n))**p/(m + S(1)), x)
1372
1373
1374def replacement6897(a, c, m, n, p, x):
1375    return Dist(n*p/(c*(m + n*p + S(1))), Int(x**m*(c*ProductLog(a*x**n))**(p + S(1))/(ProductLog(a*x**n) + S(1)), x), x) + Simp(x**(m + S(1))*(c*ProductLog(a*x**n))**p/(m + n*p + S(1)), x)
1376
1377
1378def replacement6898(a, c, m, p, x):
1379    return Dist(S(1)/c, Int(x**m*(c*ProductLog(a*x))**(p + S(1))/(ProductLog(a*x) + S(1)), x), x) + Int(x**m*(c*ProductLog(a*x))**p/(ProductLog(a*x) + S(1)), x)
1380
1381
1382def replacement6899(a, c, m, n, p, x):
1383    return -Subst(Int(x**(-m + S(-2))*(c*ProductLog(a*x**(-n)))**p, x), x, S(1)/x)
1384
1385
1386def replacement6900(a, b, d, x):
1387    return Simp((a + b*x)/(b*d*ProductLog(a + b*x)), x)
1388
1389
1390def replacement6901(a, b, d, x):
1391    return -Int(S(1)/(d*ProductLog(a + b*x) + d), x) + Simp(d*x, x)
1392
1393
1394def replacement6902(a, b, c, d, p, x):
1395    return -Dist(c*p, Int((c*ProductLog(a + b*x))**(p + S(-1))/(d*ProductLog(a + b*x) + d), x), x) + Simp(c*(c*ProductLog(a + b*x))**(p + S(-1))*(a + b*x)/(b*d), x)
1396
1397
1398def replacement6903(a, b, d, x):
1399    return Simp(ExpIntegralEi(ProductLog(a + b*x))/(b*d), x)
1400
1401
1402def replacement6904(a, b, c, d, x):
1403    return Simp(Erfi(sqrt(c*ProductLog(a + b*x))/Rt(c, S(2)))*Rt(Pi*c, S(2))/(b*c*d), x)
1404
1405
1406def replacement6905(a, b, c, d, x):
1407    return Simp(Erf(sqrt(c*ProductLog(a + b*x))/Rt(-c, S(2)))*Rt(-Pi*c, S(2))/(b*c*d), x)
1408
1409
1410def replacement6906(a, b, c, d, p, x):
1411    return -Dist(S(1)/(c*(p + S(1))), Int((c*ProductLog(a + b*x))**(p + S(1))/(d*ProductLog(a + b*x) + d), x), x) + Simp((c*ProductLog(a + b*x))**p*(a + b*x)/(b*d*(p + S(1))), x)
1412
1413
1414def replacement6907(a, b, c, d, p, x):
1415    return Simp((-ProductLog(a + b*x))**(-p)*(c*ProductLog(a + b*x))**p*Gamma(p + S(1), -ProductLog(a + b*x))/(b*d), x)
1416
1417
1418def replacement6908(a, b, d, m, x):
1419    return Dist(S(1)/b, Subst(Int(ExpandIntegrand(S(1)/(d*ProductLog(x) + d), (-a/b + x/b)**m, x), x), x, a + b*x), x)
1420
1421
1422def replacement6909(a, b, c, d, m, p, x):
1423    return Dist(S(1)/b, Subst(Int(ExpandIntegrand((c*ProductLog(x))**p/(d*ProductLog(x) + d), (-a/b + x/b)**m, x), x), x, a + b*x), x)
1424
1425
1426def replacement6910(a, d, n, x):
1427    return -Subst(Int(S(1)/(x**S(2)*(d*ProductLog(a*x**(-n)) + d)), x), x, S(1)/x)
1428
1429
1430def replacement6911(a, c, d, n, p, x):
1431    return Simp(c*x*(c*ProductLog(a*x**n))**(p + S(-1))/d, x)
1432
1433
1434def replacement6912(a, d, n, p, x):
1435    return Simp(a**p*ExpIntegralEi(-p*ProductLog(a*x**n))/(d*n), x)
1436
1437
1438def replacement6913(a, c, d, n, p, x):
1439    return Simp(a**(-S(1)/n)*c**(-S(1)/n)*Erfi(sqrt(c*ProductLog(a*x**n))/Rt(c*n, S(2)))*Rt(Pi*c*n, S(2))/(d*n), x)
1440
1441
1442def replacement6914(a, c, d, n, p, x):
1443    return Simp(a**(-S(1)/n)*c**(-S(1)/n)*Erf(sqrt(c*ProductLog(a*x**n))/Rt(-c*n, S(2)))*Rt(-Pi*c*n, S(2))/(d*n), x)
1444
1445
1446def replacement6915(a, c, d, n, p, x):
1447    return -Dist(c*(n*(p + S(-1)) + S(1)), Int((c*ProductLog(a*x**n))**(p + S(-1))/(d*ProductLog(a*x**n) + d), x), x) + Simp(c*x*(c*ProductLog(a*x**n))**(p + S(-1))/d, x)
1448
1449
1450def replacement6916(a, c, d, n, p, x):
1451    return -Dist(S(1)/(c*(n*p + S(1))), Int((c*ProductLog(a*x**n))**(p + S(1))/(d*ProductLog(a*x**n) + d), x), x) + Simp(x*(c*ProductLog(a*x**n))**p/(d*(n*p + S(1))), x)
1452
1453
1454def replacement6917(a, c, d, n, p, x):
1455    return -Subst(Int((c*ProductLog(a*x**(-n)))**p/(x**S(2)*(d*ProductLog(a*x**(-n)) + d)), x), x, S(1)/x)
1456
1457
1458def replacement6918(a, d, m, x):
1459    return -Dist(m/(m + S(1)), Int(x**m/((d*ProductLog(a*x) + d)*ProductLog(a*x)), x), x) + Simp(x**(m + S(1))/(d*(m + S(1))*ProductLog(a*x)), x)
1460
1461
1462def replacement6919(a, d, x):
1463    return Simp(log(ProductLog(a*x))/d, x)
1464
1465
1466def replacement6920(a, d, m, x):
1467    return -Int(x**m*ProductLog(a*x)/(d*ProductLog(a*x) + d), x) + Simp(x**(m + S(1))/(d*(m + S(1))), x)
1468
1469
1470def replacement6921(a, d, m, x):
1471    return Simp(x**m*(-(m + S(1))*ProductLog(a*x))**(-m)*Gamma(m + S(1), -(m + S(1))*ProductLog(a*x))*exp(-m*ProductLog(a*x))/(a*d*(m + S(1))), x)
1472
1473
1474def replacement6922(a, d, n, x):
1475    return Simp(log(ProductLog(a*x**n))/(d*n), x)
1476
1477
1478def replacement6923(a, d, m, n, x):
1479    return -Subst(Int(x**(-m + S(-2))/(d*ProductLog(a*x**(-n)) + d), x), x, S(1)/x)
1480
1481
1482def replacement6924(a, c, d, n, p, x):
1483    return Simp((c*ProductLog(a*x**n))**p/(d*n*p), x)
1484
1485
1486def replacement6925(a, c, d, m, n, p, x):
1487    return Simp(c*x**(m + S(1))*(c*ProductLog(a*x**n))**(p + S(-1))/(d*(m + S(1))), x)
1488
1489
1490def replacement6926(a, d, m, n, p, x):
1491    return Simp(a**p*ExpIntegralEi(-p*ProductLog(a*x**n))/(d*n), x)
1492
1493
1494def replacement6927(a, c, d, m, n, p, x):
1495    return Simp(a**(p + S(-1)/2)*c**(p + S(-1)/2)*Erf(sqrt(c*ProductLog(a*x**n))/Rt(c/(p + S(-1)/2), S(2)))*Rt(Pi*c/(p + S(-1)/2), S(2))/(d*n), x)
1496
1497
1498def replacement6928(a, c, d, m, n, p, x):
1499    return Simp(a**(p + S(-1)/2)*c**(p + S(-1)/2)*Erfi(sqrt(c*ProductLog(a*x**n))/Rt(-c/(p + S(-1)/2), S(2)))*Rt(-Pi*c/(p + S(-1)/2), S(2))/(d*n), x)
1500
1501
1502def replacement6929(a, c, d, m, n, p, x):
1503    return -Dist(c*(m + n*(p + S(-1)) + S(1))/(m + S(1)), Int(x**m*(c*ProductLog(a*x**n))**(p + S(-1))/(d*ProductLog(a*x**n) + d), x), x) + Simp(c*x**(m + S(1))*(c*ProductLog(a*x**n))**(p + S(-1))/(d*(m + S(1))), x)
1504
1505
1506def replacement6930(a, c, d, m, n, p, x):
1507    return -Dist((m + S(1))/(c*(m + n*p + S(1))), Int(x**m*(c*ProductLog(a*x**n))**(p + S(1))/(d*ProductLog(a*x**n) + d), x), x) + Simp(x**(m + S(1))*(c*ProductLog(a*x**n))**p/(d*(m + n*p + S(1))), x)
1508
1509
1510def replacement6931(a, c, d, m, p, x):
1511    return Simp(x**m*(c*ProductLog(a*x))**p*(-(m + S(1))*ProductLog(a*x))**(-m - p)*Gamma(m + p + S(1), -(m + S(1))*ProductLog(a*x))*exp(-m*ProductLog(a*x))/(a*d*(m + S(1))), x)
1512
1513
1514def replacement6932(a, c, d, m, n, p, x):
1515    return -Subst(Int(x**(-m + S(-2))*(c*ProductLog(a*x**(-n)))**p/(d*ProductLog(a*x**(-n)) + d), x), x, S(1)/x)
1516
1517
1518def replacement6933(u, x):
1519    return Subst(Int(SimplifyIntegrand((x + S(1))*SubstFor(ProductLog(x), u, x)*exp(x), x), x), x, ProductLog(x))
1520