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This code is automatically generated. Never edit it manually.
For details of generating the code see `rubi_parsing_guide.md` in `parsetools`.
    )import_modulematchpy)PatternReplacementRuleCustomConstraintis_match(  IntSumSetWithModuleScanMapAndFalseQZeroQ	NegativeQNonzeroQFreeQNFreeQListLog	PositiveQPositiveIntegerQNegativeIntegerQIntegerQ	IntegersQComplexNumberQPureComplexNumberQRealNumericQPositiveOrZeroQNegativeOrZeroQFractionOrNegativeQNegQEqualUnequalIntPartFracPart	RationalQProductQSumQNonsumQSubstFirstRestSqrtNumberQSqrtNumberSumQLinearQSqrtArcCoshCoefficientDenominatorHypergeometric2F1NotSimplifyFractionalPartIntegerPartAppellF1
EllipticPi	EllipticE	EllipticFArcTanArcCotArcCothArcTanhArcSinArcSinhArcCosArcCscArcSecArcCschArcSechSinhTanhCoshSechCschCoth	LessEqualLessGreaterGreaterEqual	FractionQIntLinearcQExpandIndependentQPowerQIntegerPowerQPositiveIntegerPowerQFractionalPowerQAtomQExpQLogQHeadMemberQTrigQSinQCosQTanQCotQSecQCscQSinCosTanCotSecCscHyperbolicQSinhQCoshQTanhQCothQSechQCschQInverseTrigQSinCosQ	SinhCoshQ	LeafCount	NumeratorNumberQNumericQLengthListQImReInverseHyperbolicQInverseFunctionQTrigHyperbolicFreeQInverseFunctionFreeQRealQEqQFractionalPowerFreeQComplexFreeQPolynomialQFactorSquareFreePowerOfLinearQExponent
QuadraticQLinearPairQBinomialPartsTrinomialPartsPolyQEvenQOddQPerfectSquareQNiceSqrtAuxQ	NiceSqrtQTogetherPosAuxPosQCoefficientList
ReplaceAllExpandLinearProductGCDContentFactorNumericFactorNonnumericFactorsMakeAssocListGensymSubstKernelSubstExpandExpressionApart
SmartApartMatchQPolynomialQuotientRemainderFreeFactorsNonfreeFactorsRemoveContentAuxRemoveContent	FreeTermsNonfreeTermsExpandAlgebraicFunctionCollectReciprocalsExpandCleanupAlgebraicFunctionQCoeffLeadTermRemainingTerms
LeadFactorRemainingFactorsLeadBase
LeadDegreeNumerDenom	hypergeomExponMergeMonomialsPolynomialDivide	BinomialQ
TrinomialQGeneralizedBinomialQGeneralizedTrinomialQFactorSquareFreeListPerfectPowerTestSquareFreeFactorTestRationalFunctionQRationalFunctionFactorsNonrationalFunctionFactorsReverseRationalFunctionExponentsRationalFunctionExpandExpandIntegrandSimplerQSimplerSqrtQSumSimplerQBinomialDegreeTrinomialDegreeCancelCommonFactorsSimplerIntegrandQGeneralizedBinomialDegreeGeneralizedBinomialPartsGeneralizedTrinomialDegreeGeneralizedTrinomialParts	MonomialQMonomialSumQMinimumMonomialExponentMonomialExponentLinearMatchQPowerOfLinearMatchQQuadraticMatchQCubicMatchQBinomialMatchQTrinomialMatchQGeneralizedBinomialMatchQGeneralizedTrinomialMatchQQuotientOfLinearsMatchQPolynomialTermQPolynomialTermsNonpolynomialTermsPseudoBinomialPartsNormalizePseudoBinomialPseudoBinomialPairQPseudoBinomialQPolynomialGCDPolyGCDAlgebraicFunctionFactorsNonalgebraicFunctionFactorsQuotientOfLinearsPQuotientOfLinearsPartsQuotientOfLinearsQFlattenSortAbsurdNumberQAbsurdNumberFactorsNonabsurdNumberFactorsSumSimplerAuxQPrependDropCombineExponentsFactorIntegerFactorAbsurdNumberSubstForInverseFunctionSubstForFractionalPower*SubstForFractionalPowerOfQuotientOfLinears"FractionalPowerOfQuotientOfLinearsSubstForFractionalPowerQSubstForFractionalPowerAuxQFractionalPowerOfSquareQFractionalPowerSubexpressionQApplyFactorNumericGcdMergeableFactorQMergeFactorMergeFactorsTrigSimplifyQTrigSimplifyTrigSimplifyRecurOrderFactorOrderSmallestOrderedQMinimumDegreePositiveFactorsSignNonpositiveFactorsPolynomialInAuxQPolynomialInQExponentInAux
ExponentInPolynomialInSubstAuxPolynomialInSubstDistribDistributeDegreeFunctionOfPowerDivideDegreesOfFactorsMonomialFactorFullSimplifyFunctionOfLinearSubstFunctionOfLinearNormalizeIntegrandNormalizeIntegrandAuxNormalizeIntegrandFactorNormalizeIntegrandFactorBaseNormalizeTogetherNormalizeLeadTermSignsAbsorbMinusSignNormalizeSumFactorsSignOfFactorNormalizePowerOfLinearSimplifyIntegrandSimplifyTermTogetherSimplifySmartSimplifySubstForExpnExpandToSumUnifySum
UnifyTerms	UnifyTerm	CalculusQFunctionOfInverseLinearPureFunctionOfSinhQPureFunctionOfTanhQPureFunctionOfCoshQIntegerQuotientQOddQuotientQEvenQuotientQFindTrigFactorFunctionOfSinhQFunctionOfCoshQOddHyperbolicPowerQFunctionOfTanhQFunctionOfTanhWeightFunctionOfHyperbolicQSmartNumeratorSmartDenominatorSubstForAuxActivateTrig
ExpandTrig
TrigExpandSubstForTrigSubstForHyperbolicInertTrigFreeQLCMSubstForFractionalPowerOfLinearFractionalPowerOfLinearInverseFunctionOfLinear
InertTrigQInertReciprocalQDeactivateTrigFixInertTrigFunctionDeactivateTrigAuxPowerOfInertTrigSumQPiecewiseLinearQKnownTrigIntegrandQKnownSineIntegrandQKnownTangentIntegrandQKnownCotangentIntegrandQKnownSecantIntegrandQTryPureTanSubstTryTanhSubstTryPureTanhSubstAbsurdNumberGCDAbsurdNumberGCDListExpandTrigExpandExpandTrigReduceExpandTrigReduceAuxNormalizeTrig	TrigToExpExpandTrigToExp
TrigReduceFunctionOfTrigAlgebraicTrigFunctionQFunctionOfHyperbolicFunctionOfQFunctionOfExpnQPureFunctionOfSinQPureFunctionOfCosQPureFunctionOfTanQPureFunctionOfCotQFunctionOfCosQFunctionOfSinQOddTrigPowerQFunctionOfTanQFunctionOfTanWeightFunctionOfTrigQFunctionOfDensePolynomialsQFunctionOfLogPowerVariableExpnPowerVariableDegreePowerVariableSubstEulerIntegrandQFunctionOfSquareRootOfQuadraticSquareRootOfQuadraticSubstDividesEasyDQProductOfLinearPowersQRtNthRoot	AtomBaseQSumBaseQNegSumBaseQAllNegTermQSomeNegTermQTrigSquareQRtAux
TrigSquareIntSumIntTermMap2ConstantFactorSameQReplacePartCommonFactorsMostMainFactorPositionFunctionOfExponentialQFunctionOfExponentialFunctionOfExponentialFunction FunctionOfExponentialFunctionAuxFunctionOfExponentialTestFunctionOfExponentialTestAuxstdev	rubi_testIfIntQuadraticQIntBinomialQRectifyTangentRectifyCotangent
Inequality	ConditionSimpSimpHelpSplitProductSplitSumSubstForrG  FresnelSFresnelCErfcErfiGammaFunctionOfTrigOfLinearQElementaryFunctionQComplexUnsameQ_SimpFixFactorSimpFixFactor_FixSimplifyFixSimplify_SimplifyAntiderivativeSumSimplifyAntiderivativeSum_SimplifyAntiderivativeSimplifyAntiderivative_TrigSimplifyAuxTrigSimplifyAuxCancelPartPolyLogDDistSum_doitPolynomialQuotientFloorPolynomialRemainderFactorr  CosIntegralSinIntegralLogIntegralSinhIntegralCoshIntegralRuleErf	PolyGammaExpIntegralEiExpIntegralELogGammaUtilityOperator	FactorialZeta
ProductLogDerivativeDividesHypergeometricPFQIntHideOneQNullrubi_exprubi_logDiscriminantNegativeQuotient)IntegralSsqrtAndOrIntegerFloatModIAbssimplifyMulAddPowsign
EulerGammaWC)symbolsSymbol)sincostancotcscsecr  erf)acoshasinhatanhacothacschasechcoshsinhtanhcothsechcsch)atanacscasinacotacosasecatan2)pic                 C      g | ]}t |qS  r  .0ir  r  O/usr/lib/python3/dist-packages/sympy/integrals/rubi/rules/miscellaneous_trig.py
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|| 	}t|t҃}tttdtdtttdtd tdtd  td	td tdtdtd
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|| }t|tՃ}ttttdtdtd
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|}t|tڃ}ttttdtdtd
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|||	}?t|?t
}@tttttdtd tdtd tttdtd tdtd  t|||||}At|At}Btttttdtd tdtd tttdtd tdtd  t|||||}Ct|Ct}Dtttttdtd tdtd tttdtd tdtd  t|||||}Et|Et}Ftttdtdtttdtd tdtd  t tttdtd tdtd td  t||||||||	}Gt|Gt}Htttdtdtttdtd tdtd  t tttdtd tdtd td  t||||||||	}It|It}Jtttdtdtttdtd tdtd  tdtd tttdtd tdtd tdtd  t|||||||||
}Kt|Kt}Ltttdtdtttdtd tdtd  td	td tttdtd tdtd tdtd  t|||||||||
}Mt|Mt}Ntttdtdtttdtd tdtd  t tdtdtttdtd tdtd  t  t||||||||||||}Ot|Ot}Ptttdtdtttdtd tdtd  t tdtdtttdtd tdtd  t  t||||||||||||}Qt|Qt}Rtttdtdtttdtd tdtd  tdtd tdtdtttdtd tdtd  t  t||||||||||||}St|St}Ttttdtdtttdtd tdtd  tdtd tdtdtttdtd tdtd  t  t||||||||||||}Ut|Ut}Vtttdtdtttdtd tdtd  t tdtdtttdtd tdtd  t  t||||||||||||| |!}Wt|Wt}Xtttdtdtttdtd tdtd  t tdtdtttdtd tdtd  t  t||||||||||||| |!}Yt|Yt}Ztttdtdtttdtd tdtd  t tdtdtttdtd tdtd  t  t|||||||||||"||#|$|!}[t|[t}\tttdtdtttdtd tdtd  t tdtdtttdtd tdtd  t  t|||||||||||"||#|$|!}]t|]t}^tttdtdtttdtd tdtd  t tdtdtttdtd tdtd  t  t|||||||||||%|"|&|!}_t|_t}`tttdtdtttdtd tdtd  t tdtdtttdtd tdtd  t  t|||||||||||%|"|&|!}at|at}btttdtdtttdtd tdtd  t tdtdtttdtd tdtd  t  t|||||||||||%|'|#|(|!}ct|ct}dtttdtdtttdtd tdtd  t tdtdtttdtd tdtd  t  t|||||||||||%|'|#|(|!}et|et}ftttdtdtttdtd tdtd  t tttdtd tdtd  t|||||||||)|*|+}gt|gt }htttdtdtttdtd tdtd  t tttdtd tdtd  t|||||||||)|*|+}it|it!}jtttdtdtttdtd tdtd  t tttdtd tdtd  t|||||||||)||+}kt|kt"}ltttdtdtttdtd tdtd  t tttdtd tdtd  t|||||||||)||+}mt|mt#}ntttttdtd tdtd t$tttdtd tdtd  t||||||}ot|ot%}ptttttdtd tdtd t$tttdtd tdtd  t||||||}qt|qt&}rtttdtdtttdtd tdtd  t tttdtd tdtd  t||||||||||+}st|st'}ttttdtdtttdtd tdtd  t tttdtd tdtd  t||||||||||+}ut|ut(}vtttdtdtttdtd tdtd  tdtd tdtdtttdtd tdtd  t  t|||||||||||}wt|wt)}xtttdtdtttdtd tdtd  td	td tdtdtttdtd tdtd  t  t|||||||||||}yt|yt*}ztttdtdtttdtd tdtd  t tttdtd tdtd td  tttdtd tdtd td  t||||||||	}{t|{t+}|tttdtdtttdtd tdtd  tdtd tdtdtttdtd tdtd  td	td  tttdtd tdtd tdtd  t|||||||||||}}t|}t,}~tttdtdtttdtd tdtd  tdtd tdtdtttdtd tdtd  td	td  tdtdtttdtd tdtd  t  t||||||||||||||,}t|t-}tttdtdtttdtd tdtd  t tdtdtttdtd tdtd  t.  tdtdtttdtd tdtd  t  t||||||||||||||,}t|t/}tttdtdtttdtd tdtd  tdtd tdtdtttdtd tdtd  td	td  tdtdtttdtd tdtd  t  t||||||||||||||-|(}t|t0}tttdtdtttdtd tdtd  t tdtdtttdtd tdtd  t.  tdtdtttdtd tdtd  t  t|||||||||||||.||/|0}t|t1}tttdtdtttdtd tdtd  t tdtdtttdtd tdtd  t.  tdtdtttdtd tdtd  t  t|||||||||||||.||/|0}t|t2}tttdtdtttdtd tdtd  t tdtdtttdtd tdtd  td	td  tdtdtttdtd tdtd  t  t|||||||||||||"||1|/|0|2}t|t3}tttdtdtttdtd tdtd  t tdtdtttdtd tdtd  td	td  tdtdtttdtd tdtd  t  t|||||||||||||"||1|/|0|2}t|t4}tttdtdtttdtd tdtd  t tdtdtttdtd tdtd  t.  tdtdtttdtd tdtd  t  t||||||||||||3|"|4|/|0}t|t5}tttdtdtttdtd tdtd  t tdtdtttdtd tdtd  t.  tdtdtttdtd tdtd  t  t||||||||||||3|"|4|/|0}t|t6}tttdtdtttdtd tdtd  t tdtdtttdtd tdtd  td	td  tdtdtttdtd tdtd  t  t|||||||||||||%|"|5|0}t|t7}tttdtdtttdtd tdtd  t tdtdtttdtd tdtd  td	td  tdtdtttdtd tdtd  t  t|||||||||||||%|"|5|0}t|t8}tttdtdtttdtd tdtd  t tdtdtttdtd tdtd  td	td  tdtdtttdtd tdtd  t  t|||||||||||6|'|7|*|5|0}t|t9}tttdtdtttdtd tdtd  t tdtdtttdtd tdtd  td	td  tdtdtttdtd tdtd  t  t|||||||||||6|'|7|*|5|0}t|t:}tttdtdtttdtd tdtd  t tdtdtttdtd tdtd  td	td  tdtdtttdtd tdtd  t  t|||||||||||6|'|7||1|(|0}t|t;}tttdtdtttdtd tdtd  t tdtdtttdtd tdtd  td	td  tdtdtttdtd tdtd  t  t|||||||||||6|'|7||1|(|0}t|t<}tttdtdtttdtd tdtd  t tdtdtttdtd tdtd  td	td  tdtdtttdtd tdtd  t  t|||||||||||||%|'|1|(|0}t|t=}tttdtdtttdtd tdtd  t tdtdtttdtd tdtd  td	td  tdtdtttdtd tdtd  t  t|||||||||||||%|'|1|(|0}t|t>}tttdtdtttdtd tdtd  tdtd tdtdtttdtd tdtd  td	td  tdtdtttdtd tdtd  t  t|||||||||||||}t|t?}tttdtdtttdtd tdtd  tdtd tttdtd tdtd  t||||||||8	}t|t@}tttAttdtd tdtd  tdtd tB t t|||||9|:|;}t|tC}tttdtAttdtd tdtd  tdtd tB  t|||||9|<|=}t|tD}tttdtAttdtd tdtd  tdtd tB  t|||||9|>|=}t|tE}tttFttdtd tdtd  tAttdtd tdtd  tdtd tB  t||||||?|:|=	}t|tG}tttAttdtd tdtd  tdtd t. t||||||9|@|	}t|tH}tttAttdtd tdtd  tdtd t tdtd td	td t|||||||9|@|
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}t|t|}ttttAttdtd tdtd tdtd  tdtd tz  t|||||M|>|E|BtJt}
}t|t~}tttttJt}t|t}tttttJt}t|t}tttdtdtdtdtttdtd tdtd td   tdtdtttdtd tdtd td   tdtd tdtd t|||||||N}t|t}tttdtdtdtdtttdtd tdtd td   tdtdtttdtd tdtd td   tdtd tdtd t|||||||O}t|t}tttdtdtdtdtttdtd tdtd td   tdtdtttdtd tdtd td   tdtd tdtd t|||||||O}t|t}tttt t|PtJt}t|t}ttttt  t|PtJt}t|t}tttttdtd  t||Q|PtJt}t|t}tttttdtd  ttd	td  t|||Q|PtJt}t|t}tttAttdtd tdtd  tdtd t. tdtd t||||||9|@|	}t|t}tttAttdtd tdtd  tdtd t tdtd td	td tdtd t|||||||9|@|
}t|t}tttt|RtJt}t|t}tttdtttdtd tdtd  td	td tdtd tdtdtttdtd tdtd td	td   t tdtd t|||||S}t|t}tttdtttdtd tdtd  td	td tdtd tdtttdtd tdtd  td	td tdtd  t tdtd t|||||S}t|t} ttttAttdtd tdtd  tB tAttdtd tdtd  tdtd  td	td  t||||||U|9|:|T
}t|t}ttttAttdtd tdtd  tB tAttdtd tdtd  tdtd  tAttdtd tdtd  tdtd  td	td  t|||||||U|W|9|:|T|V}t|t}ttttAttdtd tdtd  tdtd tAttdtd tdtd  tdtd  tB td	td  t|||||||U|9|:|X}t|t}tttdtdtttdtd tdtd  tdtdtttdtd tdtd   td	td tdtd t||||||Y}t|t}tttt|Z}	t|	t}
tttBtz t tdtd t||||[}t|t}tttzt t tdtd t||||[}t|t}tttztdtd ttd	td  t tdtd t|||||\}t|t}tttt|PtJt}t|t}tttt|R|]tJt}t|t}tttt|P}t|t}ttttdtd tdtd tdtd tttdtd tdtd td	td  tttdtd tdtd  t||||||^|_}t|t}ttttdtd tdtd tdtd tttdtd tdtd  tttdtd tdtd td	td  t||||||^|_}t|t}ttttdtd tdtd tdtd tttdtd tdtd td	td  tttdtd tdtd tdtd  t||||||`}t|t}ttttdtd tdtd tdtd tttdtd tdtd td	td  tttdtd tdtd tdtd  t||||||`}t|t}ttttdtd tdtd tdtd tdtttdtd tdtd  tdtd  tttdtd tdtd td	td  t||||||`}t|t} ttttdtd tdtd tdtd tdtttdtd tdtd  td	td  tttdtd tdtd tdtd  t||||||a|%|b	}!t|!t}"ttttdtd tdtd tdtd tdtttdtd tdtd  td	td  tdtttdtd tdtd  tdtd  t||||||a|%|b	}#t|#t}$ttttdtd tdtd tdtd tttdtd tdtd td	td  tttdtd tdtd td  t||||||^|_}%t|%t}&ttttdtd tdtd tdtd tdtttdtd tdtd  td	td  tttdtd tdtd td  t||||||^|_}'t|'t}(ttttdtd tdtd tdtd tttdtd tdtd t  tttdtd tdtd  t||||||})t|)t}*ttttdtd tdtd tdtd tdtttdtd tdtd  td	td  tttdtd tdtd t  t|||||||}+t|+t},ttttdtd tdtd tdtd tdtttdtd tdtd  t  tttdtd tdtd  t||||||}-t|-t}.ttttdtd tdtd tdtd tdtttdtd tdtd  td	td  tdtttdtd tdtd  t  t|||||||}/t|/t}0ttttdtd tdtd tdtd tdtttdtd tdtd  td	td  tttdtd tdtd tdtd  t|||||||^|c	}1t|1t}2ttttdtd tdtd tdtd tdtttdtd tdtd  td	td  tdtttdtd tdtd  tdtd  t|||||||^|c	}3t|3t}4ttttdtd tdtd tdtd tdtttdtd tdtd  td	td  tdtttdtd tdtd  td	td  t|||||%|:}5t|5t}6ttttdtd tdtd tdtd tdtttdtd tdtd  td	td  tdtttdtd tdtd  tdtd  t|||||S|%|b|d	}7t|7t}8tttAtz tFt  ttdtd  t||||e|f|g|h|i	}9t|9t}:tttBtdtdtttdtd tdtd   td	td ttdtd tdtd tdtd  tttdtd tdtd  t||||||||^|_
};t|;t}<tttBtdtdtttdtd tdtd   td	td ttdtd tdtd tdtd  tttdtd tdtd  t||||||||^|_
}=t|=t}>tttBtdtdtttdtd tdtd   td	td ttdtd tdtd tdtd  tttdtd tdtd td  t||||||||^|_
}?t|?t}@tttBtdtdtttdtd tdtd   td	td ttdtd tdtd tdtd  tttdtd tdtd td  t||||||||^|_
}At|At}BtttBtdtdtttdtd tdtd   td	td ttdtd tdtd tdtd  tttdtd tdtd  tttdtd tdtd  t||||||||^|_
}Ct|Ct}DtttBtdtdtttdtd tdtd   td	td ttdtd tdtd tdtd  tttdtd tdtd tttdtd tdtd   t||||||||^|_
}Et|Et}Fttttdtd tdtd tdtd tttdtd tdtd tdtd  tttdtd tdtd tdtd  t|||||||J|j	}Gt|Gt}Httttdtd tdtd tdtd tttdtd tdtd tdtd  tttdtd tdtd tdtd  t|||||||J|j	}It|It}Jttttdtd tdtd tdtd tttdtd tdtd tdtd  tttdtd tdtd tdtd  t||||||||J	}Kt|Kt}LtttAttdtd tdtd  tFttdtd tdtd   ttdtd tdtd tdtd  t||||||||k|l|J||m}Mt|Mt}NtttAttdtd tdtd tdtd  tttdtd tdtd  t|o||||||n}Ot|Ot}PtttAttdtd tdtd tdtd  tttdtd tdtd  t|o||||||n}Qt|Qt}RtttAttdtd tdtd tdtd  tttdtd tdtd t.  t|o||||||p|q|r
}St|St}TtttAttdtd tdtd tdtd  tttdtd tdtd t  t|o||||||s|%|"
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}gt|gt̃}htttAttdtd tdtd tdtd  tdtttdtd tdtd  t.  t|o||||||p|q|4
}it|it̓}jtttAttdtd tdtd tdtd  tdtttdtd tdtd  t.  t|o|||||||x|y|z}kt|kt΃}ltttAttdtd tdtd tdtd  tdtttdtd tdtd  t.  t|o|||||||x|y|z}mt|mtσ}ntttAttdtd tdtd tdtd  tdtttdtd tdtd  t.  t|o||||||{|q|r||}ot|otЃ}ptttAttdtd tdtd tdtd  tdtttdtd tdtd  t.  t|o||||||{|q|r||}qt|qtу}rtttAttdtd tdtd tdtd  tdtttdtd tdtd  td	td  t|o||||||:}st|st҃}ttttAttdtd tdtd tdtd  tdtttdtd tdtd  td	td  t|o||||||:}ut|utӃ}vtttAttdtd tdtd tdtd  tdtttdtd tdtd  td	td  t|o||||||@}wt|wtԃ}xtttAttdtd tdtd tdtd  tdtttdtd tdtd  td	td  t|o||||||@}yt|ytՃ}ztttAttdtd tdtd tdtd  ttdtdtttdtd tdtd   td	td  t|o||||||||}|~}{t|{t׃}|tttAttdtd tdtd tdtd  ttdtdtttdtd tdtd   td	td  t|o|||||||||~}}t|}t؃}~tttAttdtd tdtd tdtd  ttdtdtttdtd tdtd   td	td  t|o|||||||||~}t|tك}tttAttdtd tdtd tdtd  ttdtdtttdtd tdtd   td	td  tttdtd tdtd tdtd  t|o||||||||}||}t|tڃ}tttAttdtd tdtd tdtd  ttdtdtttdtd tdtd   td	td  tttdtd tdtd tdtd  t|o||||||||||}t|tۃ}tttAttdtd tdtd tdtd  ttdtdtttdtd tdtd   td	td  tttdtd tdtd tdtd  t|o||||||||||}t|t܃}tttAttdtd tdtd tdtd  ttdtdtttdtd tdtd    ttdtdtttdtd tdtd    t|o||||||||||}||}t|tރ}tttAttdtd tdtd tdtd  ttdtdtttdtd tdtd    ttdtdtttdtd tdtd    t|o||||||||||}||}t|t߃}tttAttdtd  tFtz  t|o|||f||}t|t}tttAttdtd tdtd tdtd  ttdtd  tttdtd tdtd td	td  t|o||||||%|b|
}t|t}tttAttdtd tdtd tdtd  ttdtd  tttdtd tdtd td	td  t|o||||||%|b|
}t|t}tttAttdtd tdtd tdtd  tttdtd tdtd tdtd  tttdtd tdtd td	td  t|o||||||||
}t|t}tttAttdtd tdtd tdtd  ttdtd  tttdtd tdtd tdtd  tttdtd tdtd td	td  t|o||||||||
}t|t}tttAttdtd tdtd tdtd  tFttdtd tdtd   tdttdtd tdtd   t|o|||||||f|
}t|t}tttAt ttztd	td  t|o|||}t|t}tttAt tĐtztd	td  t|o|||}t|t}tttAt ttztdtd  tĐtztd	td  t|o|||}t|t}ttttdtdtdtdtttd	td tdtd   t t|||||||}t|t}ttttdtdtdtdtttd	td tdtd   t t|||||||}t|t}ttttdtdtdtdtttd	td tdtd   tdtd t|||||;|}t|t}ttttdtdtdtdtttd	td tdtd   tdtd t|||||;|}t|t}ttttdtdtdtdtttd	td tdtd   t|||||}t|t}ttttdtdtdtdtttd	td tdtd   t|||||}t|t}ttttdtdtdtdtttd	td tdtd   t t|||||)||}t|t}ttttdtdtdtdtttd	td tdtd   t t|||||)||}t|t}ttttdtdtdtdtttd	td tdtd   t t|||||)|||	}t|t}ttttdtdtdtdtttd	td tdtd   t t|||||)|||	}t|t}ttttdtdtdtdtttd	td tdtd   t t||||||}t|t}ttttdtdtdtdtttd	td tdtd   t t||||||}t|t}ttttdtd ttdtdtdtdtttd	td tdtd   t  t|||||||||Q
}t|t}ttttdtd ttdtdtdtdtttd	td tdtd   t  t|||||||||Q
}t|t}ttttdtd ttdtdtdtdtttd	td tdtd   tdtd  t||||||;|}t|t}ttttdtd ttdtdtdtdtttd	td tdtd   tdtd  t||||||;|}t|t}ttttdtd ttdtdtdtdtttd	td tdtd    t||||||}t|t}ttttdtd ttdtdtdtdtttd	td tdtd    t||||||}t|t}ttttdtd ttdtdtdtdtttd	td tdtd   t  t||||||)||	}t|t}ttttdtd ttdtdtdtdtttd	td tdtd   t  t||||||)||	}t|Ðt}ttttdtd ttdtdtdtdtttd	td tdtd   t  t||||||)|||
}t|Őt}ttttdtd ttdtdtdtdtttd	td tdtd   t  t||||||)|||
}t|ǐt}ttttdtd ttdtdtdtdtttd	td tdtd   t  t|||||||}t|ɐt }ttttdtd ttdtdtdtdtttd	td tdtd   t  t|||||||}t|ːt}tttdttdtdtdtdtttd	td tdtd    t|||||}t|͐t}tttdttdtdtdtdtttd	td tdtd    t|||||}t|ϐt}tttdttdtdtdtdtttd	td tdtd    t t|||||||}t|ѐt}tttdttdtdtdtdtttd	td tdtd    t t|||||||}t|Ӑt}tttdttdtdtdtdtttd	td tdtd    t t|||||)|||	}t|Րt}tttdttdtdtdtdtttd	td tdtd    t t|||||)|||	}t|אt}tttdttdtdtdtdtttd	td tdtd    t t|||||)||}t|ِt}tttdttdtdtdtdtttd	td tdtd    t t|||||)||}t|ېt	}tttdttdtdtdtdtttd	td tdtd    tdtd t||||||}t|ݐt
}tttdttdtdtdtdtttd	td tdtd    tdtd t||||||}t|ߐt}ttttdtd ttdtdtdtdtttd	td tdtd    t||||||}t|t}ttttdtd ttdtdtdtdtttd	td tdtd    t||||||}t|t}ttttdtd tdttdtdtdtdtttd	td tdtd    t  t||||||||Q|
}t|t}ttttdtd tdttdtdtdtdtttd	td tdtd    t  t||||||||Q|
}t|t}ttttdtd tdttdtdtdtdtttd	td tdtd    t  t||||||)|||
}t|t}ttttdtd tdttdtdtdtdtttd	td tdtd    t  t||||||)|||
}t|t}ttttdtd tdttdtdtdtdtttd	td tdtd    t  t||||||)||	}t|t}ttttdtd tdttdtdtdtdtttd	td tdtd    t  t||||||)||	}t|t}ttttdtd tdttdtdtdtdtttd	td tdtd    tdtd  t|||||||}t|t}ttttdtd tdttdtdtdtdtttd	td tdtd    tdtd  t|||||||}t|t}tttttdtd tdtd tttdtd tttdtd tdtd   t|||)|*}t|t}tttttdtd tdtd tttdtd tttdtd tdtd   t|||)|*}t|t}tttttdtd tdtd ttt. tdtd tttdtd tdtd   t||||*}t|t}tttttdtd tdtd ttt. tdtd tttdtd tdtd   t||||*}t|t}ttttdtd tttdtd tdtd  tttd	td tdtd tttdtd tdtd   t||||||)|*}t|t}ttttdtd tttdtd tdtd  tttd	td tdtd tttdtd tdtd   t||||||)|*}t|t} ttttdtd tttdtd tdtd  tttd	td tdtd tttdtd tdtd   t|||||)|*|}t|t}tttt tttdtd tdtd  tttd	td tdtd tttdtd tdtd   t|||||)|*|}t|t}ttttdtdttdtd tdtd  td	td t||||}t|t}ttttdtdttdtd tdtd  td	td t||||}t|t}tttttdtd tdtd tdtd ttdtd tdtd  td	td t||||||}	t|	t }
tttttdtd tdtd tdtd ttdtd tdtd  td	td t||||||}t|t!}tttttd	td t||}t|t"}tttttd	td t||}t|t#}tttdtdttztdtd  tttdtd  t|g}t|t$}tttdtdtĐtztdtd  tĐttdtd  t|g}t|t%}ttttztdtd tttdtd  t||J}t|t&}tttĐtztdtd tĐttdtd  t||J}t|t'}ttttdtd ttztdtd  tttdtd  t||}t|t(}ttttdtd tĐtztdtd  tĐttdtd  t||}t|t)}tttdtdttztdtd  tĐttdtd  t|g|}t|t*}ttttztdtd tĐttdtd  t|J|}t|t+} ttttdtd ttztdtd  tĐttdtd  t||}!t|!t,}"ttttztttd	td  t|q|7|}#t|#t-}$tttdtt td	td tĐtz t|q|7|}%t|%t.}&tttdtt td	td ttz t|q|7|}'t|'t/}(tttĐtztttd	td  t|q|7|})t|)t0}*tttdtĐt td	td ttz t|q|7|}+t|+t1},tttdtt td	td tĐtz t|q|7|}-t|-t2}.tttdtt td	td ttz t|q|7|}/t|/t3}0tttdtĐt td	td tĐtz t|q|7|}1t|1t4}2tttBtdtdtttdtd tdtd  tttdtd tdtd   td	td ttdtd tdtd tdtd  t|||||||||
}3t|3t5}4ttttdtd tBtdtdtttdtd tdtd td   t.  t|||||||b||
}5t|5t6}6ttttdtd tBtdtdtttdtd tdtd td   t.  t|||||||b||
}7t|7t7}8ttttdtd tdtd tdtd tt8ttdtd  t. tdtd tdtd tdtd  t||||||||^|)
}9t|9t9}:ttttdtd tdtd tdtd tĐt8ttdtd  t. tdtd tdtd tdtd  t||||||||^|)
};t|;t:}<ttttdtd tdtd tdtd tdtdtdtdtttdtd tdtd td   tdtdtttdtd tdtd td    t||||||||^||}=t|=t;}>ttttdtd tdtd tdtd t<tdtdtttdtd tdtd td   tttdtd tdtd td   t|||||||^}?t|?t=}@ttttdtd tdtd tdtd tdtdtttdtd tdtd td  tdtd tdtdtttdtd tdtd td   tttdtd tdtd td   t||||||||^||}At|At>}Bttttdtd tdtd tdtd t8tdtdtttdtd tdtd td   tttdtd tdtd td   t|||||||^}Ct|Ct?}Dttttdtd tdtd tdtd tdtdtttdtd tdtd td  tdtdtttdtd tdtd td   tdtd tttdtd tdtd td   t||||||||^||}Et|Et@}Fttttdtd tdtd tdtd tttdtd tdtd  tBtdtdtttdtd tdtd    t|||||||^|	}Gt|GtA}Httttdtd tdtd tdtd tttdtd tdtd  tBtdtdtttdtd tdtd    t|||||||^|	}It|ItB}Jttttdtd tdtd tdtd tttdtd tdtd  tBtdtdtttdtd tdtd    t|||||||^|	}Kt|KtC}Lttttdtd tdtd tdtd tttdtd tdtd  tBtdtdtttdtd tdtd    t|||||||^|	}Mt|MtD}Nttttdtd tdtd tdtd tttdtd tdtd t.  tBtdtdtttdtd tdtd    t|||||||^|:|r|}Ot|OtE}Pttttdtd tdtd tdtd tttdtd tdtd t.  tBtdtdtttdtd tdtd    t|||||||^|:|r|}Qt|QtF}Rttttdtd tdtd tdtd tttdtd tdtd t.  tBtdtdtttdtd tdtd    t|||||||^|:|r|}St|StG}Tttttdtd tdtd tdtd tttdtd tdtd t.  tBtdtdtttdtd tdtd    t|||||||^|:|r|}Ut|UtH}Vttttd
tdtttdtd tdtd   ttdtd tdtd  tBtdtdtttdtd tdtd   td  t||||||||	|
}Wt|WtI}Xttttd
tdtttdtd tdtd   ttdtd tdtd  tBtdtdtttdtd tdtd   td  t||||||||	|
}Yt|YtJ}ZtttBtdtdttz  td	td tdtĐtz tdtd  t|||||}[t|[tK}\tttBtdtdttz  td	td tdttz tdtd  t|||||}]t|]tL}^tttdtdtttdtd tdtd tdtd  tttdtd tdtd td	td  t|||||}_t|_tM}`tttdtdtttdtd tdtd tdtd  tttdtd tdtd td	td  t|||||}at|atN}btttdtĐt8ttdtd  tttdtd tdtd   t||||||}ct|ctO}dtttdtt8ttdtd  tttdtd tdtd   t||||||}et|etP}ftttt8ttdtd  tttdtd tdtd  t||||||}gt|gtQ}htttdtt8ttdtd  tttdtd tdtd   t||||||}it|itR}jtttdtdtĐtz tdtdttz  td	td tdtd t||||Y}kt|ktS}lg ||||||đ|Ƒ|ȑ|ʑ|̑|Α|Б|ґ|ԑ|֑|ؑ|ڑ|ܑ|ޑ||||||||||||||||| |||||
||||||||||| |"|$|&|(|*|,|.|0|2|4|6|8|:|<|>|@|B|D|F|H|J|L|N|P|R|T|V|X|Z|\|^|`|b|d|f|h|j|l|n|p|r|t|v|x|z|||~|||||||||||||||||||||||||||||||||||đ|Ƒ|ȑ|ʑ|̑|Α|Б|ґ|ԑ|֑|ؑ|ڑ|ܑ|ޑ||||||||||||||||| |||||
||||||||||| |"|$|&|(|*|,|.|0|2|4|6|8|:|<|>|@|B|D|F|H|J|L|N|P|R|T|V|X|Z|\|^|`|b|d|f|h|j|l|n|p|r|t|v|x|z|||~|||||||||||||||||||||||||||||||||||đ|Ƒ|ȑ|ʑ|̑|Α|Б|ґ|ԑ|֑|ؑ|ڑ|ܑ|ޑ||||||||||||||||| |||||
||||||||||| 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sympy.integrals.rubi.constraintsr  r  r  r  r  r 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acement4930replacement4931replacement4932replacement4933replacement4934replacement4935replacement4936replacement4937replacement4938replacement4939replacement4940replacement4941replacement4942replacement4943replacement4944replacement4945replacement4946replacement4947replacement4948replacement4949replacement4950replacement4951replacement4952replacement4953replacement4954replacement4955replacement4956replacement4957replacement4958replacement4959replacement4960replacement4961replacement4962replacement4963replacement4964replacement4965replacement4966replacement4967replacement4968replacement4969replacement4970replacement4971replacement4972replacement4973replacement4974replacement4975replacement4976replacement4977replacement4978replacement4979replacement4980replacement4981replacement4982replacement4983replacement4984replacement4985With4986With4987replacement4988replacement4989replacement4990replacement4991replacement4992replacement4993replacement4994replacement4995replacement4996replacement4997replacement4998replacement4999replacement5000replacement5001replacement5002replacement5003replacement5004replacement5005replacement5006replacement5007c_replacement5008replacement5009replacement5010b_replacement5011replacement5012replacement5013replacement5014replacement5015replacement5016replacement5017replacement5018replacement5019replacement5020replacement5021replacement5022replacement5023replacement5024replacement5025replacement5026replacement5027replacement5028replacement5029replacement5030replacement5031replacement5032replacement5033(m  r  r  r  r  r  r 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XX\\\\PPXX^^^~~~~ff||Z\\|VtVVVVX\X\XXfVVVVZZZZZZrr"(4Jd44RFrr||~~~~zz<66Nffll\\ffhhdd||ppzz||xxddnnppnnttppvv||TTxx&&LL@@RRN@R4<<4<<<<vvbb``X``       ru  c              	   C   s   t |t| ||   | |t| ||   |   |t| ||   |  t|t| ||   ||  |t| ||   |   t| ||S Nr  r  r  r  r   rH  r  r  r  r  r  r  r  xr  r  r  r       r  c              	   C      t |t| ||   | |t| ||   |   |t| ||   |  t|t| ||   | |t| ||   | |   t| ||S rv  rw  rx  r  r  r  r    rz  r  c              	   C   s   t |t| ||   | |t| ||   |  |t| ||   |   t|t| ||   | |  |t| ||   |  t| ||S rv  rw  rx  r  r  r  r    rz  r  c              	   C   s   t |t| ||   | |t| ||   |  |t| ||   |   t|t| ||   |  |t| ||   ||   t| ||S rv  rw  rx  r  r  r  r    rz  r  c                 C   `   t |t| ||   | |t| ||   |  t|t| ||   | |  t| ||S rv  r  r  r   rH  rx  r  r  r  r       `r  c              	   C   s   t |t| ||   |  |t| ||   |  |t| ||   |  t|t| ||   | |t| ||   |   t| ||S rv  r  r  r  r  r   rH  r  r  r  r  r  ry  r  r  r  r       r  c              	   C   s   t |t| ||   | |t| ||   |   |t| ||   |  t|t| ||   |  |t| ||   |  t| ||S rv  r  r  r  r  r  r    r  r  c                 C   \   t |t| ||   | |t| ||   |  t|t| ||   |  t| ||S rv  r  r  r   rH  r  r  r  r  r       \r  c                 C   r  rv  r}  r  r  r  r  r    r  r  c              	   C   N   t |t|t|||   |td  | t|||   |  t| ||S Nr  r   r  r  rH  r  r  r  r  r  r  r  ry  r  r  r  r       Nr  c              	   C   r  r  r  r   r  r  rH  r  r  r  r  r    r  r  c                 C   6   t | t|||   | t| t|||   |S rv  r   r  rH  r  r  r  r  r  ry  r  r  r  r       6r  c                 C   r  rv  r   r  rH  r  r  r  r  r    r  r  c	           	   
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r  r  r  r  r  r  r  r  r  ry  r  r  r  r       Vr  c
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   C   r  r  r  r  r   r  rH  r  r  r  r  r  %  r  r  c	           	   
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      C   ^   t | |t|||	    |t|||	  td   tdt|||	   |  t| |	S Nr  r  r  r  r  r  r  r  M  r  r  c                 C   `   t |t| ||   | |t| ||   |  t|t| ||   | |  t| ||S rv  r}  rx  r  r  r  r  Q  r~  r  c                 C   r  rv  r  rx  r  r  r  r  U  r~  r  c              	   C   s   t |t| ||   | |t| ||   |  |t| ||   |   t|t| ||   |  |t| ||   ||   t| ||S rv  rw  rx  r  r  r  r  Y  rz  r  c              	   C   s   t |t| ||   | |t| ||   |  |t| ||   |   t|t| ||   | |  |t| ||   |  t| ||S rv  rw  rx  r  r  r  r  ]  rz  r  c              	   C   s   t |t| ||   | |t| ||   |   |t| ||   |  t|t| ||   | |t| ||   | |   t| ||S rv  rw  rx  r  r  r  r  a  rz  r  c              	   C   s   t |t| ||   | |t| ||   |   |t| ||   |  t|t| ||   ||  |t| ||   |   t| ||S rv  rw  rx  r  r  r  r  e  rz  r  c                 C   r  rv  r}  r  r  r  r  r  i  r  r  c                 C   r  rv  r  r  r  r  r  r  m  r  r  c              	   C   s   t |t| ||   | |t| ||   |   |t| ||   |  t|t| ||   |  |t| ||   |  t| ||S rv  r  r  r  r  r  r  q  r  r  c              	   C   s   t |t| ||   |  |t| ||   |  |t| ||   |  t|t| ||   | |t| ||   |   t| ||S rv  r  r  r  r  r  r  u  r  r  c              	   C   r  r  r  r  r  r  r  r   y  r  r   c              	   C   r  r  r  r  r  r  r  r!  }  r  r!  c                 C   r  rv  r  r  r  r  r  r"    r  r"  c                 C   r  rv  r  r  r  r  r  r#    r  r#  c	           	   
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      C   r  r  r  r  r  r  r  r-    r  r-  c                 C   sl   t t| | |||   td| td|   |t t| | |||   td| td|   | S r  r  r  r  r  r  r  r  ry  r  r  r  r.       lr.  c                 C   sl   t t| | |||   td| td|   |t t| | |||   td| td|   | S r  r  r  r  r  r  r/    r  r/  c                 C   sn   t t| | |||   td| td|   | t t| | |||   td| td|   | S r  r  r  r  r  r  r  r  r0       nr0  c              	   C   sl   t tdd t|t|||   | ||t tdd t|t|||   | t|||   || S Nr  r  r  r  r   r  r  r  r  r  r  r  r  ry  r  r  r  r2    r  r2  c              	   C   sl   t tdd t|t|||   | ||t tdd t|t|||   | t|||   || S r  r  r  r  r  r  r3    r  r3  c                 C   P   t td| ||   t|t| ||   ||  t| ||  |  ||S r  r  r  r   r  r  )r  r  r  r  r  r  r  ry  r  r  r  r4       Pr4  c                 C   r  r  r  )r  r  r  r  r  r  r  ry  r  r  r  r5    r  r5  c	           	      C   sn   t |td |t| ||   |td   |t|||   |td   td| | |td   |S Nr  r  r  )r  r  r  r  	r  r  r  r  r  r  r  r  ry  r  r  r  r7    r  r7  c	           	      C   sp   t |td |t| ||   |td   |t|||   |td   td| | |td   | S r  )r  r  r  r  r  r  r  r8    s   pr8  c	           	      C   sL   t |t| ||   | |t|||   |td   || |  | S Nr  )r  r  r  r  r  r  r  r  r9       Lr9  c	           	      C   sJ   t |t| ||   | |t|||   |td   || |  |S r  r  r  r  r  r  r:       Jr:  c	           	   	   C   s   t |td || td  td|td  |td   t|t| ||   |td  |t|||   |td   ||t|td |t| ||   |td   |t|||   |td   td| | |td   | S N   r  r  r  r  r  r  r   r  r  r  r  r  r  r  r;       r;  c	           	   	   C   s   t |td || td  td|td  |td   t|t| ||   |td  |t|||   |td   ||t|td |t| ||   |td   |t|||   |td   td| | |td   | S r  r  r  r   r  r  r  r  r  r  r<    r  r<  c	           	   	   C   s   t |td |td|  td  td|td  |td   t|t| ||   |td  |t|||   |td   ||t|t| ||   | |t|||   |td   td| | |td   | S Nr  r  r  r  r  r  r  r  r  r=       r=  c	           	   	   C   s   t |td |td|  td  td|td  |td   t|t| ||   |td  |t|||   |td   ||t|t| ||   | |t|||   |td   td| | |td   | S r  r  r  r  r  r  r>    r  r>  c	           	   	   C   s   t |td || td  |td|   t|t| ||   |td  |t|||   |  ||t|td |t| ||   |td   |t|||   |td   td| | |td|    | S Nr  r  r  r  r  r  r  r  r  r?       r?  c	           	   	   C   s   t |td || td  |td|   t|t| ||   |td  |t|||   |  ||t|td |t| ||   |td   |t|||   |td   td| | |td|    | S r  r  r  r  r  r  r@    r  r@  c	           	   	   C   s   t |td|  td |td || td   t|t| ||   |td  |t|||   |  ||t|t| ||   | |t|||   |td   td| | || td   | S r  r  r  r  r  r  rA       rA  c	           	   	   C   s   t |td|  td |td || td   t|t| ||   |td  |t|||   |  ||t|t| ||   | |t|||   |td   td| | || td   | S r  r  r  r  r  r  rB    r  rB  c                 C   s   t td| | td| td  t|t|||   |td  t| ||   ||ttd|t|||   |  t| ||   |td| td   | S Nr  r  r  r  r  r  r  r  rC       rC  c                 C   s   t td| | td| td  t|t|||   |td  t| ||   ||ttd |t|||   |  t| ||   |td| td   | S r  r  r  r   r  r  r  r  r  r  r  rD    s   rD  c                 C   s   t td| td td| |td   t|t|||   |td  t| ||   ||t|t|||   |td  t| ||   td| | |td   | S Nr     r  )r  r  r   r  r  r  r  r  r  r  rE       rE  c                 C   s   t td| td td| |td   t|t|||   |td  t| ||   ||t|t|||   |td  t| ||   td| | |td   | S r  r  r  r  r  r  rF  	  r  rF  c                 C   sr   t tt| ||  tt|||   t| ||   | |t tt| ||  t| ||    | | S rv  r  r
  r  r  r  r	  r  r  r  r  rG    r  rG  c                 C   st   t tt| ||  tt|||   t| ||   | | t tt| ||  t| ||    | | S rv  r  r  r  r  r  rH       trH  c                 C   sF   t td| t|t|||   |td  t| ||   ||S Nr  r  r  r  r   r  r  r  r  r  rI    r  rI  c                 C   sF   t td| t|t|||   |td  t| ||   ||S r  r  r  r  r  r  rJ    r  rJ  c	           	      C   s   t |t| ||   |  |t|||   |  t| ||  |   t|t| ||   ||  t| ||  |  ||S rv  r  r  r  r   r  r  r  r  rK       rK  c	           	      C   s   t |t| ||   |  |t|||   |  t| ||  |   t|t| ||   ||  t| ||  |  ||S rv  )r  r  r  r   )	r  r  r  r  r  r  r  r  ry  r  r  r  rL  !  r  rL  c              	   C   st   t tdd t|t|||   | ||t tdd t|t|||   | t|||  td  || S )Nr  r  r  r  r  r  r  r  rM  %  r  rM  c
           
   	   C   sb   t td| ||   ||   t|t| ||	   ||  |t| ||	   ||   |	|	S r  r  
r  r  r  r  r  r  r  r  r  ry  r  r  r  rN  )     brN  c                 C   sz   t ||t| ||
   |td   |t| ||
   |td   |t|||
   |	  || ||	 td   |
S Nr  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  ry  r  r  r  rO  -  s   zrO  c                 C   s|   t ||t| ||
   |td   |t| ||
   |td   |t|||
   |	  || ||	 td   |
 S r  )r  r  r  r  r  r  r  r  rQ  1     |rQ  c                 C   s|   t |t| ||
   |td  |t| ||
   |td   |t|||
   |	  || | ||	 td   |
 S r  r  r  r  r  r  rR  5  r  rR  c              	   C   s,  t |td ||	 td  td|td  ||	 td   t|t| ||
   |td  |t| ||
   |  |t|||
   |	td   |
|
t|td |t| ||
   |td   |t| ||
   |  |t|||
   |	td   td| | ||	 td   |
 S r  r  r  r  r  r  rS  9     . rS  c              	   C   s,  t |td ||	 td  td|td  ||	 td   t|t| ||
   |td  |t| ||
   |  |t|||
   |	td   |
|
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 S r  r  r  r  r  r  rU  A     & rU  c              	   C   s$  t |td || td|	  td  td|td  ||	 td   t|t| ||
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 S r  r  r  r  r  r  rV  E  r  rV  c              	   C   s  t |td ||	 td  |td ||	 td   t|t| ||
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 S Nr  r  r  r  r  r  r  r  r  rW  I      rW  c              	   C   s  t |td ||	 td  |td ||	 td   t|t| ||
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 S r  r  r  r  r  r  r_  i      r_  c              	   C   s  t || td|	  td |td ||	 td   t|t| ||
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 S r  r  r  r  r  r  r`  m  r  r`  c              	   C   s   t |t| ||
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|
S rv  r  r  r  r  r  ra  q  r  ra  c                 C   sd   t |t| ||   |td  |td  t| ||  |td   || |td   | S r  r  )r  r  r  r  r  r  ry  r  r  r  rb  u     drb  c                 C   s&   t ||| |||  |   | |S rv  )r   )Fr  r  r  r  r  r  ry  r  r  r  re  y  s   &re  c                 C   s   t td||  tttdtdtdtd t |  | |||  td  t| | |td    |tttd|td |S )Nr  r  r  r  )r  r  r  r   kr  r   r  r  r  r  r  r  ry  r  r  r  rf  }  r  rf  c              	   C   s0   t ttd||| |||  |    ||S r  r   rI  r  r  r  r  r  rg       0rg  c	           	   
   C   sB   t t||||  | td||| |||  |    ||S r  r  )	r  Gr  r  r  r  r  r  ry  r  r  r  ri       Bri  c                 C   s   t | |||  }t|t| |||  t|  |t|| |t|   t||| t|   tt||||  ||S rv  rH  r  r%   r&   r   r   )r  r  r  r  r  r  ry  vr  r  r  rj    s   nrj  c           	      C   sh   t | |||  }t|t| ||| |  t|  || | t|   t|| ||  ||S rv  rH  r  r%   r&   r   )	r  r  r  r  r  r  r  ry  r  r  r  r  rk    s   Trk  c                 C   Z   t |ttttfrdS tt||||   |}tt||||   | ||dr+dS dS NFT
isinstanceintr  floatr  r   r  rm  r  r  r  r  r  ry  r  r  r  r  rl       "rl  c                 C   r   t t||||   |}t|||  ttttdt||||   | ||||t||||   | |S r  r   r  r  r+   r   r  r  r  r  r  r  rm       Xrm  c                 C   r  r  r  r  r  r  r  r   r  rm  r  r  r  r  rn    r  rn  c                 C   t   t t||||   |}t|||  ttttdt||||   | ||||t||||   | | S r  r   r  r  r+   r   r  r  r  r  r  r  ro       Zro  c                 C   r  r  r  r  r  r  r  rp    r  rp  c                 C   z   t t||||   |}ttd||  ttttd| t||||   | ||||t||||   | |S r  r   r  r  r  r+   r   r  r  r  r  r  rq       `rq  c                 C   r  r  r  r  r  r  r  rr    r  rr  c                 C   |   t t||||   |}ttd||  ttttd| t||||   | ||||t||||   | | S r  r   r  r  r  r+   r   r  r  r  r  r  rs       brs  c                 C   r  r  r  r  r  r  r  r   r  rm  r  r  r  r  rt    r  rt  c                 C   r   r  r   r  r  r+   r   r  r  r  r  r  r  ru    r  ru  c                 C   sZ   t |ttttfrdS tt|| ||   |}tt|| ||   | ||dr+dS dS r  r  r  r  r  r  ry  r  r  r  r  rv    r  rv  c                 C   sr   t t|| ||   |}t|||  ttttdt|| ||   | ||||t|| ||   | |S r  r  r  r  r  r  rw    r  rw  c                 C   sj   t |ttttfrdS ttdt||||    |}ttd|t||||     ||dr3dS dS NFr  T	r  r  r  r  r  r   r  r  rm  r  r  r  r  rx       "*rx  c                 C   s   t tdt||||    |}t|||  ttttdtd|t||||     ||||td|t||||     | S r  r   r  r  r  r+   r   r  r  r  r  r  ry       "jry  c                 C   sj   t |ttttfrdS ttdt|| ||    |}ttd|t|| ||     ||dr3dS dS r  r  r  r  r  r  rz    r  rz  c                 C   s   t tdt|| ||    |}t|||  ttttdtd|t|| ||     ||||td|t|| ||     | S r  r  r  r  r  r  r{    r  r{  c              	   C   s   t |ttttfrdS tt||||   |}ttt||||   | ||dt	t
dt||||    | t| |rCdS dS )NFTr  )r  r  r  r  r  r   r  r  rm  r^  r  rH  r  r  r  r  r  r  ry  r  r  r  r  r|    s   Rr|  c                 C   s   t t||||   |}t|td|  ||  ttt||  |td |td  td  t||||   | ||||t||||   | |S r  )r   r  r  r  r+   r   r  r  r  r  r  r}    s   r}  c              	   C   s   t |ttttfrdS ttdt||||    |}tt	td|t||||     ||dt
t|t||||   |  |rGdS dS r  )r  r  r  r  r  r   r  r  r  rm  r^  rH  r  r  r  r  r~    s   "Rr~  c                 C   s   t tdt||||    |}t|td|  ||  ttt||  |td |td  td  td|t||||     ||||td|t||||     | S r  r  r  r  r  r  r  #  s   "r  c              	   C   s   t |ttttfrdS z/t| |}ttdt| |}t	t
t|tttdt| || |dtt| |}W n ttfyF   Y dS w |rKdS dS r  )r  r  r  r  r  rj  r   r  r  r  r6   r   rm  r   r^  rH  	TypeErrorAttributeErrorr  ry  r  r  resr  r  r  r  )  s   
>r  c                 C   s   t | |}ttdt| |}tttt|ttdt| |t| t	||td t
tttd|td |td  td  td|t|  | |||td|t|  ||S r  )rj  r   r  r  r  r   r   r
   r  r3   r+   r   r  r  ry  r  r  r  r  r  r  7  s   
r  c              	   C   s   t |ttttfrdS z't| |}tt||}tt	t
|ttt||| |dtt| |}W n ttfy>   Y dS w |rCdS dS r  )r  r  r  r  r  rj  r   r  r  r6   r   rm  r   r^  rH  r  r  r  r  r  r  r  >  s   
6r  c                 C   s   t | |}tt||}tttt|tt||t|t||t	d t
ttt	d|t	d |t	d  t	d  t|| | |||t|| ||S r  )rj  r   r  r  r   r   r
   r  r3   r  r+   r   r  r  r  r  r  r  L  s   
r  c	           	      C   s8   t tt| |||  | ||||  |  ||S rv  r   rd  rH  )	r  r  r  r  r  r  r  r  ry  r  r  r  r  S     8r  c                 C   sL   t tt| |||  |	 ||||  |
  ||||  |  ||S rv  r  )r  r  Hr  r  r  r  r  r  r  r  rry  r  r  r  r  W  r  r  c                 C   X   t |ttttfrdS tt||||   |}tt||||   | ||r*dS dS r  r  r  r  r  r  r  [      r  c                 C   r   r  r  r  r  r  r  r  d  r  r  c                 C   r  r  r  r  r  r  r  r  j  r   r  c                 C   r  r  r  r  r  r  r  r  s  r  r  c                 C   r  r  r  r  r  r  r  r  y  r   r  c                 C   r  r  r  r  r  r  r  r    r	  r  c                 C   r  r  r  r  r  r  r  r    r   r  c                 C   r
  r  r  r  r  r  r  r    r  r  c                 C   X   t |ttttfrdS tt||||   |}tt||||   | ||r*dS dS r  r  r  r  r  r  r    r   r  c                 C   s   t t||||   |}t|||  ttt|td  |td  td |td tdd   t||||   | ||||t||||   | |S r  r  r  r  r  r  r    s   r  c                 C   r!  r  r  r  r  r  r  r    r   r  c                 C   s   t t||||   |}t|||  ttt|td  |td  td | td tdd   t||||   | ||||t||||   | |S r  r  r  r  r  r  r       r  c                 C   r!  r  r  r  r  r  r  r    r   r  c                 C   s   t t||||   |}t|||  ttt|td  |td  td |td tdd   t||||   | ||||t||||   | | S r  r  r  r  r  r  r    r"  r  c                 C   r!  r  r  r  r  r  r  r    r   r  c                 C   s   t t||||   |}t|||  ttt|td  |td  td | td tdd   t||||   | ||||t||||   | | S r  r  r  r  r  r  r    s   r  c                 C   r!  r  r  r  r  r  r  r    r   r  c                 C   s   t t||||   |}t|td|  ||  ttt||  |td  |td  td |td tdd    t||||   | ||||t||||   | |S Nr  r  r  r  r  r  r  r  r    s   r  c                 C   r!  r  r  r  r  r  r  r    r   r  c                 C   s   t t||||   |}t|td|  ||  ttt||  |td  |td  td |td tdd    t||||   | ||||t||||   | | S r#  r  r  r  r  r  r    s   r  c	           
      C   X   t |ttttfrdS tt||||   |}	tt||||   |	 ||r*dS dS r  r  
r  r  r  r  r  r  r  r  ry  r  r  r  r  r    r   r  c	           
   	   C   Z   t t||||   |}	t|tt|t||||   |  ||tt|| | S rv  )r   r  r  r   rH  r  r%  r  r  r  r       @r  c	           
      C   r$  r  r  r%  r  r  r  r     r   r  c	           
   	   C   r&  rv  )r   r  r  r   rH  r  r%  r  r  r  r  	  r'  r  c              	   C      t |ttttfrdS z$t| |}tt||}tt	t
|ttt||| t| |}W n ttfy;   Y dS w |r@dS dS r  )r  r  r  r  r  rj  r   r  r  r6   r   rm  r   r  r  r  r  r  r  r  r       
0r  c                 C   s   t | |}tt||}tttt|tt||t|t||t	d t
ttt	dt|| | t| |||t|| ||S r  )rj  r   r  r  r   r   r
   r  r3   r  r+   r   r  r  r  r  r  r  r    s   
pr  c              	   C   r(  r  )r  r  r  r  r  rj  r   r  r  r6   r   rm  r   r  r  r  r  r  r  r  r  $  r)  r  c                 C   s   t | |}tt||}tttt|tt||t| t||t	d t
ttt	dt|| | t| |||t|| ||S r  )rj  r   r  r  r   r   r
   r  r3   r  r+   r   r  r  r  r  r  r  r  2  s   
rr  c                 C      t | | | tt|||S rv  r  r   rH  r  r  r  r  r  r  r  ry  r  r  r  r  9     r  c                 C   r*  rv  r+  r,  r  r  r  r  =  r-  r  c                 C   r*  rv  r+  r,  r  r  r  r  A  r-  r  c              	   C   s`   t |ttttfrdS ztt|t| |}tt|}W n t	t
fy)   Y dS w |r.dS dS r  r  r  r  r  r  r  rH  r6   r   r  r  )r  ry  yr  r  r  r  r  r  E     r  c                 C   s0   t t|t| |}t|ttt|| |S rv  r  rH  r  r
  r   )r  ry  r/  r  r  r  r  r  R  s   r  c              	   C   sd   t |ttttfrdS ztt|| t| |}tt|}W n t	t
fy+   Y dS w |r0dS dS r  r.  )r  wry  r/  r  r  r  r  r  r  X  s   r  c                 C   s8   t t|| t| |}t|ttt|| | |S rv  r1  )r  r2  ry  r/  r  r  r  r  r  e  s    r  c              	   C   s`   t |ttttfrdS ztt|t||}tt|}W n t	t
fy)   Y dS w |r.dS dS r  r.  )r  r  ry  r/  r  r  r  r  r  r  k  r0  r  c                 C   s>   t t|t||}t|t|| td   | td  |S r  r  rH  r  r  )r  r  ry  r/  r  r  r  r  r  x  s   *r  c              	   C   sr   t |ttttfrdS ztt|| t|||  |   |}tt|}W n t	t
fy2   Y dS w |r7dS dS r  r.  )r  r  r  ry  r/  zr  r  r  r  r  r  ~  s   &r  c                 C   s`   t t|| t|||  |   |}t|t|| td  || td    | td  |S r  r3  )r  r  r  ry  r/  r4  r  r  r  r  r    s   &:r  c           	      C   s   t | |||  }t|t| |||  t|  |t|| |t|   t||| t|   tt |t||||   ||S rv  r  )	r  r  r  r  r  r  r  ry  r  r  r  r  r    s   vr  c	           
      C   sp   t | |||  }	t|t| |||	 |  t|  ||	 | t|   t||	 ||  t | ||S rv  r  )
r  r  r  r  r  r  r  r  ry  r  r  r  r  r    s   \r  c              	   C   s|   t |ttttfrdS z t| |}tt||}tt	t
|ttt||| |}W n ttfy7   Y dS w |r<dS dS r  )r  r  r  r  r  rj  r   r  r  r6   r   rm  r   r  r  r  r  r  r  r    s   
(r  c              	   C   sl   t | |}tt||}t|t||d tttd|d |d  d  t|| | |||t|| |S r  )rj  r   r  r  r3   r+   r   r  r  r  r  r  r    s   
Tr  c                 C   N   t | t|||  |  | | tdt|||   ||   t| |S r  )r   r  r  r  rH  r  r  r  r  r  r  r  ry  r  r  r  r    r  r  c                 C   r5  r  )r   r  r  r  rH  r6  r  r  r  r    r  r  c
           
   	   C   sL   t t|||| |||	  | |    |  | |||	  ||   |	S rv  r   rH  )
r  r  r  r  r  r  r  r  r  ry  r  r  r  r    r  r  c              	   C   sj   t t|
||| |||  | |    || |||  | |	    |  | |||  ||   |S rv  r7  )r  r  r  r  r  r  r  r  r  r  r  ry  r  r  r  r       jr  c              	   C   sf   t t|	|| |||
  |   | || |||
  | |    |  | |||
  ||   |
S rv  r7  )r  r  r  r  r  r  r  r  r  r  ry  r  r  r  r       fr  c                 C   s0   t | t|  |||   |  | t| |S rv  )r   exprH  )r  r  r  r  r  r  ry  r  r  r  r    r  r  c                 C   s   t t| |S rv  )r   r  )r  ry  r  r  r  r    s   r  c                 C   sP   t |}t |}t| t| |t|   | | t|  t|||  ||S rv  r  )r  r  r  r  ry  uuvvr  r  r  r    s   @r  c                 C   sL   t |}t |}t||  t|  ||  t|  t||| |   ||S rv  rH  r  r&   r   )r  r  r  r  ry  r;  r<  r  r  r  r    s   <r  c           
      C   sz   t |}t |}t |}	t||  t|  |	| t|   ||  |	|  t|  t||| |   |	||   ||S rv  r=  )
r  r  r  r  r  r2  ry  r;  r<  Zwwr  r  r  r    s   br  c                 C   s0   t |ttttfrdS t| |}t|rdS dS r  )r  r  r  r  r  rI  r)   r  ry  r  r  r  r  r    s   
r  c                 C   s   t | |}t||S rv  )rI  r   r>  r  r  r  r    s   

r  c                 C   s   t |ttttfrdS zbttttdtt	t
tttd|td ttt| |td |td  td  tt| |td ttt| |td | | ||}t| |}tt|td |}t|t}W n ttfyy   Y dS w |r~dS dS )NFr  r  T)r  r  r  r  r  r   r   r
   	ShowStepsStepCounterr  r   r  r  r   r  rj  r   r  r  )r  ry  r2  r  r  r  r  r  r  r    s   
r  c                 C   s  t tttdttttttd|td t	t
t| |td |td  td  t
t| |td t	t
t| |td | | ||}t| |}t	t
|td |}ttd| t||d tttd|d |d  d  t
|d | | |||t
|d | ||S )NFr  r  )r   r   r
   r?  r@  r  r   r  r  r   r  rj  r  r  r3   r+   )r  ry  r2  r  r  r  r  r  r    s   
fr  c                 C   s   t | }t||S rv  )rH  r   r>  r  r  r  r  	  s   
r  c                 C      t || ||td   t|||  |td  t| ||  |td   || t|||  | t| ||  |td   ||td   | S Nr  r  r  r  r  r  r  r  r  ry  r  r  r  r  
	     r  c                 C   s   t || ||td   t|||  |td  t| ||  |td   ||t|||  | t| ||  |td   ||td   | S rB  r  r  r   r  r  rC  r  r  r  r  	  s   r  c                 C   sB   t t|||  | t| ||  | t| ||  |  ||S rv  r   rd  r  r  r  r  r  r  r  r  r  ry  r  r  r  r  	  r  r  c                 C   s   t |||  | t| ||  |  t| ||  |td   | t |||  | t| ||  |td   t| ||  |  | S )Nr  )r   r  r  r  rG  r  r  r  r  	  r  r  c                 C   s   t |||  | tdt| ||   |  t| ||  |td   |t |||  | tdt| ||   |td   t| ||  |  | S Nr  r  )r   r  r  r  rG  r  r  r  r  	     r  c              	   C   s   t || ||  t|||  |td  tdt| ||   |  || t|||  | tdt| ||   |  ||  | S r  )r  r   r  r  r  rG  r  r  r  r  	     r  c              	   C   s   t || ||  t|||  |td  tdt| ||   |  ||t|||  | tdt| ||   |  ||  | S r  )r  r   r  r  r  rG  r  r  r  r  "	     r  c                 C   rA  rB  r  r  r   r  r  rC  r  r  r  r  &	  rD  r  c              	   C   s   t || ||td   t|||  |td  tdt| ||   |td   ||t|||  | tdt| ||   |td   ||td   | S rB  rL  rC  r  r  r  r  *	  s   r  c                 C   s   t |||  | t| ||  |td   t| ||  td  |t |||  | t| ||  |td   t| ||   | S )Nr  r  )r   r  r  r  r  r  r  r  r  r  ry  r  r  r  r  .	  s   r  c                 C   s   t |||  | tdt| ||   |  t| ||  |td   | t |||  | tdt| ||   |td   t| ||  |td   | S Nr  r  r  )r   r  r  r  rG  r  r  r  r  2	  s   r  c                 C   s   t |||  | tdt| ||   |td   t| ||  td  |t |||  | tdt| ||   |td   t| ||   | S )Nr  r  r  )r   r  r  r  rM  r  r  r  r  6	  rI  r  c                 C   s   t |||  | tdt| ||   |  tdt| ||   |td   | t |||  | tdt| ||   |td   tdt| ||   |td   | S rN  )r   r  r  r  rG  r  r  r  r  :	     r  c           	      C   s~   t tdt| ||   | t| ||  |  |}t|| t||||  |td   || t|||  | || S rB  )r  r  r  r  r  r   	r  r  r  r  r  r  r  ry  r  r  r  r  r  >	  s   6Hr  c           	      C      t tdt| ||   | tdt| ||   |  |}t|| t||||  |td   || t|||  | || S rB  )r  r  r  r  r  r   rP  r  r  r  r  C	     >Hr  c              	   C   sR   t td| t|||  | tdttd|  td| |   |  ||S r  r  rC  r  r  r  r  H	  r  r  c           	      C   rQ  rB  )r  r  r  r  r  r   rP  r  r  r  r  L	  rR  r  c	           	      C   s8   t t||| | t|||  |t|||  |S rv  r   r2  )	r  r  r  r  r  r  r  r2  ry  r  r  r  r  Q	  r  r  c	           	   	   C      t || || |td   t| |t|||    |td  |||  |td   || t| |t|||    |td  |||  |  || |td   | S rB  r  	r  r  r  r  r  r  r  r  ry  r  r  r  r  U	     r  c	           	   	   C   s   t || || |td   t| |t|||    |td  |||  |td   ||t| |t|||    |td  |||  |  || |td   | S rB  rE  rU  r  r  r  r  Y	     r  c	           	   	   C   rT  rB  rL  rU  r  r  r  r  ]	  rV  r  c	           	   	   C      t || || |td   t| |t|||    |td  |||  |td   ||t| |t|||    |td  |||  |  || |td   | S rB  rL  rU  r  r  r  r  a	  rW  r  c	           	   	   C   s   t || || |td   t| |t|||    |td  |||  |td   || t| |t|||    |td  |||  |  || |td   | S rB  rE  rU  r  r  r  r  e	  rV  r  c	           	   	   C   rX  rB  r  rU  r  r  r  r  i	  rW  r  c
           
      C   B   t t|||	  | t| ||	  | t|||	  |  |	|	S rv  r   rd  r  
r  r  r  r  r  r  r  r  r  ry  r  r  r  r  m	  r  r  c
           
      C   rY  rv  r   rd  r  r[  r  r  r  r  q	  r  r  c
           
      C   sB   t t|||	  | t| ||	  | t|||	  |  |	|	S rv  rF  r[  r  r  r  r  u	  r  r  c                 C   sD   t t|||  | ||||  |
  | |||  |	|| ||S rv  )r   rc  )r  r  r  r  r  r  r  r  r  r  r  ry  r  r  r  r  y	  s   Dr  c                 C   s   t | ||||    | t|||   |td |td  t| td  |td   | t | ||||    | | t|  t|||   |td |td  t| td  |td   | S r  )r  r  r  r
  r  r  r  r  r  r  r  ry  r  r  r  r  }	     r  c                 C   s   t | ||||    | t|||   |td |td  t| td  |td   |t | ||||    | | t|  t|||   |td |td  t| td  |td   | S r  )r  r  r  r
  r  r]  r  r  r  r  	     r  c                 C   s  t |td | |td  |td |td  t| td  |td |td    t| ||||    t|||  |td   ||t| ||||    | | t|  t|||  |  |td |td  t| td  |td |td    | t| ||||    | | t|||  |td   t|||   |td |td  t| td  |td |td    | S Nr  r  r  r  r  r
  r   r  r  r  r  r  r  r  r  r  r  ry  r  r  r  r  	      r  c                 C   s  t |td | |td  |td |td  t| td  |td |td    t| ||||    t|||  |td   ||t| ||||    | | t|  t|||  |  |td |td  t| td  |td |td    | t| ||||    | | t|||   t|||  |td   |td |td  t| td  |td |td    | S r`  r  r  r
  r   r  r  r  )r  r  r  r  r  r  r  ry  r  r  r  r  	  rc  r  c                 C   s   t | ||||    t|||  |td   t|||   ||td   |t | ||||    | | t|  t|||  |td   |td |td  |td   | S r  )r  r  r  r  r
  rb  r  r  r  r  	  s   r  c                 C   s   t | ||||    t|||   t|||  |td   ||td   | t | ||||    | | t|  t|||  |td   |td |td  |td   | S r  )r  r  r  r  r
  rb  r  r  r  r  	  s   r  c                 C   sf  t |td |td  t| td  |td |td td   |td |td  |td   t| ||||    t|||  |td   ||t| ||||    t|||  |td   t|||   ||td   | t| ||||    | | t|  t|||  |td   |td |td  |td   | S r  ra  rb  r  r  r  r  	     h r  c                 C   sf  t |td |td  t| td  |td |td td   |td |td  |td   t| ||||    t|||  |td   ||t| ||||    t|||   t|||  |td   ||td   | t| ||||    | | t|  t|||  |td   |td |td  |td   | S r  rd  rb  r  r  r  r  	  re  r  c              	   C      t ttdt |||   td |  tt| |||    t|||  |  t| ||||    ttdt |||   td |  tt | |||    ||S r  )r  r:  r  r  r  r   rb  r  r  r  r  	  rO  r  c              	   C   rf  r  )r  r:  r  r  r  r   rb  r  r  r  r  	  rO  r  c                 C   s~   t t| tt| ||||    tdttdt |||    |  ttdt |||   td |   |||S r  r  r  r   r   r  r:  rb  r  r  r  r  	  s   ~r  c                 C   s   t t | tt| ||||    tdttdt |||    |   ttdt |||   td |  |||S r  rg  rb  r  r  r  r  	  s   r  c              	   C   s  t |td | |td  |td |td  t| td  |td |td    t| ||||    tdt|||   |td   ||t| ||||    | | tdt|||   |  t|  |td |td  t| td  |td |td    | t| ||||    | | tdt|||   |td   t|||   |td |td  t| td  |td |td    | S r  rd  rb  r  r  r  r  	      r  c              	   C   s  t |td | |td  |td |td  t| td  |td |td    t| ||||    tdt|||   |td   ||t| ||||    | | tdt|||   |  t|  |td |td  t| td  |td |td    | t| ||||    | | tdt|||   |td   t|||   |td |td  t| td  |td |td    | S r  ra  rb  r  r  r  r  	  rh  r  c                 C      t | ||||    tdt|||   |td   t|||   ||td   |t | ||||    | | tdt|||   |td   t|  |td |td  |td   | S Nr  r  r  r  )r  r  r  r  r
  rb  r  r  r  r  	  r_  r  c                 C   ri  rj  )r  r  r  r  r
  rb  r  r  r  r  	  r_  r  c              	   C   s~  t |td |td  t| td  |td |td td   |td |td  |td   t| ||||    tdt|||   |td   ||t| ||||    tdt|||   |td   t|||   ||td   | t| ||||    | | tdt|||   |td   t|  |td |td  |td   | S Nr  r  r  r  rd  rb  r  r  r  r  	      r  c              	   C   s~  t |td |td  t| td  |td |td td   |td |td  |td   t| ||||    tdt|||   |td   ||t| ||||    tdt|||   |td   t|||   ||td   | t| ||||    | | tdt|||   |td   t|  |td |td  |td   | S rk  ra  rb  r  r  r  r  	  rl  r  c                 C   s   t td| | ||||     t|t | | t|  td|  |td  t | | t|  td|  |td  td ttdt |||     tt| |||    || t|  t| |   |S r  )r  r  r5   r  r
  r:  rb  r  r  r  r  	  s   r  c                 C   s   t | ||||    td t |  t|t | | t|  td|  |td  t | | t|  td|  |td  td ttdt |||    tt| |||    || t|  t| |   |S r  )r  r  r  r5   r
  r:  rb  r  r  r  r  	  s   r  c              
   C   s   t ttdt |||   td | tdt|||   |  tt | |||    tt| ||||    ttdt |||   td |   tt| |||    |||S r  )r  r:  r  r  r  r   r-  rb  r  r  r  r  	  s   r  c                 C   s   t tdttd t |||    | tdt|||   |  tt| |||    tt| ||||    tdttd t |||    |   tt | |||    |||S r  )r  r  r:  r  r  r   r-  rb  r  r  r  r  	  s   r  c
           
   	   C   st   t td| ||  t| ||||	    tt | td|  |td  ||	 td  td|   |	|	S )Nr  r  )r  r  r   r  Pi
r  r  r  r  r  r  r  r  r  ry  r  r  r  r  	  r  r  c
           
   	   C   ^   t td| ||  t| ||||	    t|td ||	 td  td|   |	|	S r  r  r  r   r  rn  r  r  r  r  	  r  r  c
           
   	   C   ro  r  r  rn  r  r  r  r  	  r  r  c              	   C   sb   t ||	 t| ||||
    tt | td|  |td  ||
 td   |  |
|
S )Nr  r  )r  r   r  rm  r  r  r  r  r  r  r  r  r  r  r  ry  r  r  r  r  	  r  r  c              	   C   sJ   t ||	 t| ||||
    t|td ||
 td  |  |
|
S r  )r  r   r  r  rq  r  r  r  r  	  r  r  c              
   C   sR   t ||	 t| ||||
    tdt|td ||
 td   |  |
|
S r  )r  r   r  r  rq  r  r  r  r  	  r  r  c              
   C      t td|	 t| ||||
    t|||
   ||t|||
     |
|
t| ||||
    ||	t|||
     ||t|||
     |
 S r  r  r  r  r  r  r  r  r  r  hr  ry  r  r  r  r  	  rI  r  c              
   C   rr  r  r  rs  r  r  r  r   	  rI  r   c                 C   s*   t | |t||  |t|||  |S rv  rS  )r  r  r  r  r  r  ry  r  r  r  r  	     *r  c	           
      C   b   t | ||||    t|||  |  |}	t|t|	||td   || t|| |	| S r  )r  r  r  r   r  
r  r  r  r  r  r  r  r  ry  r  r  r  r  r  	     .4r  c	           
      C   rv  r  )r  r  r  r   r  rw  r  r  r  r  	  rx  r  c                 C   sF   t t| ||||
    t|||
  | t|||
  |	  |
|
S rv  rF  rq  r  r  r  r  
  r  r  c                 C   sN   t t| ||||    ||
  t|||  | t|||  |	  ||S rv  rF  )r  r  r  r  r  r  r  r  r  r  r  ry  r  r  r  r  
  r  r  c                 C   sF   t t| ||||
    ||||
  | ||||
  |	  |
|
S rv  )r   rh  )r  r  r  r  r  r  r  r  r  r  ry  r  r  r  r  
  r  r  c                 C      t t| | t|| ||S rv  )r   rh  r  r  r  r  r  ry  r  r  r  r  
  r-  r  c                 C   ry  rv  )r   rh  r  rz  r  r  r  r  
  r-  r  c                 C   s*   t t| | t|| t||  ||S rv  )r   rh  r  r  )r  r  r  r  r  ry  r  r  r  r	  
  ru  r	  c                 C   s   t ||td  t| |t|||    |td   |td  |t |t| |t|||    |td   || |td  t| |t|||      | S r  r  r  r  r
  r  r  r  r  r  r  ry  r  r  r  r  
  r  r  c              
   C   s   t ||td  t| |t|||    |td   |td  |t |t| |t|||    |td   t| |t|||     || |td   | S r  r  r  r  r
  r  r|  r  r  r  r  
  r  r  c                 C   s   t t|||  td||    t|  | | |  td| | |  |||  td ||   t| | | |  td| | |   | ||S r  r   r   r  r:  r|  r  r  r  r  #
  s   r  c                 C   s   t t|||  td||    t|  | | |  td |||  td ||   t| | | |  td  | ||S r  r~  r|  r  r  r  r  '
  s   r  c              
   C   s   t |t| |t|||     |td |td  td  |t || | t| |t|||     |td |td  td  | S r  )r  r  r
  r  r  r  r  r  r  ry  r  r  r  r  +
     r  c              
   C   s   t |t| |t|||     |td |td  td  |t || | t| |t|||     |td |td  td  | S r  )r  r  r
  r  r  r  r  r  r  r  /
  r  r  c              
   C   sb  t |td |td  | |td  |td |td  |td  td  tt| |t|||    |td  ||t|t| |t|||    |  |td |td  |td  td  | t|| | | t| |t|||    |td   t| |t|||     |td |td  |td  td  | S r  r  r  r   r  r
  r  r  r|  r  r  r  r  3
     d r  c              
   C   sb  t |td |td  | |td  |td |td  |td  td  tt| |t|||    |td  ||t|t| |t|||    |  |td |td  |td  td  | t|| | | t| |t|||     t| |t|||    |td   |td |td  |td  td  | S r  r  r  r   r  r
  r  r  r|  r  r  r  r  7
  r  r  c                 C   sZ  t |td |td  |td td  td |td |td  |td  |td   tt| |t|||    |td  ||t|t| |t|||    |td   |td |td  |td  |td   | t|t| |t|||    |td   || |td  t| |t|||      | S r  r  r  r   r  r
  r  r  r|  r  r  r  r  ;
     \ r  c              
   C   sZ  t |td |td  |td td  td |td |td  |td  |td   tt| |t|||    |td  ||t|t| |t|||    |td   |td |td  |td  |td   | t|t| |t|||    |td   t| |t|||     || |td   | S r  r  r  r   r  r
  r  r  r|  r  r  r  r  ?
  r  r  c              
   C   sX  t |td |||  tdt |   ttdt |   td |   t |||  t|   tt|   t|||  t |   tt |    |  t| t t | | | td  td| |  tdtt | | | td  td| |   |||  tdt |  ttdt |    t | | | td  |S r  r  r  r  r:  r5   r|  r  r  r  r  C
     Z r  c              
   C   sN  t ||||  t|  tt|   |||  t |  tt |    |  td|||  tdt |   ttdt |   td |   t| t t | | | td  td| |  tdtt | | | td  td| |   |||  tdt |   ttdt |    t | | | td  |S r  r  r  r:  r  r5   r|  r  r  r  r  G
  s   P r  c                 C   s   t ||td  |td  t| |t|||    |td   |td |td   |t ||td  t| |t|||    |td   || |td  t| |t|||      | S r  r{  r  r  r  r  r  r  ry  r  r  r  r  K
  r_  r  c              
   C   s   t ||td  |td  t| |t|||    |td   |td |td   |t ||td  t| |t|||    |td   t| |t|||     || |td   | S r  r}  r  r  r  r  r  O
  r_  r  c                 C   s   t td|  tt|| |||  |td ||    |td  t|  | | | |td   || |  |||  |td  ||   |td  t| | | | |td   || |   |  |||S r  r  r  r   r   r:  r  r  r  r  r  S
  r  r  c                 C   s   t td|  tt|| |||  |td ||    t|  | | | |td   |||  |td  ||   t| | | | |td    |  |||S r  r  r  r  r  r  r  W
  rW  r  c              
   C   s   t ||td  |td  t| |t|||     |td |td  |td td   |t || ||td   t| |t|||     |td |td  |td td   | S r  )r  r  r  r
  r  r  r  r  r  r  ry  r  r  r  r  [
  r_  r  c              
   C   s   t ||td  |td  t| |t|||     |td |td  |td td   |t || ||td   t| |t|||     |td |td  |td td   | S r  )r  r  r  r
  r  r  r  r  r  r  _
  r_  r  c                 C   s  t |td |td  | |td  |td |td  |td  |td td   t|| t| |t|||    |td   ||t||td  |td  t| |t|||    |  |td |td  |td  |td td   | t|| | ||td   t| |t|||    |td   t| |t|||     |td |td  |td  |td td   | S r  r  r  r  r  r  r  c
      r  c                 C   s  t |td |td  | |td  |td |td  |td  |td td   t|| t| |t|||    |td   ||t||td  |td  t| |t|||    |  |td |td  |td  |td td   | t|| | ||td   t| |t|||     t| |t|||    |td   |td |td  |td  |td td   | S r  r  r  r  r  r  r  g
  r  r  c                 C   s  t |td |td  |td td  |td td  |td |td  |td  |td   t|| t| |t|||    |td   ||t||td  t| |t|||    |td   || |td  t| |t|||      | t||td  |td  t| |t|||    |td   |td |td  |td  |td   | S r  r  r  r  r  r  r  k
      r  c                 C   s  t |td |td  |td td  |td td  |td |td  |td  |td   t|| t| |t|||    |td   ||t||td  t| |t|||    |td   t| |t|||     || |td   | t||td  |td  t| |t|||    |td   |td |td  |td  |td   | S r  r  r  r  r  r  r   o
  r  r   c              
   C   sp  t ||td  td |||  tdt |   ttdt |   td |   t |||  t|   tt|   t|||  t |   tt |    |  t| t t | | | | td  td| |  tdtt | | | | td  td| |   |||  tdt |  ttdt |    t | | | | td  |S r  r  r  r  r  r  r!  s
     r r!  c              
   C   sf  t ||td  |||  t|  tt|   |||  t |  tt |    |  td|||  tdt |   ttdt |   td |   t| t t | | | | td  td| |  tdtt | | | | td  td| |   |||  tdt |   ttdt |    t | | | | td  |S r  r  r  r  r  r  r"  w
  re  r"  c                 C   sj   t tdt| | |  t|||  td|  |||  td|  ttd|  | |   ||S r  r  r  r:  r   r  r  r  r  r#  {
  r8  r#  c                 C   st   t td| | t| | |  t|||  td|  |||  td|   ttd|  | |   ||S r  r  r  r  r  r  r$  
  r  r$  c                 C   s   t ||td  tdt| |t|||     |td   |td  |t |tdt| |t|||     |td   t| |t|||     || |td   | S Nr  r  r  r}  r|  r  r  r  r%  
     r%  c                 C   s   t ||td  tdt| |t|||     |td   |td  |t |tdt| |t|||     |td   || |td  t| |t|||      | S r  r{  r|  r  r  r  r&  
  r  r&  c                 C   sr  t |td |td  |td td  td |td |td  |td  |td   ttdt| |t|||     |td  ||t|tdt| |t|||     |td   |td |td  |td  |td   | t|tdt| |t|||     |td   t| |t|||     || |td   | S Nr  r  r  r  r  r|  r  r  r  r'  
     t r'  c                 C   sr  t |td |td  |td td  td |td |td  |td  |td   ttdt| |t|||     |td  ||t|tdt| |t|||     |td   |td |td  |td  |td   | t|tdt| |t|||     |td   || |td  t| |t|||      | S r  r  r|  r  r  r  r(  
  r  r(  c                 C   sz  t |td |td  | |td  |td |td  |td  td  ttdt| |t|||     |td  ||t|tdt| |t|||     |  |td |td  |td  td  | t|| | | tdt| |t|||     |td   t| |t|||     |td |td  |td  td  | S r  r  r|  r  r  r  r)  
     | r)  c                 C   sz  t |td |td  | |td  |td |td  |td  td  ttdt| |t|||     |td  ||t|tdt| |t|||     |  |td |td  |td  td  | t|| | | tdt| |t|||     |td   t| |t|||     |td |td  |td  td  | S r  r  r|  r  r  r  r*  
  r  r*  c              
   C   sX  t ||||  t|  tt|   |||  tdt |  ttdt |   td  |  td|||  tdt |   ttdt |   td |  t|t t| | | td  td| |  tdtt| | | td  td| |   |||  tdt |   ttdt |    t| | | td  |S r  r  r|  r  r  r  r+  
  r  r+  c              
   C   s`  t |t |||  t|   tt|   |||  tdt |   ttdt |   td  |  td |||  tdt |   ttdt |   td |  t|t t| | | td  td| |  tdtt| | | td  td| |   |||  tdt |  ttdt |    t| | | td  |S r  r  r|  r  r  r  r,  
  s   b r,  c              
   C   s   t tdt| | | |td   t|| |||  |td |   |||  td|td  |  ttd|  | | |td    ||S r  r  r  r  r  r  r-  
  rD  r-  c              
   C   s   t td| | t| | | |td   |td  t|| |||  |td |   |||  td|td  |   ttd|  | | |td    ||S r  r  r  r  r  r  r.  
  rW  r.  c                 C   s   t ||td  |td  tdt| |t|||     |td   |td |td   |t ||td  tdt| |t|||     |td   t| |t|||     || |td   | S Nr  r  r  r}  r  r  r  r  r/  
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  r  r0  c                 C   s  t |td |td  |td td  |td td  |td |td  |td  |td   t|| tdt| |t|||     |td   ||t||td  tdt| |t|||     |td   t| |t|||     || |td   | t||td  |td  tdt| |t|||     |td   |td |td  |td  |td   | S r  r  r  r  r  r  r1  
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     
NrC  c                 C   r  r  )r   r   r  r  r  r  r  r  r  rD     r  rD  c                 C      t |t|| |   |S rv  )r   r  r  r  r  r  r2  ry  r  r  r  rE       rE  c                 C   r  rv  )r   r  r  r  r  r  rF  	  r  rF  c                 C   $   t tt||  t||  ||S rv  rZ  r  r  r  r2  ry  r  r  r  rG       $rG  c                 C   r  rv  r\  r  r  r  r  rH    r  rH  c                 C   *   t t||  t|| t||  ||S rv  rZ  r  r  r  r  r2  ry  r  r  r  rI    ru  rI  c                 C   r  rv  r\  r  r  r  r  rJ    ru  rJ  c                 C   s0   t td|   t|ttd| |   ||S r  r  )r  r  r  r2  ry  r  r  r  rK    r  rK  c                 C   s$   t tt||  t||  ||S rv  rF  r  r  r  r  rL  !  r  rL  c                 C   s*   t t||  t|| t||  ||S rv  rF  r  r  r  r  rM  %  ru  rM  c                 C   sT   t t|| tt|| td  t| ||tt|t|| td   | S r  )r  r  r   r  r  r  r  r2  ry  r  r  r  rN  )  s   TrN  c                 C   sd   t t|| ttdt| | td  t| ||ttdt| | td  t| | S rB  )r  r  r   r  r  r  r  r  r  r  rO  -  r  rO  c                 C   sd   t t|| ttdt| | td  t| ||ttdt| | td  t| | S rB  r  r  r   r  r  r  r  r  r  r  rP  1  r  rP  c                 C   sV   t t|| tt|| td  t| || tt|t|| td   | S r  )r  r  r   r  r  r  r  r  r  r  rQ  5  r  rQ  c                 C   sl   t t|| ttdt| | td  ||t t|| ttdt| | td  t| || S rB  r  r  r   r  r  r  r  r  r  r  rR  9  r  rR  c                 C   sn   t t|| ttdt| | td  || t t|| ttdt| | td  t| || S rB  r  r  r  r  r  rS  =  r  rS  c                 C   sl   t t|| ttdt| | td  t| ||t t|| ttdt| | td  || S rB  r  r  r  r  r  rT  A  r  rT  c                 C   sn   t t|| ttdt| | td  t| || t t|| ttdt| | td  || S rB  r  r  r  r  r  rU  E  r  rU  c	           	      C   sJ   t | |ttd| td| |   td  | |||  |  |S r  )r   r  r  rU  r  r  r  rV  I  r  rV  c              
   C   sX   t td|  t|| td|  |ttd| td| |    | |  ||S r  rp  rC  r  r  r  rW  M     XrW  c              
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   
   C   b   t || td  tt| | ||  ||	  | t| ||	|   |  |	|	|||	  |	S r  r  r  r  r  r  rZ  U  r  rZ  c
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   
   C   r  r  r  r  r  r  r  r[  Y  r  r[  c	           	   
   C   ^   t tdt|||  | td|  | | || ttd| td| |     ||S r  rp  	r  r  r  r  r  r  r  r  ry  r  r  r  r\  ]  r  r\  c              
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   C   r  r  rp  r  r  r  r  r_  e  r  r_  c              
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   C   s  t |||  | tt|||    | t| tt|||     t| td |td  td  |t |||  | tt|||    | t| tt|||     t| td |td  td  | tt|||  |td   || |td   | S r  r   r:  r  r  r  r  r  r  r  r  r  r  r  ry  r  r  r  rb  q  s    rb  c                 C   s  t tt|||  | tt|||    | |tt|||     t| td |td  td  || t tt|||  | tt|||    | |tt|||     t| td |td  td  || tt|||  |td   || |td   | S r  r  r  r   r:  r  r  r  r  r  r  r  rc  u  s    rc  c                 C   s"  t tt|||  | tt|||    t|  |tt|||     t| td  |td  td  ||t tt|||  | tt|||    t|  |tt|||     t| td  |td  td  || tt|||  |td   || |td   | S r  r  r  r  r  r  rd  y  s   $ rd  c              
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   C   s   t || ||  tt|||  ||t|||     ||t||||   t|||   || ||t|||      | S rv  )r  r   r  r  r  	r  r  r  r  r  r  r  r  ry  r  r  r  rj    rK  rj  c	           	   
   C   s   t || ||  tt|||  ||t|||     || t||||   t|||   || ||t|||      | S rv  )r  r   r  r  r  r  r  r  r  rk    rJ  rk  c                 C   "   t | t| |t|  | |S rv  )r   r  r  r  r  r  r  r  ry  r  r  r  rl       "rl  c                 C   r  rv  )r   r  r  r  r  r  r  rm    r  rm  c                 C   6   t t|t| ||  | t|||  |  ||S rv  rZ  rx  r  r  r  rn    r  rn  c                 C   r  rv  r\  rx  r  r  r  ro    r  ro  c                 C   sx   t tdt|  | ||  |  tt|||  ||t tdt|  | ||  |  tt| ||  || S r  r  r  r  r   r  r  r  r  r  rp    s   xrp  c              	   C   s   t tdt|  | ||  |  ttdt| ||   ||t tdt|  | ||  |  ttdt|||   || S r  r  r  r  r  r  rq    rK  rq  c              	   C   sf   t |t|  | ||  |  | ttdt| ||  t|||    ||t|| | | S r  )r  r  r   r  r  r  r  r  r  rr    r9  rr  c              	   C   s^   t t|  | ||  | ttdt| ||  t|||    ||t|| | | S r  )r  r  r   r  r  r  r  r  r  r  rs    r  rs  c                 C   s$   t || t|  | |  |  |S rv  )r   r:  )r  r  r  r  r  ry  r  r  r  rt    r  rt  (  __doc__Zsympy.externalr   r   r   r   r   r   Z%sympy.integrals.rubi.utility_functionr   r	   r
   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r    r!   r"   r#   r$   r%   r&   r'   r(   r)   r*   r+   r,   r-   r.   r/   r0   r1   r2   r3   r4   r5   r6   r7   r8   r9   r:   r;   r<   r=   r>   r?   r@   rA   rB   rC   rD   rE   rF   rG   rH   rI   rJ   rK   rL   rM   rN   rO   rP   rQ   rR   rS   rT   rU   rV   rW   rX   rY   rZ   r[   r\   r]   r^   r_   r`   ra   rb   rc   rd   re   rf   rg   rh   ri   rj   rk   rl   rm   rn   ro   rp   rq   rr   rs   rt   ru   rv   rw   rx   ry   rz   r{   r|   r}   r~   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r  r  r  r  r  r  r  r  r	  r
  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r   r!  r"  r#  r$  r%  r&  r'  r(  r)  r*  r+  r,  r-  r.  r/  r0  r1  r2  r3  r4  r5  r6  r7  r8  r9  r:  r;  r<  r=  r>  r?  r@  rA  rB  rC  rD  rE  rF  rG  rH  rI  rJ  rK  rL  rM  rN  rO  rP  rQ  rR  rS  rT  rU  rV  rW  rX  rY  rZ  r[  r\  r]  r^  r_  r`  ra  rb  rc  rd  re  rf  rg  rh  ri  rj  rk  rl  rm  rn  ro  rp  rq  rr  rs  rt  ru  rv  rw  rx  ry  rz  r{  r|  r}  r~  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r:  r  r
  r  r  r  Zsympyr  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  Zsympy.integrals.rubi.symbolr  Zsympy.core.symbolr  r  Zsympy.functionsr  r  r  r  r  r  r  Z%sympy.functions.elementary.hyperbolicr  r  r  r  r  r   r  r  r  r  r  r  Z(sympy.functions.elementary.trigonometricr  r  r	  r
  r  r  r  r  rm  r  ZB_ZC_rc  rh  r  rd  r]  rY  Zd_Ze_r  Zg_r  Zi_Zj_Zk_Zl_r6  rP  r1  Zq_Zr_Zt_r  r  Zs_r  r  r  r  Za1_Za2_Zb1_Zb2_Zc1_Zc2_Zd1_Zd2_r  r   Ze1_Ze2_Zf1_Zf2_Zg1_Zg2_Zn3_ZPq_ZPm_ZPx_ZQm_ZQr_ZQx_Zjn_Zmn_Znon2_ZRFx_ZRGx_r  iiZPqqQRr  r  r  r  Z	_UseGammar?  r@  ru  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r	  r
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