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ClearAll[expr,exactClosedForm,numValue100,diff];(*1.DefineIntegrand*)expr=(-1)*Log[1+Sqrt[1-x]]*Log[1-x]/x;(*2.High-PrecisionNumericalIntegration(100Digits)*)numValue100=NIntegrate[expr,{x,0,1},WorkingPrecision->100,PrecisionGoal->95,MaxRecursion->50];(*3.High-PrecisionClosedFormTarget*)exactClosedForm=N[3*Zeta[2]*Log[2]-(9/4)*Zeta[3],100];(*4.DifferenceCheck*)diff=Abs[numValue100-exactClosedForm];(*OutputPrecisionAudit*)Print[""];Print[" INTEGRAL AUDIT (100 DECIMAL PLACES) "];Print[""];Print["NIntegrate (100 Digits) : ",numValue100];Print["Analytic (100 Digits) : ",exactClosedForm];Print["Residual Difference : ",ScientificForm[diff]];Print[""];

INTEGRAL AUDIT (100 DECIMAL PLACES)

NIntegrate (100 Digits) : 0.7159161998194711302743287135762855564286703663278342736294549247394226564421799104334493373215923600

Analytic (100 Digits) : 0.7159161998194711302743287135762855564286703663278342736294549247394226564421799104334493373215923600

Residual Difference : 0.×

-99

10