In[]:=
​​ClearAll[S1,sVal,closedForm]​​​​(*1.Numericaltargetto50digits*)​​sVal=2*NIntegrate[Log[(Pi*x)/Sin[Pi*x]]/x,{x,0,1},WorkingPrecision->50];​​​​(*2.SymbolicIntegrationviaMathematica*)​​closedForm=Integrate[2*Log[(Pi*x)/Sin[Pi*x]]/x,{x,0,1}];​​​​Print["Numerical Benchmark: ",sVal];​​Print["Symbolic Reduction : ",closedForm];​​Print["Evaluated Numeric : ",N[closedForm,50]];​​Print["Difference : ",Abs[sVal-N[closedForm,50]]];
Numerical Benchmark: 2.3126789504275163184869445838156156303605072959389
Symbolic Reduction :
1
∫
0
2Log[πxCsc[πx]]
x
x
Evaluated Numeric : 2.3126789504275163184869445838156156303605072959389
Difference : 0.×
-50
10