In[]:=
NIntegrate
1
1+Exp[Pix]
*
Log[1-Ix]
1-Ix
-
Log[1+Ix]
1+Ix
,{x,0,Infinity},WorkingPrecision->#&/@{10,15,20,25,30,50}//MatrixForm
Out[]//MatrixForm=
-0.08705305704
-0.0870530582583057
-0.087053058258754247162
-0.08705305825875424689636149
-0.0870530582587542468963614944500
-0.087053058258754246896361494450015251358485685422121