improper integral error?

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Always try the other available "algorithms":

sage: integral(log(x)/(x^2 - 1),x,0,infinity, algorithm='mathematica_free')
-1/2*log(x + 1)*log(x) - 1/2*polylog(2, -x) - 1/2*polylog(2, -x + 1)

See http://sagemath.org/doc/reference/cal...

With the algorithm set to mathematica_free, Sage is returning the antiderivative even though you are asking for the improper integral. You can get the improper integral by doing the following:

f=integral(log(x)/(x^2 - 1),x,algorithm='mathematica_free')
limit(f,x=oo)-limit(f,x=0)

This returns pi^2/4.