I'm really lost about how to prove the integral converges.  

I tried using the fraction expansion theorem and got to  

$$ \frac{\ln(x)}{1-x^2} = \frac{\ln(x)}{2(1+x)} + \frac{\ln(x)}{2(1-x)}$$  

But I didn't find a way to use it.  

I can't use the Dirichlet's test or Comparing tests since the function is not positive.

What am I missing?  

Thank you in advance and have a nice day!