TITLE: Need help with a definite integral $\int_0^{\infty} \frac{x-1}{\sqrt{2^x-1}\ln(2^x-1)}\,dx$
QUESTION [10 upvotes]: Evaluate:
$$\int_0^{\infty} \frac{x-1}{\sqrt{2^x-1}\ln(2^x-1)}\,dx$$

I am not sure where to start or what should be the best approach towards this problem. I tried the substitution $2^x-1=t^2$ but that seems to make things more worse. Using this substitution, I got:
$$\int_0^{\infty} \frac{1}{\ln^2 2}\frac{\ln\left(\frac{1+t^2}{2}\right)}{(1+t^2)\ln t}\,dt$$
I don't see how to proceed after this. :(
Any help is appreciated. Thanks!

REPLY [4 votes]: Try $t=\tan\theta$
Simplify everything in terms of $\sin$ and $\cos$. Substitute $\pi/2-\theta=\phi$ and add and do what you do with other definite questions. Be sure to completely simplify your numerator to break into two integrals.