Motivation for the problem here:
Is it possible to evaluate $$ \int_0^1 \tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{1+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}? $$ Or numerical value? I attempted to use Wolfram Alpha, but I didn't receive any results.
To simplify the integral let's denote $u = \tanh^{-1}(x)$ and $v= \tan^{-1}(x)$.
Rewrite the expression using $u$ and $v$: $$ \int_0^1 \tan^{-1}\left[\frac{\tanh^{-1}x - \tan^{-1}x}{1 + \tanh^{-1}x - \tan^{-1}x}\right] \frac{dx}{x} = \int_0^1 \tan^{-1}\left[\frac{u - v}{1 + u - v}\right] \frac{dx}{x} $$
I need a numerical result or a closed form since I don't have CAS or any software.