499 Followers Problem

Calculus Level 4

$\large \int_{-{1} / {\sqrt{3}}}^{{1} / {\sqrt{3}}}\frac{\cos^{-1}\left ( \frac{2x}{1+x^{2}} \right )+\tan^{-1}\left ( \frac{2x}{1-x^{2}} \right )}{e^{x}+1} \, dx$

If the integral above can be expressed in the form

$\dfrac{ \pi}{A\sqrt B} ,$

where $A$ and $B$ are positive integers with $B$ square-free, find $A+B$ .