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

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 \(A+B\).

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