Let \(\mathcal{J}\) is the expected distance of a random point from any vertex of the unit tesseract. Define

\[\large \large \mathcal{I}=\int\limits_{0}^{1} \frac{\tanh^{-1}\left(\frac{1}{\sqrt{3+x^2}}\right)}{1+x^2} dx\]

We have

\[\displaystyle \mathcal{I}-5\mathcal{J} = A\sqrt{B}\tan^{-1} \sqrt{C}+\frac{\pi}{D}\log(E+\sqrt{F})-\frac{G}{H}\pi\sqrt{K}-L\log M -N,\]

where \(G,H\) are coprime numbers, \(F\) is a prime number and \(K\) is square free.

Calculate \(A+B+C+D+E+F+G+H+K+L+M+N\).

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