Nefarious Integral

Calculus Level 5

\[\LARGE{\int_0^1}\large {\frac1x \tan^{-1} \left( \frac{\cot^{-1} \left(\frac1x\right) - \coth^{-1} \left( \frac1x\right)}{\text{cs}^{-1}\left( \frac1x | 0\right) - \coth^{-1} \left(\frac1x\right) -\pi} \right) \, dx} \]

Given that the integral above equals to \(\dfrac{\pi}{a}\ln\bigg(\dfrac{\pi^b}{c}\bigg) \) for constants \(a,b\) and \(c,\) evaluate \(a+b+c\).

Note: \(\text{cs}(x)\) is a Jacobi elliptic function.

This is part of Ordered Disorder.
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