# Hot Integral - 27

Calculus Level 5

$\large \displaystyle \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} \left[\frac{\tanh(\frac{w-x}{2})\tanh(\frac{x-y}{2})\tanh(\frac{y-z}{2})}{\cosh w +\cosh x + \cosh y + \cosh z}\right]^2 \, dw \; dx \; dy \; dz$

The above integral can be written as $\large \frac{A}{B}\pi^C - D - E\zeta(F),$

where $$A,B,C,D,E$$ and $$F$$ are positive integers with $$A,B$$ coprime. Calculate $$A+B+C+D+E+F$$.

Notation: $$\zeta(\cdot)$$ denotes the Riemann zeta function.

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