Hot Integral - 19

Calculus Level 5

$\large \int_0^\infty \dfrac{\cosh(nx)}{\cosh(\pi x) } \, dx$

For constant $$|n | < \pi$$, the value of the integral above is equal to

$\dfrac AB \sec \left ( \dfrac{n^C}D \right)$

where $$A,B,C$$ and $$D$$ are positive integers with $$A,B$$ coprime.

Calculate $$A+B+C+D$$.

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