# Inspired from Hot Integral-14

Calculus Level 5

$\large{\displaystyle \int^{\infty}_{0} \cos (\sqrt{8} x) \left(f(x) \right)^2 dx=\frac{\sqrt{A \pi}}{B} \left(\Gamma \left(\frac{C}{D} \right) \right)^{E}}$

where

$\large{f(x)= \displaystyle \int^{\infty}_{0} \cos ( x \cosh t) dt }$

Find $$A+B+C+D+E$$

$$* A,B,C,D,E$$ are positive integers need not be distinct.

$$* C,D$$ are co prime and $$A$$ being square free.

Inspiration

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