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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