\[\int_0^\infty f^2(x)\ dx = \frac{\pi^A}{B}\left(\log(C) - \psi \left(\frac{D}{E}\right)\right)\]

where \(\displaystyle f(x) = \int_0^\infty \cos(x\cosh(t))\ dt\) and \(A\), \(B\), \(C\), \(D\), and \(E\) are integers with \(D\) and \(E\) being positive coprime integers. Calculate \(A+B+C+D+E\).

**Notations:** \(\psi(n)\) is polygamma function.

This is a part of Hot Integrals

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