Yet another Integral

Calculus Level 5

\[\int_0^\infty \dfrac{\sin(2\cos^2(x)) \cosh(\sin(2x))}{1+x^2} dx=\dfrac{a\pi}{b}\sin\bigg(c+\dfrac{1}{e^d}\bigg)\]\[\] If the above equation is true for integers \(a,b,c,d,\) with \(\text{gcd}(a,b) =1 \), find \((a+b+c)^d\).

This is part of Ordered Disorder.
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