A power, a cosine, an exponential, oh my!

Calculus Level 3

\[\large \int_{-\frac{\pi}{2}}^\frac{\pi}{2}\dfrac{x^2\cos (x)}{e^x+1}\text{ }dx=\frac{\pi^A}{B}-C\]

If the equation above is true for positive integers \(A,\) \(B,\) and \(C\), find \(A+B+C\).

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