Diluting The Integrand

Calculus Level 4

\[\large \displaystyle \int_0^\pi \dfrac{x\sin 2x \sin\left(\frac \pi 2 \cos x\right)}{2x-\pi} \, dx\]

The above integral can be expressed as \(\dfrac{A}{B\pi^C}\) where \(A\), \(B\) and \(C\) are positive integers with \(A\), \(B\) coprime. Find \(A+B+C\)