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 $A+B+C$