Integrate with sin cos

Calculus Level 5

\[\large\displaystyle \int_{0}^{\frac{\pi}{4}} (\cos 2 \theta)^{\frac{3}{2}} \cos \theta \, d \theta\]

Given the integral above equals to \( \dfrac{A}{B \sqrt{C}} \pi\) where \(A,B,C\) are positive integers with \(A,B\) are coprime and \(C\) square-free. Find \(B-A-C\).