Paneer Definite Masala 4

Calculus Level 5

\[ \large\int_{-\pi /4}^{\pi /4} \dfrac{2\pi - \sin (2\pi x) }{2 - \cos2x } \, dx \]

If the integral above is equal to \( \dfrac{A\pi^B}{C\sqrt D} + E \), where \(A,B,C,D\) and \(E\) are non-negative integers with \(A,C\) coprime and \(D\) square-free, find \(A+B+C+D+E\).

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