Integration challenge 2

Calculus Level 5

\[ \large \int_0^{\frac \pi4} \frac{x^2(\sin(2x) - \cos(2x))}{\cos^2(x) (1 + \sin(2x)) } \, dx \]

If the value of the integral above equals to \( \dfrac{\pi^A}B - \dfrac{\pi^C}{D} \ln(E) \) for integers \(A,B,C,D\) and \(E\), find the minimum value of \(A+B+C+D+E\).

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