\[ \large \displaystyle \int_0^{\frac{\pi}{2}} x\ln^{2}(\sin 2x)\mathbb{d}x = \dfrac{\pi^{A}}{B}\ln^{C}(D) +\dfrac{\pi^{E}}{F} \]
If the above integral is true for positive integers:\(A,B,C,D,E,F\)

and \(D\) is a prime, find:

\[ A+B+C+D+E+F\]

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