Logs and sines and squares!

Calculus Level 5

\[\large{\int_{0}^{\pi/2}\sin^2{x}\log^{2}{(\sin^{2}x)}dx = \pi^{A}\log^{B}{C} - \pi^{D}\log{E}+ \dfrac{\pi^{F}}{G} - \dfrac{\pi}{H}}\]

If the above integral satisfies for positive, not necessarily distinct, integers \(A,B,C,D,E,F,G,H\), then submit the value of \(AB+CD+EF+GH\) as your answer.
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