\[\int _{ 0 }^{ \pi/2 }{ (\sin x) \ln { \left( \cos {x} \right) } \ln { \left( \sin {x} \right) } \, dx } =A-\frac{B\pi ^C}{D}-\ln {E}\]

Where \(A,B,C,D, E\) are positive integers, \(E\) is not a perfect power, and \(\gcd{(B,D)}=1\).

Find the value of \(A+B+C+D+E\)

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