And in the darkness bind them

Calculus Level 5

If

\[\int_0^{\pi/2} \ln( \cos x) \ln( \sin x ) \, dx\]

can be expressed in the form \(\dfrac{A\pi^M}{B}\ln^PQ - \dfrac{C\pi^N}{D}\), for positive integers \(A,B,C,D,M,N,P\) and \(Q\), with \(\gcd(A,B) = \gcd(C,D) = 1\) and \(Q\) not a perfect power, find \(A+B+C+D+M+N+P+Q\).

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