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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