Logarithm, Trignometry, Integrals

Calculus Level 5

\[\displaystyle \large \int _{ 0 }^{ \pi /2 }{ \ln(\cos(x))\ln(\sin(x)) \ dx } \]

The above integral can be expressed as

\[\dfrac { \pi ( \ln B)^A }{ C } -\dfrac { { \pi }^{ D } }{ E }\]

for positive integers \(A,B,C,D,E\), with \(B\) is not a perfect power of any integer, find \(A+B+C+D+E\).

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