Logarithm, Trignometry, Integrals

Calculus Level 5

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

The above integral can be expressed as

π(lnB)ACπDE\dfrac { \pi ( \ln B)^A }{ C } -\dfrac { { \pi }^{ D } }{ E }

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

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