Do orthodox methods always work?

Calculus Level 5

\[ \large \int_0^{\pi /2} \left( \ln( \cos^2 x ) \right)^2 \, dx =\dfrac { { \pi }^{ A } }{ B } +C\pi { \left( \ln { D } \right) }^{ E }\]

If the equation above holds true for positive integers \(A,B,C,D\) and \(E\), find the minimum value of \( A+B+C+D+E\) .

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