What's happening to the limits?

Calculus Level 5

\[\large \int _{ 0 }^{ \pi }{ { x }^{ 3 } } \left( { \ln}{ }\left| 2\sin { \dfrac { x }{ 2 } } \right| \right) ^2 \, dx=\dfrac { A }{ B } { \pi }^{ C }\zeta ( D ) -\dfrac { E }{ F } \zeta ( G ) \]

The above equation holds true for positive integers \(A\), \(B\), \(C\), \(D\), \(E\), \(F\) and \(G\), where \(\gcd(A,B) = \gcd(E,F) = 1 \).

Find the minimum value of \(A+B+C+D+E+F+G\). \[\]

Notations:


Inspiration.

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