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Calculus Level 5

\[\int _{ 0 }^{ \pi }{ \left( \ln { \left[ \left( 2\sin { \frac { x }{ 2 } } \right) \left( 2\cos { \frac { x }{ 2 } } \right) ^{ 2 } \right] } \right) ^{ 3 }\, dx } =-A\pi \zeta (B)\]

If the equation above holds true for positive integers \(A\) and \(B\), then find \(A\times B\).

Notation: \(\zeta(\cdot) \) denotes the Riemann zeta function.

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