\[\int _{ 0 }^{ \pi /3 }{ x{ \left[ \ln { \left( 2\sin { \left( \dfrac { x }{ 2 } \right) } \right) } \right] }^{ 2 } \, dx } =\dfrac { { A\pi }^{ B } }{ C } \]

This equation holds true for positive integers \(A,B,C\) with \(A\) and \(C\) being coprime. Evaluate \(A+B+C\).

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