It isn't what it looks!

Calculus Level 5

\[ \displaystyle \int_0^{2\pi} x^2\left( \ln \left| 2 \sin\dfrac{x}{2} \right| \right)^{2} \, dx= \dfrac{A\pi^C}{B} \]

If the equation above holds true for integers \(A,B\) and \(C\) with \(A,B\) coprime positive integers, find \(A+B+C\).

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