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

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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