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$\large \int _{ - \pi / 6 }^{ \pi / 6 }{ \cfrac { \sin x }{ \sin x - \cos x } } \, dx = \, \dfrac{\pi}{A} + \dfrac{1}{2} \, \ln (B-\sqrt C )$

Given $A$, $B$ and $C$ are positive integers satisfying the equation above, find $A+B+C$.

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