Roaring Integral!

Calculus Level 4

\[\large \displaystyle \int_{0}^{\pi} \dfrac{1}{(5-3\cos x)^3}\, dx = \dfrac{\alpha \pi}{\beta}\]

In the above integral, if \(\alpha\) and \(\beta\) are coprime positive integers, then find \(\alpha + \beta\) .

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