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I want to find a symbolic expression for the following integral as a function of $f$ and $g$:

$$ \int_{0}^{\pi} \sqrt{1-\frac{2}{f + g \cdot \cos \theta}} \, d\theta $$

It is guaranteed that $f$ and $g$ are real numbers such that the argument to the square root is non-negative over the integration interval, e.g. $f = 36, g = -16$.

Unfortunately, I am stuck here. I can find solutions for $g = 0$, but other cases elude me.

Numerical integration handily yields numerical solutions that I have verified to be correct against the problem I'm trying to solve, but I'd appreciate a closed-form answer. Any help or insight on this would be greatly appreciated!

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