Integrate $\int_0^{2\pi} \frac{(\cos^2{x})^{2-\beta}}{(1+\alpha \cos{x})^{4-\beta}} dx$

Question

I would like to find a solution to the following integral.

$\int_0^{2\pi} \frac{(\cos^2{x})^{2-\beta}}{(1+\alpha \cos{x})^{4-\beta}} dx$

but having some issues. $0<\alpha<1$ and $0<\beta<2$. I have tried using contour integration with residues, but I am not getting anything useful. I have similar integrals where people obtain hyper geometric functions, but I am unsure how/why thats done.

Thanks