Evaluating $\int_0^{2\pi}\frac{dt}{(a+b\cos(t))^2}=\frac{2\pi a}{(a^2-b^2)^{3/2}}$ using Residue theorem

Question

I was solving some problems in residues, I got this problem. I tried the usual way, changing the integrating path from line segment to the unit circle by substuting $z=e^{it}$, Put in denominator i got $4$ degree polynomial, It is very difficult to proceed further. Is there any easy way to do it?