Calculate $$\int_{0}^{2\pi}\frac{1}{1+\alpha\cos^2(t)}\,dt$$
with $\alpha>-1$.
I would like to use the following theorem: $$R(\cos(t),\sin(t))=\frac{p(x,y)}{q(x,y)}$$
with $p,q$ complex polynomials and no singularity on the unity circle, then:
$$\int_{0}^{2\pi}R(\cos(t),\sin(t))\,dt=2\pi*\sum_{|z|<1}\operatorname{Res}_{z}S(z)$$
with
$$S(z)=1/z*R\left(\frac{1}{z}(z+1/z),\frac{1}{2i}(z-(1/z))\right).$$
I know so far that $q=1+\alpha \cos^2(t)$ has no root, since $\cos^2 \in [0,1]$ and the condition given for $\alpha$, hence the integrand has no singularity.
Any help is appreciated.