How can I convert the integral $\int_0^{2\pi} (a^2 \cos^2 t +b^2\sin^2 t)^{-1} dt$ into an integral $\oint_\gamma z^{-1} dz$ where $z\in \mathbb C$ and $\gamma: {x^2\over a^2}+{y^2\over b^2}=1$? I can see that $|z|^2 = a^2 \cos^2 t +b^2\sin^2 t$ but the Jacobian seems very messy and I can't get the desired form.
Edit: Perhaps into a form not exactly $\oint_\gamma z^{-1} dz$ but up to a constant multiple?