I need help with the analytical solution of this integral:
$$\int_{0}^{2\pi}\frac{1}{\sqrt{a^2-b^2\cos^{2}2\phi}}\exp{\left(-\frac{(x-c\cos\phi)^2}{a+b\cos2\phi}-\frac{(y-d\sin\phi)^2}{a-b\cos2\phi}\right)}\mathrm d\phi$$
where $a^2-b^2=1$ and $c, d$ are constants. I know the solution for $a=1$, $b=0$, using the generating function of the modified Bessel functions of first kind, but I think it can't be used in this case. ¿Can it be solved using the stationary phase method?
Any help would be welcomed.