# Integral of a gaussian function of trigonometric functions

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.

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Have you tried asking a CAS for help? –  CBenni Feb 2 '13 at 12:47
Perhaps differentiating with respect to $x$ and/or $y$ helps in understanding this function... Might try some trigonometric substitution, like $u = \tan \phi$, or something based on the expression under the root. –  vonbrand Feb 2 '13 at 13:50
CBenni, I tried Mathematica but it doesn't generate a solution. –  R2D2 Feb 2 '13 at 15:34