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First an example which I know how to solve. If we have the following integral

$\int_{-\pi}^{\pi}\frac{1}{1+3~\cos^2(t)}dt$

there is a very practical way to evaluate it by interpreting it as some particular parametrization of a closed contour over a complex function. It works since the relevant residues of that underlying complex function can be readily obtained. The whole procedure is very well explained here:

http://en.wikipedia.org/wiki/Methods_of_contour_integration#Example_.28III.29_.E2.80.93_trigonometric_integrals

Now, let us make the integrand more complicated. Especially, I am interested in the following case:

$\int_{-\pi}^{\pi}\frac{1}{1+3~\cos^2(t)~\cos^2(t^2)}dt$

Since now different powers of $t$ are in the exponential functions the substitution as described in the Wikipedia article does not directly give a complex function whose residues could be easily obtained. That spoils the whole procedure. Any suggestion on how to evaluate?

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    Numerical value is 4.00883. But that $\cos^2(t^2)$ at the denominator makes things really hard.2012-09-07

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