# Integration of nontrivial trigonometric functions

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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Why do you expect complex analysis to be useful? – Siminore Sep 7 '12 at 12:06
@Siminore: Contour integration in the complex plane and residues are useful in the simpler example. – Kagaratsch Sep 7 '12 at 12:11
Of course. We know there is a formula to solve the equation $ax^2+bx+c=0$, why can't we use the same idea to solve $ax^3+bx^2+cx+d=0$? I mean that you want to integrate a very different function, which is no longer a rational function of $\sin x$ and $\cos x$. – Siminore Sep 7 '12 at 12:56
Any ideas on how to solve that integral are welcome. I do not insist on it being an application of complex analysis. It is mentioned solely to illustrate the solution to the simpler example. – Kagaratsch Sep 7 '12 at 13:01
Numerical value is 4.00883. But that $\cos^2(t^2)$ at the denominator makes things really hard. – Jon Sep 7 '12 at 13:16