With the method of Residues, we can calculate the integral \begin{equation}\int_{0}^{2\pi}\phi(\cos x,\sin x)\, dx \end{equation} where $\phi(x,y)=\frac{p(x,y)}{q(x,y)}$, ($p,q$ are polynomials of $x,y$)
In all the examples I have seen however, the antiderivative of $\phi$ is an elementary function (although very complicated) and as such, residues are an optional aid, not a "neccessity". My question is does there exist a function $\phi(x,y)$ so that \begin{equation}\int\phi(\cos x,\sin x)\, dx \end{equation} is not elementary yet \begin{equation}\int_{0}^{2\pi}\phi(\cos x,\sin x)\, dx \end{equation} can be (preferably easily) computed using the Residue Theorem?
A function is said to be elementary if it can be written in terms of polynomial, rational, exponential, logarithmic, trigonometric functions and their inverses. Whether or not a function has an elementary antiderivative is decided by the Risch algorithm.