I have a definite integral that arises when looking at Lennard-Jones potentials involving ellipses. It is of the form $$\int_{0}^{2\pi}\frac{f\left(\sin\left(t\right),\cos\left(t\right)\right)}{\left(P_{21}\left(\cos\left(t\right)\right)+P_{22}\left(\sin\left(t\right)\right)\right)^{n}}dt$$
where $P_{2i}\left(x\right)$ are quadratic polynomials and $f\left(x,y\right)$ is often the gradient of some line or area element. My question is about a simpler case of this (I hope it is simpler anyway).
Suppose you have distinct and typically unequal constants $a,b,c,d\in\mathbb{R}$ and $n\in\mathbb{N}$, then I'm looking for either a solution or a good reference to help solve $$I=\int_{0}^{2\pi}\frac{1}{\left(\left(a+b\cos\left(t\right)\right)^{2}+\left(c+d\sin\left(t\right)\right)^{2}\right)^{n}}dt.$$
I already am able to work with variants of $I$ for $c=0$ but as soon as $c$ is non-zero the $\sin\left(t\right)$ term throws a spanner in the works.
I will note that I have managed to solve $I$ using residues (by setting $n$ to values $3$ and $6$, which pertain to the Lennard-Jones potential), but it proved incredibly unwieldy and ultimately unhelpful for use.