I want to compute the (even) fourier coefficients for 
 $\dfrac 3 {5-4\cos 2\theta}$ on the interval $[0, 2\pi]$.

Namely, I want to compute the integral:
$$b\_n = \int\_0^{2\pi} \cos(n\theta) \frac 3 {5-4\cos 2\theta } \frac{d\theta}{2\pi}$$

Integrating this in mathematica, it seems like $b\_n = 0$ if $n$ is odd and $b\_n = \left(\frac 1 2 \right)^{n/2}$ for $n$ even.