Fourier coefficient of $\frac{1}{5 + 4\cos(x)}$ wrong

I’ve been trying to use Mathematica to compute the Fourier coefficient of

$ $ f(x) = \frac{1}{5 + 4\cos(x)}, $ $ which is a $ 2\pi$ periodic function. Inputting

FullSimplify[  FourierCoefficient[1/(5 + 4 Cos[x]), x, n,    FourierParameters -> {1, 1},    Assumptions -> {x \[Element] Reals, n \[Element] Integers}]] 

I get as an output

-(1/3) (-2)^Abs[n]

This is clearly wrong as the Fourier coefficient $ f$ , it being continuous and periodic, should decay with $ n$ , which the output from Mathematica does not give.

Any insight into why this is happening would be appreciated.