Bug introduced in 12.3.1 or earlier and persisting through 13.2.0 or later.
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 of $f$, it being continuous and periodic, should decay with $n$, which the output from Mathematica does not give. Not only that, the Fourier coefficients should be bounded by 1 via $$ \left|\frac{1}{2\pi}\int_{-\pi}^{\pi} f(x)e^{inx} dx\right| \leq \sup_{x\in [-\pi,\pi)}|f(x)|=1 \quad \forall n\in\mathbb{Z}. $$ Any insight into why this is happening would be appreciated.
Edit
Thank you all for your answers. To summarise, it seems that Mathematica evaluates the above correctly if instead you use either of the following:
FullSimplify[FourierCosCoefficient[1/(5 + 4 Cos[x]), x, n],
Assumptions -> {n \[Element] PositiveIntegers}]
OR
Simplify[Integrate[Cos[n*x]/(5 + 4 Cos[x]), {x, -Pi, Pi}],
Assumptions -> {n \[Element] PositiveIntegers}]
OR similar variations of the above.
The bug is strange, and hopefully it shall be fixed soon.