I am trying to find the solution(s) of the functional equation $$ f(x)+f(x+\gamma)=\frac{1}{(3-2\, \cosh(2x))(3-2\,\cosh(2(x+\gamma))}. $$ Here $\gamma=\mathrm{arcosh}\frac{\sqrt{5}}{2}$ is a constant and $x\in\mathbb{R}$. It would probably be solvable by Fourier transformation if I were able to calculate the Fourier transformation of the RHS. However, I (and Mathematica) were not able to achieve this due to the poles on the real axis and the infinite number of poles in the complex plane.
Also note that $\cosh(2 (x+\gamma))=\frac{1}{2} \left(\sqrt{5} \sinh (2 x)+3 \cosh (2 x)\right)$ which did not really help me but might be useful. I'd be thankful for any suggestions or solutions.