Let $K \geq 1$ be a real number. What are the Fourier coefficients of the function
$$\ln (K-\sin(x))$$
As it is a follow-up from this question, I am especially interested in the case $K = 5/4$ (for which the constant term is $0$), but I guess that computing these coefficients for all values of $K$ is of similar difficulty.
Note that I know how to compute any given coefficient, as it can be reduced to integrals of trigonometric rational functions. What I lack is a general formula for all coefficients.