I am trying to find the Fourier Transform of ($a,b,c\in\mathbb{R}$)
$$f(t)=\sin(at+b\sin(t))\mathrm{e}^{-ct}$$
that is, if it exists.
More precisely, I am trying to compute the integral
$$f(\omega)=\int_{-\infty}^\infty \sin(at+b\sin(t))\mathrm{e}^{-ct} \mathrm{e}^{\mathrm{i}\omega t} \mathrm{d}t $$
for which there seems to be no expressions in terms of elementary functions.
Is it nonetheless possible to get an expression for the integral? For example in terms of an infinite series?