I have the following expression I am trying to evaluate $$ \int_{0}^{2\pi}\kappa_{s}\cos(s(\theta-\mu_{s}))\exp\left(\kappa_{q}\cos(\theta-x)\right)\,d\theta $$ where $\kappa_s,\,s,\,\mu_{s},\,x$ and $\kappa_{q}$ are all constant in $\theta$, and $\theta \in [0, 2\pi)$. $s$ is an integer.
In the somewhat simpler case of
$$
\int_{\theta}\kappa_{s}\cos(\theta-x)\exp\left(\kappa_{q}\cos(\theta-x)\right)\,d\theta
$$
I can make some progress using Feynman's trick (differentiating under the integral), but when the arguments of the trigonometric functions are different I am completely lost.
This question arises when I want to take expectations of some types of circular distributions (such as a generalised von-Mises) with respect to another circular distribution (in this case a standard von-Mises. Note that the second integral is, in effect, just calculating the entropy of the vM distribution.
One final thing to note is that I will eventually be differentiating through the first expression with respect to the parameters $\kappa_s,\,\mu_{s}$ and $\kappa_{q}$, so if there is an approach whereby differentiating first and then integrating is feasible, that would also be great news!
Are there any tricks I can use here to evaluate the first integral? I'm quite okay if I can't use elementary functions, as long as reasonably good approximations exist for this hypothetical special functions.
$$I=\int_0^{2\pi}k_s\cos(s(\theta-\mu_s))\exp(k_q\cos(\theta-x))d\theta$$ Using substitution $t=\theta-x$ $$I=\int_{-x}^{2\pi-x}k_s\cos(st+sx-s\mu_s)\exp(k_q\cos t)dt$$ Using periodicity by $t$ $$I=\int_{0}^{2\pi}k_s\cos(st+sx-s\mu_s)\exp(k_q\cos t)dt$$ $$I=\int_{0}^{2\pi}k_s\cos(st)\cos(sx-s\mu_s)\exp(k_q\cos t)dt-\int_{0}^{2\pi}k_s\sin(st)\sin(sx-s\mu_s)\exp(k_q\cos t)dt$$ Using periodicity by $t$ $$I=\int_{-\pi}^{\pi}\cos(st)\cos(sx-s\mu_s)\exp(k_q\cos t)dt-\int_{-\pi}^{\pi}k_s\sin(st)\sin(sx-s\mu_s)\exp(k_q\cos t)dt$$ Using oddity of function $\sin(st)$ $$I=\int_{-\pi}^{\pi}k_s\cos(st)\cos(sx-s\mu_s)\exp(k_q\cos t)dt$$ Using evenness of function $\cos(st)$ $$I=2 k_s \cos(sx-s\mu_s)\int_{0}^{\pi}\cos(st)\exp(k_q\cos t)dt$$ Using formula $I_n(z)=1/\pi \int_0^\pi \exp(z\cos\theta)\cos n\theta\ d\theta$ (equ. 5 from https://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html) $$I=2\pi k_s \cos(sx-s\mu_s)I_{s}(k_q)$$