Integrating $\cos (x+\sin (x))$ I tried to solve $$\int\cos(x+\sin(x))\,dx$$ but it seems to be way out of my league (tried u-substitution with $u=x+\sin(x)$ and couldn't find an answer). Also, no one on the Internet seems to have tried this before and Wolfram|Alpha  and Symbolab aren't helping that much. If anyone can help me, I would apreciate it very much.

The Jacobi-Anger expansion gives
$$ e^{ix\sin\theta}=\sum_{n\in\mathbb{Z}}J_n(x) e^{ni\theta}\tag{1} $$
hence:
$$ e^{i\sin\theta+i\theta}=\sum_{n\in\mathbb{Z}} J_n(1)\,e^{(n+1)i\theta}\tag{2}$$
and by considering the real part of both sides:
$$ \cos(\theta+\sin\theta) = \sum_{n\in\mathbb{Z}} J_n(1)\,\cos((n+1)\theta)\tag{3} $$
so:
$$ \int\cos(\theta+\sin\theta)\,d\theta = \color{red}{C+\sum_{n\in\mathbb{Z}} \frac{J_{n}(1)}{n+1}\,\sin((n+1)\theta)}\tag{4} $$
where the RHS is a fast-converging series, since $J_n(1)$ behaves like $\frac{1}{2^n n!}$.
So, if you consider Bessel functions as elementary functions, there is a primitive in terms of elementary functions.
