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.
 A: If you look in wolframalpha it says that this integral can not be expressed in elementary math functions, so the only way to express is the integral itself.
This is the wolfram result:
https://www.wolframalpha.com/input/?i=integral+cos(x%2Bsin(x))
If you want to know more about this kind of functions https://en.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra)
A: 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.
