Question: > 
> Evaluate $\int \cos(\cos x)~dx$
> 
> 
> 

I tried to use chain rule but failed. Can anyone help me please?

Answer: 
This is probably too long for a comment. 

Wolfram alpha indicates that the solution has the form
$$\sum\_{n=0}^{\infty} \frac{x^{2n+1}(a\_{n}\sin(1)+b\_{n}\cos(1))}{(2n+1)!}$$

The $-a\_{n}$ appear to correspond to [oeis:A192007](https://oeis.org/A192007), e.g.f.: $\sin(\cos(x)-1)$ (even part), and the $b\_{n}$ appear to correspond to [oeis:A192060](https://oeis.org/A192060). e.g.f: $\cos(\cos(1)-1)$ (even part)
This integral doesn't have a nice closed-form solution in terms of elementary functions, so this question is impossible (assuming you're just supposed to find the antiderivative in a form simpler than $\int \cos(\cos(x)) dx$)