# Evaluate $\int \cos(\cos x) dx$

Evaluate $\int \cos(\cos x) dx$

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

• Not sure if this is relevant. Mathematica documentation claims that $\int \sin(\sin x) d x$ "can in principle be represented as an infinite sum of $_{2}F_{1}$ hypergeometric functions, or as a suitably generalized Kampé de Fériet hypergeometric function of two variables." – sdcvvc Mar 7 '12 at 17:07
• A related question. – Lucian yesterday

## 3 Answers

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, e.g.f.: $\sin(\cos(x)-1)$ (even part), and the $b_{n}$ appear to correspond to oeis:A192060. e.g.f: $\cos(\cos(1)-1)$ (even part)

The indefinite integral has no simpler form (known), but there are some definite integrals, like this $$\int_0^{\pi/2} \cos(\cos x)\,dx = \frac{\pi}{2}\;J_0(1)$$ in terms of a Bessel function.

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 $\int \cos(\cos(x)) dx$)