Evaluate $\int\sin(\sin x)~dx$

Question

I was skimming the virtual pages here and noticed a limit that made me wonder the following
question: is there any nice way to evaluate the indefinite integral below?

$$\int\sin(\sin x)~dx$$ Perhaps one way might use Taylor expansion. Thanks for any hint, suggestion.

Answer 1

Maybe you could do something like substitute $u=\sin{x}$ and get

$$\int du \: \frac{\sin{u}}{\sqrt{1-u^2}}$$

You could Taylor expand the denominator and be in a position to integrate even moments of $\sin{u}$ and see if the resulting series is useful.