Why has $\int \sin (\sin x) dx$ not been solved yet? I have Calculus 2 background, so please try to keep your answers around that level. I inly want a brief explanation.
What is it about $\sin (\sin x)$ that makes it difficult to integrate? Also, what makes us think that we express $\int \sin (\sin x) dx$ in terms of already known functions? I imagine if we didn't think we can express it in known functions, then we would've already given it a name and defined it as a special function with its own name. 

The surprising thing (beyond Calc 2, unfortunately) are definite integrals with answers as Bessel functions...
$$
\int_0^{\pi/2} \cos(\cos x)\;dx = \frac{\pi}{2}\;I_0(1)
$$
