Closed forms of $\int_A^B \sin(\sin(ax))dx$ and $\int_A^B \sin(\sin(ax)) \cdot \sin(\sin(bx))dx$

Question

I would like to know if:

1. $\int_A^B \sin(\sin(ax))dx$ has a closed-form? The solution of Maple requires the presence of Struve functions in its expression. But at least Maple is able to solve it, so even if complicated I guess it has an analytical form.

2. $\int_A^B \sin(\sin(ax)) \cdot \sin(\sin(bx))dx$ has a closed-form? this one Maple was not able to solve. I tried to find a similar expression in this tables of Bessel functions but there are none looking like these one. And with Liouville's theorem and the Risch algorithm I am a bit lost in the differential algebra to prove that it is not integrable.