This is too long for a comment.
Just out of curiosity, I used a CAS to see what could be the result of
$$I_{p,q}=\int_{0}^\infty \sin(\frac{1}{x^p})\sin(x^q)dx$$ and observed (for limited ranges of $p$ and $q$) that, if $p\neq q$, the result is either a Meijer G-function (as Phoenix already commented) or a nasty linear combination of hypergoemetric functions.
To my surprise, the very first cases where $p=q$ leads to quite simple results
$$I_{1,1}=\frac{\pi }{2}J_1(2)$$
$$I_{2,2}=\frac{1}{4} \sqrt{\frac{\pi }{2}} \left(\frac{1}{e^2}+\sin (2)-\cos (2)\right)$$
$$I_{3,3}=\frac{1}{6} \left(\pi \left(J_{\frac{1}{3}}(2)-J_{-\frac{1}{3}}(2)\right)+\sqrt{3}
K_{\frac{1}{3}}(2)\right)$$
$$I_{4,4}=-\frac{\sqrt{2+\sqrt{2}}}{16} \left(\sqrt{2} \pi
\left(J_{-\frac{1}{4}}(2)-J_{\frac{1}{4}}(2)\right)-2 K_{\frac{1}{4}}(2)\right)$$ where appear Bessel functions.