Consider integral with real $a, y$ and where $y> 1$
$$I(a, y) = \int_0^{y} \mathrm{d}x \frac{1}{1-x^2}~ \frac{\sin(ax) - (ax)\cos(ax)}{(ax)^3}$$
where singularity at $x=1$ is dealt with by taking principal value.
Mathematica spits out long and ugly result involving sine integral and cosine integral.
What methods to try to get solution in more beautiful form?
You start with the decompositions in partial fractions
$$\frac1{x^3(1-x^2)}=\frac1{x^3}+\frac1x-\frac1{2(x+1)}-\frac1{2(x-1)}$$ and $$\frac1{x^2(1-x^2)}=\frac1{x^2}+\frac1{2(x+1)}-\frac1{2(x-1)}.$$
The powers in the denominators can be decreased by parts. In the end, you get a sum of integrands of the form $\sin x/(x-a)$ or $\cos x/(x-a)$, which are indeed sine and cosine integrals.