I'm attempting to solve equation (13) from this paper
$$\int_{-\infty}^{\infty}\frac{\text{sinc}^2(t/2)}{\sqrt{1-t^2}}dt.$$
I'm attempting to do this numerically, as it there doesn't seem to be an analytical solution (even Mathematica can't find one). However, the discontinuity at $t=\pm 1$ is causing the numerical integral to be highly dependent on the step size. Does anyone know how to solve this properly, either numerically or analytically? I've read multiple papers which present results based on this or similar integrals, but none explain their methods.
You can remove the singularity analytically $$\int_{-\infty}^{\infty}\frac{\operatorname{sinc}^2(t/2)}{\sqrt{1-t^2}}dt=\int_{-\infty}^{\infty}\frac{\operatorname{sinc}^2(t/2)-\operatorname{sinc}^2(1/2)}{\sqrt{1-t^2}}dt+\int_{-\infty}^{\infty}\frac{\operatorname{sinc}^2(1/2)}{\sqrt{1-t^2}}dt$$ The left integral has the numerator going to zero, canceling the zero in the denominator, so numerical integration should work easily. The right integral can be done analytically to be $\operatorname{sinc}^2(1/2)\sin^{-1}t$, but needs to be broken at the singularity. You probably want to break the range of integration near the singularities, use this formulation near them, expand the numerator of the left integral as a Taylor series near the singularity, and use your original form away from the singularities.