I recently derived a closed form for an integral I haven't seen anywhere, $$\int_0^zY_0(\sqrt{z^2-x^2})dx=\frac2\pi\big(\operatorname{Ci}(z)\sin(z) - \operatorname{Si}(z)\cos(z) \big), \qquad z>0,$$ which, numerically, appears to be generalizable to $$\int_0^z Y_0\big(\sqrt{z^2-x^2}\big)dx = \left\{ \begin{array}{ll} \displaystyle\frac2\pi\big( \operatorname{Ci}(z)\sin(z) - \operatorname{Si}(z)\cos(z) \big), & \operatorname{Re} z\ge0 \\[15pt] \displaystyle\frac2\pi\big( \operatorname{Si}(z)\cos(z) - \operatorname{Ci}(z)\sin(z) \big), & \operatorname{Re} z<0. \end{array}\right.$$ $Y_\nu$ is the usual Bessel function of the second kind, $\operatorname{Ci}$ is the usual cosine integral, $$\operatorname{Ci} \equiv -\int_z^\infty\frac{\cos(t)}{t}dt,$$ and $\operatorname{Si}$ is the usual sine integral, $$\operatorname{Si} \equiv \int_0^z \frac{\sin(t)}{t}dt.$$
Is there a place to put such an integral for others to find? Gradshteyn and Ryzhik have a similar integral over the $J_0(z)$ Bessel function of the first kind, but nothing for $Y_0$. I'd think such an integral would naturally reside in G&R, but it doesn't seem like anyone is actively updating it?
I note for completeness that the source for G&R's integral over $J_0$, Magnus and Oberhettinger's ``Formeln und Sätze für die Speziellen Funktionen der Mathematischen Physik'' (Springer) 1948, doesn't have anything for an integral over $Y_0$, nor have rather significant internet searchings yielded any similar results. That said, the derivation was rather straightforward, so perhaps the integral is a known result in another source.
