On a family of curious integrals suggested by Stellar Dynamics

While investigating the properties of a galaxy model used in Stellar Dynamics, a curious integral identity was discovered. For a special value of a parameter, the identity reduces to a definite integral with a very simple symbolic value; but, quite surprisingly, all the consulted tables of integrals, and computer algebra systems, do not seem aware of this result. Here I show that this result is a special case ($n=0$ and $z=1$) of the following identity (established by elementary methods): $$I_n(z)\equiv {\int }_0^1\frac{\mathrm{K}(k)k}{(z+k^2{)}^{n+3\mathrm{/}2}}dk=\frac{(-2{)}^n}{(2n+1)!!}\frac{d^n}{d{z}^n}\frac{\mathrm{A}\mathrm{r}\mathrm{c}\mathrm{C}\mathrm{o}\mathrm{t}\sqrt{z}}{\sqrt{z(z+1)}},z>0,$$ where $n=0,1,2,\mathrm{3...}$, and ${\rm K}(k)$ is the complete elliptic integral of first kind.