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): $$ I_n(z)equivint_0^1{{rm K}(k) kover (z+k^2)^{n+3/2}}dk = {(-2)^nover (2n+1)!!} {d^nover dz^n} {{rm ArcCot}sqrt{z}oversqrt{z(z+1)}},quad z>0,$$ where $n=0,1,2,3...$, and ${rm K}(k)$ is the complete elliptic integral of first kind.