  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): $$ I_n(z)\equiv\int_0^1{{\rm K}(k) k\over
(z+k^2)^{n+3/2}}dk = {(-2)^n\over (2n+1)!!} {d^n\over dz^n} {{\rm
ArcCot}\sqrt{z}\over\sqrt{z(z+1)}},\quad z>0,$$ where $n=0,1,2,3...$, and ${\rm
K}(k)$ is the complete elliptic integral of first kind.
