If you have a binomial likelihood, $y|n,p\sim\textsf{Bin}(n,p)$, the Jeffreys' prior for the proportion $p$ is $\textsf{Beta}(1/2,1/2)$. If we instead reparameterize the proportion as the log odds $\phi=\log(p/1-p)$, through a change of variables we arrive at the pdf for $\phi$: $p(\phi)=\frac{1}{\pi}\sqrt{\frac{e^{\phi}}{(1+e^{\phi})^2}}$ (assuming I didn't do the math wrong). My question is does this distribution have a name? It looks related to the logistic distribution but I'm not sure how.
It's weird because most introductory sources on Jeffreys' priors use a binomial likelihood with uniform priors on $p$ and $\phi$ to motivate the desire for invariance under 1-1 transformations, find the Jefrreys' prior for $p$, but then don't actually look at what this corresponds to in the log odds parametrization.