In my research I have derived the following distributions. I am curious if they could be perhaps forms of well-known distributions. I have looked at the list of distributions here but have not gotten any hits. I'd be glad for any insights the community can provide.
\begin{equation} p_1(x) = \sqrt{\frac{2}{\pi}}\frac{1-e^{-\frac{x^2}{2}}}{x^2} I_{[0,\infty)}(x) \end{equation}
\begin{equation} p_2(x) = \frac{\ln(1+x^2)}{\pi x^2} I_{[0,\infty)}(x) \end{equation}
