the integral PDF $\sqrt {{2 \over \pi }} \int_{t\, = \,0}^\infty {e^{\, - \,{1 \over 2}\left( {v^{\,2} /t^{\,2} + 2t} \right)} dt\,} $ In  my answer  to this post I came to the conclusion that  the PDF of the volume of a parallelepiped with normal distributed coordinates,
having one vertex at the origin, is
$$
p_t (v) = \sqrt {{2 \over \pi }} \int_{t\, = \,0}^\infty
  {e^{\, - \,{1 \over 2}\left( {v^{\,2} /t^{\,2}  + 2t} \right)} dt\,} 
$$
with the corresponding CDF
$$
P_{\,t} (v) = \int_{t = 0}^\infty
  {\,\,t\;e^{\, - \,\,t} \,{\rm erf}\left( {{v \over {\sqrt 2 \,t}}} \right)dt\,} 
$$
where $0 \le v$.
I wonder whether these two integrals may have an interesting expression in terms of known functions.
 A: In terms of the Meijer G function
$$p_t (v) =\frac{\sqrt{2}}{\pi }\,\, G_{0,3}^{3,0}\left(\frac{v^2}{8}|
\begin{array}{c}
 0,\frac{1}{2},1
\end{array}
\right)$$
$$P_t (v) =\frac{2}{\pi }\,\, G_{1,4}^{3,1}\left(\frac{v^2}{8}|
\begin{array}{c}
 1 \\
 \frac{1}{2},1,\frac{3}{2},0
\end{array}
\right)$$
Edit
In terms of series, for small values of $v$
$$p_t (v)=\sqrt{\frac{2}{\pi }}-v+$$ $$\frac{v^2 \left(-2 \sqrt{2} \log (v)-2 \sqrt{2} \gamma
   +\sqrt{2}+\sqrt{2} \log (8)+\sqrt{2} \psi
   ^{(0)}\left(-\frac{1}{2}\right)\right)}{4 \sqrt{\pi
   }}+$$ $$\frac{v^3}{6}+O\left(v^4\right)$$
$$P_t (v)=\sqrt{\frac{2}{\pi }} v-\frac{v^2}{2}+\frac{v^3 (-6 \log (v)-9 \gamma +11+\log
   (8))}{18 \sqrt{2 \pi }}+O\left(v^4\right)$$
For large values of $v$, using $v=t^3$
$$p_t (v)=\frac{4t}{\sqrt{3}}\,e^{-\frac{3 t^2}{2}} \left(1+\frac{5}{18 t^2}-\frac{35}{648 t^4}+\frac{665}{34992
   t^6}+O\left(\frac{1}{t^8}\right)\right)$$
$$P_t (v)=1-\frac{34}{9 \sqrt{3}}e^{-\frac{3 t^2}{2}}\left(\frac{18 t^2}{17}+1-\frac{35}{612 t^2}+\frac{1925}{33048
   t^4}+O\left(\frac{1}{t^6}\right)\right)$$
