Computing a nasty integral (probably with computer algebra system) I'm trying to do this integral, not sure if it is possible:
$$
\int_{1}^{\infty}\int_{0}^{\infty}
\exp\left(\, -\,{x^{2} \over y^{2}}\,\right)
\exp\left(\,-\,{y^{2} \over z^{2}}\,\right)
\exp\left(\, -\left[\,z-1-\log\left(\, z\,\right)\,\right]\,\right)
\,{\rm d}y\,{\rm d}z
$$
I am happy with computer algebra solutions if they exist. So far I have had some issues with Maple, which just returns back the original integral.

Hint. Your integral may be reduced to a generalized incomplete gamma function that has been previously studied. 


*

*Step 1. Integrating with respect to $y$ gives


$$
\begin{align}
\int_0^\infty \exp\left(-\frac{y^2}{z^2}-\frac{x^2}{y^2}\right)dy&=\exp{\left(-\frac{2 x}{ z}\right)}\int_0^\infty \exp{-\left(\frac{y}{z}-\frac{x}{y}\right)^2}dy\\\\
&=z\:\exp{\left(-\frac{2 x}{ z}\right)}\int_0^\infty \exp{(-Y^2)}dY\\\\
&=\frac{\sqrt{\pi}}{2}z\:\exp{\left(-\frac{2 x}{ z}\right)}\\\\
\end{align}
$$


*

*Step 2. Integrating with respect to $z$ gives, for $x\geq 0$:


$$
\begin{align}
&\int_1^\infty \frac{\sqrt{\pi}}{2}z\:\exp{\left(-\frac{2 x}{ z}\right)}\cdot\exp(-(z-1-\log(z)))dz\\\\&=\frac{e\:\sqrt{\pi}}{2} \int_1^\infty z^2\:\exp{\left(-z-\frac{2 x}{ z}\right)} dz\\\\
&=\frac{e\:\sqrt{\pi}}{2} \Gamma (3,1;2x)
\end{align}
$$ where $\Gamma (\nu,x;b)$ is a generalized incomplete gamma function whose some properties have been considered here (M.A. Chaudhry, N.M. Temme and E.J.M. Veling) (click on the free pdf) or here (Frank E. Harris). See also various references at the end of these papers.
I'm interested in knowing the context of this integral.
