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.

As user mickep already pointed out in the comment section, the integral can be rewritten as 
$\dfrac{\sqrt\pi}2~e~I''(1)$, where $I(a)=\displaystyle\int_1^{+\infty}\exp\bigg(\!\!-az-\frac{2x}z~\bigg)~dz$. Now, if the lower integration limit 
would have been $0$ instead of $1$, then we'd have $J(a)=2~\sqrt{\dfrac{2x}a}~K_1\Big(2\sqrt{2ax}\Big)$, which, after 
being twice differentiated with regard to a, yields $J''(1)=4x\sqrt{2x}~K_3\Big(2\sqrt{2x}\Big)$. As it stands, 
however, we'd need incomplete Bessel functions to express $I(a)$, as already hinted at by Prof. 
Olivier Oloa in his answer. Unfortunately, there is no universally accepted notational con- 
vention regarding such functions.
