How to rewrite this integral that involving the Meijer-G function (a very general hypergeometric function) as other special function for example maybe Fox-H,Bessel or $Li(x)$...
$$\int\limits_{x = 0}^{x = \infty } {\frac{1}{x}\exp \left( { - \frac{1}{x} - x} \right)G_{1,3}^{3,1}\left( {x*a\left| {\begin{array}{*{20}{c}} 1 \\ {1,2,2} \end{array}} \right.} \right)dx} $$
Notice that $a$ is a positive real number. My first thought is that if we could some how make use of the parameter of the meijer G function $MeijerG[{{1},{}},{{1,2,2},{}},x]$ then this could help simplify the problem. An example would be $G_{2,1}^{1,2}\left( {z\left| {\begin{array}{*{20}{c}} {1,1} \\ {1,1} \end{array}} \right.} \right) = \frac{z}{{z + 1}}$
Thank you for your enthusiasm !