This integral comes from a toy model I was interested in, from statistical mechanics. Essentially, it involves solving an integral:
$$ \int_{-\infty}^\infty \varsigma^{-1} \exp(-\beta x)\exp\left(-\left|\frac{x}{\varsigma}\right|^m\right)\,dx $$
Where $\beta>0$, $\varsigma >0$ and $m\geq 1$. Note that $m=1$ is a special case. For specific integers values of $m$, say $m=1, 2, ...$, I am able to find a solution. However, I'd like to understand general features of this system with $m$ and this would require a better solution than simply enumerating values of $m$. Is there a known general form for this integral ? I tried looking in Gradshteyn & Ryzhik and in the Erdelyi, but didn't find it.
remark
Write $$ F(n,\beta,\zeta) = \int_{-\infty}^\infty \zeta^{-1} \exp(-\beta x)\exp\left(-\left|\frac{x}{\zeta}\right|^m\right)\,dx $$ I used zeta $\zeta$ instead of varsigma $\varsigma$. Then $$ F(m,\beta,\zeta) = \int_0^\infty \frac{e^{\beta x}+e^{-\beta x}}{\zeta} \exp\left(-\left(\frac{x}{\zeta}\right)^m\right) $$ change variables $y=x/\zeta$, $$ F(m,\beta,\zeta) = \int_0^\infty (e^{\beta\zeta y} + e^{-\beta\zeta y}) e^{-y^m}\;dy = \beta\zeta F(m,\beta\zeta,1) $$ So it suffices to do the case $\zeta=1$.
As noted it can be done in individual cases $m=1,2,3,\dots$. It seems that $F(m,b,1)$ is a linear combination of terms like $$ b^k\;{}_0F_{m-2}\left(;\cdots;\frac{b^m}{m^m}\right); $$ where $\cdots$ represents $m-2$ rationals with denominator $m$.