Let $\alpha>1$. I would like to find a closed form or an upper bound of
$$f\left(\alpha\right)=\int_{e}^{\infty}\frac{t^{1/2}}{\log^{1/2}\left(t\right)}\alpha^{-t/\log\left(t\right)}dt.$$
For the closed form I'm very skeptical but I have trouble also for an upper bound. I tried, manipulating a bit, to integrate w.r.t. $\alpha$ since $$\frac{\partial}{\partial\alpha}\alpha^{-t/\log\left(t\right)}=-\frac{t}{\alpha\log\left(t\right)}\alpha^{-t/\log\left(t\right)}$$ but it seems quite useless and at this moment I didn't see a good way to proceed. Maybe it is interesting to see, using some trivial substitutions, that $$f\left(\alpha\right)=\int_{e}^{\infty}\frac{\left(e^{3/2}\right)^{-W_{-1}\left(-1/v\right)}}{v\left(-W_{-1}\left(-\frac{1}{v}\right)\right){}^{1/2}}\frac{W_{-1}\left(-\frac{1}{v}\right)}{W_{-1}\left(-\frac{1}{v}\right)+1}\alpha^{-v}dv$$ $$=\int_{e}^{\infty}g\left(w\right)\alpha^{-v}dv$$ where $W_{-1}\left(x\right)$ is the Lambert $W$ function. So it seems that $f(\alpha)$ is somehow connected to the Mellin transform of $g(w).$
Thank you.