I am trying to estimate the growth in time of a certain quantity. In order to do this, I have to evaluate the following integral:
$\int_0^{+\infty} \exp(-b x)\exp(-c\,t \exp(-a x \log x)) dx$
with $a,b$ and $c$ positive. I only need to know what happens at large $t$. Any ideas?
I have tried to use a saddle point approximation, but failed miserably. Perhaps I am doing something silly...