integral of exponentials

Question

I have difficulties to solve the following integral:

$$ \int_{0}^{\infty}\frac{1} {x} \mathrm{e}^{-\frac{x}{a}} (\mathrm{e}^{-\mathrm{e}^{-(\frac{x-c}{b})}}) dx $$

I wanted to compute it by using Laplace Transform, but $\frac{1}{x}$ part made me confused and I could not reach to a solution.

Thank you for your help.

Answer 1

The integral won't converge, because $$\mathrm{e}^{-\frac{x}{a}} (\mathrm{e}^{-\mathrm{e}^{-(\frac{x-c}{b})}}) = \mathrm{e}^{-\mathrm{e}^{\frac{c}{b}}}+o(1)$$ Around $0$.