what is the integral of an exponential function over the square of x?

Question

What is the analytical solution of the following integral of $x$? $$\int\frac{e^{\alpha{e}^{-(x-x_0)}}}{x^2}dx$$ where $\alpha$ and $x_{0}$ are constant.

Answer 1

You may have a series solution using $$e^y=\sum_{n=0}^\infty \frac {y^n} {n!}$$ Now, make $y=\alpha{e}^{-(x-x_0)}$ to face integrals $$I_n=\int \frac{{e}^{-n(x-x_0)}}{x^2}\,dx=-\frac{e^{ nx_0} \left(n x \text{Ei}(-n x)+e^{-n x}\right)}{x}$$ where appear the exponential integral function.