Numerical evaluation of an integral "similar" to an exponential integral

Question

What is an efficient and stable numerical algorithm to evaluate the integral:

$\int_0^L e^{-\alpha x}\frac{e^{\frac{i\beta}{(x+x_0)}}}{(x+x_0)}\mbox{d}x$

with $i$ the imaginary unit, $\;(\alpha,L)>0$, $\;(\beta,x_0) \in \mathbb{R}$?

Some notes. If $\alpha=0$ the integral can be evaluated in closed form in terms of the exponential integral $E_i$ function. If $x_0=0$ and $L\rightarrow \infty$ the integral can be expressed in terms of the MeijerG function. For my purposes $L$ can be safely substituted with $\infty$ if it can help.

Answer 1

Obviously, what I should do is to replace the second exponential by sine and cosine. Then, what I should prefer is an high order self adaptative Runge-Kutta method for the evaluation of each integral. This assumes that L is not infinity. You will find details in these two links

http://en.wikipedia.org/wiki/Runge-Kutta_methods#Adaptive_Runge.E2.80.93Kutta_methods http://en.wikipedia.org/wiki/List_of_Runge-Kutta_methods

You could prefer standard methods (there are really many)
http://en.wikipedia.org/wiki/Numerical_integration