evaluating $\int_0^{\infty}\frac{e^{-t-\frac{x}{t}}}{t} dt$

Question

I got to this integral, while proving some theorem in statistics:

$$\int_0^\infty \frac{e^{-t-\frac{x}{t}}}{t} \mathop{dt}$$

I have trouble evaluating it. I tried partial integration, tried substitution with some polynomial and some trigonometric functions. None of them helped, and Wolfram can't compute it either. Do you have a hint on how to solve this?

Answer 1

With a little help from Maple, the integral is

$$\int_0^\infty \frac{e^{-t-\frac{x}{t}}}{t}\,dt = 2K_0(2\sqrt{x}),$$

where $K_0$ is the modified Bessel function of the second kind of order zero.

Answer 2

It worked on WolframAlpha after I hit the "Extended computation time" button: