I am trying to solve a definite integral of the form $$\int_c^\infty \frac{1}{x}e^{\Large -\frac{1}{2}(\frac{x-\mu}{\sigma})^2}dx$$ where $\mu$, $\sigma$, and $c$ are real constants and $c>0$. This integral appears nowhere in Wikipedia's list of integrals of Gaussian functions.
When $\mu=0$ the indefinite form of the integral is given by $$\int \frac{1}{x}e^{\Large -\frac{1}{2}(\frac{x}{\sigma})^2}dx = 0.5Ei(-\frac{0.5x^2}{\sigma^2})+C_1$$ where $Ei$ is the exponential integral. The solution to the definite integral is then $$\int_c^\infty \frac{1}{x}e^{\Large -\frac{1}{2}(\frac{x}{\sigma})^2}dx = 0.5 \Gamma(0, \frac{c^2}{2\sigma^2})$$
where $\Gamma$ is the incomplete Gamma function. However, Wolfram Alpha is unable to simplify the integral when $\mu\neq0$.
I also tried Wolfram Mathematica, which did not return any results after running for 2 hours:
(*Assume all expressions are real*)
$Assumptions = _ \[Element] Reals && c > 0
Integrate[1/(x)*Exp[-0.5*((x-mu)/sigma)^2], {x, c, Infinity}]
At this point I'm not sure whether there is a simplified solution. I have a computer code that needs to evaluate the integral thousands-to-millions-of-times for various values of $\mu$ and $\sigma$, so it would be nice to have a solution that can be calculated quickly and accurately.