I'm working on algorithm that identifies clusters of objects together in space. One of the calculations in the algorithm is of the form
$$ P = \int_0^\infty e^{-ax^2}{\rm erf}\left(\frac{x}{c} + b\right)dx$$
I am seeking an analytical solution to this integral, either in the defining bounds $(0,\infty)$, or in an approximation using bounds $(0,10)$, where I know that $x=10$ happens to be a good approximation for $x\rightarrow\infty$ given the behavior of the integrand.
I have read these posts:
Closed form for definite integral involving Erf and Gaussian?
Integral of a gaussian times ("shifted") erf squared
but these have different bounds and are of a different forms (i.e., less constants). Mathematica was not able to solve the integral, but my attempt has been to differentiate under the integral sign:
$$\frac{dP}{db} = \frac{2}{\sqrt{\pi}} e^{-b^2} \int_0^\infty e^{-2bx/c}e^{-(a+1/c^2)x^2}dx.$$
I was able to reduce this to
$$\frac{dP}{db} = \frac{1}{\sqrt{a+1/c^2}} e^{-b^2\left(1-\frac{1}{ac^2+1}\right)} {\rm erfc}\left[ \frac{b}{\sqrt{ac^2+1}} \right].$$
(pending any mistakes along the way, which is definitely possible). The problem is that this integral needed to get $P(b)$ is essentially the same as the one I started with. Do you all have any recommendations on how to proceed from this step, or better suggestions overall to solve this problem?
