# Integral Related to Exponential Function and Modified Bessel Function 1

I'm looking for a closed form solution for the following integral:

$$S(x) = \displaystyle \int\limits_0^x r \,I_0(r) \,e^{-b^2r^2} \,dr$$

where $I_0()$ is the $0$-th order modified Bessel function of first kind, $b>0$, and $x>0$.

Thanks.

Using the integral representation for $I_0$ and Fubini's theorem, the given integral equals
$$\frac{1}{\pi}\int_{0}^{\pi}\int_{0}^{x} r \exp\left(r\cos\theta-b^2 r^2\right)\,dr\,d\theta$$ or $$\frac{1-I_0(x)e^{-b^2 x^2}}{2b^2}+\frac{1}{4b^3\sqrt{\pi}}\int_{0}^{\pi}\cos(\theta)e^{\frac{\cos^2\theta}{4b^2}}\left[\text{Erf}\left(\frac{\cos\theta}{2b}\right)-\text{Erf}\left(bx-\frac{\cos\theta}{2b}\right)\right]\,d\theta.$$ The second term is not really elementary, but such representation provides the asymptotics for $r\to 0^+$ or $r\to +\infty$ in a simple way. Would you mind sharing the original probabilistic problem?