Here I have an integral derived from a heat conduction problem
$$f(r)=r*\int_0^H \frac{e^{-(r^2+h^2)} \ \ \ \ + \frac{erfc(\sqrt{\ r^2+h^2} \ \ \ )}{ \sqrt{\ r^2+h^2}}}{r^2+h^2}dh$$
I solved this one numerically and found out a regression like $f(r)=\frac{a}{r}+b$ for $H>>r$, where $a$ and $b$ are functions of $r$ and $H$. I tried to reach an analytical expression for $a$ and $b$, but can't solve the integral properly.
Do you have any suggestion?
$$\int_0^H \frac{e^{-(r^2+h^2)} + \frac{\operatorname{erfc}\sqrt{\ r^2+h^2}}{ \sqrt{\ r^2+h^2}}}{r^2+h^2}dh \,\stackrel{u=r^2+h^2}{=} \,\int_{a}^{a+b} \frac{e^{-u} + \frac{\operatorname{erfc}\sqrt u}{\sqrt u}}{2u}\frac{du}{\sqrt{u-a}}$$ Where we have simply let $a=r^2$ and $b=H^2$
We now assume that $b \gg a$, and notice that our integral decays extremely quickly. We can thus estimate this well as $$\int_a^{\infty} \frac{e^{-u} + \frac{\operatorname{erfc}\sqrt u}{\sqrt u}}{2u\sqrt{u-a}}du \;\stackrel{\text{Mathematica}}{=}\; \frac{\left(\pi- 2\pi^{1/2}\right)\operatorname{erfc}\left(a^{1/2}\right)}{2a^{1/2}}+\frac{e^{-a}}{a}$$