How to solve $\int_{a}^{b}\frac{r}{y+r} (\text{erfc}\left(\frac{y}{\sqrt{4G(t+T)}}\right)-\text{erfc}\left(\frac{y}{\sqrt{4Gt}}\right))dy$?

Question

I have the following definite integral:

[Edit: I wrote the incorrect integrand] $$\int_{a}^{b}\frac{r}{y+r} \left(\text{erfc}\left(\frac{y}{\sqrt{4G(t+T)}}\right)-\text{erfc}\left(\frac{y}{\sqrt{4Gt}}\right) \right)dy, $$

where $r,G,\text{and}\ T$ are constants and positive, and $\text{erfc}(\cdot)$ is the complementary error function. $t$ is time and it can be considered to be a constant while solving the integral.

I think I have to apply integration by parts, but I honestly have no clue how to proceed.

How can I solve this integral? Is it even possible to obtain a closed-form solution?

Thanks in advance.

Answer 1

Simplifying the notation, you want to compute $$I=\int \frac{\text{erf}(\alpha y)}{y+r}\,dy=\int \frac{\text{erf}(x)}{x+\beta}\,dx\qquad\qquad\qquad (\beta =\alpha r)$$

Even if $\beta=0$, the result is quite complex $$\int \frac{\text{erf}(x)}{x}\,dx=\frac{2 }{\sqrt{\pi}}\,x \,\,\, _2F_2\left(\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};-x^2\right)$$ If $\beta$ is small, you could use $$\int\frac{\text{erf}(x)}{x+\beta}\,dx=\sum_{n=0}^\infty (-1)^n \beta^n \int x^{-(n+1)}\,\text{erf}(x) \,dx$$ and, for $n>1$ $$\int x^{-(n+1)}\,\text{erf}(x) \,dx=\frac 1{1-n}\left(x^{1-n}\,\text{erf}(x) +\frac{1}{\sqrt{\pi }}\Gamma \left(1-\frac{n}{2},x^2\right)\right)$$

It is probably reasonable to stay with numerical integration.