I'm trying to solve the following integral
$$ \int_{-\infty}^{\infty} e^{-(\alpha t + \beta)^2}\operatorname{erf}(at + b)\operatorname{erf}(ct + d)\text{d}t $$
I've tried with differentiation under the integration sign and integration by parts with no success.
Thanks in advance!
This is not a complete answer, but I hope it will be helpful. I assume $\alpha, a, c$ are real and nonzero.
Let $$J(\alpha,\beta,a,b,c,d) = \int_{-\infty}^\infty e^{-(\alpha t + \beta)^2} \text{erf}(at+b)\; \text{erf}(ct+d)\; dt$$ Now $$ \dfrac{\partial J}{\partial b} = \dfrac{2}{\sqrt{\pi}} \int_{-\infty}^\infty e^{-(\alpha t + \beta)^2 - (a t + b)^2} \; \text{erf}(ct+d)\; dt $$ Write $$\eqalign{(\alpha t + \beta)^2 + (a t + b)^2 &= A^2 s^2 + r^2\cr ct + d &= cs + \delta\cr A &= \sqrt{\alpha^2 + a^2}\cr s &= t + \dfrac{\alpha \beta + a b}{\alpha^2 + a^2}\cr r &= \dfrac{\alpha b - a \beta}{\sqrt{a^2+\alpha^2}}\cr \delta &= d - c \dfrac{\alpha \beta + ab}{\alpha^2 + a^2}}$$ Then $$\dfrac{\partial J}{\partial b} = \dfrac{2}{\sqrt{\pi}} e^{-r^2} \int_{-\infty}^\infty e^{-A^2 s^2} \text{erf}(cs + \delta)\; ds = \dfrac{2}{A} e^{-r^2} \; \text{erf}\left( \dfrac{A\delta}{c^2+A^2}\right)$$ and $J$ is an antiderivative of this with respect to $b$ (note that $r$ and $\delta$ are affine functions of $b$). Unfortunately the antiderivative has no closed form AFAIK.
I worked long and hard on similar integrals (you can check my posts) and it seems that such an integral cannot be evaluated analytically. I can give you my approximations though. Hope it will help somehow.
$$ \int_{-\infty}^{\infty}\exp\!\left(-b^{2}\left(x-c\right)^{2}\right)\mathrm{erf}\Bigl(a\left(x-d\right)\Bigr)\mathrm{erf}\Bigl(a\left(x-d\right)\Bigr)\,\mathrm{d}x\approx $$ $$ \approx\frac{\sqrt{\pi}}{b}-\frac{\sqrt{\pi}}{\sqrt{b^{2}+\frac{a^{2}\pi^{2}}{8}}}\exp\!\left(-\frac{\pi^{2}a^{2}b^{2}\left(c-d\right)^{2}}{8b^{2}+a^{2}\pi^{2}}\right) $$
and
$$ \int_{-\infty}^{\infty}\exp\!\left(-b^{2}\left(x-c\right)^{2}\right)\mathrm{erf}\Bigl(b\left(x-c\right)\Bigr)\mathrm{erf}\Bigl(a\left(x-d\right)\Bigr)\,\mathrm{d}x\approx $$ $$ \approx\frac{a}{b\sqrt{\frac{4a^{2}}{\pi}+\frac{b^{2}\pi}{2}}}\exp\left(-\frac{\pi^{2}a^{2}b^{2}\left(c-d\right)^{2}}{8a^{2}+b^{2}\pi^{2}}\right) $$
for $a,b>0$.
