I'm am trying to solve the following integral
$$\int\limits_{-\infty}^{+\infty}dx \; e^{-(ax+b)^2}\mathrm{Erf}(cx+d)\mathrm{Erf}(ex+f)$$
I tried the same reasoning as for these integrals that can be solved analytically but it is not as straightforward... $$\int\limits_{-\infty}^{+\infty}dx \; e^{-ax^2}\mathrm{Erf}(cx)\mathrm{Erf}(dx)=\frac{2}{\sqrt{\pi a}}\arctan(\frac{cd}{\sqrt{a(c^2+d^2+a)}})$$ $$\int\limits_{-\infty}^{+\infty}dx \; e^{-(ax+b)^2}\mathrm{Erf}(cx+d)=\frac{\sqrt{\pi}}{a}\mathrm{Erf}(\frac{ad-bc}{\sqrt{a^2+c^2}})$$
The idea is to express the Erf functions as integrals, proceed to a change of variable so that the "x" does not appear in the integration bound anymore. Then one can perform the integration on x (which is a Gaussian integration) and then on the variables of the Erf integrals.
I hope this is not too confusing...