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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...

Alexandre Krajenbrink
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  • Such itegrals were nightmare in my research, and to the best of my knowledge and experience this one cannot be evaluated analytically. (though I'd be happy to learn that I'm wrong). – petru Oct 27 '16 at 19:18
  • The answer is expressed through the Owen's T function and is given here https://math.stackexchange.com/questions/3183848/a-definite-integral-involving-a-gaussian-and-shifted-error-functions?rq=1 . – Przemo Apr 16 '19 at 11:03

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