Integrating a function involved error functions and a Gaussian kernel

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I am currently facing a problem of integrating
exp(-ax^2)*erf(bx+c)*erf(dx+f) with the integral boundaries 0 and
infinity.

I have gone through some handbooks but what I could locate is the integration of exp(-ax^2)*erf(bx)*erf(cx) from 0 to infinity which yields arctan(b*c/sqrt(a*(b^2+c^2+a)))/sqrt(a*pi).

I would really appreciate if anyone could guide me through solving this problem. Thanks.
 
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I was trying perform sort of educated guess based on arctan(b*c/sqrt(a*(b^2+c^2+a)))/sqrt(a*pi) - solution for the integration of exp(-ax^2)*erf(bx)*erf(cx) from 0 to infinity but no luck, since the one I need is a translated version for the above.

I am wondering is that any systematic back tracking procedure for that?
 
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