I am trying to see if I can have a closed form solution for the following integrals (I give two forms in case one of them is more usefull than the other).
$$I_1 = \int_{0}^{t} e^{-(a + b y)^{2}} \mathrm{erf}(a-by) \; dy \sim \int_{a-b t}^{a} e^{-(2a - y)^{2}} \mathrm{erf}(y) dy$$ $$I_2 = \int_{0}^{t} y \; e^{-(a + b y)^{2}} \mathrm{erf}(a-by) \; dy \sim \int_{a-b t}^{a} (a-y) e^{-(2a - y)^{2}} \mathrm{erf}(y) \;dy$$ where $a,b>0$.
I have looked the nist table of integrals, the book of Abramowitz and Stegun and of Gradshteyn and Ryzhik, but I can find something useful for definite integrals. The only slightly relevant integrals I found are: $$\int_{0}^{p} e^{-x^{2}}\mathrm{erf}(p-x) = \sqrt{\pi}(\mathrm{erf}(p/\sqrt{2}))^{2}/2$$ $$\int_{0}^{p} e^{-x^{2}}\mathrm{erf}(x) = \sqrt{\pi}(\mathrm{erf}(p))^{2}/4$$ But the integrals (A) and (B) are not the same.
Do you have an idea of how could they be solved anallytically (if they can be solved)?
Thanks!