While studying the communication signal processing, it was proved that the following integrals play an important role in the block error rate approximation of the MPSK signals: \begin{equation} \int_{0}^{\infty}\exp(-ax^{2}+bx)\mathrm{erfc}(x)dx， \end{equation} \begin{equation} \int_{-\infty}^{0}\exp(-ax^{2}+bx)\mathrm{erfc}(x)dx， \end{equation} where $\mathrm{erfc}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{\infty}e^{-x^2}dx$ is the complementary error function.
It is a famous result that $$\int_{-\infty}^{\infty}\exp(-ax^{2}+bx)\mathrm{erfc}(x)dx=\sqrt{\frac{\pi}{a}}\exp({\frac{b^{2}}{4a}})\mathrm{erfc}(\frac{b}{\sqrt{a(a+1)}})$$ And I use mathematica to calculte the above two integrals, it gives me the following results while I could not understand how to get them through some substuition or some integral skills? \begin{equation} \int_{0}^{\infty}\exp(-ax^{2}+bx)\mathrm{erfc}(x)dx=\frac{\mathrm{erfc}(2\sqrt{a}+be^{\frac{b^{2}}{4a}}\sqrt{\pi}(1+\mathrm{erf}(\frac{b}{2\sqrt{a}})))}{4a^{3/2}}, \end{equation} if $$Re[a]=0 \quad \& \quad Re[b]<0 \quad or \quad Re[a]>0,$$ where $Re[\cdot]$ is the real part of a real number. \begin{equation} \int_{-\infty}^{0}\exp(-ax^{2}+bx)\mathrm{erfc}(x)dx=\frac{\mathrm{erfc}(-2\sqrt{a}+be^{\frac{b^{2}}{4a}}\sqrt{\pi}\mathrm{erfc}(\frac{b}{2\sqrt{a}}))}{4a^{3/2}}, \end{equation} provided that $$Re[a]=0 \quad \& \quad Re[b]>0 \quad or \quad Re[a]>0.$$
I would be extremely grateful if anyone could give me some advice. Have a nice and pleasure day!
@ Przemo If it is convenient for you to give me some slight suggestions about how to calculate, thanks a lot for your generous help ~