\(u=\text{erfc}\left(\frac{G}{\sqrt{t}}+H\sqrt{t}\right)\) (16)
\(dv=e^{H^{2}t}\text{\ dt}\) (17)
\(\therefore v=\int e^{H^{2}t}dt=\frac{e^{H^{2}t}}{H^{2}}\) (18)
\(\therefore\frac{\text{du}}{\text{dt}}=\frac{d}{\text{dt}}\left[\text{erfc}\left(\frac{G}{\sqrt{t}}+H\sqrt{t}\right)\right]=-\frac{2\text{\ e}^{-\left(\frac{G}{\sqrt{t}}+H\sqrt{t}\right)^{2}}\ }{\sqrt{\pi}}\ \frac{d}{\text{dt}}\left(\frac{G}{\sqrt{t}}+H\sqrt{t}\right)\) (19)
\(\frac{d}{\text{dt}}\left(\frac{G}{\sqrt{t}}+H\sqrt{t}\right)=\frac{d}{\text{dt}}\left(\text{G\ }t^{-1/2}+H\ t^{1/2}\right)=-\frac{1}{2}\text{G\ }t^{-3/2}\ +\frac{1}{2}\text{H\ t}^{-1/2}=-\frac{1}{2\ t}\left(\frac{G}{\sqrt{t}}-H\sqrt{t}\right)\) (20)
\(\therefore\frac{\text{du}}{\text{dt}}=\frac{\left(\frac{G}{\sqrt{t}}-H\sqrt{t}\right)\ e^{-\left(\frac{G}{\sqrt{t}}+H\sqrt{t}\right)^{2}}\ }{\sqrt{\pi}\text{\ t}}\) (21)
\(\therefore du=\frac{\left(\frac{G}{\sqrt{t}}-H\sqrt{t}\right)\ e^{-\left(\frac{G}{\sqrt{t}}+H\sqrt{t}\right)^{2}}\ }{\sqrt{\pi}\text{\ t}}\text{\ dt}\) (22)
\(\therefore\int_{t_{0}}^{t_{f}}{e^{B^{2}t}\ \left[\text{erfc}\left(\frac{G}{\sqrt{t}}+H\sqrt{t}\right)\right]\text{\ dt}}=\left[\text{erfc}\left(\frac{G}{\sqrt{t}}+H\sqrt{t}\right)\right]\left.\ \frac{e^{H^{2}t}}{H^{2}}\right|_{t_{0}}^{t_{f}}-\int_{t_{0}}^{t_{f}}{\frac{e^{H^{2}t}}{H^{2}}\ \left[\frac{\left(\frac{G}{\sqrt{t}}-H\sqrt{t}\right)\ e^{-\left(\frac{G}{\sqrt{t}}+H\sqrt{t}\right)^{2}}\ }{\sqrt{\pi}\text{\ t}}\right]\text{\ dt}}\)
\(=\frac{1}{H^{2}}\left[e^{H^{2}t_{f}}\text{\ erfc}\left(\frac{G}{\sqrt{t_{f}}}+H\sqrt{t_{f}}\right)-e^{H^{2}t_{0}}\text{\ erfc}\left(\frac{G}{\sqrt{t_{0}}}+H\sqrt{t_{0}}\right)\right]-\frac{1}{H^{2}\ \sqrt{\pi}}\int_{t_{0}}^{t_{f}}{\ \frac{\left(\frac{G}{\sqrt{t}}-H\sqrt{t}\right)\ e^{H^{2}t}\ e^{-\left(\frac{G}{\sqrt{t}}+H\sqrt{t}\right)^{2}}\ }{t}\text{dt}}\) (23)