\(\frac{e^{2\ G\ H}}{H^{2}\ \sqrt{\pi}}\int_{t_{0}}^{t_{f}}{\frac{1}{t}\ \left(\frac{G}{\sqrt{t}}-H\sqrt{t}\right)\ e^{H^{2}t-\left(\frac{G}{\sqrt{t}}+H\sqrt{t}\right)^{2}}\text{\ dt}}=\frac{e^{2\ G\ H}}{H^{2}\ \sqrt{\pi}}\int_{\frac{\text{G\ }}{\sqrt{t_{0}}}}^{\frac{\text{G\ }}{\sqrt{t_{f}}}}{\left(\frac{w}{G}\right)^{2}\ \left(w-\frac{\text{G\ H}}{w}\right)e^{\left(G\ H/w\right)^{2}-\left(w+G\ H/w\right)^{2}}\ \left(-\frac{2G^{2}}{w^{3}}\right)\text{\ dw}}\)
\(=-\frac{2e^{2\ G\ H}}{H^{2}\ \sqrt{\pi}}\int_{\frac{\text{G\ }}{\sqrt{t_{0}}}}^{\frac{\text{G\ }}{\sqrt{t_{f}}}}{\left(1-\frac{\text{G\ H}}{w^{2}}\right)\ e^{\left(G\ H/w\right)^{2}-\left(w+G\ H/w\right)^{2}}\text{\ dw}}\)
\(=-\frac{2e^{2\ G\ H}}{H^{2}\ \sqrt{\pi}}\int_{\frac{\text{G\ }}{\sqrt{t_{0}}}}^{\frac{\text{G\ }}{\sqrt{t_{f}}}}{\left(1-\frac{\text{G\ H}}{w^{2}}\right)\ e^{\left(G\ H/w\right)^{2}-{w^{2}-2\ G\ H-\left(G\ H/w\right)}^{2}}\text{\ dw}}\)
\(=-\frac{2e^{2\ G\ H}}{H^{2}\ \sqrt{\pi}}\int_{\frac{\text{G\ }}{\sqrt{t_{0}}}}^{\frac{\text{G\ }}{\sqrt{t_{f}}}}{\left(1-\frac{\text{G\ H}}{w^{2}}\right)\ e^{-w^{2}-2\ G\ H}\text{\ dw}}\)
\(=-\frac{2}{H^{2}\ \sqrt{\pi}}\int_{\frac{\text{G\ }}{\sqrt{t_{0}}}}^{\frac{\text{G\ }}{\sqrt{t_{f}}}}{\left(1-\frac{\text{G\ H}}{w^{2}}\right)\ e^{-w^{2}}\text{\ dw}}\) (33)