\[\int_{0}^{\infty}e^{-at}\mathbf{H}_{1}\left(t\right)\,\mathrm{d}t=\frac{2}{\pi
a%
}-\frac{2a}{\pi\sqrt{1+a^{2}}}\ln\left(\frac{1+\sqrt{1+a^{2}}}{a}\right),\]