\[\int_{x}^{\infty}\frac{K_{0}\left(t\right)}{t}\,\mathrm{d}t=\frac{1}{2}\left(%
\ln\left(\tfrac{1}{2}x\right)+\gamma\right)^{2}+\frac{\pi^{2}}{24}-\sum_{k=1}^%
{\infty}\left(\psi\left(k+1\right)+\frac{1}{2k}-\ln\left(\tfrac{1}{2}x\right)%
\right)\frac{(\tfrac{1}{2}x)^{2k}}{2k(k!)^{2}},\]