\[\int_{0}^{x}\frac{I_{0}\left(t\right)-1}{t}\,\mathrm{d}t=\frac{1}{2}\sum_{k=1}%
^{\infty}(-1)^{k-1}\frac{\psi\left(k+1\right)-\psi\left(1\right)}{k!}(\tfrac{1%
}{2}x)^{k}I_{k}\left(x\right)=\frac{2}{x}\sum_{k=0}^{\infty}(-1)^{k}(2k+3)(%
\psi\left(k+2\right)-\psi\left(1\right))I_{2k+3}\left(x\right).\]