\[\int_{0}^{z}\operatorname{Gi}\left(t\right)\,\mathrm{d}t\sim\frac{1}{\pi}\ln z%
+\frac{2\gamma+\ln 3}{3\pi}-\frac{1}{\pi}\sum_{k=1}^{\infty}\frac{(3k-1)!}{k!(%
3z^{3})^{k}},\]