Closed form for $\int_{-\infty}^0\operatorname{Ei}^3x\,dx$ Let $\operatorname{Ei}x$ denote the exponential integral:
$$\operatorname{Ei}x=-\int_{-x}^\infty\frac{e^{-t}}tdt.\tag1$$
It's not difficult to find that
$$\int\operatorname{Ei}x\,dx=x\,\operatorname{Ei}x-e^x,\tag2$$
$$\int\operatorname{Ei}^2x\,dx=x\,\operatorname{Ei}^2x-2\,e^x\operatorname{Ei}x+2\,\operatorname{Ei}(2x)\tag3$$
and
$$\int_{-\infty}^0\operatorname{Ei}x\,dx=-1,\tag4$$
$$\int_{-\infty}^0\operatorname{Ei}^2x\,dx=\ln4.\tag5$$

Is it possible generalize these results for higher powers of $\operatorname{Ei}x$?
In particular, are there closed forms for
$$\int\operatorname{Ei}^3x\,dx\tag6$$
or
$$\int_{-\infty}^0\operatorname{Ei}^3x\,dx\ ?\tag7$$

$$\begin{align}\int_0^\infty\operatorname{Ei}^3(-x)\,dx&=-3\operatorname{Li}_2\left(\frac14\right)-6\ln^22.\\\\\int_0^\infty\operatorname{Ei}^4(-x)\,dx&=24\operatorname{Li}_3\left(\frac14\right)-48\operatorname{Li}_2\left(\frac13\right)\ln2-13\,\zeta(3)\\&-32\ln^32+48\ln^22\,\ln3-24\ln2\,\ln^23+6\pi^2\ln2.\end{align}$$
