One of the ways of calculating the integral in closed form is to think of crafitly using the geometric series, but even so it seems evil enough.
$$\int_0^1\int_0^1\int_0^1\int_0^1\frac{1}{(1+x) (1+y) (1+z)(1+w) (1+ x y z w)} \ dx \ dy \ dz \ dw$$
Maybe you can guide me, bless me with another precious hints, clues. Thanks MSE users!
Supplementary question: How about the generalization?
$$\int_0^1\int_0^1\cdots\int_0^1\frac{1}{(1+x_1) (1+x_2)\cdots (1+x_n)(1+ x_1 x_2 \cdots x_n)} \ dx_1 \ dx_2 \cdots \ dx_n$$
This response will only address the $n=4$ case,
$$I_{4}:=\int_{[0,1]^{4}}\frac{\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\,\mathrm{d}w}{\left(1+x\right)\left(1+y\right)\left(1+z\right)\left(1+w\right)\left(1+xyzw\right)}.\tag{1}$$
According to WolframAlpha, the multiple integral $(1)$ above has the approximate numerical value $I_{4}\approx0.223076.$
Starting with the substitution $w=\frac{1-t}{1+xyzt}$, we can whittle the multiple integral down to the following double integral:
$$\begin{align} I_{4} &=\small{\int_{0}^{1}\mathrm{d}x\int_{0}^{1}\mathrm{d}y\int_{0}^{1}\mathrm{d}z\int_{0}^{1}\frac{\mathrm{d}w}{\left(1+x\right)\left(1+y\right)\left(1+z\right)\left(1+w\right)\left(1+xyzw\right)}}\\ &=\small{\int_{0}^{1}\mathrm{d}x\int_{0}^{1}\mathrm{d}y\int_{0}^{1}\mathrm{d}z\int_{0}^{1}\frac{\mathrm{d}t}{\left(1+x\right)\left(1+y\right)\left(1+z\right)\left(2-t+xyzt\right)}}\\ &=\int_{0}^{1}\mathrm{d}x\int_{0}^{1}\mathrm{d}y\int_{0}^{1}\mathrm{d}z\,\frac{\ln{(2)}-\ln{\left(1+xyz\right)}}{\left(1+x\right)\left(1+y\right)\left(1+z\right)\left(1-xyz\right)}\\ &=\int_{0}^{1}\mathrm{d}x\int_{0}^{1}\mathrm{d}y\int_{0}^{xy}\mathrm{d}v\,\frac{\ln{\left(\frac{2}{1+v}\right)}}{\left(1+x\right)\left(1+y\right)\left(xy+v\right)\left(1-v\right)};~~~\small{\left[xyz=v\right]}\\ &=\int_{0}^{1}\mathrm{d}x\int_{0}^{x}\mathrm{d}u\int_{0}^{u}\mathrm{d}v\,\frac{\ln{\left(\frac{2}{1+v}\right)}}{\left(1+x\right)\left(x+u\right)\left(u+v\right)\left(1-v\right)};~~~\small{\left[xy=u\right]}\\ &=\int_{0}^{1}\mathrm{d}x\int_{0}^{x}\mathrm{d}v\int_{v}^{x}\mathrm{d}u\,\frac{\ln{\left(\frac{2}{1+v}\right)}}{\left(1+x\right)\left(x+u\right)\left(u+v\right)\left(1-v\right)}\\ &=\int_{0}^{1}\mathrm{d}v\int_{v}^{1}\mathrm{d}x\int_{v}^{x}\mathrm{d}u\,\frac{\ln{\left(\frac{2}{1+v}\right)}}{\left(1+x\right)\left(x+u\right)\left(u+v\right)\left(1-v\right)}\\ &=\int_{0}^{1}\mathrm{d}v\int_{v}^{1}\mathrm{d}u\int_{u}^{1}\mathrm{d}x\,\frac{\ln{\left(\frac{2}{1+v}\right)}}{\left(1+x\right)\left(x+u\right)\left(u+v\right)\left(1-v\right)}\\ &=\int_{0}^{1}\mathrm{d}v\int_{v}^{1}\mathrm{d}u\,\frac{\ln{\left(\frac{(1+u)^2}{4u}\right)}\ln{\left(\frac{2}{1+v}\right)}}{\left(1-u\right)\left(u+v\right)\left(1-v\right)}\\ &=\int_{0}^{1}\mathrm{d}u\int_{0}^{u}\mathrm{d}v\,\frac{\ln{\left(\frac{(1+u)^2}{4u}\right)}\ln{\left(\frac{2}{1+v}\right)}}{\left(1-u\right)\left(u+v\right)\left(1-v\right)}.\tag{2}\\ \end{align}$$
WolframAlpha's numerical approximation of the iterated integral obtained in the last line of $(2)$ is consistent with the original approximation stated above, so I am reasonably confident that I haven't made any errors so far.
Continuing, transforming variables and changing the order of integration yields the following equivalent double integral representation of $I_{4}$:
$$\begin{align} I_{4} &=\int_{0}^{1}\mathrm{d}u\int_{0}^{u}\mathrm{d}v\,\frac{\ln{\left(\frac{(1+u)^2}{4u}\right)}\ln{\left(\frac{2}{1+v}\right)}}{\left(1-u\right)\left(u+v\right)\left(1-v\right)}\\ &=\int_{0}^{1}\mathrm{d}u\int_{\frac{1-u}{1+u}}^{1}\mathrm{d}y\,\frac{\ln{\left(\frac{(1+u)^2}{4u}\right)}\ln{\left(1+y\right)}}{\left(1-u\right)\left(u+\frac{1-y}{1+y}\right)y\left(1+y\right)};~~~\small{\left[\frac{1-v}{1+v}=y\right]}\\ &=-\frac12\int_{0}^{1}\mathrm{d}x\int_{x}^{1}\mathrm{d}y\,\frac{\ln{\left(1-x^2\right)}\ln{\left(1+y\right)}}{xy\left(1-xy\right)};~~~\small{\left[\frac{1-u}{1+u}=x\right]}\\ &=-\frac12\int_{0}^{1}\mathrm{d}y\int_{0}^{y}\mathrm{d}x\,\frac{\ln{\left(1-x^2\right)}\ln{\left(1+y\right)}}{xy\left(1-xy\right)}.\tag{3}\\ \end{align}$$
Now, the dilogarithm function $\operatorname{Li}_{2}{\left(z\right)}$ for complex argument is traditionally defined via the integral representation
$$\operatorname{Li}_{2}{\left(z\right)}:=-\int_{0}^{z}\frac{\ln{\left(1-t\right)}}{t}\,\mathrm{d}t;~~~\small{z\in\mathbb{C}\setminus(1,\infty)}.\tag{4}$$
The following indefinite integral may then be confirmed by differentiated both sides of the equation:
$$\small{\int\frac{\ln{\left(c+dx\right)}}{a+bx}\,\mathrm{d}x=\frac{\operatorname{Li}_{2}{\left(\frac{b\left(c+dx\right)}{bc-ad}\right)}+\ln{\left(c+dx\right)}\ln{\left(\frac{d\left(a+bx\right)}{ad-bc}\right)}}{b}+\color{grey}{constant}.}\tag{5}$$
Next, splitting up the logarithm function of $x$ in the numerator and applying partial fraction decomposition to the rational part, we find
$$\begin{align} I_{4} &=-\frac12\int_{0}^{1}\mathrm{d}y\int_{0}^{y}\mathrm{d}x\,\frac{\ln{\left(1-x^2\right)}\ln{\left(1+y\right)}}{xy\left(1-xy\right)}\\ &=-\frac12\int_{0}^{1}\mathrm{d}y\int_{0}^{y}\mathrm{d}x\,\frac{\ln{\left(1+x\right)}\ln{\left(1+y\right)}}{xy\left(1-xy\right)}\\ &~~~~~-\frac12\int_{0}^{1}\mathrm{d}y\int_{0}^{y}\mathrm{d}x\,\frac{\ln{\left(1-x\right)}\ln{\left(1+y\right)}}{xy\left(1-xy\right)}\\ &=-\frac12\int_{0}^{1}\mathrm{d}y\,\ln{\left(1+y\right)}\int_{0}^{y}\mathrm{d}x\,\left[\frac{1}{1-xy}+\frac{1}{xy}\right]\ln{\left(1+x\right)}\\ &~~~~~-\frac12\int_{0}^{1}\mathrm{d}y\,\ln{\left(1+y\right)}\int_{0}^{y}\mathrm{d}x\,\left[\frac{1}{1-xy}+\frac{1}{xy}\right]\ln{\left(1-x\right)}\\ &=-\frac12\int_{0}^{1}\mathrm{d}y\,\ln{\left(1+y\right)}\int_{0}^{y}\mathrm{d}x\,\frac{\ln{\left(1+x\right)}}{1-xy}\\ &~~~~~-\frac12\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1+y\right)}}{y}\int_{0}^{y}\mathrm{d}x\,\frac{\ln{\left(1+x\right)}}{x}\\ &~~~~~-\frac12\int_{0}^{1}\mathrm{d}y\,\ln{\left(1+y\right)}\int_{0}^{y}\mathrm{d}x\,\frac{\ln{\left(1-x\right)}}{1-xy}\\ &~~~~~-\frac12\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1+y\right)}}{y}\int_{0}^{y}\mathrm{d}x\,\frac{\ln{\left(1-x\right)}}{x}\\ &=\frac12\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1+y\right)}}{y}\left[-\int_{0}^{y}\mathrm{d}x\,\frac{y\ln{\left(1+x\right)}}{1-xy}\right]\\ &~~~~~+\frac12\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1+y\right)}\operatorname{Li}_{2}{\left(-y\right)}}{y}\\ &~~~~~-\frac12\int_{0}^{1}\mathrm{d}y\,\ln{\left(1+y\right)}\int_{1-y}^{1}\mathrm{d}t\,\frac{\ln{\left(t\right)}}{1-y\left(1-t\right)};~~~\small{\left[1-x=t\right]}\\ &~~~~~+\frac12\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1+y\right)}\operatorname{Li}_{2}{\left(y\right)}}{y}\\ &=\frac12\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1+y\right)}}{y}\left[\operatorname{Li}_{2}{\left(y\right)}+\ln{\left(1-y\right)}\ln{\left(1+y\right)}-\operatorname{Li}_{2}{\left(\frac{y}{1+y}\right)}\right]\\ &~~~~~-\frac12\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1+y\right)}}{y}\int_{1-y}^{1}\mathrm{d}t\,\frac{\left(\frac{y}{1-y}\right)\ln{\left(t\right)}}{1+\left(\frac{y}{1-y}\right)t}\\ &~~~~~+\frac12\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1+y\right)}\operatorname{Li}_{2}{\left(-y\right)}}{y}+\frac12\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1+y\right)}\operatorname{Li}_{2}{\left(y\right)}}{y}\\ &=\small{\frac12\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1+y\right)}}{y}\left[\operatorname{Li}_{2}{\left(y\right)}+\ln{\left(1-y\right)}\ln{\left(1+y\right)}+\operatorname{Li}_{2}{\left(-y\right)}+\frac12\ln^{2}{\left(1+y\right)}\right]}\\ &~~~~~\small{-\frac12\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1+y\right)}}{y}\left[\operatorname{Li}_{2}{\left(\frac{y}{y-1}\right)}-\operatorname{Li}_{2}{\left(-y\right)}-\ln{\left(1-y\right)}\ln{\left(1+y\right)}\right]}\\ &~~~~~+\frac12\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1+y\right)}\operatorname{Li}_{2}{\left(-y\right)}}{y}+\frac12\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1+y\right)}\operatorname{Li}_{2}{\left(y\right)}}{y}\\ &=\frac12\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1+y\right)}}{y}\left[\operatorname{Li}_{2}{\left(y\right)}+\operatorname{Li}_{2}{\left(-y\right)}+\ln{\left(1-y\right)}\ln{\left(1+y\right)}\right]\\ &~~~~~+\frac14\int_{0}^{1}\mathrm{d}y\,\frac{\ln^{3}{\left(1+y\right)}}{y}\\ &~~~~~\small{+\frac12\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1+y\right)}}{y}\left[\operatorname{Li}_{2}{\left(y\right)}+\frac12\ln^{2}{\left(1-y\right)}+\operatorname{Li}_{2}{\left(-y\right)}+\ln{\left(1-y\right)}\ln{\left(1+y\right)}\right]}\\ &~~~~~+\frac12\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1+y\right)}\operatorname{Li}_{2}{\left(-y\right)}}{y}+\frac12\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1+y\right)}\operatorname{Li}_{2}{\left(y\right)}}{y}\\ &=\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1+y\right)}}{y}\left[\operatorname{Li}_{2}{\left(y\right)}+\operatorname{Li}_{2}{\left(-y\right)}+\ln{\left(1-y\right)}\ln{\left(1+y\right)}\right]\\ &~~~~~+\frac14\int_{0}^{1}\mathrm{d}y\,\frac{\ln^{3}{\left(1+y\right)}}{y}+\frac14\int_{0}^{1}\mathrm{d}y\,\frac{\ln^{2}{\left(1-y\right)}\ln{\left(1+y\right)}}{y}\\ &~~~~~+\frac12\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1+y\right)}\operatorname{Li}_{2}{\left(-y\right)}}{y}+\frac12\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1+y\right)}\operatorname{Li}_{2}{\left(y\right)}}{y}\\ &=\frac32\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1+y\right)}\operatorname{Li}_{2}{\left(y\right)}}{y}+\frac32\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1+y\right)}\operatorname{Li}_{2}{\left(-y\right)}}{y}\\ &~~~~~+\frac14\int_{0}^{1}\mathrm{d}y\,\frac{\ln^{3}{\left(1+y\right)}}{y}+\frac14\int_{0}^{1}\mathrm{d}y\,\frac{\ln^{2}{\left(1-y\right)}\ln{\left(1+y\right)}}{y}\\ &~~~~~+\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1-y\right)}\ln^{2}{\left(1+y\right)}}{y}.\tag{6}\\ \end{align}$$
And so we have reduced our multiple integral to a sum of five single-variable polylogarithmic integrals. Instead of attempting to evaluate each of these in turn, we'll save much energy if we make a few rearrangements first.
$$\begin{align} I_{4} &=\frac32\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1+y\right)}\operatorname{Li}_{2}{\left(y\right)}}{y}+\frac32\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1+y\right)}\operatorname{Li}_{2}{\left(-y\right)}}{y}\\ &~~~~~+\frac14\int_{0}^{1}\mathrm{d}y\,\frac{\ln^{3}{\left(1+y\right)}}{y}+\frac14\int_{0}^{1}\mathrm{d}y\,\frac{\ln^{2}{\left(1-y\right)}\ln{\left(1+y\right)}}{y}\\ &~~~~~+\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1-y\right)}\ln^{2}{\left(1+y\right)}}{y}\\ &=\frac32\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1+y\right)}\operatorname{Li}_{2}{\left(-y\right)}}{y}+\frac32\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1+y\right)}\operatorname{Li}_{2}{\left(y\right)}}{y}\\ &~~~~~+\frac14\int_{0}^{1}\mathrm{d}y\,\frac{\ln^{3}{\left(1+y\right)}}{y}+\frac14\int_{0}^{1}\mathrm{d}y\,\frac{\ln^{2}{\left(1-y\right)}\ln{\left(1+y\right)}}{y}\\ &~~~~~\small{+\int_{0}^{1}\mathrm{d}y\,\frac{\ln^{3}{\left(1-y^2\right)}-\ln^{3}{\left(1-y\right)}-\ln^{3}{\left(1+y\right)}-3\ln^{2}{\left(1-y\right)}\ln{\left(1+y\right)}}{3y}}\\ &=-\frac34\int_{0}^{1}\mathrm{d}y\,\frac{(-2)\ln{\left(1+y\right)}\operatorname{Li}_{2}{\left(-y\right)}}{y}+\frac32\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1+y\right)}\operatorname{Li}_{2}{\left(y\right)}}{y}\\ &~~~~~-\frac13\int_{0}^{1}\mathrm{d}y\,\frac{\ln^{3}{\left(1-y\right)}}{y}-\frac{1}{12}\int_{0}^{1}\mathrm{d}y\,\frac{\ln^{3}{\left(1+y\right)}}{y}\\ &~~~~~+\frac13\int_{0}^{1}\mathrm{d}y\,\frac{\ln^{3}{\left(1-y^2\right)}}{y}-\frac34\int_{0}^{1}\mathrm{d}y\,\frac{\ln^{2}{\left(1-y\right)}\ln{\left(1+y\right)}}{y}\\ &=-\frac34\left[\operatorname{Li}_{2}{\left(-y\right)}^{2}\right]_{0}^{1}+\frac32\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1+y\right)}\operatorname{Li}_{2}{\left(y\right)}}{y}\\ &~~~~~\small{-\frac13\int_{0}^{1}\mathrm{d}y\,\frac{\ln^{3}{\left(1-y\right)}}{y}-\frac{1}{12}\int_{0}^{1}\mathrm{d}y\,\frac{\ln^{3}{\left(1+y\right)}}{y}+\frac13\int_{0}^{1}\mathrm{d}y\,\frac{\ln^{3}{\left(1-y^2\right)}}{y}}\\ &~~~~~-\frac18\int_{0}^{1}\mathrm{d}y\,\frac{\ln^{3}{\left(1-y^2\right)}-\ln^{3}{\left(\frac{1-y}{1+y}\right)}-2\ln^{3}{\left(1+y\right)}}{y}\\ &=-\frac34\left[\operatorname{Li}_{2}{\left(-1\right)}\right]^{2}+\frac32\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1+y\right)}\operatorname{Li}_{2}{\left(y\right)}}{y}\\ &~~~~~-\frac13\int_{0}^{1}\mathrm{d}y\,\frac{\ln^{3}{\left(1-y\right)}}{y}+\frac16\int_{0}^{1}\mathrm{d}y\,\frac{\ln^{3}{\left(1+y\right)}}{y}\\ &~~~~~+\frac{5}{24}\int_{0}^{1}\mathrm{d}y\,\frac{\ln^{3}{\left(1-y^2\right)}}{y}+\frac18\int_{0}^{1}\mathrm{d}y\,\frac{\ln^{3}{\left(\frac{1-y}{1+y}\right)}}{y}\\ &=-\frac34\left[\operatorname{Li}_{2}{\left(-1\right)}\right]^{2}+\frac32\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1+y\right)}\operatorname{Li}_{2}{\left(y\right)}}{y}\\ &~~~~~-\frac13\int_{0}^{1}\mathrm{d}y\,\frac{\ln^{3}{\left(1-y\right)}}{y}+\frac16\int_{0}^{1}\mathrm{d}y\,\frac{\ln^{3}{\left(1+y\right)}}{y}\\ &~~~~~+\frac{5}{48}\int_{0}^{1}\mathrm{d}z\,\frac{\ln^{3}{\left(1-z\right)}}{z};~~~\small{\left[y=\sqrt{z}\right]}\\ &~~~~~-\int_{0}^{1}\mathrm{d}y\,\frac{\left[\frac12\ln{\left(\frac{1+y}{1-y}\right)}\right]^{3}}{y}\\ &=-\frac34\left[\operatorname{Li}_{2}{\left(-1\right)}\right]^{2}-\frac32\operatorname{Li}_{2}{\left(1\right)}\operatorname{Li}_{2}{\left(-1\right)}-\frac32\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1-y\right)}\operatorname{Li}_{2}{\left(-y\right)}}{y}\\ &~~~~~-\frac{11}{48}\int_{0}^{1}\mathrm{d}y\,\frac{\ln^{3}{\left(1-y\right)}}{y}+\frac16\int_{0}^{1}\mathrm{d}y\,\frac{\ln^{3}{\left(1+y\right)}}{y}-\int_{0}^{1}\mathrm{d}y\,\frac{\left[\operatorname{arctanh}{\left(y\right)}\right]^{3}}{y}.\tag{7}\\ \end{align}$$
The first two logarithmic integrals can immediately be written as Nielsen generalized polylogarithms. It's also not difficult to reduce the third logarithmic integral to Nielsen polylogarithms:
$$\begin{align} \int_{0}^{1}\mathrm{d}y\,\frac{\left[\operatorname{arctanh}{\left(y\right)}\right]^{3}}{y} &=\int_{0}^{1}\mathrm{d}y\,\frac{\left[\frac12\ln{\left(\frac{1+y}{1-y}\right)}\right]^{3}}{y}\\ &=-\int_{0}^{1}\mathrm{d}y\,\frac{\ln^{3}{\left(\frac{1-y}{1+y}\right)}}{8y}\\ &=-\frac14\int_{0}^{1}\mathrm{d}x\,\frac{\ln^{3}{\left(x\right)}}{1-x^2};~~~\small{\left[\frac{1-y}{1+y}=x\right]}\\ &=-\frac18\int_{0}^{1}\mathrm{d}x\,\frac{\ln^{3}{\left(x\right)}}{1-x}-\frac18\int_{0}^{1}\mathrm{d}x\,\frac{\ln^{3}{\left(x\right)}}{1+x}\\ &=-\frac38\int_{0}^{1}\mathrm{d}x\,\frac{\ln^{2}{\left(x\right)}\ln{\left(1-x\right)}}{x}+\frac38\int_{0}^{1}\mathrm{d}x\,\frac{\ln^{2}{\left(x\right)}\ln{\left(1+x\right)}}{x}\\ &=\frac34\,S_{3,1}{\left(1\right)}-\frac34\,S_{3,1}{\left(-1\right)}.\tag{8}\\ \end{align}$$
This just leaves the dilogarithmic integral to evaluate.
$$\begin{align} \int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1-y\right)}\operatorname{Li}_{2}{\left(-y\right)}}{y} &=-\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1-y\right)}}{y}\int_{0}^{1}\mathrm{d}x\,\frac{\ln{\left(1+yx\right)}}{x}\\ &=-\int_{0}^{1}\mathrm{d}x\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1-y\right)}\ln{\left(1+xy\right)}}{xy}\\ &=:-\int_{0}^{1}\mathrm{d}x\,\frac{J{\left(-x\right)}}{x}\\ &=-\int_{0}^{1}\mathrm{d}x\,\frac{S_{1,2}{\left(-x\right)}}{x}-\int_{0}^{1}\mathrm{d}x\,\frac{\operatorname{Li}_{3}{\left(-x\right)}}{x}\\ &=-S_{2,2}{\left(-1\right)}-\operatorname{Li}_{4}{\left(-1\right)}.\tag{9}\\ \end{align}$$
(See Appendix 2 for definition and evaluation of the auxiliary function $J{(a)}$ used above.)
Putting everything together, we arrive at
$$\begin{align} I_{4} &=-\frac34\left[\operatorname{Li}_{2}{\left(-1\right)}\right]^{2}-\frac32\operatorname{Li}_{2}{\left(1\right)}\operatorname{Li}_{2}{\left(-1\right)}\\ &~~~~~-\frac32\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1-y\right)}\operatorname{Li}_{2}{\left(-y\right)}}{y}\\ &~~~~~-\frac{11}{48}\int_{0}^{1}\mathrm{d}y\,\frac{\ln^{3}{\left(1-y\right)}}{y}+\frac16\int_{0}^{1}\mathrm{d}y\,\frac{\ln^{3}{\left(1+y\right)}}{y}\\ &~~~~~-\int_{0}^{1}\mathrm{d}y\,\frac{\left[\operatorname{arctanh}{\left(y\right)}\right]^{3}}{y}\\ &=-\frac34\left[\operatorname{Li}_{2}{\left(-1\right)}\right]^{2}-\frac32\operatorname{Li}_{2}{\left(1\right)}\operatorname{Li}_{2}{\left(-1\right)}\\ &~~~~~+\frac32\,S_{2,2}{\left(-1\right)}+\frac32\operatorname{Li}_{4}{\left(-1\right)}\\ &~~~~~+\frac{11}{8}\,S_{1,3}{\left(1\right)}-S_{1,3}{\left(-1\right)}\\ &~~~~~-\frac34\,S_{3,1}{\left(1\right)}+\frac34\,S_{3,1}{\left(-1\right)}\\ &=\frac32\,S_{2,2}{\left(-1\right)}+\frac{11}{8}\,S_{1,3}{\left(1\right)}-S_{1,3}{\left(-1\right)}-\frac{7\pi^4}{480}.\\ \end{align}$$
Appendix 1.
The Nielsen generalized polylogarithm may be defined for positive integer indices via the integral representation
$$S_{n,p}{\left(z\right)}:=\frac{\left(-1\right)^{n+p-1}n}{n!\,p!}\int_{0}^{1}\frac{\ln^{n-1}{\left(t\right)}\ln^{p}{\left(1-zt\right)}}{t}\,\mathrm{d}t;~~~\small{n,p\in\mathbb{N}^{+}}.$$
Setting $n=1$,
$$S_{1,p}{\left(z\right)}:=\frac{\left(-1\right)^{p}}{p!}\int_{0}^{1}\frac{\ln^{p}{\left(1-zt\right)}}{t}\,\mathrm{d}t;~~~\small{p\in\mathbb{N}^{+}}.$$
Setting $p=1$,
$$S_{n,1}{\left(z\right)}=\frac{\left(-1\right)^{n}n}{n!}\int_{0}^{1}\frac{\ln^{n-1}{\left(t\right)}\ln{\left(1-zt\right)}}{t}\,\mathrm{d}t;~~~\small{n\in\mathbb{N}^{+}}.$$
Appendix 2.
Define the real function $J:(-\infty,1]\to\mathbb{R}$ via the integral representation
$$J{\left(a\right)}:=\int_{0}^{1}\frac{\ln{\left(1-y\right)}\ln{\left(1-ay\right)}}{y}\,\mathrm{d}y;~~~\small{a\le1}.$$
Then, for $a\le1$ we have
$$\begin{align} J{\left(a\right)} &=\int_{0}^{1}\frac{\ln{\left(1-y\right)}\ln{\left(1-ay\right)}}{y}\,\mathrm{d}y\\ &=\int_{0}^{1}\mathrm{d}y\,\frac{\ln{\left(1-y\right)}}{y}\int_{0}^{1}\mathrm{d}x\,\frac{ay}{ayx-1}\\ &=-a\int_{0}^{1}\mathrm{d}y\int_{0}^{1}\mathrm{d}x\,\frac{\ln{\left(1-y\right)}}{1-ayx}\\ &=-\int_{0}^{1}\mathrm{d}x\int_{0}^{1}\mathrm{d}y\,\frac{a\ln{\left(1-y\right)}}{1-axy}\\ &=-\int_{0}^{1}\mathrm{d}x\,\frac{\operatorname{Li}_{2}{\left(\frac{ax}{ax-1}\right)}}{x}\\ &=\int_{0}^{1}\mathrm{d}x\,\frac{\frac12\ln^{2}{\left(1-ax\right)}+\operatorname{Li}_{2}{\left(ax\right)}}{x}\\ &=\frac12\int_{0}^{1}\mathrm{d}x\,\frac{\ln^{2}{\left(1-ax\right)}}{x}+\int_{0}^{1}\mathrm{d}x\,\frac{\operatorname{Li}_{2}{\left(ax\right)}}{x}\\ &=S_{1,2}{\left(a\right)}+\operatorname{Li}_{3}{\left(a\right)}.\\ \end{align}$$