Compute $\int_0^\infty \frac{\operatorname{Li}_3(x)}{1+x^2}\ dx$
Finally I got the answer:
Using the generalized integral expression of the polylogrithmic function which can be found in the book (Almost) Impossible Integrals, Sums and series page 4.
$$\int_0^1\frac{x\ln^n(u)}{1-xu}\ du=(-1)^n n!\operatorname{Li}_{n+1}(x)$$ and by setting $n=2$ we get
$$\operatorname{Li}_{3}(x)=\frac12\int_0^1\frac{x\ln^2 u}{1-xu}\ du$$
we can write
$$\int_0^\infty\frac{\operatorname{Li}_{3}(x)}{1+x^2}\ dx=\frac12\int_0^1\ln^2u\left(\int_0^\infty\frac{x}{(1-ux)(1+x^2)}\ dx\right)\ du$$ $$=\frac12\int_0^1\ln^2u\left(-\frac12\left(\frac{\pi u}{1+u^2}+\frac{2\ln(-u)}{1+u^2}\right)\right)\ du,\quad \color{red}{\ln(-u)=\ln u+i\pi}$$
$$=-\frac{\pi}{4}\underbrace{\int_0^1\frac{u\ln^2u}{1+u^2}\ du}_{\frac3{16}\zeta(3)}-\frac12\underbrace{\int_0^1\frac{\ln^3u}{1+u^2}\ du}_{-6\beta(4)}-i\frac{\pi}2\underbrace{\int_0^1\frac{\ln^2u}{1+u^2}\ du}_{2\beta(3)}$$
Then
$$\int_0^\infty\frac{\operatorname{Li}_{3}(x)}{1+x^2}\ dx=-\frac{3\pi}{64}\zeta(3)+3\beta(4)-i\pi\beta(3)\tag{2}$$
Bonus:
By combining $(1)$ in the question body and $(2)$, the imaginary part $i\pi\beta(3)$ nicely cancels out and we get
$$\int_0^1 \frac{\operatorname{Li}_3(x)}{1+x^2}\ dx=2\beta(4)-\zeta(2)G-\frac{3\pi}{128}\zeta(3)$$
where $\beta(4)$ $=\frac{1}{768}\psi^{(3)}(1/4)-\frac{\pi^4}{96}$
For a different solution, use the first result from A simple idea to calculate a class of polylogarithmic integrals by using the Cauchy product of squared Polylogarithm function by Cornel Ioan Valean.
Essentially, the main new results in the presentation are:
Let $a\le1$ be a real number. The following equalities hold: \begin{equation*} i) \ \int_0^1 \frac{\log (x)\operatorname{Li}_2(x) }{1-a x} \textrm{d}x=\frac{(\operatorname{Li}_2(a))^2}{2 a}+3\frac{\operatorname{Li}_4(a)}{a}-2\zeta(2)\frac{\operatorname{Li}_2(a)}{a}; \end{equation*} \begin{equation*} ii) \ \int_0^1 \frac{\log^2(x)\operatorname{Li}_3(x) }{1-a x} \textrm{d}x=20\frac{\operatorname{Li}_6(a)}{a}-12 \zeta(2)\frac{\operatorname{Li}_4(a)}{ a}+\frac{(\operatorname{Li}_3(a))^2}{a}. \end{equation*} For a fast proof, see the paper above (series expansion combined with the Cauchy product of squared Polylogarithms)
The use of these new results with integrals allows you to obtain your result elegantly, but also other results that are (very) difficult to obtain by other means, including results from the book, (Almost) Impossible Integrals, Sums, and Series.
BONUS: Using these results you may also establish that (or the versions with integration by parts applied).
$$i) \ \int_0^1 \frac{\arctan(x) \operatorname{Li}_2(x)}{x}\textrm{d}x$$ $$=\frac{1}{384}\left(720\zeta(4)+105\pi\zeta(3)+384\zeta(2)G-\psi^{(3)}\left(\frac{1}{4}\right)\right),$$ $$ii)\ \int_0^1 \frac{\arctan(x) \operatorname{Li}_2(-x)}{x}\textrm{d}x$$ $$=\frac{1}{768}\left(\psi^{(3)}\left(\frac{1}{4}\right)-384\zeta(2)G-126\pi\zeta(3)-720\zeta(4)\right).$$
EXPLANATIONS (OP's request): The following way in large steps shows the amazing possible creativity in such calculations.
We'll want to focus on the integral, $\displaystyle \int_0^1 \frac{\arctan(x)\operatorname{Li}_2(x)}{x}\textrm{d}x$ which is a translated form of the main integral.
Now, based on $i)$ where we plug in $a=i$ and then consider the real part, we obtain an integral which by a simple integration by parts reveals that
$$\int_0^1 \frac{\arctan(x)\operatorname{Li}_2(x)}{x}\textrm{d}x=\int_0^1 \frac{\arctan(x)\log(1-x) \log(x)}{x}\textrm{d}x+\frac{17}{48}\pi^2 G+\frac{\pi^4}{32}-\frac{1}{256}\psi^{(3)}\left(\frac{1}{4}\right).$$
Looks like we need to evaluate one more integral and we're done. Well, if you read the book (Almost) Impossible Integrals, Sums, and Series (did you?), particularly the solutions in the sections 3.24 & 3.25 you probably observed the powerful trick of splitting the nonnegative real line at $x=1$ with the hope of getting the same integral in the other side but with an opposite sign. Therefore, with such a careful approach (since we need to avoid the divergence issues), we obtain immediately that $$\int_0^1 \frac{\arctan(x)\log(1-x) \log(x)}{x}\textrm{d}x$$ $$=\frac{1}{2} \underbrace{\int_0^1 \frac{\arctan(x)\log^2(x)}{x}\textrm{d}x}_{\text{Trivial}}+\frac{\pi}{4}\underbrace{\int_0^1 \frac{\log(x)\log(1-x)}{x}\textrm{d}x}_{\text{Trivial}}$$ $$-\frac{1}{2}\Re\left \{\int_0^{\infty}\frac{\arctan(1/x) \log(1-x)\log(x)}{x}\textrm{d}x\right \},$$
and the last integral works simply nice with Cornel's strategy described in the second part of this post (it involves the use of Cauchy Principal Value) https://math.stackexchange.com/q/3488566.