Are we aware of an elementary way of proving that?
$$\int_0^1 \frac{\log \left(\frac{1}{t}\right) \log (t+2)}{t+1} \, dt=\frac{13}{24} \zeta (3)$$
Of course, with the help of Mathematica it can be done, but I wonder if there exists an elementary, simple, easy way of finishing it.
$$\int \frac{\log \left(\frac{1}{t}\right) \log (t+2)}{t+1} \, dt=$$
$$=\text{Li}_3\left(\frac{t+2}{t}\right)-\text{Li}_3\left(-\frac{t+2}{t}\right)+\text{Li}_3(-t)+\text{Li}_3(t+2)-\left(\text{Li}_2\left(\frac{t+2}{t}\right)-\text{Li}_2\left(-\frac{t+2}{t}\right)\right) \log \left(-\frac{t+2}{t}\right)-\text{Li}_2(-t-1) \left(\log \left(\frac{1}{t}\right)+\log (t)\right)-\text{Li}_2(-t) \left(\log (t+2)-\log \left(-\frac{t+2}{t}\right)\right)-\text{Li}_2(t+2) \left(\log \left(-\frac{t+2}{t}\right)+\log (t)\right)-\frac{1}{2} \left(-\log \left(\frac{2 (t+1)}{t}\right)+\log \left(-\frac{2}{t}\right)+\log (t+1)\right) \log ^2\left(-\frac{t+2}{t}\right)-(\log (-t-1)-\log (t+1)) \log (t+2) \log \left(-\frac{t+2}{t}\right)-\log (t) \log (t+1) \log (t+2)-\frac{1}{2} (\log (t+1)-\log (-t-1)) \log (t+2) (\log (t+2)-2 \log (t))$$
that is obtain by Mathematica.
EDIT: I think once we rewrite all as
$$ \frac{1}{2}\int_0^1 \left(\frac{\log ^2\left(\frac{t+2}{t}\right)}{t+1}-\frac{\log ^2(t+2)}{t+1}-\frac{\log ^2\left(\frac{1}{t}\right)}{t+1}\right) \, dt$$
and employ the proper variable change, we're almost done, the idea that just crossed my mind.
The hardest part is probably the integral
$$\int_0^1 \frac{\log ^2\left(\frac{t+2}{t}\right)}{t+1} \ dt$$ where
by letting $t/(t+2)\mapsto t$, we obtain the far nicer form
$$2\int_0^{1/3} \frac{\log ^2(t)}{1-t^2} \, dt$$
$$=\int_0^{1/3} \frac{\log ^2(t)}{ 1-t} \ dt+\int_0^{1/3}\frac{\log ^2(t)}{ 1+t} \, dt$$
where then using geometric series and swapping the integration and summation, we obtain that$$\int_0^1 \frac{\log ^2\left(\frac{t+2}{t}\right)}{t+1} \ dt=$$
$$2\text{Li}_3\left(\frac{1}{3}\right)-2\text{Li}_3\left(-\frac{1}{3}\right)+ \log (3) \left(4 \text{Li}_2\left(\frac{1}{3}\right)-\text{Li}_2\left(\frac{1}{9}\right)+\log (2) \log (3)\right) \tag1$$
Using geometric series again, we easily get that
$$\int_0^1 \frac{\log ^2\left(\frac{1}{t}\right)}{t+1} \, dt=\frac{3}{2} \zeta (3) \tag2$$
The last integral can be reduced by variable change to a well-known integral treated by Leonard Lewin in his book about polylogarithms, that is $$\int_0^1 \frac{\log ^2(t+2)}{t+1} \, dt=\int_1^2 \frac{\log ^2(t+1)}{t} \, dt$$
So, we can use the fact that
$$\int_0^t \frac{\log ^2(1+t)}{t} \, dt$$
$$=\log(t)\log^2(1+t)-2/3\log^3(1+t)-2\log(1+t)\operatorname{Li}_2\left(\frac{1}{1+t}\right)-2\operatorname{Li}_3\left(\frac{1}{1+t}\right)+2\zeta(3)$$
So, using the foregoing result, we immediately get that
$$\int_1^2 \frac{\log ^2(t+1)}{t} \, dt$$
$$=-2 \text{Li}_3\left(\frac{1}{3}\right)-2 \text{Li}_2\left(\frac{1}{3}\right) \log (3)+\frac{7 \zeta (3)}{4}-\frac{1}{3} 2 \log ^3(3)+\log (2) \log ^2(3) \tag3$$
Combining $(1) (2) (3)$, we obtain that
$$\int_0^1 \frac{\log \left(\frac{1}{t}\right) \log (t+2)}{t+1} \, dt$$
$$=2 \text{Li}_3\left(\frac{1}{3}\right)-\text{Li}_3\left(-\frac{1}{3}\right)-\frac{1}{2} \left(\text{Li}_2\left(\frac{1}{9}\right)-6 \text{Li}_2\left(\frac{1}{3}\right)\right) \log (3)-\frac{13 \zeta (3)}{8}+\frac{\log ^3(3)}{3}$$
Using $(20)$ from here http://mathworld.wolfram.com/Dilogarithm.html and then the identity and $(1)$ from this question/answer Proving $\text{Li}_3\left(-\frac{1}{3}\right)-2 \text{Li}_3\left(\frac{1}{3}\right)= -\frac{\log^33}{6}+\frac{\pi^2}{6}\log 3-\frac{13\zeta(3)}{6}$?, we get that
$$\int_0^1 \frac{\log \left(\frac{1}{t}\right) \log (t+2)}{t+1} \, dt=\frac{13}{24} \zeta (3).$$
A SPECIAL ADDENDUM:
The present integral can be viewed as the key core of calculating a pretty tough integral that was posted on MSE some time go,
$$\int_0^1 \frac{\text{Li}_2 \left(-\frac{1}{1-z}\right)-\text{Li}_2 \left(-\frac{1}{1+z}\right)}{z}dz$$ at this link Evaluating $\int_0^1 \frac{\text{Li}_2 \left(-\frac{1}{1-z}\right)-\text{Li}_2 \left(-\frac{1}{1+z}\right)}{z}dz$.
Simple integration by parts combined with the use of the geometric series shows that
$$\int_0^1 \frac{\text{Li}_2 \left(-\frac{1}{1-z}\right)-\text{Li}_2 \left(-\frac{1}{1+z}\right)}{z}dz$$
$$=\int_0^1 \frac{\log (z+2) \log (z)}{z+1}dz+\underbrace{\int_0^1\frac{\log (2-z) \log (z)}{1-z} \ dz}_{\large \sum _{n=1}^{\infty } \frac{(-1)^n H_n}{n^2}=-5/8 \zeta (3)}-\underbrace{\int_0^1\frac{\log (1-z) \log (z)}{1-z}dz}_{\large\sum _{n=1}^{\infty } \frac{H_n}{(n+1)^2}=\zeta (3)}-\underbrace{\int_0^1\frac{\log (z+1) \log (z)}{z+1} \, dz}_{\large \sum _{n=1}^{\infty } \frac{(-1)^n H_n}{(n+1)^2}=-1/8\zeta (3)}$$
where since the integral unknown is calculated in this post, we conclude that
$$\int_0^1 \frac{\text{Li}_2 \left(-\frac{1}{1-z}\right)-\text{Li}_2 \left(-\frac{1}{1+z}\right)}{z}dz$$
$$=-\frac{49}{24}\zeta(3).$$
A SPECIAL ADDENDUM $2$: By a careful approach of the previous results, we also obtain the value of the very beautiful series
$$\sum _{k=1}^{\infty } \sum _{n=1}^{\infty } \frac{(-1)^k }{k2^k n 2^n (k+n)}=\frac{1}{12} \left(3 \text{Li}_3\left(\frac{1}{4}\right)+6\log (2) \text{Li}_2\left(\frac{1}{4}\right) +4 \log ^3(2)-4 \zeta (3)\right)$$
Q.E.D.
Best Answer
The last series Special Addendum in the OP looks quite interesting! (just an observation, not an answer)
We have by symmetry of the series:
$$S = \sum _{k=1}^{\infty } \sum _{n=1}^{\infty } \frac{(-1)^k }{k2^k n 2^n (k+n)} = \sum _{k=1}^{\infty } \sum _{n=1}^{\infty } \frac{(-1)^n}{k2^k n 2^n (k+n)}$$
Hence, $$\begin{align}\sum _{k=1}^{\infty } \sum _{n=1}^{\infty } \frac{1+(-1)^k+(-1)^n+(-1)^{k+n} }{k2^k n 2^n (k+n)} &= \sum _{k=1}^{\infty } \sum _{n=1}^{\infty } \frac{(1+(-1)^k)(1+(-1)^n)}{k2^k n 2^n (k+n)}\\&= \frac{1}{2}\sum _{k=1}^{\infty } \sum _{n=1}^{\infty } \frac{1}{k4^k n 4^n (k+n)}\end{align}$$
Since, the term $(1+(-1)^k)(1+(-1)^n)$ survives only when both $k,n$ are even.
Since, $$\sum _{k=1}^{\infty } \sum _{n=1}^{\infty } \frac{a^{n+k}}{kn(k+n)} = \sum _{k=1}^{\infty } \sum _{n=1}^{\infty } \int_0^1 \frac{a^{n+k}x^{n+k}}{kn}\frac{dx}{x} = \int_0^1 \frac{\log^2 (1-ax)}{x}\,dx$$
We have: $$2S = -\int_0^1 \frac{\log^2(1-\frac{x}{2})}{x}\,dx-\int_0^1 \frac{\log^2(1+\frac{x}{2})}{x}\,dx+\frac{1}{2}\int_0^1 \frac{\log^2(1-\frac{x}{4})}{x}\,dx$$
The integrals for $0 < a < 1$:
$\displaystyle \begin{align} \int_0^1 \frac{\log^2 (1-ax)}{x}\,dx &= \int_0^a \frac{\log^2 (1-x)}{x}\,dx \\&= \log a \log^2(1-a)+2\int_0^a \frac{\log x \log(1-x)}{1-x}\,dx\\&= \log a \log^2(1-a)+2\log (1-a) \operatorname{Li}_2(1-a) - 2\operatorname{Li}_3(1-a)+2\operatorname{Li}_3(1)\end{align}$
and the integrals for $0 < a < 1$:
$\displaystyle \begin{align} \int_0^1 \frac{\log^2 (1+ax)}{x}\,dx &= \int_0^{a} \frac{\log^2 (1+x)}{x}\,dx \\&= \log a \log^2(1+a)-2\int_0^a \frac{\log x \log(1+x)}{1+x}\,dx\\&= \log a \log^2 (1+a) - \frac{2}{3}\log^3 (1+a)-2\log (1+a)\operatorname{Li}_2\left(\frac{1}{1+a}\right)-2\operatorname{Li}_3\left(\frac{1}{1+a}\right)+2\operatorname{Li}_3(1)\end{align}$
Wolfram alpha computes these three integrals as:
[1] $$\displaystyle \int_0^1 \log^2 \left(1+\frac{x}{2}\right)\,\frac{dx}{x}=-2 \text{Li}_3\left(\frac{2}{3}\right)-\text{Li}_2\left(\frac{2}{3}\right) \log \left(\frac{9}{4}\right)+2 \zeta (3)\\+\frac{4}{3} \log (18) \coth ^{-1}(5)^2$$
[2] $$\int_0^1 \log^2\left(1-\frac{x}{2}\right)\,\frac{dx}{x} = \frac{1}{4}\zeta(3) - \frac{1}{3}\log^3 (2)$$
[3] $$\int_0^1 \log^2\left(1-\frac{x}{4}\right)\,\frac{dx}{x} = -2 \text{Li}_3\left(\frac{3}{4}\right)-\text{Li}_2\left(\frac{3}{4}\right) \log \left(\frac{16}{9}\right)+2 \zeta (3)\\-\log^2(3)\log (4)+\log^2(4)\log\left(\frac{9}{4}\right)$$