I dabbled with this a little and managed to get it in terms of some series involving digamma and its derivatives.
Yet again, series involving digamma. How in the world can we evaluate these?. They do indeed have interesting closed forms in terms of polylogs, zeta, log, etc.
There has got to be a way, rather it be Euler sums or whatever.
One of them is
$$\sum_{k=1}^{\infty}\frac{1}{2k(k+1)^{2}}\left(\psi_{1}\left(k/2+3/2\right)-\psi_{1}\left(k/2+1\right)\right)$$.
there are some already known results that could be used. i.e. $$\sum_{k=1}^{\infty}\psi_{1}\left(k/2+1\right)x^{k}=\frac{\pi^{2}x(x+3)}{6(1-x^{2})}-\frac{4Li_{2}(x)}{1-x^{2}}$$.
I think I can derive the other ones, but it would be tedious, and right now I don't have the time nor the gumption to do so. If anyone would like to delve, please go ahead. I find these polygamma sums interesting and have found little on them. This is why I would like to do some research and write something up. It may have to wait a while though.
I think I managed to derive $$\sum_{n=1}^{\infty}\psi_{1}\left(\frac{n}{2}+\frac{3}{2}\right)x^{n}=\frac{x\pi^{2}}{12(x+1)}-\frac{Li_{2}(x)}{x(x^{2}-1)}-\frac{\pi^{2}x}{12(1-x)}$$
Another, and perhaps better, approach would be to write it as:
$\displaystyle 2ab=(a+b)^{2}-a^{2}-b^{2}$
$$2I=\int_{0}^{1}\log^{2}(x)\log^{2}(1-x^{2})dx-\int_{0}^{1}\log^{2}(x)\log^{2}(1+x)dx-\int_{0}^{1}\log^{2}(x)\log^{2}(1-x)dx$$, then perhaps use known euler sum generating functions.
i.e. $$\log^{2}(1-x)=2\sum_{k=1}^{\infty}\frac{H_{n}}{n+1}x^{n+1}$$
and $$\log^{2}(1+x)=2\sum_{n=1}^{\infty}\frac{(-1)^{n}H_{n}}{n}x^{n}+2Li_{2}(1+x)$$
and $$\log^{2}(1-x^{2})=2\sum_{n=1}^{\infty}\frac{H_{n}}{n+1}x^{2n+2}$$
which, for the far right integral, results in $$4\sum_{n=1}^{\infty}\frac{H_{n}}{(n+1)(n+2)^{3}}=-8\zeta(3)+24-\frac{4\pi^{2}}{3}-\zeta(4)$$.
do the others in a similar fashion, then sum them up.
Like you super, I had trouble with one sum. Other than that it all works out nicely. I know what you mean by an error. It sure would be easy in all this mess. One little negative or anything and it's wrong. Then, you have to root it out :)
$\displaystyle 2ab=(a+b)^{2}-a^{2}-b^{2}$
$$2I=\int_{0}^{1}\log^{2}(x)\log^{2}(1-x^{2})dx-\int_{0}^{1}\log^{2}(x)\log^{2}(1+x)dx-\int_{0}^{1}\log^{2}(x)\log^{2}(1-x)dx$$
Using the designated gen funcs on the appropriate log integral above:
$$\int_{0}^{1}\log^{2}(x)\log^{2}(1-x^{2})dx\tag{1}$$
$$=4\sum_{n=1}^{\infty}\frac{H_{n}}{(n+1)(2n+3)^{3}}=14\zeta(3)\log(2)-28\zeta(3)+8\log^{2}(2)+2\pi^{2}\log(2)-48\log(2)-\frac{\pi^{4}}{8}-\frac{14\pi^{2}}{3}+96$$
$$\int_{0}^{1}\log^{2}(x)\log^{2}(1-x)dx=4\sum_{n=1}^{\infty}\frac{H_{n}}{(n+1)(n+2)^{3}}=-8\zeta(3)+24-\frac{4}{3}\pi^{2}-\frac{\pi^{4}}{90}\tag{2}$$
Here is the ickiest one:
$$\int_{0}^{1}\log^{2}(x)\log^{2}(1+x)dx=4\sum_{n=1}^{\infty}\frac{(-1)^{n}H_{n}}{n(n+1)^{3}}-2\int_{0}^{1}\log^{2}(x)Li_{2}(x+1)dx\tag{3}$$
$$=7\log(2)\zeta(3)-\frac{5}{2}\zeta(3)+8Li_{4}(1/2)+1/3\log^{4}(2)-\frac{\pi^{2}}{3}\log^{2}(2)+4\log^{2}(2)-24\log(2)-\frac{\pi^{4}}{12}-\frac{2\pi^{2}}{3}+24$$
combining them all and dividing by 2, returns the result.
$\displaystyle 2I=(1)-(3)-(2)$
The above Euler sums can be done using residues. Though, (3) would probably prove the hardest to get along with.
The sum could probably be done using $\pi csc(\pi z)$ in its kernel. The dilog integral I am not sure about. I did not attempt it. I ran this one through Mathematica.
Take for example: $$4\sum_{n=1}^{\infty}\frac{H_{n}}{(n+1)(n+2)^{3}}$$
Use the kernel $\displaystyle (\gamma+\psi(-z))^{2}$ with $\displaystyle r(n)=\frac{1}{(n+1)(n+2)^{3}}$
$\displaystyle Res(0)=-5/16$
$\displaystyle Res(-1)=0$
$\displaystyle Res(-2)=-6+\frac{2\pi^{2}}{3}+2\zeta(3)-\frac{\pi^{4}}{36}$
$\displaystyle Res(n)=\frac{1}{(z-n)^{2}}+\frac{2H_{n}}{z-n}+\cdot\cdot\cdot $
$\displaystyle =-\sum_{n=1}^{\infty}\frac{4n+5}{(n+1)^{2}(n+2)^{4}}+2\sum_{n=1}^{\infty}\frac{H_{n}}{(n+1)(n+2)^{3}}+\cdot\cdot\cdot $
$\displaystyle =\frac{-91}{16}+2\zeta(3)+\frac{\pi^{4}}{30}+2H$
sum up the residues:
$$4\left(-5/16-6+\frac{2\pi^{2}}{3}+2\zeta(3)-\frac{\pi^{4}}{36}-91/16+2\zeta(3)+\frac{\pi^{4}}{30}+2H\right)=0$$
$$H=-8\zeta(3)+24-\frac{4\pi^{2}}{3}-\frac{\pi^{4}}{90}$$