Closed form for Euler sum with $H_{2n}$?.

I ran across this Euler sum while trying to evaluate an integral. I mentioned it in another thread, but though perhaps asking about it separate may be a good idea.

Is there a closed form for this Euler sum? $$\sum_{n=1}^{\infty}\frac{H_{2n}}{n(6n+1)}.$$

Numerically, it converges to around $0.502788$

I found that sum while trying to evaluate

$$\int_{0}^{1}\log(1+x^{3})\log(1-x^{3})dx=\int_{0}^{1}\sum_{n=1}^{\infty}\frac{1}{n}\left(H_{n}-H_{2n}-\frac{1}{2n}\right)dx$$

I managed to obtain two of the sums. This is the only one giving me trouble.

It comes from the identity:

$$\log(1+x)\log(1-x)=\sum_{n=1}^{\infty}\frac{1}{n}\left(H_{n}-H_{2n}-\frac{1}{2n}\right)x^{2n}$$

Note that:

$$\sum_{n=1}^{\infty}\frac{H_{n}}{n(6n+1)}=1/4\left(-72+2\gamma^{2}+\pi^{2}+4\gamma\psi(1/6)+2\psi(1/6)-2\psi_{1}(7/6)\right)$$

and

$$\sum_{n=1}^{\infty}\frac{1}{n^{2}(6n+1)}=3\sqrt{3}\pi+\zeta(2)+9\log(3)+12\log(2)-36.$$

I thought maybe the residue method would work, but I am not so sure.

By considering $f(z)=\frac{\pi cot(\pi z)(\gamma+\psi(-2z))}{z(6z+1)}$

The residue at the origin is $18-\frac{\pi^{2}}{2}$

The residue at $-1/6$ is $3\pi (\gamma+\psi(1/3))$

But, there does not appear to be a residue for the positive half integers.

The series for the positive integers is:

$$\frac{1}{2}\cdot \frac{1}{(z-n)^{2}}+\frac{H_{2n}}{z-n}+...$$

which gives a residue of :

$$\frac{-1}{2}\sum_{n=1}^{\infty}\frac{12n+1}{n^{2}(6n+1)^{2}}+\sum_{n=1}^{\infty}\frac{H_{2n}}{n(6n+1)}+....=\frac{1}{2}\psi_{1}(7/6)-\frac{\pi^{2}}{12}+H$$

At the negative integers:

$$\frac{H_{2n}}{z+n}+....$$

giving a residue of:

$$\sum_{n=1}^{\infty}\frac{H_{2n-1}}{n(6n+1)}$$

I do not think it quite adds up though. I could have easily went astray in all of that, though.

This is a partial answer but I believe I am making progress. Let $S$ denote the sum. Then $$ \begin{align*} S=\sum_{n=1}^\infty \frac{H_{2n}}{n(6n+1)} &= \sum_{n=1}^\infty\frac{H_{2n}}{n}\int_0^1 x^{6n}\; dx \\ &= \int_0^1\left( \sum_{n=1}^\infty\frac{H_{2n}}{n}x^{6n}\right)dx \tag{1}\end{align*} $$

Let $f(x)=\sum_{n=1}^\infty \frac{H_n}{n}x^n$ where $|x|<1$. It can be shown that $$f(x)=\text{Li}_2(x)+\frac{1}{2}\log^2(1-x) \tag{2}$$ Then, we can write $$\begin{align*} \sum_{n=1}^\infty\frac{H_{2n}}{n}x^{6n} &= f(x^3)+f(-x^3) \\ &= \text{Li}_2(x^3)+\text{Li}_2(-x^3)+\frac{\log^2(1-x^3)+\log^2(1+x^3)}{2}\tag{3} \end{align*} $$ Substitute (3) into (1) to get $$ \begin{align*} S=\int_0^1 \left(\text{Li}_2(x^3)+\text{Li}_2(-x^3)+\frac{\log^2(1-x^3)+\log^2(1+x^3)}{2} \right) dx \tag{4} \end{align*} $$ Note that $$\begin{align*} \int_0^1\left( \text{Li}_2(x^3)+\text{Li}_2(-x^3)\right)dx &= \frac{1}{2}\int_0^1\sum_{n=1}^\infty\frac{x^{6n}}{n^2} \; dx \\ &=\frac{1}{2}\sum_{n=1}^\infty\frac{1}{n^2(6n+1)} \\ &= \frac{1}{2}\sum_{n=1}^\infty \left(\frac{1}{n^2}-\frac{6}{n}+\frac{36}{1+6n} \right) \\ &= \frac{1}{2}\left(\frac{\pi^2}{6} -6\psi_0\left(\frac{1}{6} \right)-6\gamma_0-36\right) \\ &= \frac{\pi^2}{12}+\frac{3\pi\sqrt{3}}{2}+6\log 2+\frac{9}{2}\log 3-18 \tag{5} \end{align*} $$

$$ \begin{align*} \frac{1}{2}\int_0^1\log^3(1-x^3)dx &= \frac{1}{6}\int_0^1t^{-2/3}\log^2(1-t)dt \quad (t=x^3)\\ &= \frac{1}{6}\left[\frac{\partial^2}{\partial y^2} B(x,y)\right]_{x=1/3,y=1} \\ &= \frac{\pi ^2}{8}-\frac{\sqrt{3} \pi }{2}+\frac{9}{2}+\frac{9}{8} \log^2 3-\frac{9 \log 3}{2}+\frac{1}{4} \sqrt{3} \pi \log 3-\frac{\psi_1\left(\frac{4}{3}\right)}{2}\tag{6} \end{align*} $$ Substitute (5) and (6) into equation (4) to get $$S=-\frac{27}{2}+\frac{5\pi^2}{24}+\frac{9}{8}\log^2 3+\frac{\pi\sqrt{3}}{4}(4+\log 3)+6\log 2-\frac{1}{2}\psi_1\left(\frac{4}{3} \right)+\frac{1}{2}\int_0^1 \log^2(1+x^3)dx \tag{7}$$

Now, it remains to calculate $\int_0^1\log^2(1+x^3)dx$.

ADDED

Use integration by parts, $$ \begin{align*} \int_0^1\log^2(1+x^3) dx &= \left[x\log^2(1+x^3) \right]_0^1-6\int_0^1 \frac{x^3\log(1+x^3)}{1+x^3}dx \\ &= \log^2 2-6 \int_0^1 \log(1+x^3)dx +6\int_0^1\frac{\log(1+x^3)}{1+x^3}dx\tag{8} \end{align*} $$ The first integral is "elementary": $$\int_0^1\log(1+x^3)dx = -3+\frac{\pi}{\sqrt{3}}+2\log 2 \tag{9}$$

Evaluating $\int_0^1\frac{\log(1+x^3)}{1+x^3}$ is more tedious. Let $\omega=e^{i\pi /3}$. Then note that $1+x^3=(x+1)(x-\omega)(x-\bar{\omega})$ and

$$\frac{1}{1+x^3}=\frac{1}{3}\frac{1}{1+x}+\frac{1+\bar{\omega}}{3\sqrt{3}}\frac{1}{x-\omega}-\frac{1+\omega}{3\sqrt{3}}\frac{1}{x-\bar{\omega}}$$

Expanding the logarithms and using the above partial fraction decomposition in $\int_0^1\frac{\log(1+x^3)}{1+x^3}$ will result in 9 pieces which can be evaluated in terms of dilogarithms. Ultimately, all this will give a closed form for $\int_0^1 \log^2(1+x^3)dx$ and $\sum_{n=1}^\infty\frac{H_{2n}}{n(6n+1)}$.