As a follow-up to this question, I would like to know if there exists a closed form for this sum:
$$f(x) \equiv \sum_{k=1}^\infty \frac{H_k}{2k + 1} x^{2k+1}. \tag{1}$$
with $x$ real and positive. There are some similar identities in the wikipedia article, but I could not relate them to this sum. I would imagine that $\tanh^{-1}$ and/or $\log$ would be involved, but that's just a guess.
As mentioned in the comments, we have:
$$f'(x) = \frac{\log (1+x^2)}{x^2-1}, \tag{2}$$
which Mathematica can integrate and gives:
$$f(x) = \frac{1}{4} \left\lbrace 2 \log^2 (1-x) - 2\log (1-x)\log(x-1) + \left( \log(x-1) + \log(x+1) \right) \left( \log (x-1) + \log(x+1) + 2\log 2 \right) - 2 \text{Li}_2 \left( \frac{1-x}{2} \right) - 2 \text{Li}_2 \left( \frac{1+x}{2} \right) + \text{constant} \right\rbrace. \tag{3}$$
I fixed the constant numerically by comparing $(1)$ and $(3)$ at some value of $x$. It was easy to guess the closed form of the constant:
$$\text{constant} = \pi^2 - 2 i \pi \log 2. \tag{4}$$