This morning I try to create interesting integrals involving harmonic numbers. See this Wikipedia. And look at, also if you need it, the definition of the Harmonic number $H_x$ using the digamma function from this MathWorld.
After I've asked to Wolfram Alpha online calculator this code
int -log(1-Harmonic(x))log(1+Harmonic(2x))dx, from x=0 to 1
I wondered this
Question. Can you justify a very good approximation of this $$\text{constant}=-\int_0^1\log\left(1-H_x\right)\log\left(1+H_{2x}\right)\,dx\tag{1}$$ using numerical analysis? Or well, can you provide me the expression of previous constant written as a series with a good rate of convergence? Many thanks.
You can to choose some of the two previous approaches. But if a different approach (yours) gets a very good approximation/information of the constant defined in $(1)$, feel free to add it as an answer.
As said in comments, the expansion around $x=\frac 12$ is more than ugly. Moroever, the convergence does not look to ve very fast $$\left( \begin{array}{cc} n & \text{result} \\ 2 & 0.847997 \\ 4 & 0.901222 \\ 6 & 0.925063 \\ 8 & 0.938477 \\ 10 &0.947086 \end{array} \right)$$
A quick and dirty nonlinear regression $$y=\frac{0.619974+0.809459\, n}{1+0.820115\, n}$$ leads to an asymptotic value of $0.987006$.