This question has an interesting aspect that merits being commented, which is that we can compute a closed form for the sum using Mellin transforms and harmonic sums.
Introduce the sum $S(x)$ given by $S(x) = \sum_{n\ge 1} \left(1- xn \log\frac{2xn+1}{2xn-1}\right)$ so that we are interested in $S(1).$
As mentioned before the sum term is harmonic and may be evaluated by inverting its Mellin transform.
Recall the harmonic sum identity $\mathfrak{M}\left(\sum_{k\ge 1} \lambda_k g(\mu_k x);s\right) = \left(\sum_{k\ge 1} \frac{\lambda_k}{\mu_k^s} \right) g^*(s)$ where $g^*(s)$ is the Mellin transform of $g(x).$
In the present case we have $\lambda_k = 1, \quad \mu_k = k \quad \text{and} \quad g(x) = 1-x\log\frac{2x+1}{2x-1} = 1-x\log\left(1+\frac{2}{2x-1}\right).$
The abscissa of convergence of $\sum_{k\ge 1} \frac{\lambda_k}{\mu_k^s} = \zeta(s)$ is $\Re(s)>1.$
Now to find the Mellin transform $g^*(s)$ of $g(x)$ the fundamental strip being $\langle 0,2\rangle$ (which contains the abscissa of convergence of the Dirichlet series sum term) we first use integration by parts to get $\int_0^\infty \left(-x\log\left(1+\frac{2}{2x-1}\right)\right) x^{s-1} dx \\= \left[\left(-x\log\left(1+\frac{2}{2x-1}\right)\right) \frac{x^{s+1}}{s+1}\right]_0^\infty - \int_0^\infty \frac{4}{4x^2-1} \frac{x^{s+1}}{s+1} dx = - \int_0^\infty \frac{4}{4x^2-1} \frac{x^{s+1}}{s+1} dx.$ with the fundamental strip being $\langle -1, 0\rangle$. It becomes evident that we require the following Mellin transform: $h^*(s) = \int_0^\infty h(x) x^{s-1} dx$ where $h(x) = \frac{4}{4x^2-1}.$ This transform integral is not strictly speaking convergent but we can compute its principal value by using a semicircular contour in the upper half plane that is traversed clockwise and picks up half the residues at the two poles at $\pm 1/2.$ This gives $h^*(s) (1-e^{i\pi s}) = \frac{1}{2} \times 2 \pi i \left(\mathrm{Res}(h(x) x^{s-1}; x=1/2) + \mathrm{Res}(h(x) x^{s-1}; x=-1/2)\right)$ which gives $h^*(s) (1-e^{i\pi s}) = \frac{1}{2} \times 2 \pi i ((1/2)^{s-1}-(-1/2)^{s-1}) = 2^{-s} \times 2 \pi i (1+(-1)^s).$ This yields $h^*(s) = 2^{-s} \times 2 \pi i \frac{1+e^{i\pi s}}{1-e^{i\pi s}} = 2^{-s} \times 2 \pi i \frac{e^{-i\pi s/2}+e^{i\pi s/2}}{e^{-i\pi s/2}-e^{i\pi s/2}} = -\frac{2}{2^s} \pi \cot(\pi s/2).$ Returning to $g^*(s)$ we have $g^*(s) = \frac{2}{2^{s+2}} \frac{1}{s+1} \pi \cot(\pi (s+2)/2) = \frac{1}{2^{s+1}} \frac{1}{s+1} \pi \cot(\pi s / 2).$
Therefore the Mellin transform $Q(s)$ of $S(x)$ is given by $\frac{1}{2^{s+1}} \frac{1}{s+1} \pi \cot(\pi s / 2)\zeta(s).$
The Mellin inversion integral here is $\frac{1}{2\pi i} \int_{3/2-i\infty}^{3/2+i\infty} Q(s)/x^s ds$ which we evaluate by shifting it to the right for an expansion about infinity.
We must sum the residues at the poles at the positive even integers of $Q(s)/x^s$ which are $\mathrm{Res}(Q(s)/x^s; s=2q) = \frac{1}{2^{2q+1}} \frac{1}{2q+1} \times 2 \times \zeta(2q) \frac{1}{x^{2q}} \\ =\frac{1}{2^{2q+1}} \frac{1}{2q+1} \times 2 \times \frac{(-1)^{q+1} B_{2q} (2\pi)^{2q}}{2\times (2q)!} \frac{1}{x^{2q}} = \frac{1}{2} \times \frac{(-1)^{q+1} B_{2q} \pi^{2q}}{(2q+1)!} \frac{1}{x^{2q}}.$
We thus require the sum $ - \frac{1}{2} \sum_{q\ge 1} \frac{i^{2q} B_{2q} \pi^{2q}}{(2q+1)!} \frac{1}{x^{2q}}.$
The exponential generating function of the even Bernoulli numbers is $-1 + \frac{1}{2} t + \frac{t}{e^t-1} = \sum_{q\ge 1} B_{2q} \frac{t^{2q}}{(2q)!}.$ Integrate this to obtain $-t - \frac{t^2}{4} + t\log(1-e^t) + \mathrm{Li}_2(e^t).$ The constant that appeared during the integration was the well-known zeta function value $\mathrm{Li}_2(1) = \zeta(2) = \pi^2/6$ so that we finally have $ \sum_{q\ge 1} B_{2q} \frac{t^{2q}}{(2q+1)!} = \frac{1}{t} \left(-\frac{\pi^2}{6}-t - \frac{t^2}{4} + t\log(1-e^t) + \mathrm{Li}_2(e^t)\right).$ To conclude use another well known zeta function value which is $\mathrm{Li}_2(-1) = -\pi^2/12$ (derived from $\mathrm{Li}_2(1)$) and put $t=i\pi /x = i\pi$ to obtain that (we lose the minus sign because we are shifting to the right) $S(1) = \frac{1}{2} \frac{1}{i\pi} \left(-\frac{\pi^2}{6}-i\pi + \frac{\pi^2}{4} + i\pi\log 2 + \mathrm{Li}_2(-1)\right) \\ = \frac{1}{2} \frac{1}{i\pi} \left(-i\pi + i\pi \log 2 \right) = \frac{1}{2} (\log 2 - 1).$
Remark. The integral of the generating function of the Bernoulli numbers is easily verified by differentiation: $\left(-\frac{t^2}{2} + t\log(1-e^t) + \mathrm{Li}_2(e^t)\right)' = -t + \frac{t}{1-e^t} (-e^t) + \log(1-e^t) + \left(\sum_{n\ge 1} \frac{e^{nt}}{n^2}\right)' \\ = \frac{-t+te^t}{1-e^t} - \frac{te^t}{1-e^t} + \log(1-e^t) + \sum_{n\ge 1} \frac{e^{tn}}{n} \\ = \frac{-t}{1-e^t} + \log(1-e^t) - \log(1-e^t) = \frac{t}{e^t-1}.$