Beauty of mathematical constants

For $x\in\mathbb(0,1)$ , if $f(x)=n-k^x$ and $g(x)=1+k^x$ for $n>0,k\geq 0$ some consecutive integers, $n>k$ $\sum_{n=1}^{\infty}\frac{(-1)^n}{2n^2}\int_0^1\left(\ln f(x)+2\ln g(x)\right)dx=-\frac{\eta(2)}{\zeta(2)}\ln\left(\frac{1}{2^{\zeta(4)}}\prod_{m\geq 1}\sqrt[m^2]{m^2}\right)=\eta(2)\ln\left(\frac{2e^{\gamma}\pi}{A^{22}}\right)$ Is the above closed form is correct?

where $\zeta(.)$ is Riemann zeta function $\eta(.)$ is Dirichlet eta funtion , $A$ is Glashier-kinkelin constant , $\gamma$ is Euler-Mascheroni constant and $e$ is Euler-number

Inspiration .