Integral involving Glaisher- Kinkelin & Catalan constant

[Tolaso J Kos]

Show that: \( \displaystyle J=\int_{1}^{\infty}\frac{1-x+\ln x}{x\left ( 1+x^2 \right )\ln^2 x}\,dx=\frac{\ln 2}{6}+\frac{\ln \pi}{2}-6\ln \mathbb{A}+\frac{2}{\pi}G \).