I continue my work about the integral as $\int_{0}^{\infty }\!{\frac {t\ln \left( {t}^{2}+{z}^{2} \right) }{{ {\rm e}^{a\pi\,t}}-1}}\,{\rm d}t$.
Recently, i find the general formula for $\int_{0}^{\infty }\!{\frac {t\ln \left( {t}^{2}+{z}^{2} \right) }{{ {\rm e}^{3\,\pi\,t}}-1}}\,{\rm d}t$ in terms of Barnes G-function
And for example i have $\int_{0}^{\infty }\!{\frac {t}{{{\rm e}^{3\,\pi\,t}}-1}\ln\left( {t} ^{2}+{\frac{1}{4}} \right) }\,{\rm d}t$ = $-{\frac {\ln \left( \pi \right) }{6}}-{\frac {23\,\ln \left( 2 \right) }{108}}-{\frac {{\it G}}{9\,\pi}}-{\frac{83}{432}}+{ \frac {\ln \left( A \right) }{18}}+{\frac {19\,\ln \left( 3 \right) }{216}}+{\frac {\ln \left( \Gamma \left( {\frac{1}{4}} \right) \right) }{3}}$ where G is the Catalan's constant and A is the Glaisher-Kinkelin's constant.
But i don't have any idea if we can prove this closed form with the residue theorem.
Please if someone can prove it ?