On Some Doubly Logarithmic Integrals in Gradshteyn and Ryzhik

Duc Van Khanh Tran

Published 2023-07-18Version 1

In the paper "Integrals, an Introduction to Analytic Number Theory," Vardi proved the integral identity $\int_{\pi /4}^{\pi/2} \ln{\ln{\tan{x}}}\,dx = \frac{\pi}{2}\ln{\left(\frac{\Gamma\left(\frac{3}{4}\right)}{\Gamma\left(\frac{1}{4}\right)}\sqrt{2\pi}\right)}$ in the Gradshteyn and Ryzhik table using an analytic number theoretical method. In this paper, we prove the integral identity above and some other similar identities in the Gradshteyn and Ryzhik table analytically without involving number theoretical methods.