@Article{ATA-40-422,
author = {Lima , F. M. S.},
title = {Generalization of Certain Hyperbolic Integrals and a Dilogarithm Functional Relation},
journal = {Analysis in Theory and Applications},
year = {2025},
volume = {40},
number = {4},
pages = {422--434},
abstract = {<p style="text-align: justify;">In a previous work [Indag. Math., 23(1) (2012)], I did employ a hyperbolic
version of the Beukers, Calabi, and Kolk change of variables to solve $$\int_0^1\int_0^1(1-x^2y^2)^{-1}dxdy,$$ which yielded exact closed-form expressions for some definite integrals and, from one
of them, I proved a two-term dilogarithm identity. Here in this note, I derive closed-form expressions for $$\int_0^b[{\rm sinh}^{-1}({\rm cosh} \ x)-x]dx, b\ge 0 \ {\rm and} \ \int_{\alpha/2}^{\beta/2} {\rm ln}({\rm tanh} \ x)dx, \ \ b\in \mathbb{R},$$ where $\alpha := {\rm sinh}^{−1} (1)$ and $β := b + {\rm sinh}^{−1} ({\rm cosh} \ b).$ From these general results, I derive
a dilogarithm functional relation.</p>},
issn = {1573-8175},
doi = {https://doi.org/10.4208/ata.OA-2018-0011},
url = {http://global-sci.org/intro/article_detail/ata/23863.html}
}