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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.
}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-2018-0011}, url = {http://global-sci.org/intro/article_detail/ata/23863.html} }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.