(untitled) (Analysis in Theory and Applications, 2025)
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 _0^1_0^1(1-x^2y^2)^-1dxdy, 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 _0^b[sinh^-1(cosh x)-x]dx, b0 and _α/2^β/2 ln(tanh x)dx, bR, where α := sinh^−1 (1) and β := b + sinh^−1 (cosh b). From these general results, I derive a dilogarithm functional relation.