New definite integrals and a two-term dilogarithm identity

Among the several proofs known for ∑n=1∞1/n2=π2/6, the one given by Beukers, Calabi, and Kolk involves the evaluation of ∫01∫011/(1−x2y2)dxdy. It starts by showing that this double integral is equivalent to 34∑n=1∞1/n2, and then a non-trivial trigonometric   change of variables is applied which transforms that integral into ∫∫T1dudv, where TT is a triangular domain whose area is simply π2/8π2/8. Here in this note, I introduce a hyperbolic version of this change of variables and, by applying it to the above integral, I find exact closed-form expressions for ∫0∞[sinh−1(coshu)−u]du, ∫α∞[u−cosh−1(sinhu)]du, and ∫α/2∞ln(tanhu)du, where α=sinh−1(1)α=sinh−1(1). From the latter integral, I also derive a two-term dilogarithm identity.