This paper deals with the calculation of the period and action integral of second-order autonomous nonlinear oscillators. We show that the Laplace transform of these energy-dependent quantities can be simply expressed as a function of the potential function which makes it possible to obtain a number of new results in an analytical closed-form. In addition to their computational interest, these results allow us to revisit the characterization and properties of classical isoperiodic and isochronous potentials. Applications in quantum mechanics for the semiclassical Einstein-Brillouin-Keller quantization and in statistical mechanics for Gibbs ensembles of nonlinear oscillators are also proposed.