The exact analytical expression of the period of a conservative nonlinear
oscillator with a non-polynomial potential, is obtained. Such an oscillatory
system corresponds to the transverse vibration of a particle attached to the center of a
stretched elastic wire. The result is given in terms of elliptic functions and validates the
approximate formulae derived from various approximation procedures as the
harmonic balance method and the rational harmonic balance method usually
implemented for solving such a nonlinear problem.