Summary: In this article we solve in closed form a nonlinear differential equation modelling the planar, non-inflectional elastica (deflection \(y\)) of a thin, flexible, simply supported, \(x\)-straight rod, which withdraws bending and doesn’t retain its original length \(L\) when deformed under a compressive thrust. We solve a Dirichlet two-point boundary value problem providing an explicit inverse function \(x(y)\) through elliptic integrals of I and II kind. Integration is performed twice via the “shooting” getting both branches of elastica. In such a way \(y(x)\) is required to be invertible, and then our solution is found within each of two monotonicity \(x\)-ranges for \(y\), before and beyond the bifurcation value \(x^*= L/2\). An auxiliary unknown is then introduced and successively computed through a suitable welding condition. In such a way \(x= x(y)\) is completely known in its two symmetrical branches.