A semi-algorithm to find elementary first order invariants of rational second order ordinary differential equations

Here we present a method to find elementary first integrals of rational second order ordinary differential equations (SOODEs) based on a Darboux type procedure [L.G.S. Duarte, S.E.S. Duarte, L.A.C.P. da Mota, A method to tackle first order ordinary differential equations with Liouvillian functions in the solution, J. Phys. A: Math. Gen. 35 (2002) 3899-3910, L.G.S. Duarte, S.E.S. Duarte, L.A.C.P. da Mota, Analyzing the structure of the integrating factors for first order ordinary differential equations with Liouvillian functions in the solution, J. Phys. A: Math. Gen. 35 (2002) 1001-1006]. Apart from practical computational considerations, the method will be capable of telling us (up to a certain polynomial degree) if the SOODE has an elementary first integral and, in the positive case, finds it via quadratures.