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 \cite{ManMac,firsTHEOps1,secondTHEOps1}. 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 positive case, finds it via quadratures. 
J. Avellar, L.G.S. Duarte, S.E.S. Duarte and L.A.C.P. da Mota