Finding Elementary First Integrals for Rational Second Order Ordinary   Differential Equations
  Here we present an algorithm to find elementary first integrals of rational second order ordinary differential equations (SOODEs). In \cite{PS2}, we have presented the first algorithmic way to deal with SOODEs, introducing the basis for the present work. In \cite{royal}, the authors used these results and developed a method to deal with SOODEs and a classification of those. Our present algorithm is based on a much more solid theoretical basis (many theorems are presented) and covers a much broader family of SOODEs than before since we do not work with restricted ansatz. Furthermore, our present approach allows for an easy integrability analysis of SOODEs and much faster actual calculations. 
J. Avellar, L.G.S. Duarte, S.E.S. Duarte and L.A.C.P. da Mota