Finding first order differential invariants through the S-function. The method presented in Duarte and da Mota (2009) and Avellar et al. (2014) to search for first order invariants of second order ordinary differential equation (2ODEs) makes use of the so called Darboux polynomials. The main difficulty involved in this process is the determination of the Darboux polynomials, which is computationally very expensive. Here, we introduce an optional argument in the main routine that enables a shortcut in the calculations through the use of the -function associated with the 2ODE