  We have already dealt with the problem of solving First Order Differential
Equations (1ODEs) presenting elementary functions before in [1, 2]. In this
present paper, we have established solid theoretical basis through a relation
between the 1ODE we are dealing with and a rational second order ordinary
differential equation, presenting a Liouvillian first Integral. Here, we have
expanded the results in [3], where we have establish a theoretical background
to deal with rational second order ordinary differential equations (2ODEs) via
the S-function method. Using this generalisation and other results hereby
introduced, we have produced a method to integrate the 1ODE under scrutiny. Our
methods and algorithm are capable to deal efficiently with chaotic systems,
determining regions of integrability.
