  Here we present a very efficient method to search for Liouvillian first
integrals of second order rational ordinary differential equations (rational
2ODEs). This new algorithm can be seen as an improvement to the S-function
method we have developed [24]. Here, we show how to further use the knowledge
of the S-function to find an integrating factor of a set of first order
rational ordinary differential equations (rational 1ODEs) which is shared by
the original 2ODE, without having to actually solving these 1ODEs. This new use
of the S-function, that is the theoretical basis of our new method to compute
the integrating factor, proved to be a linear process of computation for a vast
class of non-linear rational 2ODEs, making it much more efficient.
