At the end of this 90 minute lecture by Stephen Wolfram on the history of the development of Mathematica, Wolfram takes questions from the audience. One of those questions asked how the
Integrate function chooses the method to compute the integral, whether substitution, integration-by-parts, and so on.
Stephen Wolfram answered "None of those". He then went on to briefly mention a higher level method and I did not quite understand the name of the method nor the way this does Integration. But, it was an all too brief answer.
Does anyone know what the details are on the methods employed by Mathematica to do this integration operation. Wolfram also mentioned that the methods displayed in the "step-by-step" details is a "complete fake" (his words), invented purely for pedagogical reasons. Wolfram also mentioned that the problem of converting the answer to the Integrate function into something meaningful and useful to the user was the bigger part of the problem.