On 3/4/2014 9:51 AM, email@example.com wrote: > Users may want to use more forms, > and integration program should express (if possible) answer > in terms of functions appearing in users input.
I agree with this.
There is a larger question, which is the context of an actual integral: to what extent does it matter if one has a closed form or not? It is pressing if the integral is actually a multiple (nested) integral as appeared in evaluating Feynman diagrams, in which case evaluating some of the inner ones in closed form reduced the computation substantially. The Reduce system was used (heavily?) for this. But for a single integral, the case is not so clear.
Today, I think that most computational scientists will first turn to a numerical quadrature library for a (definite) integral. The insight that comes from viewing a graph with respect to some parameter may not be a total substitute for the insight in using a formula, but when the formula is full of unfamiliar functions, it is hard to understand. From this perspective, an approximation by series (taylor, asymptotic) might be preferable.
Also, it may be the case that (after considerable computation) the computer algebra system can't figure out some formula, and the computation must be done numerically anyway.
It is also possible that the formula is difficult to evaluate numerically (if that is its ultimate use)... large, unstable...
So the integration "problem" as envisioned by Liouville may be more pure mathematics than applied mathematics. That is, as the subject line suggests, integral for fun. RJF