On Tuesday, March 4, 2014 7:31:02 PM UTC-5, Richard Fateman wrote: > 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.
You seem to want numbers, in such case using numerical methods is appropriate. I see researches that spend considerable effort to obtain formulas. For them symbolic integral will be valuable and even getting messy or no answer may help indicating that there are no simple answer (or messy answer may be simplified with some manual help). To put this differently, having symbolic integral is rare event (even when using enlarged class of integrands) which when happens have some significance (for example some kind of singularities can not appear in elementary functions, larger classes allow more singularities but still of restricted kind). So it makes sense to search for closed forms, even if they are rare.
Concerning applications: given current industrial secrecy it it hard to say what is really applied -- there are things which are claimed to be used but my impression is that play no real role and there are used things kept secret. On research side a lot of "applied" research looks like bad theory to me: bad because it solve neither real applied problem nor any fundamental question. Compared to this symbolic integration has _some_ applications and background theory tell something about algebraic nature of functions. This to me looks better than average "applied" research.