This work is part of the continuing activity at Berkeley in computer systems for symbolic mathematics (computer algebra). This research involves the establishment and extension of TILU, our online table-look-up symbolic integration service. This table-look-up service currently includes only about 700 forms, although we have data to extend it to the majority of about 13,000 forms in current tables, including many definite integrals involving special functions.
Why bother with table look-up, when there are alleged to be algorithms for symbolic integration? There are several reasons. The best of the algorithms are for indefinite integration only; even so, they somewhat ignore special functions, and often provide a useless result: that no answer exists in closed form in terms of elementary functions. The algebraic algorithms generally ignore the domain of validity of the answer. The results of some of the more powerful techniques are, in many cases, woefully unsimplified. Our table look-up can do the task in about 6.5 milliseconds CPU time on an HP-9000/715 at http://torte.cs.berkeley.edu:8010/tilu. If we can't find it, you will know soon. By comparison, typical algorithms can be orders of magnitude slower.
TILU serves an average of 125 requests a day, and is now tied to a button on the new release of the "Graphing Calculator" shipped with every Apple Macintosh computer.
In addition to studying extensions of this system to more integrals, we are also considering applications to other domains such as symbolic solution of ordinary differential equations.