From: Bill Dubuque <wgd@orion.ai.mit.edu>
Subject: Re: Closed form integration: algebraic case
Date: 18 Oct 1999 23:55:34 -0400
Newsgroups: sci.math

hwatheod@leland.Stanford.EDU (theodore hwa) wrote:
| 
| Matthew Wiener's posting here on closed form integration some years
| back fascinated me [...] The main theorem there was Liouville's theorem
| [...] But the algebraic case appears to be far more complicated [...]

That is indeed the case. The algebraic case involves some non-trivial
algebraic geometry, whereas the transcendental case is easily accessible
to a bright undergrad. In fact the algebraic case is so difficult that
it is only very recently that full implementations have begun to appear
in computer algebra systems. However, there are various special cases
that have simple implementations. If you're interested in the details 
you should refer to the original research papers in the Jnl. of Symbolic
Computation, and various conferences (SYMSAC, EUROSAM, etc), or 
lookup research papers in Math Reviews by Robert Risch, Barry Trager, 
James Davenport, Michael Singer, Manuel Bronstein et. al. 

For much prettier mathematics I highly recommend the beautiful
differential Galois theory involved in the higher-dimensional
generalizations - namely solving LDEs in closed form via solutions 
of lower-order LDEs (e.g. solving third-order LDEs via Bessel
functions). Michael Singer has written many papers on such topics.
See also Irving Kaplansky's beautiful primer: "An introduction to 
differential algebra".

By the way, you'll reach more computer algebraists on sci.math.symbolic.

-Bill Dubuque