Asking if something is the derivative of a (combo of) elementary
functions is -exactly- the same kind of question as "ruler and
compass constructions", and thus the topic should be called
"differential galois theory", which it is.
-----------------
When are indefinite integrals/power series/solutions to differential
equations writeable in "closed form", i.e., as an algebraic
combination of elementary transcendental functions (i.e., $\exp$ and $\log$,
hence all trig); we call this an EL function, as "elementary
functions" also allow things algebraic over EL.

Liouville (1835) was the first to prove that stuff like
$\int e^{-x^2}$ wasn't elementary.
other EGs: $(\sin x)/x$ (Dirichlet function?), $x^x$, $\sqrt{1-x^4}$
Meta-theorem: If you can't see how to integrate it easily, it probably 
      can't be integrated in closed form.

Recognizing the geometry inherent in these integrals (i.e., when can
you and can't you find an elementary integral) is what led to the
study of Riemann surfaces, and a lot of algebraic geometry.
EG, elliptic integrals exist b/c elliptic curves exist -- and these
don't have rational parametrization, hence they don't have elementary
anti-derivatives.

These days, this topic goes by the name of "symbolic algebra" and
"differential algebra".

-------

There's an "algorithm" by Risch to do this; the problems with it are:
- it relies on determining if an EL function is zero,
  which is not recursively solvable
- the algebraic solution may only hold on a set of measure 0
- it's often unsatisfying: it just ends with "nope, can't be done"
  (i.e., w/o giving a "good reason")
- it's not useful in applications: definite integrals are easy enough

Hard-core References:
Michael F. Singer: Formal Solutions of Differential Equations; J.
Symb.
Comput. 10 (1990) 59-94
Yiu-Kwong Man: On Computing Closed Forms for Indefinite Summations;
J.Symb.
Comput. 16 (1993) 355-376
R. Moenck: On Computing Closed Forms for Summation. MACSYMA (1977)
225-236
R.H. Risch: Algebraic Properties of Elementary Functions of Analysis;
Am. J.
Math. (1979) 101
Liouville's Theorem (1835, analytic proof)
Risch's Algorithm:
  R.H.Risch: Trans.A.M.S. (1969) 139
  R.H.Risch: Bull.A.M.S. (1970) 76
  R.H.Risch: Proc.AMS (1976) 57
  R.H.Risch: Am.J.Math. (1979) 101
  M.F.Singer: J.Symbolic Comput. (1990) (10) 59-94
  M.Karr: J.Assoc.Comp.Mach. 28 (1981) (2) 305-350
  M.Karr: J.Symbolic Comput. (1985) (1) 303-315
  J.C.Lafon: Computing, Suppl. 4 (1982)  71-77
Indefinite summation: see Gosper, Zeilberger, and Wilf.




(easy algebraic proof these days of Liouville's Theorem:
Maxwell Rosenlicht, Pacific Journal of Mathematics, 54, (1968) 153-161 and 65,
(1976), 485-492)

That integral is non elementary by Chebyshev's theorem. In general,
the integral of  $x^p (a + b x^r)^q dx$  is elementary if and only if
at least one of  (p+1)/r,  q,  or  (p+1)/r + q  is an integer
(see Marchisoto \& Zakeri: "An Invitation to Integration in Finite
Terms", The College Mathematics Journal, Vol. 25, No. 4, Sept.1994,
pp.295-308).

---------------------

From: ajung@informatik.uni-rostock.de (Andreas Jung)
Subject: Re: Risch Algorithm
Date: 12 Mar 1999 18:41:40 +0100
Newsgroups: sci.math.symbolic
Keywords: Why isn't the Risch algorithm implemented?

G. A. Edgar (edgar@math.ohio-state.edu.nospam) wrote:
: In article <19990312013001.25053.00001011@ng-cf1.aol.com>, Jemfinch02 <jemfinch02@aol.com> wrote:

: > Why doesn't any CAS implement the full algorithm?

: Axiom claims to, I believe.

But it's only a claim. For at least the following integrals,
the 1996 version of Axiom failed to find out if a symbolic 
integral exists or not, but printed an error message instead:

  integrate(asin(exp(x)), x)
  integrate(acos(exp(x)), x)
  integrate(asin(log(x)), x)
  integrate(acos(log(x)), x)
  integrate(log(1+sqrt(1-q^2*sin(x)^2)), x)

Perhaps, someone can feed these examples into an actual version
of Axiom and tell us if things have changed.

Greetings,
						Andreas Jung.
--
Andreas Gisbert Jung     DL9AAI Tel:0381/498-3364 Fax:0381/498-3366
Theoretische Informatik  mailto:ajung@informatik.uni-rostock.de
Universitaet Rostock     http://www.informatik.uni-rostock.de/~ajung/
PGP fingerprint =  8A 0B 05 CA EE AB 7B 01  D9 07 6A D0 84 38 BB 82
==============================================================================

From: fateman@peoplesparc.cs.berkeley.edu (Richard J. Fateman)
Subject: Re: Risch Algorithm
Date: 12 Mar 1999 21:58:48 GMT
Newsgroups: sci.math.symbolic

In article <36e951d4@news.uni-rostock.de>,
Andreas Jung <ajung@informatik.uni-rostock.de> wrote:
>G. A. Edgar (edgar@math.ohio-state.edu.nospam) wrote:
>: In article <19990312013001.25053.00001011@ng-cf1.aol.com>, Jemfinch02 <jemfinch02@aol.com> wrote:
>
>: > Why doesn't any CAS implement the full algorithm?
>
>: Axiom claims to, I believe.
>
>But it's only a claim. For at least the following integrals,

There is a fallacy in claiming the Risch "algorithm" is an
algorithm at all:  it depends, for solution of subproblems,
on heuristics to tell if expressions are equivalent to zero.
It was shown in 1968 or so by Daniel Richardson, that the
zero equivalence problem over a class of expressions including
the Risch target class, is recursively unsolvable.  You
can always tell if a polynomial is identically zero. But you cannot
tell by a uniform algorithm if a rational expression in 1 variable
including exp sin sqrt, composition, pi, +-/, constants is zero.
[I think I got that about right..].

But that is not the problem with most implementations of the
Risch algorithm.  They mostly have not been programmed completely,
because, even assuming you can solve the zero-equivalence problem,
the procedure is hard to program.  Why aren't people pushing
to do this more?...

Another reason is, it is not nearly as useful as you might think,
because it returns algebraic antiderivatives whose validity may
be on a set of measure zero.  It may also, in the vast majority
of problems, simply say, after an impressive pause, nope. can't do it.

Also, most people are interested in definite, not INdefinite integrals,
at least once they've finished with Freshman calculus.

And in cases where approximate solutions are easily obtained 
for the corresponding definite integral, the Risch algorithm
may grind on for a while and then say there is no closed form.

The theory of algebraic integration, and the history of this
problem is interesting, but the connection with computer algebra
systems aimed at solving applied mathematics problems in
complex analysis is not nearly as central as one might think.

  My opinion, anyway.


-- 
Richard J. Fateman
fateman@cs.berkeley.edu   http://http.cs.berkeley.edu/~fateman/


