From: Raymond Manzoni <raymman@club-internet.fr>
Subject: Re: Impossible Integral
Date: Fri, 15 Dec 2000 13:43:42 +0100
Newsgroups: sci.physics,sci.math
Summary: [missing]

  Hi,

 
  Concerning the lack of symmetry between computing a derivative and an
integral let's look at a simpler example. Consider all quotients of
polynomials and compute derivatives, sums, products, quotients and so on
: you'll always get quotients of polynomials. Now integrate the quite
simple quotient 1/x, what will you get? Something new, something you
can't write with only finite polynomials (infinite one do the work!).
Now let's accept too logarithmic, exponential, and algebraic expressions
(common acceptance of 'closed form') and some could hope we have
everything since derivatives, sum... are of this kind too but let's
consider the simple 1/ln(x) and ask : what's the integral?
  Derivative often conserve some trace of its integral origin
(exponential for example can't disappear during derivation!). If none is
found (in the field under study) its something new! End of the intuitive
explanation. 

  Powerful methods were developed by Abel, Liouville, Hermite, Hardy,
Mordukhai-Boltovskoi, Ritt ('Integration in finite terms':1948),
Rosenlicht, R.Risch ('The Solution of Problem of Integration in Finite
Terms':1970),Rothstein,Trager,Bronstein and many others (more general
functions were considered with success : error function, polylogarithms,
hypergeometric functions..).
 
  It may be proved that some simple integrals don't admit a closed form
but this becomes rather elaborate for complicated integrals
(computations much more adapted to computers!). But note too that most
of the symbolic packages don't cover all the cases of Risch (and
followers) algorithm (the complete algebraic part is often missing) so
that they can't always certificate there's no closed form.
  A search of 'Risch algorithm will give you more information. 

  This was about indefinite integration. Definite integration is not as
well handled (some definite integrals may be written in closed form
while their corresponding indefinite integral don't) and a narrow view
of current packages shows hidden tables of definite integrals which are
parsed after (or before) the powerful methods failed!

  Online you'll find some fine and first-hand information : Manuel
Bronstein's 'Symbolic Integration Tutorial' available here :
 
http://www-sop.inria.fr/safir/WHOSWHO/Manuel.Bronstein/bronstein-fr.html
  (he wrote a fine book too for a direct implementation)
  
  Another tutorial is Andreas Neubacher's 'An Introduction to the
Symbolic Integration of Elementary Functions' here :
http://www.risc.uni-linz.ac.at/library/DataBase/search.cgi?refkey=10196&OF=0

  A classical reference (Maple's mathematical content survey) is 
'Algorithms for Computer Algebra' by Stephen R. Czapor, George Labahn,
Keith O. Geddes at
http://www.amazon.com/exec/obidos/ASIN/0792392590/qid=976873667/sr=1-20/102-2690704-2616928

  Enjoy all that,

                    Raymond