Symbolic Integration: The Algorithms

The pioneering paper about symbolic integration is:

Joel Moses:
Symbolic Integration: The Stormy Decade
Communications of the ACM, Vol 14, No 8 August 1971, pp. 548-650

This paper is an updated resumee of Moses' thesis 'Symbolic Integration', which is available at these addresses:

• http://theses.mit.edu/Dienst/UI/2.0/Describe/0018.mit.theses/1967-24.
• http://www.lcs.mit.edu/publications/specpub.php?id=615

A modern text about symbolic integration is:

Manueal Bronstein:
Symbolic Integration I Transcendental Functions
1997 Springer Berlin Heidelberg New York

Fatemans paper about algebraic simplification:

The Integration Algorithm

The following description of the integration algorithm is taken from the ACM article (with substancial omissions):

First Stage

The first stage is a simple test to determine whether derivatives of a subexpression of the integrand divide the rest of the integrand. This test determines whether the integrand is of the form:

∫ c* op(u(x))*u'(x) dx

where

• c is a constant
• u(x) is some function of x
• u'(x) is its derivative
• op is an elementary function, namely one of
  • sin
  • cos
  • tan
  • cot
  • sec
  • csc
  • asin
  • atan
  • asec
  • log
  In addition, op(u(x)) can have the forms

  u(x)	(op being the identity)
  1/u(x)	(op being the reciprocal)
  u(x)d	where d <> -1
  du(x)	where d is a constant.

For integrands of this form, a table lookup and a substitution are sufficient to answer the integral.

Experience shows that this simple algorithm solves many integrals. The first stage examples demonstrates the power of the algorithm.

Second Stage

The algorithm enters the second stage when the first stage cannot solve the integral. The second stage contains eleven methods which might be applicable to a given problem. A pattern matching routine determines which methods should be attempted.

second stage examples.

Third Stage

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