# The Risch Integration Algorithm

• K. O. Geddes
• S. R. Czapor
• G. Labahn
Chapter

## Abstract

When solving for an indefinite integral, it is not enough simply to ask to find an antiderivative of a given function f(x). After all, the fundamental theorem of integral calculus gives the area function A(x)=∭ x a f(t) dt as an antiderivative of f (x). One really wishes to have some sort of closed expression for the antiderivative in terms of well-known functions (e.g. sin(x), e x, log(x)) allowing for common function operations (e.g. addition, multiplication, composition). This is known as the problem of integration in closed form or integration in finite terms. Thus, one is given an elementary function f(x), and asks to find if there exists an elementary function g(x) which is the antiderivative of f(x) and, if so, to determine g(x)

## Keywords

Computer Algebra Rational Part Integration Algorithm Constant Field Integral Basis
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
J. Baddoura, “Integration in Finite Terms and Simplification with Dilogarithms: a Progress Report,” pp. 166–171 in Proc. Computers and Math., ed. E. Kaltofen, S.M. Watt, Springer-Verlag (1989).Google Scholar
2. 2.
R.J. Bradford, “On the Computation of Integral Bases and Defects of Integrity,” Ph.D. Thesis, Univ. of Bath, England (1988).Google Scholar
3. 3.
M. Bronstein, “Simplification of Real Elementary Functions,” pp. 207–211 in Proc. ISSAC’ 89, ed. G.H. Gnnet, ACM Press (1989).Google Scholar
4. 4.
M. Bronstein, “The Transcendental Risch Differential Equation,” J. Symbolic Comp., 9(1) pp. 49–60 (1990).
5. 5.
M. Bronstein, “Integration of Elementary Functions,” J. Symbolic Comp., 9(2) pp. 117–173 (1990).
6. 6.
M. Bronstein, “A Unification of Liouvillian Extensions,” Appl. Alg. in E.C.C., 1(1) pp. 5–24 (1990).
7. 7.
G.W. Cherry, “Integration in Finite Terms with Special Functions: the Error Function,” J. Symbolic Comp., 1 pp. 283–302 (1985).
8. 8.
G.W. Cherry, “Integration in Finite Terms with Special Functions: the Logarithmic Integral,” SIAM J. Computing, 15 pp. 1–21 (1986).
9. 9.
J.H. Davenport, “Integration Formelle,” IMAG Res. Rep. 375, Univ. de Grenoble (1983).Google Scholar
10. 10.
J.H. Davenport, “The Risch Differential Equation Problem,” SIAM J. Computing, 15 pp. 903–918 (1986).
11. 11.
E. Hecke, Lectures on the Theory of Algebraic Numbers, Springer-Verlag (1980).Google Scholar
12. 12.
E. Kaltofen, “A Note on the Risch Differential Equation,” pp. 359–366 in Proc. EUROSAM’ 84, Lecture Notes in Computer Science 174, ed. J. Fitch, Springer-Verlag (1984).Google Scholar
13. 13.
A. Ostrowski, “Sur l’integrabilite elementaire de quelques classes d’expressions,” Commentaai Math. Helv., 18 pp. 283–308 (1946).
14. 14.
R. Riscn, “On the Integration of Elementary Functions which are built up using Algebraic Operations,” Report SP-2801/002/00. Sys. Dev. Corp., Santa Monica, CA (1968).Google Scholar
15. 15.
R. Riscn, “The Problem of Integration in Finite Terms,” Trans. AMS, 139 pp. 167–189 (1969).Google Scholar
16. 16.
R. Risch, “The Solution of the Problem of Integration in Finite Terms,” Bull. AMS, 76 pp. 605–608 (1970).
17. 17.
R. Risch, “Algebraic Properties of the Elementary Functions of Analysis,” Amer. Jour. of Math., 101 pp. 743–759 (1979).
18. 18.
J.F. Ritt, Integration in Finite Terms, Columbia University Press, New York (1948).
19. 19.
M. Rosenlicht, “Integration in Finite Terms,” Amer. Math. Monthly, 79 pp. 963–972 (1972).
20. 20.
M. Rosenlicht, “On Liouville’s Theory of Elementary Functions,” Pacific J. Math, 65 pp. 485–492 (1976).
21. 21.
M. Rothstein, “Aspects of Symbolic Integration and Simplification of Exponential and Primitive Functions,” Ph.D. Thesis, Univ. of Wisconsin, Madison (1976).Google Scholar
22. 22.
M. Singer, B. Saunders, and B.F. Caviness, “An Extension of Liouville’s Theorem on Integration in Finite Terms,” SIAM J. Computing, pp. 966–990 (1985).Google Scholar
23. 23.
B. Trager, “Integration of Algebraic Functions,” Ph.D. Thesis, Dept. of EECS, M.l.T. (1984).Google Scholar

• K. O. Geddes
• 1
• S. R. Czapor
• 2
• G. Labahn
• 1