The Risch algorithm is a decision procedure for indefinite integration that determines whether a given integral is elementary, and if so, returns a closed-form result for the integral. It builds a tower of logarithmic, exponential, and algebraic extensions. The case of algebraic extensions is quite complicated and is therefore not completely implemented in any computer algebra system. Liouville's principle, which dates back to the 19th century, is an important part of the Risch algorithm. There are extensions to the Risch algorithm, notably by Cherry, to be able to handle some special functions.

# Risch Algorithm

## See also

Elementary Function, Horowitz Reduction, Indefinite Integral, Liouville's Principle
*This entry contributed by Bhuvanesh
Bhatt*

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## References

Bronstein, M.*Symbolic Integration I: Transcendental Functions.*New York: Springer-Verlag, 1997.Cherry, G. W.

*Algorithms for Integrating Elementary Functions in Terms of Logarithmic Integrals and Error Functions.*Ph.D. thesis. University of Delaware, 1983.Cherry, G. W. "Integration in Finite Terms with Special Functions: The Logarithmic Integral."

*SIAM J. Computing*

**15**, 1-12, 1986.Cherry, G. W. "An Analysis of the Rational Exponential Integral."

*SIAM J. Computing*

**18**, 893-905, 1989.Davenport, J. H.

*On the Integration of Algebraic Functions.*Berlin: Springer-Verlag, 1981.Geddes, K. O.; Czapor, S. R.; and Labahn, G. "The Risch Integration Algorithm." Ch. 12 in

*Algorithms for Computer Algebra.*Amsterdam, Netherlands: Kluwer, pp. 511-573, 1992.Risch, R. "On the Integration of Elementary Functions Which are Built Up using Algebraic Operations." Report SP-2801/002/00. Santa Monica, CA: Sys. Dev. Corp., 1968.Risch, R. "The Problem of Integration in Finite Terms."

*Trans. Amer. Math. Soc.*

**139**, 167-189, 1969.Risch, R. "The Solution of the Problem of Integration in Finite Terms."

*Bull. Amer. Math. Soc.*, 1-76, 605-608, 1970.Risch, R. "Algebraic Properties of Elementary Functions of Analysis."

*Amer. J. Math.*

**101**, 743-759, 1979.

## Referenced on Wolfram|Alpha

Risch Algorithm## Cite this as:

Bhatt, Bhuvanesh. "Risch Algorithm." From *MathWorld*--A Wolfram Web Resource, created by Eric
W. Weisstein. https://mathworld.wolfram.com/RischAlgorithm.html