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 (Trager 1976). Structured special cases, such as pseudo-elliptic integrals, may nevertheless admit separate constructive criteria (Blake 2026).

Desmond (2026) used exhaustive symbolic integration to exhibit elementary antiderivatives involving square-root algebraic extensions over exponential-logarithmic towers that were not found by several tested computer algebra systems. For example, the integrand

has elementary antiderivative , although it resisted SymPy, Mathematica, RUBI, FriCAS, Maxima, and Giac under the strategies tested by Desmond (2026).

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 (1983, 1986, 1989), to be able to handle some special functions.

See also

Elementary Function, Horowitz Reduction, Indefinite Integral, Liouville's Principle, Pseudo-Elliptic Integral, Symbolic Integration

Portions of this entry contributed by Bhuvanesh Bhatt

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References

Blake, S. "IntegrateAlgebraic." Wolfram Function Repository. https://resources.wolframcloud.com/FunctionRepository/resources/IntegrateAlgebraic/.Blake, S. "A Generalisation of Goursat's Algorithm for Integration in Finite Terms." 30 Apr 2026. https://arxiv.org/abs/2604.27806.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." PhD thesis. Newark, DE: 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, Germany: Springer-Verlag, 1981.Desmond, H. "Exhaustive Symbolic Integration: Integration by Differentiation and the Landscape of Symbolic Integrability." 6 May 2026. https://arxiv.org/abs/2605.04978.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.Trager, B. M. "Algebraic Factoring and Rational Function Integration." Master's thesis. Cambridge, MA: Massachusetts Institute of Technology, 1976.

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Risch Algorithm

Cite this as:

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

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