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.
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. Ph.D. thesis. University of Delaware, 1983.Cherry,
G. W. "Integration in Finite Terms with Special Functions: The Logarithmic
Integral." SIAM J. Computing15, 1-12, 1986.Cherry,
G. W. "An Analysis of the Rational Exponential Integral." SIAM
J. Computing18, 893-905, 1989.Davenport, J. H. On
the Integration of Algebraic Functions. Berlin: 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. S.M. thesis.
Massachusetts Institute of Technology, 1976.
, although it resisted SymPy, Mathematica,
RUBI, FriCAS, Maxima, and Giac under the strategies tested by Desmond (2026).
Blake, S. "IntegrateAlgebraic." Wolfram
Function Repository.