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Pseudo-Elliptic Integral

A pseudo-elliptic integral is an integral of the form

int(F(t)dt)/(sqrt(R(t))),

where F(t) is a rational function and R(t) is a polynomial of degree 3 or 4, which nevertheless has an indefinite integral in elementary functions. The integrand naturally defines a one-form on the algebraic curve y^2=R(t), which has genus one when R has distinct roots and degree 3 or 4, so generic integrals of this form are elliptic integrals rather than elementary functions.

Goursat (1887) gave criteria for such integrals in terms of linear fractional transformations permuting the roots of R. Blake (2026) gives a modern exposition of Goursat's algorithm and develops a cube-root analog for integrals of the form

int(F(t)dt)/(RadicalBox[{R, {(, t, )}}, 3]),

with R(t) cubic.

See also

Algebraic Curve, Elementary Function, Elliptic Integral, Indefinite Integral, Linear Fractional Transformation, Liouville's Principle, Risch Algorithm

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References

Blake, S. "A Generalisation of Goursat's Algorithm for Integration in Finite Terms." 30 Apr 2026. https://arxiv.org/abs/2604.27806.Goursat, E. "Note sur quelques intégrales pseudo-elliptiques." Bull. Soc. Math. France 15, 106-120, 1887.Pappalardi, F. and van der Poorten, A. J. "Pseudo-Elliptic Integrals, Units, and Torsion." 14 Mar 2004. https://arxiv.org/abs/math/0403228.

Cite this as:

Weisstein, Eric W. "Pseudo-Elliptic Integral." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Pseudo-EllipticIntegral.html

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