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We can integrate arbitrary rational functions of the trigonometric functions because of the tangent half-angle substitution (https://en.m.wikipedia.org/wiki/Tangent_half-angle_substitution). This led me to the following question:

Is every integral of a rational function of Jacobian elliptic functions (https://dlmf.nist.gov/22) of a fixed modulus expressible as a finite composition of elementary functions, Jacobian elliptic functions, elliptic integrals and the Jacobi epsilon (https://dlmf.nist.gov/22.16#ii)? If so, does there exist an algorithm for it?

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    $\begingroup$ What happens if you try samples of this on Maple or Mathematica? $\endgroup$
    – Gerald Edgar
    Commented Nov 18, 2023 at 22:36
  • $\begingroup$ @GeraldEdgar It takes some time but it looks promising. $\endgroup$
    – Nomas2
    Commented Nov 18, 2023 at 22:54

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The answer is positive. Rational function of Jacobi elliptic function is elliptic, and for every elliptic function $f$, $f(z)dz$ is an Abelian differential, and integral of it is an Abelian integral. Abelian integrals can be decomposed into a sum of the standard elliptic integrals (of first, second and third kind), and each of them is a special function, one of those that you listed. This decomposition is an analog of the decomposition of a rational function into simple fraction. The algorithm of decomposition is described in all standard books on elliptic functions, for example,

N. I. Akhiezer, Elements of the theory of elliptic functions. Translated from the second Russian edition by H. H. McFaden. Translations of Mathematical Monographs, 79. American Mathematical Society, Providence, RI, 1990. viii+237 pp. Chapter 5, section 29.

I am sure that standard computer algebra programs do this.

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  • $\begingroup$ "For every elliptic function $f$, $f(z)dz$ is an Abelian integral" – can this statement be found in the Akhiezer reference? $\endgroup$
    – Nomas2
    Commented Nov 19, 2023 at 11:39
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    $\begingroup$ @Nomas2: I missed the integral sign, now corrected. Yes, all what I wrote can be found in any book on elliptic functions, including Akhiezer. $\endgroup$
    – Alexandre Eremenko
    Commented Nov 19, 2023 at 15:38

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