The answer is positive. Rational function of Jacobi elliptic function is elliptic, and for every elliptic function $f$, $f(z)dz$ 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.

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.