Elliptic integrals

[salvomic]

These integrals are related to the Jacobi Elliptic Function (see in Wikipedia).

See here in the Library my version for the Prime.

About the two version of 1st kind integral I used in the two programs, they are related, being the same integral: x = sin(φ), then φ=ASIN(x), m=k^2...
The relation involve also sn(), cn(), dn() [sometimes called amplitude sine, amplitude cosine, delta amplitude

However I get values a bit different (see attachment): what kind of rounding is responsible of the difference, in this case?

A part of the examples in the attached image, another example: in the first case for the function ell1F() I test x=1 k=0.4 then ell1F(1,0.4) that returns 1.63999977306; in the second case (Jacobi) for the function Jacobi_fn() I test φ=ASIN(x), m=0.4^2 that returns 1.63999986587...

First case: the integral is int(1/(SQRT((1-t^2)*(1-(k^2)*t^2))),t,0,x);
Second case: the integral is int(1/(SQRT(1-m*SIN(θ)*SIN(θ))),θ,0,φ).
The *should* be the same value.
laTeX form (Wikipedia):
1st form
2nd form

What about it?

Salvo