Look at phi vs u in terms of types in the elliptic functions w.r.t. type

[emsr]

Look at phi vs u in terms of types in the elliptic functions.
It would be more consistent with ellint_3 to have phi be a scalar and u be an object.

[emsr]

sn(k,u) = sin(phi)
u(k,phi) = asn(k, sin(phi))
phi(k,u) = asin(sn(k,u))

[emsr]

I need a name for u.
u is ellint_1!
u(k,phi) = F(k,phi) = \ integ(0,phi)1/sqrt(1-k2 sin2(phi))

[emsr]

The boostorg.math people had a big discussion on this very issue. At least about arguments of jacobi_zeta. They use the same as me and in the same order. They struggles with the same complexity with these functions as me.

Anyway, I think the only difference between u(k,phi) and F(k,phi) is that the former wraps:
Maybe like this:
u(k,phi) = int(phi/(pi/2))*K(k) + F(k,phi%(pi/2));
F is periodic.
Conversely, phi(k,u) should be able to do the modular thing on phi:
phi(k, u) = phi(k, u%K(k))