> Maple uses the conventions in Abramowitz & Stegun or DLMF, https://dlmf.nist.gov/
Maple uses sin(phi) instead of phi as input in elliptic integrals.
And Maple doesn't support properties 19.2.10.
To recover them one should use something like
(-1)^floor(z/Pi + 1/2)*EllipticF(sin(z), k) + 2*floor(z/Pi + 1/2)*EllipticK(k)
instead of EllipticF(sin(z), k) respectively.
Mathematica does 19.2.10.
Mathematica has another advantage in conventions - uses k^2 instead of k. It's useful when you have integral with + sign before k^2. So in Maple you need to put imaginary k*I parameter and then get complex answer with -0.*I imaginary part. Answer that completely real:
In Mathematica you just put real -k^2 and get real answer.
The same issue with some Jacobi functions in Maple.