Elliptic Integrals of the First Kind

Question

Suppose I have $$F(\phi(x), k) = x$$ where the elliptic integral of the first kind is defined to be $$F(\phi, k) = \int_{0}^{\phi} \frac{1}{\sqrt{1-k^2\sin(\theta)}} \, d\theta $$

How could I invert this in order to make $\phi$ the subject?

Answer 1

Maple has this (in terms of the elliptic function sn):

Answer 2

Just to complement GEdgar's answer (+1):

This Wikipedia article gives the definition of the Jacobi elliptic function $sn(x)$ as the inverse of the incomplete elliptic integral of the first kind.

The article gives lots of other definitions of $sn(x)$ by means of either doubly periodic meromorphic functions or alternatively in terms of theta functions, but I believe you re going to be disappointed if you want an inverse function which is easily expressible in terms of elementary functions - sorry.