Incomplete Elliptic Integrals of the second kind -- evaluation

I am programming as a hobby and trying something with elliptical orbits of satellites.
I need to understand how to measure the length of an arc on an ellipse, e.g. from 0 to pi/3.
Now I've come upon Elliptic Integrals and an associated brick wall.

For the "Complete Elliptic Integral of the second kind" the length of an arc from 0 to pi/2, I've found a series the program can use to approximate the value of this kind of arc.

However, for the INCOMPLETE sort, there is nowhere a description of how the value is calculated, no series, no formula. Nothing.

They tell you in most places either :
1. Look it up
or
2. use a website to calculate it, such as https://keisan.casio.com/

The casio.com website can indeed calculate the value where k = 0.6 and the limits are 0 to pi/3 bringing the value:
0.98922159...... so there must be an infinite series one utilizes to calculate this.

It is most certainly not an elementary function. So, what is the series?