I tried to understand the integral form of the AGM and its connection to the complete elliptic integral of first kind. (similar to this question)
The main point is to show that
$$\int_0^{\frac{\pi}{2}} \frac{\mathrm{d}\theta}{\sqrt{x^2 \cos^2 \theta + y^2 \sin^2 \theta}}=\int_0^{\frac{\pi}{2}} \frac{\mathrm{d}\phi}{\sqrt{(\frac{x+y}{2})^2 \cos^2 \theta + x y \sin^2 \phi}}$$
as, e.g., in the Wikipedia article and using the substitution
$$ \sin \theta = \frac{2 x \sin \phi }{ (x+y)+(x-y) \sin^2 \phi} $$
It was not difficult but quite cumbersome to make all the simplifications to show eventually that all this is true.
But then, how on earth do you come up with this substitution in the first place.
Do you need to be a genius like Gauss or is there a somewhat simple motivation for the substitution?
A reasonable goal probably was trying to find a substitution that gets $x \rightarrow x'$ and $y\rightarrow y'$ with $x'=y'$. Which you actually get in the sense of a limit to the AGM, but still makes me wonder how to imagine the choice of substitution.
