In the case of the circle I can hardly make any conclusions from the integral $(1)$, most of the theorems come from geometrical considerations. It's not clear how to prove periodicity from this integral or derive the addition theorem.

$$\arcsin(y) = \displaystyle \int_{0}^{y} \frac{\mathrm dy}{\sqrt{1 - y^2}} (1)$$

In the case of the lemniscate $(2)$ we (Fagnano) can derive a doubling theorem by trying out substitutions in analogy with those which rationalize the first integral. (I learned this from Siegel - Topics in Complex Function Theory). One result (of the sort I wish I could find more) which does come nicely from this integral is via the substitution $y \mapsto iy$ we notice it has double periodicity.

$$\text{sl}^{-1}(y) = \displaystyle \int_{0}^{y} \frac{\mathrm dy}{\sqrt{1 - y^4}} (2)$$

Euler and others were able to produce theorems about the elliptic integral $(3)$ by analogy with the lemniscate (I read this in Stillwell - Mathematics and its History) - Just as ideas from $\arcsin$ helped to produce theorems about the lemniscate integral. Still, these theorems are very hard earned and it appears that you have to be a master like Euler to derive them.

$$F(y,k) = \displaystyle\int_0^y\frac{\mathrm dy}{\sqrt{(1-k^2 y^2)(1-y^2)}} (3)$$

What I would really like to know is, can we derive more results about the functions from these integrals - maybe using integral manipulation techniques I just don't know about?

Another question that has been bothering me deeply is the integrands (which I believe are called invariant differentials of the lie groups for the algebraic curve) - What sort of coincidence is it that allows the integrands to be of this particular form?

Thank you!