Hmmm I thought the closed form solutions are Jacobi elliptics?

https://cococubed.com/code_pages/pendulum.shtml

Yes. That's what the author means by

> There is a closed-form solution, but only if you extend “closed-form” to mean more than the elementary functions a student would see in a calculus class.

My bad.

I should have encouraged more curiosity by pointing out that these functions actually have additive as well as multiplicative identities

https://msp.org/pjm/1955/5-2/pjm-v5-n2-p02-p.pdf

(Might even be relevant to the pendulum, see the comment below on the Arithmetic-Geometric Mean ;)

Never saw that AGM expression for K0 before in the earlier linked post. Nor had I heard of the AGM, very cool.