Recent research on the perimeter of an ellipse

Question

I've recently been reading up on the problem of finding a closed-form solution to the perimeter of an ellipse. I am aware that, in the case of an ellipse, the standard formula used to derive arc-length results in a non-elementary integral. Would anybody be able to point me in the direction of recent research that has been done on the problem, recent breakthroughs related to the problem, or areas of mathematics that would be useful to look into in order to understand the problem better? PDF links to research would be ideal.

Thank you!

Do you want references for various closed form formulas (using elementary functions) approximating the perimeter of an ellipse, or exact formulas that involve elliptic functions, or proofs that the perimeter gives rise to a non-elementary integral? Your question is a little strange, because there is no problem here that would require "recent research" -- it's been well known for a long time that no formula involving elementary functions exist, and that elliptic functions can be used to give a formula, and the integral is non-elementary. — Dave L. Renfro, Aug 23 '22 at 19:46
For some approximate formulas using elementary functions, see Inequalities for the perimeter of an ellipse by Barnard/Pearce/Schovanec (published in 2001). For some exact formulas using elliptic functions, see An eloquent formula for the perimeter of an ellipse (and its references) by Semjon Adlaj (2012), and this paper. For non-elementary integrals, see this question. — Dave L. Renfro, Aug 23 '22 at 20:03

Recent research on the perimeter of an ellipse

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