Here we derive a bunch of explicit formulas for the circumferences of all types of Cassinian ovals in terms of the complete elliptic integrals of the first kind and their equivalent expressions in terms of the hypergeometric functions.
"The Rectifications of the Cassinian Ovals: Third Round." J. Geom. Symmetry Phys. 57 87 - 98, 2020. https://doi.org/10.7546/jgsp-57-2020-87-98