How to calculate $$ \int_{0}^{2K(k)} dn(u,k)^2\;du?$$ Where $dn$ is the Jacobi Elliptical function dnoidal and $k \in (0,1)$ is the modulus. I know from the Fórmula $(110.07)$ of [1] (see page 10) that $$ \int_{0}^{K(k)} dn(u,k)^2\;du=E(k),$$ where $E$ is the normal elliptic integral of the second kind complete. For this I can conclude that $$ \int_{0}^{2K(k)} dn(u,k)^2\;du=2E(k)?$$

[1] P. F. Byrd. M. D. Friedman. Hand Book of Elliptical Integrals for Engineers and Scientis. Springer-Verlag New York Heidelberg Berlim, $1971$.