I want to calculate $$ \int_{0}^{2K(k)} dn(u,k)^4\;du.$$ Where $dn$ is the Jacobi Elliptical function dnoidal and $k \in (0,1)$ is the modulus. I know from the Fórmula $(314.04)$ of [1] (see page 194) that $$ \int dn(u,k)^4\;du= \frac{1}{3}[k^2 sn (u) cn (u) dn (u) - k'^2 u + 2(1 + k'^2) E(u)]. $$ where $E$ is the normal elliptic integral of the second kind complete and $E(u)=E(am(u),k)$. From this, how to calculate $$\int_{0}^{2K(k)} dn(u,k)^4\;du?$$
I am not able to evaluate the expression above at the extremes $0$ and $2K(k)$.
[1] P. F. Byrd. M. D. Friedman. Hand Book of Elliptical Integrals for Engineers and Scientis. Springer-Verlag New York Heidelberg Berlim, $1971$.