  We give closed form evaluations for many families of integrals, whose
integrands contain algebraic functions of the complete elliptic integrals $K$
and $E$. Our methods exploit the rich structures connecting complete elliptic
integrals, Jacobi theta functions, lattice sums, and Eisenstein series. Various
examples are given, and along the way new (including 10-dimensional) lattice
sum evaluations are produced.
