Some integrals of the Dedekind $eta$ function

Let $eta$ be the weight $1/2$ Dedekind function. A unification and generalization of the integrals $int_0^infty f(x)eta^n(ix)dx$, $n=1,3$, of Glasser cite{glasser2009} is presented. Simple integral inequalities as well as some $n=2$, $4$, $6$, $8$, $9$, and $14$ examples are also given. A prominent result is that $$int_0^infty eta^6 (ix)dx= int_0^infty xeta^6 (ix)dx ={1 over {8pi}}left({{Gamma(1/4)} over {Gamma(3/4)}}right)^2,$$ where $Gamma$ is the Gamma function. The integral $int_0^1 x^{-1} ln x ~eta(ix)dx$ is evaluated in terms of a reducible difference of pairs of the first Stieltjes constant $gamma_1(a)$.