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 $$\int_0^\infty \eta^6 (ix)dx= \int_0^\infty x\eta^6 (ix)dx ={1 \over {8\pi}}\left({{\Gamma(1/4)} \over {\Gamma(3/4)}}\right)^2,$$ 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 $\gamma_1(a)$.