Closed form for a family of integrals of the form $\int_{0}^{\infty}\Phi(x)x^{m}\cosh(x)dx$ and $\int_{0}^{\infty}\Phi(x)x^{m}\sinh(x)dx$

Question

The above $\Phi(x)$ is related to the Riemann's $\xi(s)$ function.

Specifically $$ \Phi(x)=\sum_{n=0}^{\infty}(2\pi^{2}n^{4}e^{9x}-3\pi n^{2}e^{5x})\exp(-\pi n^{2}e^{4x}) $$ As stated above, I'm looking for a closed form for the family of integrals $$ \begin{align*} S_{m}&=\int_{0}^{\infty}\Phi(x) x^{m}\sinh(x)dx\\ C_{n}&=\int_{0}^{\infty}\Phi(x) x^{n}\cosh(x)dx\\ \end{align*} $$ Where $n\in \mathbb{N}_{0}$

I was able to find $C_0$, i.e. $$ \int_{0}^{\infty}\Phi(x) \cosh(x)dx=\frac{1}{16} $$

It'd be interesting to see if there are closed forms for the other cases, $S_m, S_n$ ;).