A very odd-looking statement about $\zeta(3)$ and $\text{Li}_2\left(\frac{1}{\varphi^2}\right)$

There is something about the results that reminds me of work done on Euler double sums by Borwein, Boersma and another author whose name evades me. There was also some work by Peter Jordan on infinite sums of Psi functions that had application to wing theory.

I apologize if that sends you off on the wrong path!

Comment becomes an answer

There are prepublication notes by Tim Jameson on $\zeta(3)$ and $\zeta(4)$ http://www.maths.lancs.ac.uk/~jameson/polylog.pdf

There is also the paper by Yue and Williams which may help you. https://projecteuclid.org/euclid.pjm/1102620561

It would be really interesting if you could work to draw out all the connections in their simplest and clearest form and then publish your work.

Lengthy comment which may be of some help to you:

The simplest integrals I could find (repeatedly transforming by substitution from the basic $\coth(x)$ integral you give above) for $\zeta(2)$ and $\zeta(3)$ involving the Golden Ratio $\phi$ as one of the limits were

$$\zeta(2)=\frac{10}{3} \int_1^{\phi} \log(x) \left( \frac{1}{x-1} +\frac{1}{x+1}-\frac{1}{x} \right) dx$$

$$\zeta(3)={10} \int_1^{\phi} \log(x)^2 \left( \frac{1}{x-1} +\frac{1}{x+1}-\frac{1}{x} \right) dx$$

from the partial fraction expansion $$\frac{1}{x}\frac{(x^2+1)}{(x^2-1)}=\left( \frac{1}{x-1} +\frac{1}{x+1}-\frac{1}{x} \right)$$

These are not a million miles away from the standard integrals used to calculate to di- and tri-logarithms.