Note: This question has been brought here from MSE.

I have been working on various sums involving the zeta function (which come up frequently in my research), and it turned out that many of them had nice closed forms. Today, I was trying to evaluate $$\sum_{r\ge2}\frac{\zeta(r)}{r^2}$$ which turned out to be a bit harder than the others. Wolfram Alpha and Mathematica both cannot give the answer, they only give an approximate value of $0.835998$. After half an hour of work, I turned this sum to $$-\int_{0}^{\infty}\frac{t}{e^t}\psi(1-e^{-t})dt-\gamma$$ where $\gamma$ is the Euler–Mascherni constant and $\psi$ is the Digamma function. Now I don't know what to do further. Does this integral have any closed form? Any help would be appreciated. By the way, some of my ideas to evaluate the integral would be to use some integral or sum representation of the digamma function, and in this case we can interchange the sums and integrals safely.
Note: My work is too long to be stated here, so I cannot write it.
Update: This integral can be further turned to $$\lim_{n\to\infty}H_n-\gamma-\sum_{k=1}^{n}\mathrm{Li}_2\left(\frac1k\right)$$ where $\mathrm{Li}$ is the polylogarithm. I turned this to $$\lim_{n\to\infty}H_n-\gamma-\sum_{r=1}^{\infty}\frac{H_{n,r}}{r^2}$$ where $H_{n,r}$ are generalized harmonic numbers.

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    $\begingroup$ I'm not sure about a closed-form, but an arbitrarily good analytic approximation is $\sum_{k=2}^N\frac{\zeta(k)-1}{k^2}+\pi^2/6 -1$, which should be rather accurate even for relatively small values of $N$. Of course the exact result is recovered in the limit $N\to\infty$. $\endgroup$ – Zachary Dec 11 '20 at 19:36

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