Can this series be expressed in closed form, and if so, what is it?

Can this series be expressed in closed form, and if so, what is it?
$$
\sum_{n=1}^\infty\frac{1}{9^{n+1}-1}
$$

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Let us denote by $f$ the following function

$$ f(a):=\sum_{n=1}^\infty\frac{1}{a^{n+1}-1}. $$

Then

$$f(a) =
\sum_{n=1}^\infty\frac{1}{a^{n+1}-1}=
\frac{1}{1-a}-\sum_{n=1}^\infty\frac{1}{1-a^n}\stackrel{\left(\spadesuit\right)}{=}
\frac{1}{1-a}-\frac{\psi_{1/a}(1)+\ln(a-1)+\ln(1/a)}{\ln(a)},$$

where $\psi_q(z)$ denotes the $q$-polygamma function, and in $\left(\spadesuit\right)$ we used the equation $(4)$ from here.

$$
f(9) = \frac78 – \frac{\ln(8)}{\ln(9)} – \frac{\psi_{1/9}(1)}{\ln(9)} \approx 0.014045117662188129358728474369089\dots
$$
Note that $f(2)=\mathcal{C}_{\textrm{EB}}-1 \approx 0.60669515241529\dots,$ where $\mathcal{C}_{\textrm{EB}}$ is the Erdős–Borwein constant.