Specifically, I would like compute the following in closed form for some $\alpha \in [0,1]$
$$ \sum_{\ell \geq 1} \arcsin( \alpha^\ell ) \left( 1 + 2 \alpha^{2\ell} \right) \,. $$
I have tried using the Taylor expansion for $\arcsin$ and summing over the geometric series over $\ell$, it just looks more complicated
$$ \sum_{k \geq 0} \frac{ (2k)! }{ 2^{2k} (k!)^2 (2k+1) } \left( \frac{ \alpha^{2k+1} }{ 1 - \alpha^{2k+1} } + \frac{ 2 \alpha^{2k+3} }{ 1 - \alpha^{2k+3} } \right) \,, $$
and I don't see any further simplifications beyond this point.
Any help would be appreciated!