# Is such transformed beta a known distribution?

I have a distribution $Y \sim 1/(1-Beta(\alpha,\beta))$. I would like to understand its properties.

I was able to write down its density function, but I was not able to integrate it to get its mean, for example (except through simulation).

The Beta is mirror-symmetric $1-\mathrm{Beta}(\alpha, \beta) =\mathrm{Beta}(\beta, \alpha)$, so you are essentially looking for the distribution of the reciprocal of a Beta random variable.
Also, if $X\sim \mathrm{Beta}(\beta, \alpha)$, then $1/X-1\sim\mathrm{BetaPrime}(\alpha, \beta)$, where $\mathrm{BetaPrime}$ is the Beta prime distribution.
So, I would say that your $Y-1$ is distributed $\mathrm{BetaPrime}(\alpha, \beta)$.