\arctan (\frac{x+1}{x-1}) to power series I want to find an expression

${\displaystyle \arctan \left(\frac{x+1}{x-1}\right)}$ to power series
I want to find an expression for ${\displaystyle \arctan \left(\frac{x+1}{x-1}\right)}$ as a power series, with ${\displaystyle x_0=0}$, for every ${\displaystyle x\ne 1}$
My initial thought was to use the known ${\displaystyle \arctan \left(x\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{x}^{2n+1}}{2n+1}}$ but I don't know how to keep going if I replace x with ${\displaystyle \frac{x+1}{x-1}}$