Natural logarithm 2F1,hypergeometric

Question

I find out that $\ln (1+z) = z _2F_1(1,1,2,-z)$ and that $\ln (1-z) = -z _2F_1(1,1,2,z)$, but what is $\ln\Big( \dfrac{1+z}{1-z} \Big)$? Is there a possibility to add two $_2F_1$?

I mean what can I do with this $\ln\Big( \dfrac{1+z}{1-z} \Big) = z(_2F_1(1,1,2,-z) + _2F_1(1,1,2,z))$ ?

Answer 1

$$ \log\left(\frac{1+z}{1-z}\right) = 2 z \;{}_2F_1\left(\frac{1}{2},1;\frac{3}{2};z^2\right) $$