$\sum_{m=1}^\infty \frac{1}{m} \bigg[{}_2F_1(-m,m,2,u)\bigg]^2 = \frac 1 4 -\frac 1 2 \log u$

I have very strong numerical support that this is true when 
$0<u\le1$. 
Can anyone help proving or disproving ?