Has anybody computed in closed form the elliptic integral of the first kind $K(k)$ when $\frac{K'}{K}=\sqrt{2}-1$?

I tried to search the literature, but nothing has turned up. This page http://mathworld.wolfram.com/EllipticIntegralSingularValue.html cites several cases $\frac{K'}{K}=\sqrt{r}$, when $r$ is integer.

Update: This question has been answered here.