I'm trying to derive this result and so far I have gotten to the integral

$$L=4\int_0^1\sqrt{\frac{1+t^2}{1-t^2}}dt.$$

I'm aware that this is a complete elliptic integral of the second kind and that $\frac{\Gamma^2\left(1/4\right)}{\sqrt{2\pi}}$ is the Lemniscate Constant, but that is really all I know. My guesses of how to continue so far are to try to transform the integral into lemniscatic integrals and apply the beta function. I'm stuck in a rut because I don't know any identities that can help me go on, so if you guys have any hints, substitutions, transformations, or identities to share that would be great.