In a previous post we proved that
\[\int_0^\alpha \frac{du}{\sqrt{1+u^4}}=\frac{3\Gamma^2\left(\frac{1}{4}\right)}{16\sqrt{\pi}}\]where $\alpha=\sqrt{1+\sqrt{2}+\sqrt{2+2\sqrt{2}}}$. Here we will show that
\[\begin{align*}\int_0^\beta \frac{du}{\sqrt{1+u^4}}=\frac{\Gamma^2\left(\frac{1}{4}\right)}{16\sqrt{\pi}}\tag{1}\end{align*}\]where $\beta=\sqrt{1+\sqrt{2}-\sqrt{2+2\sqrt{2}}}$, which actually ends up being much easier to do.
Proof
We start by proving the following,
Fagnano’s lemniscatic doubling theorem:
Let $0<s<\sqrt{\sqrt{2}-1}$, then
\[\displaystyle{2\int_0^s\frac{\mathrm dx}{\sqrt{1-x^4}}=\int_0^t\frac{\mathrm dx}{\sqrt{1-x^4}}}\]where
\[\displaystyle{t=\frac{2s\sqrt{1-s^4}}{1+s^4}.}\] \[\,\]
Proof:
First observe that
\[t'(s)=\frac{2(s^4+2s^2-1)(s^4-2s^2-1)}{(1+s^4)^2\sqrt{1-s^4}}\]which is continuous and increasing on the interval $[0,\,\sqrt{\sqrt{2}-1}]$ where $\sqrt{\sqrt{2}-1}$ is the smallest solution of $t’(s)=0.$ We also have
\[1-t^4(s)=\frac{(s^4+2s^2-1)(s^4-2s^2-1)}{(1+s^4)^2}\]therefore
\[\frac{\mathrm dt}{\sqrt{1-t^4(s)}}=2\frac{\mathrm ds}{\sqrt{1-s^4}}.\]Integrating on the appropriate interval then yields the theorem.1 $\blacksquare$
Corollary:
\[\displaystyle{2\int_0^s\frac{\mathrm dx}{\sqrt{1+x^4}}=\int_0^t\frac{\mathrm dx}{\sqrt{1+x^4}}}\]where
\[\displaystyle{t=\frac{2s\sqrt{1+s^4}}{1-s^4}.}\] \[\,\]
This can be easily shown by letting $x\mapsto e^{i\pi/4}x$ in Fagnano’s theorem. $\blacksquare$
To derive $(1)$ first let $t=1$ then solve for $s$, from which we find that $s=\beta$. Then we can show $\int_0^1\frac{\mathrm dx}{\sqrt{1+x^4}}=\Gamma^3(1/4)/8$ (which can be proven using the beta function). As a consequence, we get an absurd expression for $3$ that only uses the numbers $0,\,1,\,2,$ and $4$.
\[\int_0^{\sqrt{1+\sqrt{2}+\sqrt{2+2\sqrt{2}}}}\frac{\mathrm dx}{\sqrt{1+x^4}}\Bigg/\int_0^{\sqrt{1+\sqrt{2}-\sqrt{2+2\sqrt{2}}}}\frac{\mathrm dx}{\sqrt{1+x^4}}=3\]This quotient of integrals should have good applications in nerd sniping.
References
Elliptic Functions Introduction Course, Vladimir G. Tkachev, https://users.mai.liu.se/vlatk48/teaching/lect2-agm.pdf ↩