<s>[INST] Arc length of the squircle The squircle is given by the equation $x^4+y^4=r^4$. Apparently, its circumference or arc length $c$ is given by 
$$c=-\frac{\sqrt[4]{3} r G_{5,5}^{5,5}\left(1\left|
\begin{array}{c}
 \frac{1}{3},\frac{2}{3},\frac{5}{6},1,\frac{4}{3} \\
 \frac{1}{12},\frac{5}{12},\frac{7}{12},\frac{3}{4},\frac{13}{12} \\
\end{array}
\right.\right)}{16 \sqrt{2} \pi ^{7/2} \Gamma \left(\frac{5}{4}\right)}$$
Where $G$ is the Meijer $G$ function. Where can I find the derivation of this result? Searching for any combination of squircle and arc length or circumference has led to nowhere. 
 [/INST] You can do this using this empirical formula to find perimeter of general super-ellipse
$$L=a+b\times\left(\frac{2.5}{n+0.5}\right)^\frac{1}{n}\times \left( b+a\times(n-1)\times\frac{\frac{0.566}{n^2}}{b+a\times\left(\frac{4.5}{0.5+n^2}\right)}\right).$$
</s>