For comparison, let's keep the same parameters and try the more simple P(x)=x^2. Then, Equal[ Integrate[Hypergeometric2F1[1/3, 2/3, 1, x^2], {x, 0, 1}], 2 (3 ArcSinh[1/(2 Sqrt[2])] + Log[8])/(Sqrt[3] Pi)] N[% /. Equal -> List, 20] Maybe other list readers do not find so much fun when "chasing a trail of smoke and reason" (though it does go even higher). So let's look at another geometry: Show[ContourPlot[ -p^2 + q^2 - (4/27)*(-3*q*p^2 - q^3)^2 == 0, {q, -2, 2}, {p, -2, 2}]] https://0x0.st/zbVV.png Should we be surprised to find out that the area interior to sextic lemniscate zbVV, up to harmonic scale factor pi/sqrt(3), equals a rational number 3/2? --Brad On Fri, Apr 26, 2019 at 11:01 AM Brad Klee <bradklee@gmail.com> wrote:
Does anyone else have an opinion? --Brad