More generally, Integrate[((a + b/Sqrt[3 + t])*ArcSec[t])/(Sqrt[1 + t]*(2 + t)), {t, 1, 2}] == (a*Pi^2)/72 + (b*Pi^2)/180 + (1/6)*b*Pi*ArcSin[(1/16)*(7 - 3*Sqrt[5])] But AargH! Now I can't reconstruct the derivation from the Mathematica and ries transcripts! Mathematica overwrote a _crucial_ command line, and ries can't handle the b≠0 case, except when b magically = -5a/2. The breakthrough came when ries quickly (level 6) identified 0.411233516712190204444136 = 1/24 (π + Cos[π Log[GoldenRatio]]^10)^2 but I can't figure out what the heck I integrated to get .4112... ! --rwg On Wed, Jul 4, 2018 at 5:34 AM Bill Gosper <billgosper@gmail.com> wrote:
Integrate[((2 + 1/Sqrt[3 + t])*ArcSec[t])/(Sqrt[1 + t]*(2 + t)), {t, 1, 2}] == Pi^2/30 + (1/6)*Pi*ArcSin[(1/16)*(7 - 3*Sqrt[5])]
The integral {t,1,6} = 2 π^2/15 = Coxeter's still-mysterious orthoscheme integral. --rwg