I suspect there are more of these, but they seem just out of reach. E.g., Integrate[ArcSec@t/Sqrt[2 + t]/Sqrt[(t + 1)^3], {t, 1, 2}] = (1/Sqrt[2] - 4/(3 Sqrt[3])) π + Sqrt[2] ArcSin[17/81], which Mathematica can do indefinitely(!), but -l8 runs me out of memory(?): ries/ries -l8 .1020759204661688653138 [. . .] ln(tanpi(6 x)^2)-1 = e-e^(1/ln(2 pi)) for x = T - 4.14884e-15 {175} sqrt(tanpi(sqrt(9"/2 x))) = 1-ln(ln(sqrt(e+phi))) for x = T + 1.38825e-15 {180} ln(tanpi(sqrt(x)+1/9^2)) = 1/((log_(5 phi)(6))+1) for x = T + 1.69075e-16 {179} Killed: 9 -l7 reaches the first of these, but can't quite express the answer. --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