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%I #14 Jan 06 2014 03:20:57
%S 5,1,0,0,9,4,9,4,1,8,0,3,4,7,0,2,7,4,2,5,0,7,3,2,6,2,3,7,3,3,5,3,4,8,
%T 8,2,9,0,5,3,5,4,6,9,8,3,0,5,2,5,1,4,7,9,0,8,8,6,0,4,2,4,4,6,1,0,8,7,
%U 3,1,9,1,3,7,4,7,8,8,1,7,4,5,4,3,1,2,9,5,4,2,2,1,9,8,0,1,4,1,5,2,1,6,7,2,3
%N Decimal expansion of arc length of the Keratoid Cusp curve (loop arc length).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KeratoidCusp.html">Keratoid Cusp</a>
%e 0.5100949418...
%t eq = y^2 == x^2*y + x^5; f1[x_] = y /. Solve[eq, y][[1]]; f2[x_] = y /. Solve[eq, y][[2]]; x1 = (x - 1/4)/2 /. Solve[f2'[x] == 0][[1]]; y1 = f1[x1]; y2 = f2[x1]; g[y_] = x /. Solve[eq, x][[1]]; gp[y_] := (2*y - g[y]^2)/(2*y*g[y] + 5*g[y]^4); ni[a_, b_] := NIntegrate[a, b, WorkingPrecision -> 120]; i1 = ni[Sqrt[1 + f1'[x]^2], {x, x1, 0}]; i2 = ni[Sqrt[1 + f2'[x]^2], {x, x1, 0}]; i3 = ni[Sqrt[1 + gp[y]^2], {y, y1, y2}]; Take[RealDigits[i1 + i2 + i3][[1]], 105] (* _Jean-François Alcover_, Jan 17 2012 *)
%K nonn,cons
%O 0,1
%A _Jean-François Alcover_, Jan 17 2012
%E Offset corrected by _Rick L. Shepherd_, Jan 05 2014
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