OFFSET
1,2
LINKS
D. H. Bailey, J. M. Borwein, R. E. Crandall, Advances in the theory of box integrals (2010) p. 18.
Eric Weisstein's MathWorld, Clausen's Integral
FORMULA
K_1 = integral_[3..4] arcsec(x)/sqrt(x^2 - 4*x + 3) dx.
K_1 = Cl_2(th) - Cl_2(th + Pi/3) - Cl_2(th - Pi/2) + Cl_2(th - Pi/6) - Cl_2(3*th + Pi/3) + Cl_2(3*th + 2*(Pi/3)) - Cl_2(3*th - 5*(Pi/6)) + Cl_2(3*th + 5*(Pi/6)) + (6*th - 5*(Pi/2))*log(2 - sqrt(3)), where Cl_2 is the Clausen function and th = (arctan((16 - 3*sqrt(15))/11) + Pi)/3.
EXAMPLE
1.6619078747381233774065816861630594973488686732512589...
MAPLE
evalf(int(arcsec(x)/sqrt(x^2 - 4*x + 3), x=3..4), 120); # Vaclav Kotesovec, Jan 26 2015
MATHEMATICA
Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]); th = (ArcTan[(16 - 3*Sqrt[15])/11] + Pi)/3; K1 = Cl2[th] - Cl2[th + Pi/3] - Cl2[th - Pi/2] + Cl2[th - Pi/6] - Cl2[3*th + Pi/3] + Cl2[3*th + 2*(Pi/3)] - Cl2[3*th - 5*(Pi/6)] + Cl2[3*th + 5*(Pi/6)] + (6*th - 5*(Pi/2))*Log[2 - Sqrt[3]] // Re; RealDigits[K1, 10, 103] // First
CROSSREFS
KEYWORD
AUTHOR
Jean-François Alcover, Jan 26 2015
STATUS
approved