OFFSET
0,1
LINKS
D. H. Bailey, J. M. Borwein, and R. E. Crandall, Advances in the theory of box integrals (2010) p. 18.
Eric Weisstein's World of Mathematics, Inverse Tangent Integral.
FORMULA
K_2 = integral_[0..Pi/4] sqrt(1 + sec(x)^2)*arctan(1/sqrt(1 + sec(x)^2)) dx.
K_2 = (1/2)*Ti_2(-2 + sqrt(3)) + (Pi/8)*log(2 + sqrt(3)) + Pi^2/32, where Ti_2 is Lewin's arctan integral, Ti_2(x) = (i/2)*(Li_2(-i*x) - Li_2(i*x)).
EXAMPLE
0.69266081515264750650943118588427245846713483280766884258...
MAPLE
evalf(int(sqrt(1 + sec(x)^2)*arctan(1/sqrt(1 + sec(x)^2)), x=0..Pi/4), 120); # Vaclav Kotesovec, Jan 26 2015
MATHEMATICA
Ti2[x_] := (I/2)* (PolyLog[2, -I *x] - PolyLog[2, I *x]); K2 = (1/2)*Ti2[-2 + Sqrt[3]] + (Pi/8)*Log[2 + Sqrt[3]] + Pi^2/32 // Re; RealDigits[K2, 10, 103] // First
CROSSREFS
KEYWORD
AUTHOR
Jean-François Alcover, Jan 26 2015
STATUS
approved