Vardi's Integral

Author

Eric W. Weisstein
February 5, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/VardisIntegral.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/VardisIntegral.html.

©2006 Wolfram Research, Inc. except for portions noted otherwise

First Integral

Analytic

Definite

v5.2

Integrate[Log[Log[Tan[x]]],{x,π/4,π/2}]//Timing

v6

Integrate[Log[Log[Tan[x]]],{x,π/4,π/2}]//Timing

Indefinite

Integrate[Log[Log[Tan[x]]],x]

Exact

<<Utilities`Simplify`

GammaSimplify[(π/2)Log[((Gamma[(3/4)])/(Gamma[(1/4)]))(Sqrt[2π])]]//FullSimplify

Numerical

NIntegrate[Log[Log[Tan[x]]],{x,Pi/4,Pi/2},WorkingPrecision->20]

N[(π/2)Log[((Gamma[(3/4)])/(Gamma[(1/4)]))(Sqrt[2π])],20]

Second Integral

Integrate[Log[Log[(1/x)]](1/(1+x+(x^2))),{x,0,1}]//FullSimplify

(π/(Sqrt[3]))Log[((Gamma[(2/3)])/(Gamma[(1/3)]))(((2π))^(1/3))]//FullSimplify

N[%]

NIntegrate[Log[Log[(1/x)]](1/(1+x+(x^2))),{x,0,1}]

Third Integral

Integrate[Log[Log[(1/x)]](1/(1-x+(x^2))),{x,0,1}]//FullSimplify

N[%]

((2π)/(Sqrt[3]))((5/6)Log[2π]-Log[Gamma[(1/6)]])//FullSimplify

N[%]