A188141 Decimal expansion of integral ((arctan(1/x))^3,x=0..infinity).

1, 9, 7, 5, 4, 1, 6, 9, 7, 7, 0, 9, 8, 9, 0, 2, 4, 0, 9, 4, 6, 1, 2, 9, 6, 6, 9, 1, 4, 9, 8, 0, 1, 5, 8, 2, 7, 7, 1, 6, 7, 4, 5, 2, 6, 8, 7, 4, 7, 1, 2, 5, 5, 7, 1, 7, 8, 8, 3, 8, 6, 0, 5, 3, 6, 1, 5, 5, 1, 2, 6, 3, 9, 0, 0, 3, 0, 0, 4, 6, 8, 3, 2, 9, 0, 0, 1, 5, 9, 1, 1, 1, 8, 9, 3, 8, 9, 9, 8, 3, 6, 6, 9, 3, 2, 1, 2, 2, 0, 9

Offset: 1

Author

Jean-François Alcover, Mar 23 2011

Keywords

• nonn
• cons

Comments

The computation of this integral was mentioned as a challenge by Robert Israel on the newsgroup sci.math (Dec 22 2010), a closed form solution being given by Valeri Astanoff.

Examples

			1.9754169..
		

Links

• The Math Forum at Drexel - Valeri Astanoff Re: Nice Integral, Dec 22, 2010.

Crossrefs

Cf. A086054 (int(arctan(1/x)^2, x=0..infinity)).

Programs

Mathematica

RealDigits[N[(3/8)*(Pi^2*Log[4] - 7*Zeta[3]) , 110]][[1]]
(* or as a numerical check : *)
RealDigits[NIntegrate[ArcTan[1/x]^3, {x, 0, Infinity}, WorkingPrecision -> 110]][[1]] (* Jean-François Alcover, Mar 23 2011 *)
RealDigits[ N[ Integrate[ ArcTan[1/x]^3, {x, 0, Infinity}], 110]][[1]] (* Jean-François Alcover, Oct 19 2012, since version 6.0 *)