A188141 - OEIS

%I #22 May 19 2014 15:22:39

%S 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,

%T 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,

%U 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

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

%C 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.

%H The Math Forum at Drexel - Valeri Astanoff <a href="http://mathforum.org/kb/message.jspa?messageID=7340370">Re: Nice Integral</a>, Dec 22, 2010.

%e 1.9754169..

%t RealDigits[N[(3/8)*(Pi^2*Log[4] - 7*Zeta[3]) , 110]][[1]]

%t (* or as a numerical check : *)

%t RealDigits[NIntegrate[ArcTan[1/x]^3, {x, 0, Infinity}, WorkingPrecision -> 110]][[1]] (* _Jean-François Alcover_, Mar 23 2011 *)

%t RealDigits[ N[ Integrate[ ArcTan[1/x]^3, {x, 0, Infinity}], 110]][[1]] (* _Jean-François Alcover_, Oct 19 2012, since version 6.0 *)

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

%K nonn,cons

%O 1,2

%A _Jean-François Alcover_, Mar 23 2011