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%I #21 May 09 2016 03:00:44
%S 1,0,2,5,7,6,0,5,1,0,9,3,1,3,3,0,4,5,0,3,9,8,5,4,8,6,6,0,9,6,9,5,5,2,
%T 7,9,5,3,3,4,8,7,1,8,5,6,2,1,5,0,6,9,3,9,4,2,2,3,3,8,6,8,4,4,0,1,5,8,
%U 5,1,9,2,0,8,9,9,0,7,0,9,4,2,2,2,6,7,8,7,8,7,9,1,9,7,7,9,5,3,0,7,1,3,2,9,6
%N Decimal expansion of the integral_{x=0..1} arctan(arctanh(x))/x.
%C "The arctangent of the hyperbolic arctangent is analytic in the whole disk |x| < 1, and therefore, can be expanded into the MacLaurin series", see the first reference.
%H Iaroslav V. Blagouchine, <a href="http://dx.doi.org/10.1016/j.jmaa.2016.04.032">Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/pi</a>, Journal of Mathematical Analysis and Applications (Elsevier), 2016. <a href="http://arxiv.org/abs/1408.3902">arXiv version</a>, arXiv:1408.3902 [math.NT], 2014-2016.
%H Iaroslav V. Blagouchine, <a href="http://link.springer.com/article/10.1007%2Fs11139-013-9528-5">Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results</a>, The Ramanujan Journal, Volume 35, Issue 1, pp. 21-110, 2014, DOI: 10.1007/s11139-013-9528-5 <a href="http://www.researchgate.net/publication/257381156_Rediscovery_of_Malmsten%27s_integrals_their_evaluation_by_contour_integration_methods_and_some_related_results">PDF file</a>
%F The integral is equivalent to Pi*(log(Gamma(1/Pi)) - log(Gamma(1/2 + 1/Pi)) - log(Pi)/2), see page 82 of the second reference.
%e = 1.02576051093133045039854866096955279533487185621506939422338684401585192089...
%p evalf(Pi*(log(GAMMA(1/Pi)) - log(GAMMA(1/2 + 1/Pi)) - log(Pi)/2),120); # _Vaclav Kotesovec_, May 17 2015
%t nn = 111; RealDigits[ NIntegrate[ ArcTan[ ArcTanh[ x]]/x, {x, 0, 1}, AccuracyGoal -> nn, WorkingPrecision -> nn], 10, nn][[1]] (* or *)
%t RealDigits[Pi (Log[Gamma[1/Pi]] - Log[Gamma[1/2 + 1/Pi]] - Log[Pi]/2), 10, 111][[1]] (* _Robert G. Wilson v_, May 14 2015 *)
%Y Cf. A257957, A257955, A257958, A257959, A155968, A049541, A000796.
%K nonn,cons
%O 1,3
%A _Robert G. Wilson v_, May 14 2015
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