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%I #29 Aug 31 2019 03:03:29
%S 5,0,6,6,7,0,9,0,3,2,1,6,6,2,2,9,8,1,9,8,5,2,5,5,8,0,4,7,8,3,5,8,1,5,
%T 1,2,4,7,2,8,4,3,5,4,7,3,4,7,0,2,0,5,8,2,9,2,0,0,0,2,4,5,8,6,5,9,4,7,
%U 0,5,1,4,5,1,3,2,2,6,9,3,1,5,0,3
%N Decimal expansion of Integral_{x=0..1} sin(log(x))/((x+1)*log(x)) dx.
%C This integral has an elegant evaluation in terms of the gamma function (see below formula). There is an interesting "symmetry" between the expressions involving the gamma function in this evaluation.
%H John M. Campbell, <a href="http://www.mathematica-journal.com/2017/10/an-algorithm-for-trigonometric-logarithmic-definite-integrals/">An Algorithm for Trigonometric-Logarithmic Definite Integrals</a>, in the Mathematica Journal, Vol. 19.10 (2017).
%H W. M. Gosper, <a href="/A309204/a309204.pdf">Material from Bill Gosper's Computers & Math talk, M.I.T., 1989</a>, i+38+1 pages, annotated and scanned, included with the author's permission. (There are many blank pages because about half of the original pages were two-sided, half were one-sided.) See page 8.
%F Equals log(2) + log(((Gamma(1 - i/2)^2*Gamma(1 + i))/(Gamma(1 + i/2)^2*Gamma(1 - i)))^(i/2)), where i = sqrt(-1) denotes the imaginary unit.
%F Equals Sum_{n >= 0} (-1)^n*arctan(1/(n+1)).
%e This integral is equal to 0.50667090321662298198525580478358151247...
%t Print[RealDigits[Re[Log[2] + Log[((Gamma[1 - I/2]^2 Gamma[1 + I])/(Gamma[1 + I/2]^2 Gamma[1 - I]))^(I/2)]], 10, 100]] ;
%t NIntegrate[Sin[Log[x]]/(x + 1)/Log[x], {x, 0, 1}]
%o (PARI) intnum(x=0,1,sin(log(x))/(x+1)/log(x))
%Y Decimal expansions of definite integrals over elementary functions: A256127, A256128, A256129, A204067, A204068, A205885, A206161, A206160, A206769, A229174, A083648, A094691, A098687, A177218, A188141, A233382, A256273, A258086.
%Y Cf. A309209 (continued fraction of the negation of this constant).
%K cons,nonn
%O 0,1
%A _John M. Campbell_, Apr 06 2016
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