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%I #6 Sep 12 2014 10:47:20
%S 6,4,5,3,6,5,8,4,6,3,3,4,3,4,1,0,8,1,8,9,6,2,3,3,6,0,5,1,3,1,9,1,2,9,
%T 5,3,6,0,6,5,3,0,4,1,1,6,3,6,5,2,6,0,0,7,4,6,3,4,0,6,2,3,2,2,1,0,0,7,
%U 9,4,6,5,8,8,8,4,9,6,7,1,6,9,8,2,8,7,5,7,3,3,3,5,4,8,7,0,5,7,4,9,9,8,4,7
%N Decimal expansion of integral_{0..Pi} x*log(2*cos(x/2))^2 dx, the first of two definite integrals studied by Rutledge and Douglas, and using the constant A_4 (A214508).
%H G. E. Raynor, <a href="http://www.projecteuclid.org/euclid.bams/1183502262">On Serret's integral formula</a>, Bull. Amer. Math. Soc. Volume 45, Number 12, Part 1 (1939), 911-917
%H G. Rutledge, R. D. Douglass, <a href="http://www.jstor.org/stable/2303745">Table of definite integrals</a>, Am. Math. Monthly 45 (1938) 525
%F A_4 + 31*Pi^4/480, where A_4 is A214508.
%e 6.453658463343410818962336051319129536065304116365260074634...
%t A4 = (13*Pi^4)/288 + (1/6)*Pi^2*Log[2]^2 - Log[2]^4/6 - 4*PolyLog[4, 1/2] - (7/2)*Log[2]*Zeta[3]; RealDigits[A4 + 31*Pi^4/480, 10, 104] // First
%Y Cf. A214508, A247320.
%K nonn,cons,easy
%O 1,1
%A _Jean-François Alcover_, Sep 12 2014
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