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A244843 Decimal expansion of the integral of log(2+x^2+y^2)/((1+x^2)*(1+y^2)) dx dy over the square [0,1]x[0,1].

%I A244843 #27 Feb 16 2025 08:33:23
%S A244843 5,6,9,5,9,6,1,5,8,1,8,3,6,1,4,5,0,6,2,3,6,4,5,5,5,3,6,7,2,7,1,7,4,6,
%T A244843 9,0,1,0,7,8,7,6,1,2,6,8,2,1,2,2,8,7,8,3,6,8,2,8,1,8,4,0,8,1,2,4,8,5,
%U A244843 2,3,0,0,2,5,0,2,9,9,1,8,1,1,6,1,4,0,5,6,5,7,4,2,2,2,7,2,4,5,8,6,8
%N A244843 Decimal expansion of the integral of log(2+x^2+y^2)/((1+x^2)*(1+y^2)) dx dy over the square [0,1]x[0,1].
%C A244843 The computation of this integral is given by Bailey & Borwein as an example of the use of CAS packages (and additional tools) to simplify large symbolic expressions.
%H A244843 Vincenzo Librandi, <a href="/A244843/b244843.txt">Table of n, a(n) for n = 0..1000</a>
%H A244843 D. H. Bailey and J. M. Borwein, <a href="http://moodle.thecarma.net/jon/ontology.pdf">Experimental computation as an ontological game changer</a>, 2014, see p. 5.
%H A244843 D. H. Bailey, J. M. Borwein and A. D. Kaiser, <a href="http://carmamaths.org/resources/jon/auto.pdf">Automated Simplification of Large Symbolic Expressions</a>
%H A244843 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/ClausensIntegral.html">Clausen's Integral</a>.
%H A244843 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/Polylogarithm.html">Polylogarithm</a>.
%F A244843 Pi^2/8*log(2) - 7/48*zeta(3) + 11/24*Pi*Cl2(Pi/6) - 29/24*Pi*Cl2(5*Pi/6), where Cl2 is the Clausen function Cl2(t) = Sum_{n>0} sin(n*t)/n^2.
%e A244843 0.56959615818361450623645553672717469010787612682122878368281840812485230025...
%t A244843 Clausen2[x_] := Im[PolyLog[2, Exp[x*I]]]; Pi^2/8*Log[2] - 7/48*Zeta[3] + 11/24*Pi*Clausen2[Pi/6] - 29/24*Pi*Clausen2[5*Pi/6] // RealDigits[#, 10, 101]& // First
%o A244843 (PARI) Cl2(x)=imag(polylog(2,exp(x*I)));
%o A244843 Pi^2/8*log(2) - 7/48*zeta(3) + 11/24*Pi*Cl2(Pi/6) - 29/24*Pi*Cl2(5*Pi/6) \\ _Charles R Greathouse IV_, Aug 27 2014
%Y A244843 Cf. A261027 (Cl_2(Pi/6)), A261028 (Cl_2(5*Pi/6)).
%K A244843 cons,nonn
%O A244843 0,1
%A A244843 _Jean-François Alcover_, Jul 07 2014