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Decimal expansion of the fourth Malmsten integral: int_{x=1..infinity} log(log(x))/(1 + x)^2 dx, negated.
5

%I #29 Jan 18 2017 02:32:44

%S 0,6,2,8,1,6,4,7,9,8,0,6,0,3,8,9,9,7,9,4,0,1,5,8,4,3,0,0,9,3,7,6,0,1,

%T 4,3,7,3,5,1,8,2,3,2,8,6,9,2,4,3,3,6,4,0,7,0,6,4,1,2,0,8,6,4,5,3,0,6,

%U 1,7,8,9,4,3,1,2,6,6,6,5,3,3,7,9,5,9,3,5,6,0,0,0,6,3,3,7,8,6,4,6,7,7,3,1,1,5,5,8

%N Decimal expansion of the fourth Malmsten integral: int_{x=1..infinity} log(log(x))/(1 + x)^2 dx, negated.

%H G. C. Greubel, <a href="/A256129/b256129.txt">Table of n, a(n) for n = 0..5000</a>

%H Iaroslav V. Blagouchine, <a href="http://dx.doi.org/10.1007/s11139-013-9528-5">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>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Carl_Johan_Malmsten">Carl Malmsten</a>

%F Equals integral_{x=0..1} log(log(1/x))/(1 + x)^2 dx.

%F Equals integral_{x=0..infinity} 0.5*log(x)/(1 + cosh(x)) dx.

%F Equals (log(Pi) - log(2) - gamma)/2.

%e -0.0628164798060389979401584300937601437351823286924336...

%p evalf((log(Pi/2)-gamma)/2,120); # _Vaclav Kotesovec_, Mar 17 2015

%t RealDigits[(Log[Pi/2]-EulerGamma)/2,10,105][[1]] (* _Vaclav Kotesovec_, Mar 17 2015 *)

%o (PARI) (-Euler+log(Pi)-log(2))/2 \\ _Michel Marcus_, Mar 18 2015

%Y A115252 (first Malmsten integral), A256127 (second Malmsten integral), A256128 (third Malmsten integral), A002162 (log 2), A053510 (log Pi), A001620 (Euler's constant, gamma).

%K nonn,cons

%O 0,2

%A _Iaroslav V. Blagouchine_, Mar 15 2015