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A225125 Decimal expansion of Integral_{x=0..Pi/2} x^3*cosec(x) dx.

%I A225125 #33 Jun 16 2025 12:16:44
%S A225125 1,6,9,2,9,9,2,4,6,8,4,1,3,6,0,1,2,4,4,6,7,8,0,1,3,8,3,4,8,9,8,1,0,8,
%T A225125 7,0,8,0,7,8,6,9,8,6,7,1,5,6,8,0,7,2,3,4,9,5,6,8,8,0,1,5,7,7,8,9,4,7,
%U A225125 6,4,3,7,2,1,3,1,9,8,7,9,8,7,2,7,9,1,8,7,3,6,3,9,6,3,5,4,4,9,4,2
%N A225125 Decimal expansion of Integral_{x=0..Pi/2} x^3*cosec(x) dx.
%C A225125 The simpler Integral_{x=0..Pi/2} x*cosec(x) dx evaluates as 2*Catalan.
%H A225125 G. C. Greubel, <a href="/A225125/b225125.txt">Table of n, a(n) for n = 1..10000</a>
%H A225125 StackExchange, <a href="http://math.stackexchange.com/questions/302087/a-integral-with-polygamma">An integral with PolyGamma.</a>
%F A225125 Equals 3*Catalan*Pi^2/2-1/128*(polygamma(3, 1/4)-polygamma(3, 3/4)).
%e A225125 1.6929924684136012446780138348981087080786986715680723495688...
%t A225125 3*Catalan*Pi^2/2-1/128*(PolyGamma[3, 1/4]-PolyGamma[3, 3/4]); (* or *)
%t A225125 3*Catalan*Pi^2/2-3/64*(Zeta[4, 1/4]-Zeta[4, 3/4]) // RealDigits[#, 10, 100] & // First
%t A225125 RealDigits[Integrate[x^3 Csc[x],{x,0,Pi/2}],10,120][[1]] (* _Harvey P. Dale_, Jun 16 2025 *)
%o A225125 (PARI) 3*Catalan*Pi^2/2-3/64*(zetahurwitz(4,1/4)-zetahurwitz(4,3/4)) \\ _Charles R Greathouse IV_, Jan 31 2018
%Y A225125 Cf. A006752, A221209.
%K A225125 nonn,cons,easy
%O A225125 1,2
%A A225125 _Jean-François Alcover_, Apr 30 2013