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%I #22 Sep 30 2014 01:59:01
%S 5,9,7,3,8,1,8,0,9,4,5,1,8,0,3,4,8,4,6,1,3,1,1,3,2,3,5,0,9,0,8,7,3,7,
%T 6,4,3,0,6,4,3,8,5,9,0,4,2,5,5,5,6,7,3,0,7,7,0,3,2,0,7,1,6,1,5,5,0,3,
%U 1,1,0,3,3,2,4,9,8,2,4,1,2,1,7,8,9,0,9,8,9,9,0,4,0,4,4,7,4,4,4,3,7,3,3,0,0,9
%N Decimal expansion of the integral over dx/((1+x^2)(1+tan x)) in the limits 0 and Pi/2.
%H juantheron, <a href="http://math.stackexchange.com/questions/364452/how-to-evaluate-int-0-frac-pi2-frac11x21-tan-x-mathrm-d">How to evaluate int_0^(pi/2) dx/(1+x^2)/(1+tan x)</a>, math.stackexchange, Apr 14 2013
%e 0.59738180945180348461311323509087376430643859042555673077032071615503110332498…
%p Digits := 60 :
%p # Expand 1/(1+tan x) in a Taylor series around Pi/4 and exchange
%p # summation and integration.
%p for dd from 80 to 100 by 10 do
%p taylor(1/(1+tan(z)),z=Pi/4,dd) ;
%p gfun[seriestolist](%) ;
%p c := evalf(%) ;
%p x := 0.0 ;
%p for i from 0 to nops(c)-1 do
%p 1/(1+zz^2)*op(i+1,c)*(zz-Pi/4)^i ;
%p int(%,zz=0..Pi/2) ;
%p x := x+evalf(%) ;
%p end do:
%p print(x) ;
%p end do:
%t RealDigits[ NIntegrate[ 1/((1+x^2)(1+Tan[x])),{x, 0, Pi/2}, WorkingPrecision -> 110], 10, 105][[1]] (* _Robert G. Wilson v_, Sep 29 2014 *)
%K cons,less,nonn
%O 0,1
%A _R. J. Mathar_, Dec 08 2013
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