? f(x) = (1-x)^(-2/3) * (1+x)^(-1/3) / (x^2+4);
? J = intnum( x=-1, 1, f(x));
? v = ( 1+ sqrt(-3) ) / 2;
? u = v^2;
? c = ( (3+4*I)/5 )^(1/3)
%130 = 0.952608221822056413578700298090 + 0.304199894340908282509646316476*I
?
? 2*Pi / sqrt(3) / 10 * real( (2+I)/c )
%131 = 0.801487589833185015198191025384
? J
%132 = 0.801487589833185015171411244041

sage: s = (2+i) / ( (3+4*i)/5 )^(1/3)
sage: s.minpoly()
x^6 - 20*x^3 + 125
sage: K.<S> = NumberField(s.minpoly())
sage: s.minpoly().base_extend(K).factor()
(x - S) * (x + 1/25*S^5 - 4/5*S^2) * (x^2 + S*x + S^2) * (x^2 + (-1/25*S^5 + 4/5*S^2)*x - 1/5*S^4 + 4*S)
sage: 5/S
-1/25*S^5 + 4/5*S^2
sage: (S+5/S).minpoly()
x^3 - 15*x - 20

sage: _.roots(ring=RR, multiplicities=False)
[-2.80560283257759, -1.61322984339245, 4.41883267597004]

sage: root1, root2, root3 = ( S + 5/S ).minpoly().roots(ring=RR, multiplicities=False)
sage: root3
4.41883267597004
sage: ( 2*pi / sqrt(3) / 20 * root3 ).n()
0.801487589833185
sage: numerical_integral( lambda x: 1 / (1-x)^(2/3) / (1+x)^(1/3) / (4+x^2), (-1,1) )[0]
0.801485689398172