> Using SageMathCloud I was surprised to find this bug:
>
> integral(sqrt(cos(x)-cos(x)^3), x, 0, pi/2)
> -2/3
>
> The correct result is 2/3 (luckily I noticed that something was wrong
> because the result must be positive!)
>
> numerical_integral gets it right:
>
> numerical_integral(sqrt(cos(x)-cos(x)^3), 0, pi/2)
> (0.6666666687973078, 4.409680787972339e-07)
>
>
Usually such things are due to abs_integrate (see e.g.
http://trac.sagemath.org/ticket/12731 and
http://trac.sagemath.org/wiki/symbolics), but interestingly, abs_integrate
comes to the *rescue* when domain:real. The problem here is due to
domain:complex (which is our default).
sage -maxima
...
(%i1) integrate(sqrt(cos(x)-cos(x)^3),x,0,%pi/2);
(%o1) 0
(%i3) integrate(sqrt(cos(x)-cos(x)^3),x,0,%pi);
2
(%o3) -
3
(%i4) load(abs_integrate);
(%o4)
/Users/karl.crisman/Downloads/sage/local/share/maxima/5.35.1/share/contr\
ib/integration/abs_integrate.mac
(%i5) integrate(sqrt(cos(x)-cos(x)^3),x,0,%pi/2);
2
(%o5) -
3
(%i6) integrate(sqrt(cos(x)-cos(x)^3),x,0,%pi);
(%o6) 0
(%i7) domain:complex;
(%o7) complex
(%i8) integrate(sqrt(cos(x)-cos(x)^3),x,0,%pi/2);
2
(%o8) - -
3
(%i9) integrate(sqrt(cos(x)-cos(x)^3),x,0,%pi);
2 %i 2
(%o9) ---- - -
3 3
new session:
(%i1) domain:complex;
(%o1) complex
(%i2) integrate(sqrt(cos(x)-cos(x)^3),x,0,%pi/2);
2
(%o2) - -
3
Nils or Ralf or others, any thoughts?
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