For int(2*cot(2*x),x)

Maple 2024: -1/2*ln(cot(2*x)^2+1)
MMA 14 : ln(sin(2*x))
Fricas : 1/2*ln( (1-cos(4*x))/2)
giac: log(abs(sin(2*x)))
maxima: ln(sin(2*x))
sympy: ln(sin(2*x))

The plot of these over say x=-4*Pi..4*Pi is not the same.

Maple and Fricas give same plot. But Mathematica gives
differet plot. There are gaps in the plot of ln(sin(2*x))
compared to Maple's and Fricas version.

When differentiating these, I get back the integrand. So
they are all correct in this sense.

This PDF file has the code and the plots.

<https://12000.org/tmp/03182024/different_anti.pdf>

Any comment on which is the better anti-derivative here?

--Nasser

"Nasser M. Abbasi" schrieb:
>
> For int(2*cot(2*x),x)
>
> Maple 2024: -1/2*ln(cot(2*x)^2+1)
> MMA 14 : ln(sin(2*x))
> Fricas : 1/2*ln( (1-cos(4*x))/2)
> giac: log(abs(sin(2*x)))
> maxima: ln(sin(2*x))
> sympy: ln(sin(2*x))
>
> The plot of these over say x=-4*Pi..4*Pi is not the same.
>
> Maple and Fricas give same plot. But Mathematica gives
> differet plot. There are gaps in the plot of ln(sin(2*x))
> compared to Maple's and Fricas version.
>
> When differentiating these, I get back the integrand. So
> they are all correct in this sense.
>
> This PDF file has the code and the plots.
>
> <https://12000.org/tmp/03182024/different_anti.pdf>
>
> Any comment on which is the better anti-derivative here?
>

Derive 6.10 returns LN(SIN(2*x)), like MMA 14, Maxima, and SymPy do.
Along the real axis, the imaginary part of this antiderivative jumps
between zero and pi at each 1/x type pole of the integrand. This is
entirely acceptable in my view.

Maple's -1/2*ln(cot(2*x)^2+1) avoids the jumps on the real axis by
effectively squaring the logarithmic argument. This is nice, but
definite integrals that straddle one of the poles are still meaningless
of course.

The reciprocal of Maple's logarithmic argument is used in the FriCAS
result 1/2*ln((1-cos(4*x))/2), where the trigonometric argument is
doubled rather than the trigonometric function squared. This is just an
equivalent formulation of the Maple antiderivative.

Giac's log(abs(sin(2*x))) needlessly introduces the abs() function
which is not analytic. This should be avoided when the integrand itself
is analytic, as the antiderivative then will not be valid everywhere
on the complex plane. I would prefer it to be avoided even if the
integration variable had been declared real, but others may prefer that
their students not be exposed to an incomprehensible complex unit.

So, all of the results are good in my view, with the exception of that
by Giac.

Martin.