Bug?

[OHappyDay]

I recently stumbled upon something that looks like a bug:

Try to integrate the following function:

f(x)=log(1-4*cos(x)+4)
integrate(f,x,0,pi)

According to the following video (https://www.youtube.com/watch?v=nscSDYApAjM) the result should be

2*pi*log(2)

But my local Sage as well as online I get this result:

1/12*I*pi^2 + pi*log(3) - 1/2*pi*log(9/4) + 1/2*I*log(2)^2 - I*dilog(2) + I*dilog(-1/2) + I*dilog(-2)

which is a complex number instead.

Similarly this function

g(x)=log(1-cos(x)+1/4)
integrate(g,x,0,pi)

results in

1/12*I*pi^2 - pi*log(3) + 1/2*pi*log(9/4) + 1/2*I*log(2)^2 - I*dilog(2) + I*dilog(-1/2) + I*dilog(-2)

although it should be ZERO.

[David Joyner]

On Thu, Sep 5, 2024 at 9:16 AM 'OHappyDay' via sage-support <sage-s...@googlegroups.com> wrote:

I recently stumbled upon something that looks like a bug:

Try to integrate the following function:

f(x)=log(1-4*cos(x)+4)
integrate(f,x,0,pi)

According to the following video (https://www.youtube.com/watch?v=nscSDYApAjM) the result should be

2*pi*log(2)

But my local Sage as well as online I get this result:

1/12*I*pi^2 + pi*log(3) - 1/2*pi*log(9/4) + 1/2*I*log(2)^2 - I*dilog(2) + I*dilog(-1/2) + I*dilog(-2)

which is a complex number instead.

This could be a bug in Maxima:

sage: CC(integrate(f,x,0,pi, algorithm='sympy'))

4.35517218060720

sage: CC(integrate(f,x,0,pi, algorithm='maxima'))

4.44089209850063e-16 - 3.28986813369645*I

sage: CC(2*pi*log(2))

4.35517218060720

 

Similarly this function

g(x)=log(1-cos(x)+1/4)
integrate(g,x,0,pi)

results in

1/12*I*pi^2 - pi*log(3) + 1/2*pi*log(9/4) + 1/2*I*log(2)^2 - I*dilog(2) + I*dilog(-1/2) + I*dilog(-2)

although it should be ZERO.

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[OHappyDay]

I have reported this to the Maxima team.

[OHappyDay]

The Maxima team has opened two bug reports about this issue:

incorrect dilogarithm limit in definite integral & incorrect trig substitution:

https://sourceforge.net/p/maxima/bugs/4368/ and https://sourceforge.net/p/maxima/bugs/4369

[OHappyDay]

For the curious: the Maxima team opened a third bug report: https://sourceforge.net/p/maxima/bugs/4373/ which discusses a possible fix (evaluating  conjugate(li[n](x)) does not consider that li[n](x) can also be complex :-( ).