Incorrect integral in Maxima

[kcrisman]

See ​this ask.sagemath.org question.

sage: integral(log(cot(x)-1),x,0,pi/4)

ValueError: Integral is divergent.

But according to the poster it is pi log(2) / 8. And indeed in the most recent Maxima (5.28) we have

(%i5) display2d:false;

(%o5) false
(%i6) integrate(log(cot(x)-1),x,0,%pi/4);

Is %pi/8 an ?integer?

n;
Is %pi/4 an ?integer?

n;
Is 2*%pi an ?integer?

n;
(%o6) -(%i*(2*li[2](%i+1)-2*li[2](1-%i))+%pi*log(2))/4
 +%i*li[2]((%i+1)/2)/2-%i*li[2](-(%i-1)/2)/2

though with the interaction this is not an improvement we can immediately use.

[kcrisman]

According to ​the upstream report, that problem is also fixed ​here and there is also a commit of the integral in question as a test.

expand(integrate(log(cot(x)-1),x,0,%pi/4));
-%i*li[2](%i+1)/2+%i*li[2](%i/2+1/2)/2-%i*li[2](1/2-%i/2)/2+%i*li[2](1-%i)/2-%pi*log(2)/4$

So I guess when the next Maxima after 5.29.1 is ready and we put it in, we can try to close this report with a doctest.

[kcrisman]

See #13973 for where we might fix this.

[pbruin]

Maxima 5.33 returns the answer without asking any questions.

[kcrisman]

Great, then all that's needed is a doctest once #13973 is ready.

Maxima 5.33 returns the answer without asking any questions.

[pbruin]

Unfortunately the fact that Sage loads the abs_integrate package causes this to fail when called from within Sage with the message PSLOG: internal error. So we still have to look into that.

[kcrisman]

See also #12731.

[rws]

Replying to pbruin:

Maxima 5.33 returns the answer without asking any questions.

Maybe but 5.35 is back to original (PSLOG: internal error).

[rws]

Replying to pbruin:

Unfortunately the fact that Sage loads the abs_integrate package causes this to fail when called from within Sage with the message PSLOG: internal error. So we still have to look into that.

Maybe but Sage+Maxima-5.35 without abs_integrate gives -1/4*pi*log(2) - 1/2*I*polylog(2, I + 1) + 1/2*I*polylog(2, -I + 1) + 1/2*I*polylog(2, 1/2*I + 1/2) - 1/2*I*polylog(2, -1/2*I + 1/2) so this is not resolved either way.

[pbruin]

In SageMath 7.6.beta2, the answer appears to be correct up to simplification (probably thanks to Maxima commit ce7c53f1):

sage: integral(log(cot(x)-1),x,0,pi/4)
-1/4*pi*log(2) - 1/2*I*dilog(I + 1) + 1/2*I*dilog(-I + 1) + 1/2*I*dilog(1/2*I + 1/2) - 1/2*I*dilog(-1/2*I + 1/2)
sage: N(_)
0.272198261287950
sage: N(pi*log(2)/8)
0.272198261287950

Should we add this as a doctest and declare the bug to be fixed?

[kcrisman]

I vote yes.

[vbraun]

On one of the 32-bit buildbots:

sage -t --long src/sage/symbolic/integration/integral.py
6771**********************************************************************
6772File "src/sage/symbolic/integration/integral.py", line 771, in sage.symbolic.integration.integral.integrate
6773Failed example:
6774    N(a)  # long time
6775Expected:
6776    0.272198261287950
6777Got:
6778    0.272198261287950 + 5.55111512312578e-17*I
6779**********************************************************************
67801 item had failures:
6781   1 of 125 in sage.symbolic.integration.integral.integrate
6782    [168 tests, 1 failure, 16.86 s]

[pbruin]

The doctest should now be more robust.

[tmonteil]

Follow-up: a bug with this integral reappared in recent Maxima release, see #30063