amusingly, one can see that some power of 2 is lurking around :
(%i36) expand(diff(integrate(x*log(x)**13/(1+x**2),x),x));
13
4096 x log (x)
(%o36) ---------------
2
13 x + 13
Le dimanche 30 octobre 2022 à 23:49:31 UTC+1, pvit...@gmail.com a écrit :
> Ok, thanks a lot. I already reported the bug.
>
> El domingo, 30 de octubre de 2022 a las 10:27:23 UTC+1, Frédéric Chapoton
> escribió:
>
>> The factor-2 problem is also present for the integral with no bounds.
>> Please report in maxima's bug reporting site.
>>
>> (%i20) integrate(x*log(x)^4/(x^2 + 1),x);
>> 4 2
>> 2 log (x) log(x + 1) 2 2
>> (%o20) - (3 ((- ---------------------) + li (- x ) - 2 log(x) li (- x )
>> 3 5 4
>> 3 2
>> 4 log (x) li (- x
>> )
>> 2 2 2
>> + 2 log (x) li (- x ) -
>> -------------------))/2
>> 3 3
>> (%i21) diff(%,x);
>> 4
>> 2 x log (x)
>> (%o21) -----------
>> 2
>> x + 1
>>
>> Le mardi 25 octobre 2022 à 00:17:07 UTC+2, pvit...@gmail.com a écrit :
>>
>>> Thank you very much for the analysis. If I understand correctly, the bug
>>> is in this part of the integral: x*log(x)^4/(x^2 + 1)
>>>
>>> I will try to report the bug to maxima. But, as far I can see
>>> https://sourceforge.net/p/maxima/bugs/ is very quiet, with few answers
>>> to the reports.
>>>
>>> El lunes, 24 de octubre de 2022 a las 21:33:41 UTC+2, Frédéric Chapoton
>>> escribió:
>>>
>>>> where we can see that there is a factor 2 between the wrong symbolic
>>>> value and the correct numeric value
>>>>
>>>> This should be filed as a bug in maxima.
>>>>
>>>> Le lundi 24 octobre 2022 à 21:24:30 UTC+2, Frédéric Chapoton a écrit :
>>>>
>>>>> and one more step :
>>>>>
>>>>> sage: integrate(x*log(x)^4/(x^2 + 1), x,0,1).n()
>>>>> 1.45817965567036
>>>>> sage: (x*log(x)^4/(x^2 + 1)).nintegral(x,0,1)
>>>>> (0.7290898278351722, 2.48288156701193e-09, 357, 0)
>>>>> sage: integrate(-log(x)^4/(x^2 + 1), x,0,1).n()
>>>>> -23.9077878738501
>>>>> sage: (-log(x)^4/(x^2 + 1)).nintegral(x,0,1)
>>>>> (-23.90778787384685, 1.267767046897461e-08, 483, 0)
>>>>>
>>>>>
>>>>> Le lundi 24 octobre 2022 à 21:19:46 UTC+2, Frédéric Chapoton a écrit :
>>>>>
>>>>>> more study of the bug (coming from maxima)
>>>>>>
>>>>>> sage: C=x^2*(log(x))^4/((x+1)*(1+x^2))
>>>>>> sage: aa,bb=C.partial_fraction_decomposition()
>>>>>> sage: integral(aa,x,0,1)
>>>>>> -5/128*pi^5 + 45/64*zeta(5)
>>>>>> sage: integral(bb,x,0,1)
>>>>>> 45/4*zeta(5)
>>>>>> sage: _+__
>>>>>> -5/128*pi^5 + 765/64*zeta(5)
>>>>>> sage: _.n()
>>>>>> 0.440633136273039
>>>>>> sage: aa.nintegral(x,0,1)
>>>>>> (-11.58934902297507, 5.068708119893017e-08, 525, 0)
>>>>>> sage: bb.nintegral(x,0,1)
>>>>>> (11.66543724536065, 4.943314557692702e-08, 525, 0)
>>>>>> sage: integral(aa,x,0,1).n()
>>>>>> -11.2248041090899
>>>>>> sage: integral(bb,x,0,1).n()
>>>>>> 11.6654372453629
>>>>>>
>>>>>>
>>>>>>
>>>>>> Le lundi 24 octobre 2022 à 00:14:09 UTC+2, pvit...@gmail.com a
>>>>>> écrit :
>>>>>>
>>>>>>> I am using Sage 9.7 running in Arch linux over WSL2
>>>>>>>
>>>>>>> I get different results for an integral using numerical integration
>>>>>>> (which seems to agree with Mathematica) and symbolic integration:
>>>>>>>
>>>>>>> numerical_integral(x^2*(log(x))^4/((x+1)*(1+x^2)),0,1)
>>>>>>> (0.07608822217400527, 1.981757967172001e-07)
>>>>>>>
>>>>>>> integral(x^2*(log(x))^4/((x+1)*(1+x^2)),x,0,1)
>>>>>>> 6*I*polylog(5, I) - 6*I*polylog(5, -I) + 765/64*zeta(5)
>>>>>>> integral(x^2*(log(x))^4/((x+1)*(1+x^2)),x,0,1).n()
>>>>>>> 0.440633136273036
>>>>>>>
>>>>>>
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