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# integrate x^3/(exp(x)-1) between 0 and infinity

If I type

integrate(x^3/(exp(x)-1),x,0,infinity) I get -1/15pi^4 + limit(-1/4x^4 + x^3log(-e^x + 1) + 3x^2dilog(e^x) - 6xpolylog(3, e^x) + 6polylog(4, e^x), x, +Infinity, minus)

The command numerical_integral(x^3/(exp(x)-1),0,infinity) gives 6.4939394075

I have two questions :

1. How do I evaluate the limit ?
2. The correct answer is pi^4/15(=6.49393940226683) : why SageMath does not give it with symbolic integration ?

Thanks

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## Comments

This seems closely related to https://groups.google.com/forum/#!top...

( 2017-06-08 20:02:20 +0200 )edit

## 3 Answers

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To complement Eric's answer, you will can use giac integration algorithm from Sage version 8.0.beta4, so you should either install the latest development version, or wait for the official 8.0 version. Also, it is not enough to install giac package, you should also install the giacpy_sage package.

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## Comments

Will the latter be standard as a package too in 8.0?

( 2017-06-09 20:34:00 +0200 )edit

Before proceeding, I have installed the giac package 1.2.3.25

integral(x^3/(exp(x)-1),x,0,infinity, algorithm='giac')

returns

ValueError: Unknown algorithm: giac

even after restarting SageMath

What would you advice ?

Thank you.

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## Comments

Either wait for the release of Sage 8.0 (quite soon I think) or install the latest development version of Sage, since it works only for Sage >= 8.0.beta5.

( 2017-06-08 22:24:16 +0200 )edit

if is_package_installed("giac") is True, then i guess giac("integrate(x^3/(exp(x)-1), x, 0, inf)") will also work in v.7.6.

( 2017-06-08 23:59:38 +0200 )edit

For this type of computation, it is worth trying with some algorithm different from the default one (Maxima):

sage: integrate(x^3/(exp(x)-1),x,0,infinity, algorithm='giac')
1/15*pi^4


EDIT: the above works only for Sage 8.0.beta5 and higher versions (after the ticket https://trac.sagemath.org/ticket/22891 has been merged). This will therefore be available in the next stable release of Sage (8.0).

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Asked: 2017-06-08 19:56:33 +0200

Seen: 2,923 times

Last updated: Jun 08 '17