#2412 Problems with integral (x/(exp(x)+1),x,0,inf)

[christoph reineke]

The correct result of integrate(x/(exp(x)+1),x,0,inf); should be pi^2/12.

Maxima returns:
(%o2) limit(-x*log(%e^x+1)-li[2](-%e^x)+x^2/2,x,inf,minus)-%pi^2/12
build_info("5.27.0","2012-04-24 08:52:03","i686-pc-mingw32","GNU Common Lisp (GCL)","GCL 2.6.8")

regards

chris

[Aleksas]

my solution:

(%i1) S:'integrate(x/(exp(x)+1),x,0,inf)$

(%i2) intparts(S,u):=block([f,var,v,a,b],
f:part(S,1),var:part(S,2),v:integrate(f/u,var),
if last(S)#var then
(a:part(S,3),b:part(S,4),
limit(u*v,var,b,minus)-limit(u*v,var,a,plus)
-'integrate(v*diff(u,var),var,a,b))
else u*v-'integrate(v*diff(u,var),var)
)$

(%i3) intparts(S,1/(exp(x)+1));
(%o3) integrate((x^2*%e^x)/(%e^x+1)^2,x,0,inf)/2
(%i4) ev(%, nouns);
(%o4) %pi^2/12

(%i5) float(%), numer;
(%o5) 0.82246703342411
(%i6) quad_qagi(x/(%e^x+1), x, 0, inf);
(%o6) [0.82246703342411,5.9689712369638092*10^-10,135,0]

Aleksas D

[christoph reineke]

Thank you very much!
By the way, we have the same problem with the similar expression
integrate(x/(exp(x)-1),x,0,inf)=pi^2/6.
Sorry, I noticed that after submitting my first message!

Regards

Chris

[Aleksas]

Solution of the second problem:
(%i1) S:'integrate(x/(exp(x)-1),x,0,inf);
(%o1) integrate(x/(%e^x-1),x,0,inf)

(%i2) intparts(S,u):=block([f,var,v,a,b],
f:part(S,1),var:part(S,2),v:integrate(f/u,var),
if last(S)#var then
(a:part(S,3),b:part(S,4),
limit(u*v,var,b,minus)-limit(u*v,var,a,plus)
-'integrate(v*diff(u,var),var,a,b))
else u*v-'integrate(v*diff(u,var),var)
)$

(%i3) changevar(S, y=exp(x)-1, y, x);
(%o3) integrate(log(y+1)/(y^2+y),y,0,inf)

(%i4) intparts(%,log(y+1));
(%o4) -integrate((log(y)-log(y+1))/(y+1),y,0,inf)

(%i5) logcontract(%);
(%o5) -integrate(log(y/(y+1))/(y+1),y,0,inf)

(%i6) ev(%, nouns);
(%o6) %pi^2/6

Aleksas D

[christoph reineke]

Yes, it works. Thanks again!
Shall we close this bug report?

Regards

Chris

[Dan Gildea]

added asymptotic expansion for polylogarithm - can compute limit now.