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