Results from commercial macsyma
integrate(sech(x)^2*x^2,x,minf,inf); => %pi^2/6 (might be right, I
didn't check)
SAME
integrate(sech(x/2)^2*x^2,x,minf,inf); => 0 (must be wrong)
4/3*%pi^2/3
integrate(sech(x/3)^2*x^2,x,minf,inf); => 0 (must be wrong)
'LIMIT(-12*X^2/(2*%E^(2*X/3)+2)+6*X^2-18*LOG(%E^(2*X/3)+1)*X-27*LI[2](-%E^(2
*X/3)),X,INF)
integrate(sech(2*x)^2*x^2,x,minf,inf); => - 47/48 %pi^2 (must be wrong)
%pi^2/48
This last one is interesting -- Maxima 5.9.3 => - 47/48 %pi^2 while
5.9.1 and 5.9.2 both yield - 11/192 %pi^2, which also must be wrong.