kill(all)$

van=facebook.com/LearningCalculus$

F(x):=integrate(x/(%e^x-1)^(1/3),x,0,inf);

     goo.gl/YsOjO0

q(u,n):=prod(u+i,i,0,n-1);

F21(a,b,c,z):=sum(q(a,n)*q(b,n)/q(c,n)*(z^n/n!),n,0,inf);

F32(a,b,c,d,e,z):=sum(q(a,n)*q(b,n)*q(c,n)/(q(d,n)*q(e,n))*(z^n/n!),n,0,inf)$

F(x):=-3*(1-%e^(-x))^(1/3)/(%e^(x)-1)^(1/3)*
      (3*F32(1/3,1/3,1/3,4/3,4/3,%e^(-x))+
       x*F21(1/3,1/3,4/3,%e^(-x)));

     goo.gl/F93Arr

'F(inf)-'F(0)=1/3*%pi*(%pi+sqrt(3)*log(27));float(rhs(%));