1/(x-1)-1/(1+x)-2/(1+x^2)-4/(1+x^4)-8/(1+x^8)
=2/(1-x^2)-2/(1+x^2)-4/(1+x^4)-8/(1+x^8)
=4/(1-x^4)-4/(1+x^4)-8/(1+x^8)
=8/(1-x^8)-8/(1+x^8)
=16/(1+x^16)