1/(1-x) -1/(1+x)-2x/(1+x^2)+(-4x^3)/(1+x^4)
= (1+x)/(1-x) (1+x)- (1-x)/ (1-x)(1+x)-2x/(1+x^2)+(-4x^3)/(1+x^4)
=[ (1+x)- (1-x)]/ (1-x^2)-2x/(1+x^2)+(-4x^3)/(1+x^4)
=(1+x- 1+x)/ (1-x^2)-2x/(1+x^2)+(-4x^3)/(1+x^4)
=2x/ (1-x^2)-2x/(1+x^2)+(-4x^3)/(1+x^4)
=2x(1+x^2)/ (1-x^2)(1+x^2)-2x(1-x^2)/(1+x^2)(1-x^2)+(-4x^3)/(1+x^4)
=[2x(1+x^2)-2x(1-x^2)]/(1-x^4)+(-4x^3)/(1+x^4)
=2x(1+x^2-1+x^2)/(1-x^4)+(-4x^3)/(1+x^4)
=2x*2x^2/(1-x^4)+(-4x^3)/(1+x^4)
=4x^3/(1-x^4)+(-4x^3)/(1+x^4)
=4x^3(1+x^4)/(1-x^4)(1+x^4)-4x^3(1-x^4)/(1+x^4)(1-x^4)
=[4x^3(1+x^4)-4x^3(1-x^4)]/(1+x^4)(1-x^4)
=[4x^3(1+x^4-1+x^4)]/(1+x^4)(1-x^4)
=4x^3*2x^4/(1-x^8)
=8x^7/(1-x^8)