Before considering partial fractions we need to divide the bottom into the top because it's an improper fraction (top has a higher degree than the bottom):
(x4+1)/[x2(x-1)]=x+1+(x2+1)/[x2(x-1)].
Now reduce the fraction part to partial fractions:
(x2+1)/[x2(x-1)]≡A/x2+B/x+C/(x-1), where A, B, C are constants to be found by matching coefficients,
x2+1≡A(x-1)+Bx(x-1)+Cx2=Ax-A+Bx2-Bx+Cx2,
So A=B=-1, B+C=1, C=2.
(x4+1)/[x2(x-1)]=x+1-1/x-1/x2+2/(x-1).