Firstly, we need to factorise the denominator:

x⁴ - 1 = (x² + 1)(x² - 1) = (x² + 1)(x + 1)(x - 1)

So

The partial fraction decomposition will be of the form:

We multiply throughout by (x² + 1)(x + 1)(x - 1):

x³ - 2 = (Ax + B)(x + 1)(x - 1) + C(x² + 1)(x - 1) + D(x² + 1)(x + 1)

Let x = 1:

LHS = -1

RHS = 4D

So D = -1/4

Let x = -1:

LHS = -3

RHS = -4C

So C = 3/4

Coefficient of x³ on LHS = 1

Coefficient of x³ on RHS = A + C + D

Since D = -1/4 and C = 3/4, then A = 1/2.

Constant term on LHS = -2

Constant term on RHS = -B - C + D

This gives us B = 1.

So