Evaluate: `int(x^4\ dx)/((x-1)(x^2+1))`

Explanation

`I=int frac{x^{4} d x}{(x-1)(x^{2}+1)}`

`=int[frac{x^{4}-1+1}{(x-1)(x^{2}+1)}] d x`

`=int frac{(x^{4}-1) d x}{(x-1)(x^{2}+1)}+int frac{d x}{(x-1)(x^{2}+1)}`

`=int frac{(x^{2}-1)(x^{2}+1) d x}{(x-1)(x^{2}+1)}+int frac{d x}{(x-1)(x^{2}+1)}`

`=int frac{(x-1)(x+1) d x}{(x-1)}+int frac{d x}{(x-1)(x^{2}+1)}`

`=int(x+1) d x+int frac{d x}{(x-1)(x^{2}+1)} ldots cdots cdots cdots cdots(1)`

Let `frac{A x+C}{(x-1)(x^{2}+1)}=frac{A-1}{x-1}+frac{x^{2}+1}{A(x^{2}+1)+(B x+C)(x-1)}`

`Rightarrow frac{1}{(x-1)(x^{2}+1)}=frac{(x-1)(x^{2}+1)}{(3)}`

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