$\int \frac{3 x + 1}{(x^{3} - x^{2} - x + 1)} d x$

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Solution

$I = \int \frac{3 x + 1}{x^{3} - x^{2} - x + 1} d x$
Using partial fraction
$\frac{3 x + 1}{x^{3} - x^{2} - x + 1} = \frac{1}{2 (x - 1)} + \frac{2}{(x - 1^{2})} - \frac{1}{2 (x + 1)}$
$I = \int [\frac{1}{2 (x - 1)} + \frac{2}{(x - 1)^{2}} - \frac{1}{2 (x + 1)}] d x$
$I = \frac{1}{2} log (x - 1) - \frac{2}{(x - 1)^{2}} - \frac{1}{2} log (x + 1)$

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