Let \[ I = \int {\frac{x}{{{x^4} - 1}}} dx\]
Let's put \[ {x^2} = t \Rightarrow 2xdx = dt \Rightarrow xdx = \frac{1}{2}dt\]

therefore,\[ I = \frac{1}{2}\int {\frac{{dt}}{{{t^2} - 1}}} = \frac{1}{2} \cdot \frac{1}{2}\log \left| {\frac{{t - 1}}{{t + 1}}} \right| + C\]

\[ = \frac{1}{4}\left[ {\log \left| {{x^2} - 1} \right| - \log \left| {{x^2} + 1} \right|} \right] + C\]

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