Integral de x^4/(x^4-3*x^2+1) dx

                                                   //            /    ___          \                       \     //            /    ___           \                        \                   
                                                   ||   ___      |2*\/ 5 *(1/2 + x)|                       |     ||   ___      |2*\/ 5 *(-1/2 + x)|                        |                   
  /                                                ||-\/ 5 *acoth|-----------------|                       |     ||-\/ 5 *acoth|------------------|                        |                   
 |                                                 ||            \        5        /                2      |     ||            \        5         /                 2      |                   
 |        4                      /      2    \     ||--------------------------------  for (1/2 + x)  > 5/4|     ||---------------------------------  for (-1/2 + x)  > 5/4|      /          2\
 |       x                    log\-1 + x  - x/     ||               10                                     |     ||                10                                      |   log\-1 + x + x /
 | ------------- dx = C + x + ---------------- + 4*|<                                                      | + 4*|<                                                        | - ----------------
 |  4      2                         2             ||            /    ___          \                       |     ||            /    ___           \                        |          2        
 | x  - 3*x  + 1                                   ||   ___      |2*\/ 5 *(1/2 + x)|                       |     ||   ___      |2*\/ 5 *(-1/2 + x)|                        |                   
 |                                                 ||-\/ 5 *atanh|-----------------|                       |     ||-\/ 5 *atanh|------------------|                        |                   
/                                                  ||            \        5        /                2      |     ||            \        5         /                 2      |                   
                                                   ||--------------------------------  for (1/2 + x)  < 5/4|     ||---------------------------------  for (-1/2 + x)  < 5/4|                   
                                                   \\               10                                     /     \\                10                                      /                   

$$\int \frac{x^{4}}{\left(x^{4} - 3 x^{2}\right) + 1}\, dx = C + x + 4 \left(\begin{cases} - \frac{\sqrt{5} \operatorname{acoth}{\left(\frac{2 \sqrt{5} \left(x - \frac{1}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(x - \frac{1}{2}\right)^{2} > \frac{5}{4} \\- \frac{\sqrt{5} \operatorname{atanh}{\left(\frac{2 \sqrt{5} \left(x - \frac{1}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(x - \frac{1}{2}\right)^{2} < \frac{5}{4} \end{cases}\right) + 4 \left(\begin{cases} - \frac{\sqrt{5} \operatorname{acoth}{\left(\frac{2 \sqrt{5} \left(x + \frac{1}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(x + \frac{1}{2}\right)^{2} > \frac{5}{4} \\- \frac{\sqrt{5} \operatorname{atanh}{\left(\frac{2 \sqrt{5} \left(x + \frac{1}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(x + \frac{1}{2}\right)^{2} < \frac{5}{4} \end{cases}\right) + \frac{\log{\left(x^{2} - x - 1 \right)}}{2} - \frac{\log{\left(x^{2} + x - 1 \right)}}{2}$$

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