Matematica

Con $\small x^4+a^4$ o $\small x^4-a^4$

1.: $\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle4}+a^{\displaystyle4}}=\displaystyle \frac{1}{4a^{\displaystyle3}\displaystyle \sqrt{2}}\ln\left(\displaystyle \frac{x^{\displaystyle2}+ax\displaystyle \sqrt{2}+a^{\displaystyle2}}{x^{\displaystyle2}-ax\displaystyle \sqrt{2}+a^{\displaystyle2}}\right)\,-\,\displaystyle \frac{1}{2a^{\displaystyle3}\displaystyle \sqrt{2}}\tan^{\displaystyle-1}\displaystyle \frac{ax\displaystyle \sqrt{2}}{x^{\displaystyle2}-a^{\displaystyle2}}$

2.: $\displaystyle \int\displaystyle \frac{x\,dx}{x^{\displaystyle4}+a^{\displaystyle4}}=\displaystyle \frac{1}{2a^{\displaystyle2}}\tan^{\displaystyle-1}\displaystyle \frac{x^{\displaystyle2}}{a^{\displaystyle2}}$

3.: $\displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{x^{\displaystyle4}+a^{\displaystyle4}}=\displaystyle \frac{1}{4a\displaystyle \sqrt{2}}\ln\left(\displaystyle \frac{x^{\displaystyle2}-ax\displaystyle \sqrt{2}+a^{\displaystyle2}}{x^{\displaystyle2}+ax\displaystyle \sqrt{2}+a^{\displaystyle2}}\right)\,-\,\displaystyle \frac{1}{2a\displaystyle \sqrt{2}}\tan^{\displaystyle-1}\displaystyle \frac{ax\displaystyle \sqrt{2}}{x^{\displaystyle2}-a^{\displaystyle2}}$

4.: $\displaystyle \int\displaystyle \frac{x^{\displaystyle3}\,dx}{x^{\displaystyle4}+a^{\displaystyle4}}=\displaystyle \frac{1}{4}ln\left(x^{\displaystyle4}+a^{\displaystyle4}\right)$

5.: $\displaystyle \int\displaystyle \frac{dx}{x\left(x^{\displaystyle4}+a^{\displaystyle4}\right)}=\displaystyle \frac{1}{4a^{\displaystyle4}}\ln\left(\displaystyle \frac{x^{\displaystyle4}}{x^{\displaystyle4}+a^{\displaystyle4}}\right)$

6.: $\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle2}\left(x^{\displaystyle4}+a^{\displaystyle4}\right)}=-\displaystyle \frac{1}{a^{\displaystyle4}x}\,-\,\displaystyle \frac{1}{4a^{\displaystyle5}\displaystyle \sqrt{2}}\ln\left(\displaystyle \frac{x^{\displaystyle2}-ax\displaystyle \sqrt{2}+a^{\displaystyle2}}{x^{\displaystyle2}+ax\displaystyle \sqrt{2}+a^{\displaystyle2}}\right)\,+\,\displaystyle \frac{1}{2a^{\displaystyle5}\displaystyle \sqrt{2}}\tan^{\displaystyle-1}\displaystyle \frac{ax\displaystyle \sqrt{2}}{x^{\displaystyle2}-a^{\displaystyle2}}$

7.: $\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle3}\left(x^{\displaystyle4}+a^{\displaystyle4}\right)}=-\displaystyle \frac{1}{2a^{\displaystyle4}x^{\displaystyle2}}\,-\,\displaystyle \frac{1}{2a^{\displaystyle6}}\tan^{\displaystyle-1}\displaystyle \frac{x^{\displaystyle2}}{a^{\displaystyle2}}$

8.: $\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle4}-a^{\displaystyle4}}=\displaystyle \frac{1}{4a^{\displaystyle3}}\ln\left(\displaystyle \frac{x-a}{x+a}\right)\,-\,\displaystyle \frac{1}{2a^{\displaystyle3}}\tan^{\displaystyle-1}\displaystyle \frac{x}{a}$

9.: $\displaystyle \int\displaystyle \frac{x\,dx}{x^{\displaystyle4}-a^{\displaystyle4}}=\displaystyle \frac{1}{4a^{\displaystyle2}}\ln\left(\displaystyle \frac{x^{\displaystyle2}-a^{\displaystyle2}}{x^{\displaystyle2}+a^{\displaystyle2}}\right)$

10.: $\displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{x^{\displaystyle4}-a^{\displaystyle4}}=\displaystyle \frac{1}{4a}\ln\left(\displaystyle \frac{x-a}{x+a}\right)\,+\,\displaystyle \frac{1}{2a}\tan^{\displaystyle-1}\displaystyle \frac{x}{a}$

11.: $\displaystyle \int\displaystyle \frac{x^{\displaystyle3}\,dx}{x^{\displaystyle4}-a^{\displaystyle4}}=\displaystyle \frac{1}{4}\ln\left(x^{\displaystyle4}+a^{\displaystyle4}\right)$

12.: $\displaystyle \int\displaystyle \frac{dx}{x\left(x^{\displaystyle4}-a^{\displaystyle4}\right)}=\displaystyle \frac{1}{4a^{\displaystyle4}}\ln\left(\displaystyle \frac{x^{\displaystyle4}-a^{\displaystyle4}}{x^{\displaystyle4}}\right)$

13.: $\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle2}\left(x^{\displaystyle4}-a^{\displaystyle4}\right)}=\displaystyle \frac{1}{a^{\displaystyle4}x}\,+\,\displaystyle \frac{1}{4a^{\displaystyle5}}\ln\left(\displaystyle \frac{x-a}{x+a}\right)\,+\,\displaystyle \frac{1}{2a^{\displaystyle5}}\tan^{\displaystyle-1}\displaystyle \frac{x}{a}$

14.: $\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle3}\left(x^{\displaystyle4}-a^{\displaystyle4}\right)}=\displaystyle \frac{1}{2a^{\displaystyle4}x^2 }+ \frac{1}{4a^6} \ln\left(\frac{x^2-a^2}{x^2+a^2}\right)$