\[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^8} \, dx \]
Optimal antiderivative \[ \frac {\arctan \! \left (\frac {2^{\frac {1}{8}} x}{\left (x^{6}+x^{2}\right )^{\frac {1}{4}}}\right ) 2^{\frac {5}{8}}}{4}+\frac {\arctan \! \left (\frac {2^{\frac {5}{8}} x \left (x^{6}+x^{2}\right )^{\frac {1}{4}}}{x^{2} 2^{\frac {1}{4}}-\sqrt {x^{6}+x^{2}}}\right ) 2^{\frac {1}{8}}}{4}-\frac {\arctanh \! \left (\frac {2^{\frac {1}{8}} x}{\left (x^{6}+x^{2}\right )^{\frac {1}{4}}}\right ) 2^{\frac {5}{8}}}{4}+\frac {\arctanh \! \left (\frac {\frac {x^{2} 2^{\frac {5}{8}}}{2}+\frac {\sqrt {x^{6}+x^{2}}\, 2^{\frac {3}{8}}}{2}}{x \left (x^{6}+x^{2}\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{8}}}{4} \]
command
int((x^4-1)*(x^6+x^2)^(1/4)/(x^8+1),x)
Maple 2022.1 output
\[\int \frac {\left (x^{4}-1\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{4}}}{x^{8}+1}\, dx\]
Maple 2021.1 output
\[\text {output too large to display}\]