\[ \int \frac {1}{\sqrt [4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}} \, dx \]
Optimal antiderivative \[ -\frac {\arctan \! \left (\frac {\left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x \left (x^{4}+1\right )}\right ) 2^{\frac {3}{4}}}{4}+\frac {\arctanh \! \left (\frac {\left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x \left (x^{4}+1\right )}\right ) 2^{\frac {3}{4}}}{4} \]
command
integrate(1/(x^20+3*x^16+2*x^12-2*x^8-3*x^4-1)^(1/4),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \frac {1}{4} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {2^{\frac {3}{4}} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{2 \, x}\right ) - \frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} + \frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} - \frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) \]