\[ \int \frac {1}{a+b \cos ^6(x)} \, dx \]
Optimal antiderivative \[ -\frac {\arctan \left (\frac {\cot \left (x \right ) \sqrt {a^{\frac {1}{3}}+b^{\frac {1}{3}}}}{a^{\frac {1}{6}}}\right )}{3 a^{\frac {5}{6}} \sqrt {a^{\frac {1}{3}}+b^{\frac {1}{3}}}}-\frac {\arctan \left (\frac {\cot \left (x \right ) \sqrt {a^{\frac {1}{3}}-\left (-1\right )^{\frac {1}{3}} b^{\frac {1}{3}}}}{a^{\frac {1}{6}}}\right )}{3 a^{\frac {5}{6}} \sqrt {a^{\frac {1}{3}}-\left (-1\right )^{\frac {1}{3}} b^{\frac {1}{3}}}}-\frac {\arctan \left (\frac {\cot \left (x \right ) \sqrt {a^{\frac {1}{3}}+\left (-1\right )^{\frac {2}{3}} b^{\frac {1}{3}}}}{a^{\frac {1}{6}}}\right )}{3 a^{\frac {5}{6}} \sqrt {a^{\frac {1}{3}}+\left (-1\right )^{\frac {2}{3}} b^{\frac {1}{3}}}} \]
command
integrate(1/(a+b*cos(x)^6),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ \text {Timed out} \]