\[ \int \frac {1}{a-b \cos ^8(x)} \, dx \]
Optimal antiderivative \[ -\frac {\arctan \left (\frac {\cot \left (x \right ) \sqrt {a^{\frac {1}{4}}-b^{\frac {1}{4}}}}{a^{\frac {1}{8}}}\right )}{4 a^{\frac {7}{8}} \sqrt {a^{\frac {1}{4}}-b^{\frac {1}{4}}}}-\frac {\arctan \left (\frac {\cot \left (x \right ) \sqrt {a^{\frac {1}{4}}-i b^{\frac {1}{4}}}}{a^{\frac {1}{8}}}\right )}{4 a^{\frac {7}{8}} \sqrt {a^{\frac {1}{4}}-i b^{\frac {1}{4}}}}-\frac {\arctan \left (\frac {\cot \left (x \right ) \sqrt {a^{\frac {1}{4}}+i b^{\frac {1}{4}}}}{a^{\frac {1}{8}}}\right )}{4 a^{\frac {7}{8}} \sqrt {a^{\frac {1}{4}}+i b^{\frac {1}{4}}}}-\frac {\arctan \left (\frac {\cot \left (x \right ) \sqrt {a^{\frac {1}{4}}+b^{\frac {1}{4}}}}{a^{\frac {1}{8}}}\right )}{4 a^{\frac {7}{8}} \sqrt {a^{\frac {1}{4}}+b^{\frac {1}{4}}}} \]
command
integrate(1/(a-b*cos(x)^8),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 \[ \text {Timed out} \]