\[ \int \frac {(c+d \sec (e+f x))^4}{a+b \cos (e+f x)} \, dx \]
Optimal antiderivative \[ \frac {d^{3} \left (4 a c -b d \right ) \arctanh \left (\sin \left (f x +e \right )\right )}{2 a^{2} f}+\frac {d \left (2 a c -b d \right ) \left (2 a^{2} c^{2}-2 a b c d +b^{2} d^{2}\right ) \arctanh \left (\sin \left (f x +e \right )\right )}{a^{4} f}+\frac {2 \left (a c -b d \right )^{4} \arctan \left (\frac {\sqrt {a -b}\, \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {a +b}}\right )}{a^{4} f \sqrt {a -b}\, \sqrt {a +b}}+\frac {d^{4} \tan \left (f x +e \right )}{a f}+\frac {d^{2} \left (6 a^{2} c^{2}-4 a b c d +b^{2} d^{2}\right ) \tan \left (f x +e \right )}{a^{3} f}+\frac {d^{3} \left (4 a c -b d \right ) \sec \left (f x +e \right ) \tan \left (f x +e \right )}{2 a^{2} f}+\frac {d^{4} \left (\tan ^{3}\left (f x +e \right )\right )}{3 a f} \]
command
integrate((c+d*sec(f*x+e))^4/(a+b*cos(f*x+e)),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \left [-\frac {6 \, {\left (a^{4} c^{4} - 4 \, a^{3} b c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a b^{3} c d^{3} + b^{4} d^{4}\right )} \sqrt {-a^{2} + b^{2}} \cos \left (f x + e\right )^{3} \log \left (\frac {2 \, a b \cos \left (f x + e\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (f x + e\right ) + b\right )} \sin \left (f x + e\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (f x + e\right )^{2} + 2 \, a b \cos \left (f x + e\right ) + a^{2}}\right ) - 3 \, {\left (8 \, {\left (a^{5} - a^{3} b^{2}\right )} c^{3} d - 12 \, {\left (a^{4} b - a^{2} b^{3}\right )} c^{2} d^{2} + 4 \, {\left (a^{5} + a^{3} b^{2} - 2 \, a b^{4}\right )} c d^{3} - {\left (a^{4} b + a^{2} b^{3} - 2 \, b^{5}\right )} d^{4}\right )} \cos \left (f x + e\right )^{3} \log \left (\sin \left (f x + e\right ) + 1\right ) + 3 \, {\left (8 \, {\left (a^{5} - a^{3} b^{2}\right )} c^{3} d - 12 \, {\left (a^{4} b - a^{2} b^{3}\right )} c^{2} d^{2} + 4 \, {\left (a^{5} + a^{3} b^{2} - 2 \, a b^{4}\right )} c d^{3} - {\left (a^{4} b + a^{2} b^{3} - 2 \, b^{5}\right )} d^{4}\right )} \cos \left (f x + e\right )^{3} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (2 \, {\left (a^{5} - a^{3} b^{2}\right )} d^{4} + 2 \, {\left (18 \, {\left (a^{5} - a^{3} b^{2}\right )} c^{2} d^{2} - 12 \, {\left (a^{4} b - a^{2} b^{3}\right )} c d^{3} + {\left (2 \, a^{5} + a^{3} b^{2} - 3 \, a b^{4}\right )} d^{4}\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (4 \, {\left (a^{5} - a^{3} b^{2}\right )} c d^{3} - {\left (a^{4} b - a^{2} b^{3}\right )} d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{12 \, {\left (a^{6} - a^{4} b^{2}\right )} f \cos \left (f x + e\right )^{3}}, \frac {12 \, {\left (a^{4} c^{4} - 4 \, a^{3} b c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a b^{3} c d^{3} + b^{4} d^{4}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cos \left (f x + e\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (f x + e\right )}\right ) \cos \left (f x + e\right )^{3} + 3 \, {\left (8 \, {\left (a^{5} - a^{3} b^{2}\right )} c^{3} d - 12 \, {\left (a^{4} b - a^{2} b^{3}\right )} c^{2} d^{2} + 4 \, {\left (a^{5} + a^{3} b^{2} - 2 \, a b^{4}\right )} c d^{3} - {\left (a^{4} b + a^{2} b^{3} - 2 \, b^{5}\right )} d^{4}\right )} \cos \left (f x + e\right )^{3} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \, {\left (8 \, {\left (a^{5} - a^{3} b^{2}\right )} c^{3} d - 12 \, {\left (a^{4} b - a^{2} b^{3}\right )} c^{2} d^{2} + 4 \, {\left (a^{5} + a^{3} b^{2} - 2 \, a b^{4}\right )} c d^{3} - {\left (a^{4} b + a^{2} b^{3} - 2 \, b^{5}\right )} d^{4}\right )} \cos \left (f x + e\right )^{3} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (2 \, {\left (a^{5} - a^{3} b^{2}\right )} d^{4} + 2 \, {\left (18 \, {\left (a^{5} - a^{3} b^{2}\right )} c^{2} d^{2} - 12 \, {\left (a^{4} b - a^{2} b^{3}\right )} c d^{3} + {\left (2 \, a^{5} + a^{3} b^{2} - 3 \, a b^{4}\right )} d^{4}\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (4 \, {\left (a^{5} - a^{3} b^{2}\right )} c d^{3} - {\left (a^{4} b - a^{2} b^{3}\right )} d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{12 \, {\left (a^{6} - a^{4} b^{2}\right )} f \cos \left (f x + e\right )^{3}}\right ] \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Timed out} \]