\[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^5}{(a+b \sec (e+f x))^2} \, dx \]
Optimal antiderivative \[ \frac {d^{4} \left (-2 a d +5 b c \right ) \arctanh \left (\sin \left (f x +e \right )\right )}{2 b^{3} f}+\frac {d^{2} \left (-4 a^{3} d^{3}+15 a^{2} b c \,d^{2}-20 a \,b^{2} c^{2} d +10 b^{3} c^{3}\right ) \arctanh \left (\sin \left (f x +e \right )\right )}{b^{5} f}+\frac {2 \left (-a d +b c \right )^{5} \arctanh \left (\frac {\sqrt {a -b}\, \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {a +b}}\right )}{a \left (a -b \right )^{\frac {3}{2}} b^{3} \left (a +b \right )^{\frac {3}{2}} f}-\frac {\left (-a d +b c \right )^{5} \sin \left (f x +e \right )}{b^{4} \left (a^{2}-b^{2}\right ) f \left (b +a \cos \left (f x +e \right )\right )}+\frac {2 \left (-a d +b c \right )^{4} \left (4 a d +b c \right ) \arctanh \left (\frac {\sqrt {a -b}\, \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {a +b}}\right )}{a \,b^{5} f \sqrt {a -b}\, \sqrt {a +b}}+\frac {d^{5} \tan \left (f x +e \right )}{b^{2} f}+\frac {d^{3} \left (3 a^{2} d^{2}-10 a b c d +10 b^{2} c^{2}\right ) \tan \left (f x +e \right )}{b^{4} f}+\frac {d^{4} \left (-2 a d +5 b c \right ) \sec \left (f x +e \right ) \tan \left (f x +e \right )}{2 b^{3} f}+\frac {d^{5} \left (\tan ^{3}\left (f x +e \right )\right )}{3 b^{2} f} \]
command
integrate(sec(f*x+e)*(c+d*sec(f*x+e))^5/(a+b*sec(f*x+e))^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________