\[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{(a+a \sec (e+f x))^2} \, dx \]
Optimal antiderivative \[ \frac {d^{2} \left (12 c^{2}-16 c d +7 d^{2}\right ) \arctanh \left (\sin \left (f x +e \right )\right )}{2 a^{2} f}+\frac {\left (c -d \right ) \left (c +8 d \right ) \left (c +d \sec \left (f x +e \right )\right )^{2} \tan \left (f x +e \right )}{3 f \left (a^{2}+a^{2} \sec \left (f x +e \right )\right )}+\frac {\left (c -d \right ) \left (c +d \sec \left (f x +e \right )\right )^{3} \tan \left (f x +e \right )}{3 f \left (a +a \sec \left (f x +e \right )\right )^{2}}-\frac {d \left (4 c^{3}+32 c^{2} d -80 c \,d^{2}+32 d^{3}+d \left (2 c^{2}+16 c d -21 d^{2}\right ) \sec \left (f x +e \right )\right ) \tan \left (f x +e \right )}{6 a^{2} f} \]
command
integrate(sec(f*x+e)*(c+d*sec(f*x+e))^4/(a+a*sec(f*x+e))^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {3 \, {\left (12 \, c^{2} d^{2} - 16 \, c d^{3} + 7 \, d^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a^{2}} - \frac {3 \, {\left (12 \, c^{2} d^{2} - 16 \, c d^{3} + 7 \, d^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a^{2}} - \frac {6 \, {\left (8 \, c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 5 \, d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 8 \, c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{2} a^{2}} - \frac {a^{4} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, a^{4} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6 \, a^{4} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, a^{4} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + a^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, a^{4} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 12 \, a^{4} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 54 \, a^{4} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 60 \, a^{4} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 21 \, a^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a^{6}}}{6 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________