\[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{(a+a \sec (e+f x))^3} \, dx \]
Optimal antiderivative \[ \frac {\left (4 c -3 d \right ) d^{3} \arctanh \left (\sin \left (f x +e \right )\right )}{a^{3} f}+\frac {\left (c -d \right ) \left (2 c +9 d \right ) \left (c +d \sec \left (f x +e \right )\right )^{2} \tan \left (f x +e \right )}{15 a f \left (a +a \sec \left (f x +e \right )\right )^{2}}+\frac {\left (c -d \right ) \left (c +d \sec \left (f x +e \right )\right )^{3} \tan \left (f x +e \right )}{5 f \left (a +a \sec \left (f x +e \right )\right )^{3}}+\frac {\left (2 c^{4}+8 c^{3} d +21 c^{2} d^{2}-88 c \,d^{3}+72 d^{4}-d^{2} \left (2 c^{2}+10 c d -27 d^{2}\right ) \sec \left (f x +e \right )\right ) \tan \left (f x +e \right )}{15 f \left (a^{3}+a^{3} \sec \left (f x +e \right )\right )} \]
command
integrate(sec(f*x+e)*(c+d*sec(f*x+e))^4/(a+a*sec(f*x+e))^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\frac {120 \, d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )} a^{3}} - \frac {60 \, {\left (4 \, c d^{3} - 3 \, d^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a^{3}} + \frac {60 \, {\left (4 \, c d^{3} - 3 \, d^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a^{3}} - \frac {3 \, a^{12} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 12 \, a^{12} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 18 \, a^{12} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 12 \, a^{12} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3 \, a^{12} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 10 \, a^{12} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 60 \, a^{12} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 80 \, a^{12} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 30 \, a^{12} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15 \, a^{12} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 60 \, a^{12} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 90 \, a^{12} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 420 \, a^{12} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 255 \, a^{12} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a^{15}}}{60 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________