\[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^5}{(a+a \sec (e+f x))^2} \, dx \]
Optimal antiderivative \[ \frac {5 \left (2 c -d \right ) d^{2} \left (2 c^{2}-3 c d +2 d^{2}\right ) \arctanh \left (\sin \left (f x +e \right )\right )}{2 a^{2} f}-\frac {d \left (c^{2}+10 c d -12 d^{2}\right ) \left (c +d \sec \left (f x +e \right )\right )^{2} \tan \left (f x +e \right )}{3 a^{2} f}+\frac {\left (c -d \right ) \left (c +10 d \right ) \left (c +d \sec \left (f x +e \right )\right )^{3} \tan \left (f x +e \right )}{3 f \left (a^{2}+a^{2} \sec \left (f x +e \right )\right )}+\frac {\left (c -d \right ) \left (c +d \sec \left (f x +e \right )\right )^{4} \tan \left (f x +e \right )}{3 f \left (a +a \sec \left (f x +e \right )\right )^{2}}-\frac {d \left (4 c^{4}+40 c^{3} d -176 c^{2} d^{2}+160 c \,d^{3}-48 d^{4}+d \left (2 c^{3}+20 c^{2} d -57 c \,d^{2}+30 d^{3}\right ) \sec \left (f x +e \right )\right ) \tan \left (f x +e \right )}{6 a^{2} f} \]
command
integrate(sec(f*x+e)*(c+d*sec(f*x+e))^5/(a+a*sec(f*x+e))^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {15 \, {\left (4 \, c^{3} d^{2} - 8 \, c^{2} d^{3} + 7 \, c d^{4} - 2 \, d^{5}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a^{2}} - \frac {15 \, {\left (4 \, c^{3} d^{2} - 8 \, c^{2} d^{3} + 7 \, c d^{4} - 2 \, d^{5}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a^{2}} - \frac {2 \, {\left (60 \, c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 75 \, c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 30 \, d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 120 \, c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 120 \, c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 40 \, d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 60 \, c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 45 \, c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 18 \, d^{5} \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 )}^{3} a^{2}} - \frac {a^{4} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 5 \, a^{4} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 10 \, a^{4} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 10 \, a^{4} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 5 \, a^{4} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - a^{4} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, a^{4} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 15 \, a^{4} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 90 \, a^{4} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 150 \, a^{4} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 105 \, a^{4} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 27 \, a^{4} d^{5} \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
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________