\[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx \]
Optimal antiderivative \[ \frac {d^{3} \left (20 c^{2}-30 c d +13 d^{2}\right ) \arctanh \left (\sin \left (f x +e \right )\right )}{2 a^{3} f}-\frac {2 d \left (2 c^{4}+15 c^{3} d +72 c^{2} d^{2}-180 c \,d^{3}+76 d^{4}\right ) \tan \left (f x +e \right )}{15 a^{3} f}-\frac {d^{2} \left (4 c^{3}+30 c^{2} d +146 c \,d^{2}-195 d^{3}\right ) \sec \left (f x +e \right ) \tan \left (f x +e \right )}{30 a^{3} f}+\frac {\left (c -d \right ) \left (2 c^{2}+15 c d +76 d^{2}\right ) \left (c +d \sec \left (f x +e \right )\right )^{2} \tan \left (f x +e \right )}{15 f \left (a^{3}+a^{3} \sec \left (f x +e \right )\right )}+\frac {\left (c -d \right ) \left (2 c +11 d \right ) \left (c +d \sec \left (f x +e \right )\right )^{3} \tan \left (f x +e \right )}{15 a f \left (a +a \sec \left (f x +e \right )\right )^{2}}+\frac {\left (c -d \right ) \left (c +d \sec \left (f x +e \right )\right )^{4} \tan \left (f x +e \right )}{5 f \left (a +a \sec \left (f x +e \right )\right )^{3}} \]
command
integrate(sec(f*x+e)*(c+d*sec(f*x+e))^5/(a+a*sec(f*x+e))^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {30 \, {\left (20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a^{3}} - \frac {30 \, {\left (20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a^{3}} - \frac {60 \, {\left (10 \, c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 7 \, d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 10 \, c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 5 \, 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 )}^{2} a^{3}} + \frac {3 \, a^{12} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 15 \, a^{12} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 30 \, a^{12} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 30 \, a^{12} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 15 \, a^{12} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 3 \, a^{12} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 10 \, a^{12} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 100 \, a^{12} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 200 \, a^{12} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 150 \, a^{12} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 40 \, a^{12} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15 \, a^{12} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 75 \, a^{12} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 150 \, a^{12} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1050 \, a^{12} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1275 \, a^{12} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 465 \, a^{12} d^{5} \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
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________