\[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3 \, dx \]
Optimal antiderivative \[ \frac {a^{3} \left (40 c^{3}+90 c^{2} d +78 c \,d^{2}+23 d^{3}\right ) \arctanh \left (\sin \left (f x +e \right )\right )}{16 f}+\frac {a^{3} \left (40 c^{3}+90 c^{2} d +78 c \,d^{2}+23 d^{3}\right ) \tan \left (f x +e \right )}{16 f}+\frac {\left (40 c^{3}+90 c^{2} d +78 c \,d^{2}+23 d^{3}\right ) \left (a^{3}+a^{3} \sec \left (f x +e \right )\right ) \tan \left (f x +e \right )}{48 f}+\frac {a \left (3 c +8 d \right ) \left (a +a \sec \left (f x +e \right )\right )^{2} \left (c +d \sec \left (f x +e \right )\right )^{2} \tan \left (f x +e \right )}{30 f}+\frac {a \left (a +a \sec \left (f x +e \right )\right )^{2} \left (c +d \sec \left (f x +e \right )\right )^{3} \tan \left (f x +e \right )}{6 f}+\frac {a \left (a +a \sec \left (f x +e \right )\right )^{2} \left (8 c^{3}+148 c^{2} d +132 c \,d^{2}+42 d^{3}+d \left (6 c^{2}+62 c d +31 d^{2}\right ) \sec \left (f x +e \right )\right ) \tan \left (f x +e \right )}{120 f} \]
command
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^3*(c+d*sec(f*x+e))^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {15 \, {\left (40 \, a^{3} c^{3} + 90 \, a^{3} c^{2} d + 78 \, a^{3} c d^{2} + 23 \, a^{3} d^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) - 15 \, {\left (40 \, a^{3} c^{3} + 90 \, a^{3} c^{2} d + 78 \, a^{3} c d^{2} + 23 \, a^{3} d^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) - \frac {2 \, {\left (600 \, a^{3} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} + 1350 \, a^{3} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} + 1170 \, a^{3} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} + 345 \, a^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} - 3400 \, a^{3} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} - 7650 \, a^{3} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} - 6630 \, a^{3} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} - 1955 \, a^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 7920 \, a^{3} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 17820 \, a^{3} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 15444 \, a^{3} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 4554 \, a^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 9360 \, a^{3} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 22500 \, a^{3} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 17964 \, a^{3} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 5814 \, a^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 5560 \, a^{3} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15390 \, a^{3} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 12570 \, a^{3} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 3165 \, a^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 1320 \, a^{3} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 4410 \, a^{3} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 4590 \, a^{3} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1575 \, a^{3} d^{3} \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 )}^{6}}}{240 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________