\[ \int \left (a+b \sec ^3(e+f x)\right ) \tan ^5(e+f x) \, dx \]
Optimal antiderivative \[ -\frac {a \ln \left (\cos \left (f x +e \right )\right )}{f}-\frac {a \left (\sec ^{2}\left (f x +e \right )\right )}{f}+\frac {b \left (\sec ^{3}\left (f x +e \right )\right )}{3 f}+\frac {a \left (\sec ^{4}\left (f x +e \right )\right )}{4 f}-\frac {2 b \left (\sec ^{5}\left (f x +e \right )\right )}{5 f}+\frac {b \left (\sec ^{7}\left (f x +e \right )\right )}{7 f} \]
command
integrate((a+b*sec(f*x+e)^3)*tan(f*x+e)^5,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {420 \, a \log \left ({\left | -\frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1} + 1 \right |}\right ) - 420 \, a \log \left ({\left | -\frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1} - 1 \right |}\right ) + \frac {1089 \, a + 64 \, b + \frac {8463 \, a {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {448 \, b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {28749 \, a {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {1344 \, b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {51555 \, a {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {2240 \, b {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {51555 \, a {\left (\cos \left (f x + e\right ) - 1\right )}^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {4480 \, b {\left (\cos \left (f x + e\right ) - 1\right )}^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {28749 \, a {\left (\cos \left (f x + e\right ) - 1\right )}^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {8463 \, a {\left (\cos \left (f x + e\right ) - 1\right )}^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {1089 \, a {\left (\cos \left (f x + e\right ) - 1\right )}^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}}}{{\left (\frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1} + 1\right )}^{7}}}{420 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________