\[ \int \frac {\tan ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {\arctanh \left (\frac {\sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}}{\sqrt {a}}\right )}{a^{\frac {5}{2}} f}+\frac {\left (a +b \right )^{2}}{3 a \,b^{2} f \left (a +b \left (\sec ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}+\frac {\frac {1}{a^{2}}-\frac {1}{b^{2}}}{f \sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}} \]
command
integrate(tan(f*x+e)^5/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {{\left ({\left (\frac {{\left (2 \, a^{11} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + a^{10} b \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - 4 \, a^{9} b^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - 3 \, a^{8} b^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}{a^{10} b^{2}} - \frac {3 \, {\left (2 \, a^{11} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - 3 \, a^{10} b \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - 4 \, a^{9} b^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + a^{8} b^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )\right )}}{a^{10} b^{2}}\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + \frac {3 \, {\left (2 \, a^{11} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - 3 \, a^{10} b \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - 4 \, a^{9} b^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + a^{8} b^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )\right )}}{a^{10} b^{2}}\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - \frac {2 \, a^{11} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + a^{10} b \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - 4 \, a^{9} b^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - 3 \, a^{8} b^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{a^{10} b^{2}}}{{\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a + b\right )}^{\frac {3}{2}}} + \frac {6 \, \arctan \left (-\frac {\sqrt {a + b} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a + b} + \sqrt {a + b}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} a^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}}{3 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________