\[ \int \frac {1}{\left (b \tan ^2(e+f x)\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {\cot \left (f x +e \right )}{2 b^{2} f \sqrt {b \left (\tan ^{2}\left (f x +e \right )\right )}}-\frac {\cot ^{3}\left (f x +e \right )}{4 b^{2} f \sqrt {b \left (\tan ^{2}\left (f x +e \right )\right )}}+\frac {\ln \left (\sin \left (f x +e \right )\right ) \tan \left (f x +e \right )}{b^{2} f \sqrt {b \left (\tan ^{2}\left (f x +e \right )\right )}} \]
command
integrate(1/(b*tan(f*x+e)^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {32 \, \log \left (\frac {{\left | -\cos \left (f x + e\right ) + 1 \right |}}{{\left | \cos \left (f x + e\right ) + 1 \right |}}\right )}{b^{\frac {5}{2}} \mathrm {sgn}\left (\tan \left (f x + e\right )\right )} - \frac {64 \, \log \left ({\left | -\frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1} + 1 \right |}\right )}{b^{\frac {5}{2}} \mathrm {sgn}\left (\tan \left (f x + e\right )\right )} - \frac {{\left (\sqrt {b} + \frac {12 \, \sqrt {b} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {48 \, \sqrt {b} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{2}}{b^{3} {\left (\cos \left (f x + e\right ) - 1\right )}^{2} \mathrm {sgn}\left (\tan \left (f x + e\right )\right )} - \frac {\frac {12 \, b^{\frac {7}{2}} {\left (\cos \left (f x + e\right ) - 1\right )} \mathrm {sgn}\left (\tan \left (f x + e\right )\right )}{\cos \left (f x + e\right ) + 1} + \frac {b^{\frac {7}{2}} {\left (\cos \left (f x + e\right ) - 1\right )}^{2} \mathrm {sgn}\left (\tan \left (f x + e\right )\right )}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}}{b^{6}}}{64 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________