\[ \int \frac {1}{\left (b \tan ^3(e+f x)\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2}{3 b f \sqrt {b \left (\tan ^{3}\left (f x +e \right )\right )}}-\frac {2 \left (\cot ^{2}\left (f x +e \right )\right )}{7 b f \sqrt {b \left (\tan ^{3}\left (f x +e \right )\right )}}+\frac {\arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (f x +e \right )\right )\right ) \left (\tan ^{\frac {3}{2}}\left (f x +e \right )\right ) \sqrt {2}}{2 b f \sqrt {b \left (\tan ^{3}\left (f x +e \right )\right )}}+\frac {\arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (f x +e \right )\right )\right ) \left (\tan ^{\frac {3}{2}}\left (f x +e \right )\right ) \sqrt {2}}{2 b f \sqrt {b \left (\tan ^{3}\left (f x +e \right )\right )}}-\frac {\ln \left (1-\sqrt {2}\, \left (\sqrt {\tan }\left (f x +e \right )\right )+\tan \left (f x +e \right )\right ) \left (\tan ^{\frac {3}{2}}\left (f x +e \right )\right ) \sqrt {2}}{4 b f \sqrt {b \left (\tan ^{3}\left (f x +e \right )\right )}}+\frac {\ln \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (f x +e \right )\right )+\tan \left (f x +e \right )\right ) \left (\tan ^{\frac {3}{2}}\left (f x +e \right )\right ) \sqrt {2}}{4 b f \sqrt {b \left (\tan ^{3}\left (f x +e \right )\right )}} \]
command
integrate(1/(b*tan(f*x+e)^3)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{84} \, b^{4} {\left (\frac {42 \, \sqrt {2} \sqrt {{\left | b \right |}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} + 2 \, \sqrt {b \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{b^{6} f \mathrm {sgn}\left (\tan \left (f x + e\right )\right )} + \frac {42 \, \sqrt {2} \sqrt {{\left | b \right |}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} - 2 \, \sqrt {b \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{b^{6} f \mathrm {sgn}\left (\tan \left (f x + e\right )\right )} + \frac {21 \, \sqrt {2} \sqrt {{\left | b \right |}} \log \left (b \tan \left (f x + e\right ) + \sqrt {2} \sqrt {b \tan \left (f x + e\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{b^{6} f \mathrm {sgn}\left (\tan \left (f x + e\right )\right )} - \frac {21 \, \sqrt {2} \sqrt {{\left | b \right |}} \log \left (b \tan \left (f x + e\right ) - \sqrt {2} \sqrt {b \tan \left (f x + e\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{b^{6} f \mathrm {sgn}\left (\tan \left (f x + e\right )\right )} + \frac {8 \, {\left (7 \, b^{2} \tan \left (f x + e\right )^{2} - 3 \, b^{2}\right )}}{\sqrt {b \tan \left (f x + e\right )} b^{7} f \mathrm {sgn}\left (\tan \left (f x + e\right )\right ) \tan \left (f x + e\right )^{3}}\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{\left (b \tan \left (f x + e\right )^{3}\right )^{\frac {3}{2}}}\,{d x} \]________________________________________________________________________________________