\[ \int \cot ^2(e+f x) (1+\tan (e+f x))^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {3 \arctanh \left (\sqrt {1+\tan \left (f x +e \right )}\right )}{f}+\frac {\arctan \left (\frac {3-2 \sqrt {2}+\left (1-\sqrt {2}\right ) \tan \left (f x +e \right )}{\sqrt {-14+10 \sqrt {2}}\, \sqrt {1+\tan \left (f x +e \right )}}\right ) \sqrt {\sqrt {2}-1}}{f}+\frac {\arctanh \left (\frac {3+2 \sqrt {2}+\left (1+\sqrt {2}\right ) \tan \left (f x +e \right )}{\sqrt {14+10 \sqrt {2}}\, \sqrt {1+\tan \left (f x +e \right )}}\right ) \sqrt {1+\sqrt {2}}}{f}-\frac {\cot \left (f x +e \right ) \sqrt {1+\tan \left (f x +e \right )}}{f} \]
command
integrate(cot(f*x+e)^2*(1+tan(f*x+e))^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {3 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} + 1\right )}{2 \, f} + \frac {3 \, \log \left ({\left | \sqrt {\tan \left (f x + e\right ) + 1} - 1 \right |}\right )}{2 \, f} - \frac {{\left (f^{2} \sqrt {2 \, \sqrt {2} + 2} - f \sqrt {2 \, \sqrt {2} - 2} {\left | f \right |}\right )} \arctan \left (\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right )}{2 \, f^{3}} - \frac {{\left (f^{2} \sqrt {2 \, \sqrt {2} + 2} - f \sqrt {2 \, \sqrt {2} - 2} {\left | f \right |}\right )} \arctan \left (-\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} - 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right )}{2 \, f^{3}} + \frac {{\left (f^{2} \sqrt {2 \, \sqrt {2} - 2} + f \sqrt {2 \, \sqrt {2} + 2} {\left | f \right |}\right )} \log \left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} \sqrt {\tan \left (f x + e\right ) + 1} + \sqrt {2} + \tan \left (f x + e\right ) + 1\right )}{4 \, f^{3}} - \frac {{\left (f^{2} \sqrt {2 \, \sqrt {2} - 2} + f \sqrt {2 \, \sqrt {2} + 2} {\left | f \right |}\right )} \log \left (-2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} \sqrt {\tan \left (f x + e\right ) + 1} + \sqrt {2} + \tan \left (f x + e\right ) + 1\right )}{4 \, f^{3}} - \frac {\sqrt {\tan \left (f x + e\right ) + 1}}{f \tan \left (f x + e\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________