\[ \int \cot ^4(e+f x) \sqrt {1+\tan (e+f x)} \, dx \]
Optimal antiderivative \[ \frac {7 \arctanh \left (\sqrt {1+\tan \left (f x +e \right )}\right )}{8 f}-\frac {\arctan \left (\frac {\sqrt {2+2 \sqrt {2}}-2 \sqrt {1+\tan \left (f x +e \right )}}{\sqrt {-2+2 \sqrt {2}}}\right ) \sqrt {2+2 \sqrt {2}}}{2 f}+\frac {\arctan \left (\frac {\sqrt {2+2 \sqrt {2}}+2 \sqrt {1+\tan \left (f x +e \right )}}{\sqrt {-2+2 \sqrt {2}}}\right ) \sqrt {2+2 \sqrt {2}}}{2 f}+\frac {\ln \left (1+\sqrt {2}-\sqrt {2+2 \sqrt {2}}\, \sqrt {1+\tan \left (f x +e \right )}+\tan \left (f x +e \right )\right )}{2 f \sqrt {2+2 \sqrt {2}}}-\frac {\ln \left (1+\sqrt {2}+\sqrt {2+2 \sqrt {2}}\, \sqrt {1+\tan \left (f x +e \right )}+\tan \left (f x +e \right )\right )}{2 f \sqrt {2+2 \sqrt {2}}}+\frac {9 \cot \left (f x +e \right ) \sqrt {1+\tan \left (f x +e \right )}}{8 f}-\frac {\left (\cot ^{2}\left (f x +e \right )\right ) \sqrt {1+\tan \left (f x +e \right )}}{12 f}-\frac {\left (\cot ^{3}\left (f x +e \right )\right ) \sqrt {1+\tan \left (f x +e \right )}}{3 f} \]
command
integrate(cot(f*x+e)^4*(1+tan(f*x+e))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {7 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} + 1\right )}{16 \, f} - \frac {7 \, \log \left ({\left | \sqrt {\tan \left (f x + e\right ) + 1} - 1 \right |}\right )}{16 \, f} + \frac {{\left (f^{2} \sqrt {\sqrt {2} + 1} + f \sqrt {\sqrt {2} - 1} {\left | f \right |}\right )} \arctan \left (\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right )}{2 \, f^{3}} + \frac {{\left (f^{2} \sqrt {\sqrt {2} + 1} + f \sqrt {\sqrt {2} - 1} {\left | f \right |}\right )} \arctan \left (-\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} - 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right )}{2 \, f^{3}} + \frac {{\left (f^{2} \sqrt {\sqrt {2} - 1} - f \sqrt {\sqrt {2} + 1} {\left | f \right |}\right )} \log \left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} \sqrt {\tan \left (f x + e\right ) + 1} + \sqrt {2} + \tan \left (f x + e\right ) + 1\right )}{4 \, f^{3}} - \frac {{\left (f^{2} \sqrt {\sqrt {2} - 1} - f \sqrt {\sqrt {2} + 1} {\left | f \right |}\right )} \log \left (-2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} \sqrt {\tan \left (f x + e\right ) + 1} + \sqrt {2} + \tan \left (f x + e\right ) + 1\right )}{4 \, f^{3}} + \frac {27 \, {\left (\tan \left (f x + e\right ) + 1\right )}^{\frac {5}{2}} - 56 \, {\left (\tan \left (f x + e\right ) + 1\right )}^{\frac {3}{2}} + 21 \, \sqrt {\tan \left (f x + e\right ) + 1}}{24 \, f \tan \left (f x + e\right )^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________