\[ \int \frac {1}{\sqrt [3]{\tan (5 x)}} \, dx \]
Optimal antiderivative \[ \frac {3 \ln \left (1+\tan ^{\frac {2}{3}}\left (5 x \right )\right )}{20}-\frac {\ln \left (1+\tan ^{2}\left (5 x \right )\right )}{20}-\frac {\arctan \left (\frac {\left (1-2 \left (\tan ^{\frac {2}{3}}\left (5 x \right )\right )\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{10} \]
command
integrate(1/tan(5*x)^(1/3),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{10} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \tan \left (5 \, x\right )^{\frac {2}{3}} - 1\right )}\right ) - \frac {1}{20} \, \log \left (\tan \left (5 \, x\right )^{\frac {4}{3}} - \tan \left (5 \, x\right )^{\frac {2}{3}} + 1\right ) + \frac {1}{10} \, \log \left (\tan \left (5 \, x\right )^{\frac {2}{3}} + 1\right ) \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{\tan \left (5 \, x\right )^{\frac {1}{3}}}\,{d x} \]________________________________________________________________________________________