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