\[ \int \frac {\tan ^{\frac {2}{3}}(c+d x)}{a+i a \tan (c+d x)} \, dx \]
Optimal antiderivative \[ \frac {\arctan \left (-\sqrt {3}+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right )}{12 a d}+\frac {\arctan \left (\sqrt {3}+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right )}{12 a d}+\frac {\arctan \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )}{6 a d}-\frac {i \ln \left (1+\tan ^{\frac {2}{3}}\left (d x +c \right )\right )}{6 a d}+\frac {i \ln \left (1-\left (\tan ^{\frac {2}{3}}\left (d x +c \right )\right )+\tan ^{\frac {4}{3}}\left (d x +c \right )\right )}{12 a d}+\frac {i \arctan \left (\frac {\left (1-2 \left (\tan ^{\frac {2}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{6 a d}+\frac {\ln \left (1-\sqrt {3}\, \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )\right ) \sqrt {3}}{24 a d}-\frac {\ln \left (1+\sqrt {3}\, \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )\right ) \sqrt {3}}{24 a d}+\frac {i \left (\tan ^{\frac {2}{3}}\left (d x +c \right )\right )}{2 d \left (a +i a \tan \left (d x +c \right )\right )} \]
command
integrate(tan(d*x+c)^(2/3)/(a+I*a*tan(d*x+c)),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\sqrt {3} \log \left (-\frac {\sqrt {3} - 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} + i}{\sqrt {3} + 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} - i}\right )}{24 \, a d} + \frac {\sqrt {3} \log \left (-\frac {\sqrt {3} - 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} - i}{\sqrt {3} + 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} + i}\right )}{8 \, a d} + \frac {i \, \log \left (\tan \left (d x + c\right )^{\frac {2}{3}} + i \, \tan \left (d x + c\right )^{\frac {1}{3}} - 1\right )}{8 \, a d} + \frac {i \, \log \left (\tan \left (d x + c\right )^{\frac {2}{3}} - i \, \tan \left (d x + c\right )^{\frac {1}{3}} - 1\right )}{24 \, a d} - \frac {i \, \log \left (\tan \left (d x + c\right )^{\frac {1}{3}} + i\right )}{12 \, a d} - \frac {i \, \log \left (\tan \left (d x + c\right )^{\frac {1}{3}} - i\right )}{4 \, a d} + \frac {\tan \left (d x + c\right )^{\frac {2}{3}}}{2 \, a d {\left (\tan \left (d x + c\right ) - i\right )}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {\tan \left (d x + c\right )^{\frac {2}{3}}}{i \, a \tan \left (d x + c\right ) + a}\,{d x} \]________________________________________________________________________________________