\[ \int \frac {1}{\sqrt [3]{\tan (c+d x)} (a+b \tan (c+d x))} \, dx \]
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="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ \text {Timed out} \]