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