\[ \int \frac {1}{(b \tan (c+d x))^{4/3}} \, dx \]
Optimal antiderivative \[ -\frac {\arctan \left (\frac {\left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}\right )}{b^{\frac {4}{3}} d}-\frac {\arctan \left (-\sqrt {3}+\frac {2 \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}\right )}{2 b^{\frac {4}{3}} d}-\frac {\arctan \left (\sqrt {3}+\frac {2 \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}\right )}{2 b^{\frac {4}{3}} d}-\frac {\ln \left (b^{\frac {2}{3}}-b^{\frac {1}{3}} \sqrt {3}\, \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}\right ) \sqrt {3}}{4 b^{\frac {4}{3}} d}+\frac {\ln \left (b^{\frac {2}{3}}+b^{\frac {1}{3}} \sqrt {3}\, \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}\right ) \sqrt {3}}{4 b^{\frac {4}{3}} d}-\frac {3}{b d \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}} \]
command
integrate(1/(b*tan(d*x+c))^(4/3),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{4} \, b {\left (\frac {\sqrt {3} {\left | b \right |}^{\frac {5}{3}} \log \left (\sqrt {3} \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} {\left | b \right |}^{\frac {1}{3}} + \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} + {\left | b \right |}^{\frac {2}{3}}\right )}{b^{4} d} - \frac {\sqrt {3} {\left | b \right |}^{\frac {5}{3}} \log \left (-\sqrt {3} \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} {\left | b \right |}^{\frac {1}{3}} + \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} + {\left | b \right |}^{\frac {2}{3}}\right )}{b^{4} d} - \frac {2 \, {\left | b \right |}^{\frac {5}{3}} \arctan \left (\frac {\sqrt {3} {\left | b \right |}^{\frac {1}{3}} + 2 \, \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}}{{\left | b \right |}^{\frac {1}{3}}}\right )}{b^{4} d} - \frac {2 \, {\left | b \right |}^{\frac {5}{3}} \arctan \left (-\frac {\sqrt {3} {\left | b \right |}^{\frac {1}{3}} - 2 \, \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}}{{\left | b \right |}^{\frac {1}{3}}}\right )}{b^{4} d} - \frac {4 \, {\left | b \right |}^{\frac {5}{3}} \arctan \left (\frac {\left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}}{{\left | b \right |}^{\frac {1}{3}}}\right )}{b^{4} d} - \frac {12}{\left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} b^{2} d}\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{\left (b \tan \left (d x + c\right )\right )^{\frac {4}{3}}}\,{d x} \]________________________________________________________________________________________