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