\[ \int \sqrt [3]{a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx \]
Optimal antiderivative \[ -\frac {\left (-i b +a \right )^{\frac {1}{3}} \left (-i B +A \right ) x}{4}-\frac {\left (i b +a \right )^{\frac {1}{3}} \left (i B +A \right ) x}{4}-\frac {\left (i b +a \right )^{\frac {1}{3}} \left (i A -B \right ) \ln \left (\cos \left (d x +c \right )\right )}{4 d}+\frac {\left (-i b +a \right )^{\frac {1}{3}} \left (i A +B \right ) \ln \left (\cos \left (d x +c \right )\right )}{4 d}+\frac {3 \left (-i b +a \right )^{\frac {1}{3}} \left (i A +B \right ) \ln \left (\left (-i b +a \right )^{\frac {1}{3}}-\left (a +b \tan \left (d x +c \right )\right )^{\frac {1}{3}}\right )}{4 d}-\frac {3 \left (i b +a \right )^{\frac {1}{3}} \left (i A -B \right ) \ln \left (\left (i b +a \right )^{\frac {1}{3}}-\left (a +b \tan \left (d x +c \right )\right )^{\frac {1}{3}}\right )}{4 d}-\frac {\left (-i b +a \right )^{\frac {1}{3}} \left (i A +B \right ) \arctan \left (\frac {\left (1+\frac {2 \left (a +b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{\left (-i b +a \right )^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{2 d}+\frac {\left (i b +a \right )^{\frac {1}{3}} \left (i A -B \right ) \arctan \left (\frac {\left (1+\frac {2 \left (a +b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{\left (i b +a \right )^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{2 d}+\frac {3 B \left (a +b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{d} \]
command
integrate((a+b*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 \[ \text {Timed out} \]