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