\[ \int (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {2 \,\mathrm {I} a^{\frac {3}{2}} \left (c -\mathrm {I} d \right )^{\frac {3}{2}} \arctanh \! \left (\frac {\sqrt {2}\, \sqrt {a}\, \sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {c -\mathrm {I} d}\, \sqrt {a +\mathrm {I} a \tan \left (f x +e \right )}}\right ) \sqrt {2}}{f}-\frac {\left (-1\right )^{\frac {1}{4}} a^{\frac {3}{2}} \left (3 \,\mathrm {I} c^{2}+18 c d -11 \,\mathrm {I} d^{2}\right ) \arctanh \! \left (\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {d}\, \sqrt {a +\mathrm {I} a \tan \left (f x +e \right )}}{\sqrt {a}\, \sqrt {c +d \tan \left (f x +e \right )}}\right )}{4 f \sqrt {d}}+\frac {a \left (3 \,\mathrm {I} c +5 d \right ) \sqrt {a +\mathrm {I} a \tan \! \left (f x +e \right )}\, \sqrt {c +d \tan \! \left (f x +e \right )}}{4 f}+\frac {a^{2} \left (c +\mathrm {I} d \right ) \left (c +d \tan \! \left (f x +e \right )\right )^{\frac {3}{2}}}{2 d f \sqrt {a +\mathrm {I} a \tan \! \left (f x +e \right )}}-\frac {a^{2} \left (c +d \tan \! \left (f x +e \right )\right )^{\frac {5}{2}}}{2 d f \sqrt {a +\mathrm {I} a \tan \! \left (f x +e \right )}} \]
command
integrate((a+I*a*tan(f*x+e))^(3/2)*(c+d*tan(f*x+e))^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {Exception raised: TypeError} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \frac {\sqrt {2 \, a d^{2} + 2 \, \sqrt {{\left (d \tan \left (f x + e\right ) + c\right )}^{2} - 2 \, {\left (d \tan \left (f x + e\right ) + c\right )} c + c^{2} + d^{2}} a d} {\left (d \tan \left (f x + e\right ) + c\right )}^{2} a {\left (\frac {i \, {\left (d \tan \left (f x + e\right ) + c\right )} a d - i \, a c d}{a d^{2} + \sqrt {{\left (d \tan \left (f x + e\right ) + c\right )}^{2} a^{2} d^{2} - 2 \, {\left (d \tan \left (f x + e\right ) + c\right )} a^{2} c d^{2} + a^{2} c^{2} d^{2} + a^{2} d^{4}}} + 1\right )} \log \left ({\left | d \tan \left (f x + e\right ) + c \right |}\right )}{2 \, {\left ({\left (-i \, d \tan \left (f x + e\right ) - i \, c\right )} d + i \, c d + d^{2}\right )}} \]