\[ \int (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)} \, dx \]
Optimal antiderivative \[ \frac {12 i a \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}} \left (\sec ^{2}\left (d x +c \right )\right )}{35 d \sqrt {a +i a \tan \left (d x +c \right )}}+\frac {32 i a \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}} \left (\sec ^{4}\left (d x +c \right )\right )}{35 d \sqrt {a +i a \tan \left (d x +c \right )}}-\frac {2 i \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}} \sqrt {a +i a \tan \left (d x +c \right )}}{7 d}-\frac {16 i \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}} \left (\sec ^{2}\left (d x +c \right )\right ) \sqrt {a +i a \tan \left (d x +c \right )}}{35 d} \]
command
integrate((e*cos(d*x+c))^(7/2)*(a+I*a*tan(d*x+c))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (-5 i \, \sqrt {a} e^{\left (\frac {7}{2} i \, d x + \frac {7}{2} i \, c\right )} - 35 i \, \sqrt {a} e^{\left (\frac {3}{2} i \, d x + \frac {3}{2} i \, c\right )} + 105 i \, \sqrt {a} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )} + 7 i \, \sqrt {a} e^{\left (-\frac {5}{2} i \, d x - \frac {5}{2} i \, c\right )}\right )} e^{\frac {7}{2}}}{140 \, d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________