\[ \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^4 \, dx \]
Optimal antiderivative \[ \frac {78 i a^{4} \sqrt {e \sec \left (d x +c \right )}}{7 d}+\frac {78 a^{4} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \sqrt {e \sec \left (d x +c \right )}}{7 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 i a \sqrt {e \sec \left (d x +c \right )}\, \left (a +i a \tan \left (d x +c \right )\right )^{3}}{7 d}+\frac {26 i \sqrt {e \sec \left (d x +c \right )}\, \left (a^{2}+i a^{2} \tan \left (d x +c \right )\right )^{2}}{35 d}+\frac {78 i \sqrt {e \sec \left (d x +c \right )}\, \left (a^{4}+i a^{4} \tan \left (d x +c \right )\right )}{35 d} \]
command
integrate((e*sec(d*x+c))^(1/2)*(a+I*a*tan(d*x+c))^4,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (\frac {\sqrt {2} {\left (-195 i \, a^{4} e^{\frac {1}{2}} - 365 i \, a^{4} e^{\left (6 i \, d x + 6 i \, c + \frac {1}{2}\right )} - 793 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c + \frac {1}{2}\right )} - 663 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c + \frac {1}{2}\right )}\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{\sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} + 195 \, {\left (i \, \sqrt {2} a^{4} e^{\frac {1}{2}} + i \, \sqrt {2} a^{4} e^{\left (6 i \, d x + 6 i \, c + \frac {1}{2}\right )} + 3 i \, \sqrt {2} a^{4} e^{\left (4 i \, d x + 4 i \, c + \frac {1}{2}\right )} + 3 i \, \sqrt {2} a^{4} e^{\left (2 i \, d x + 2 i \, c + \frac {1}{2}\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )}}{35 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \frac {\sqrt {2} {\left (730 i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 1586 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 1326 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 390 i \, a^{4}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 35 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} {\rm integral}\left (-\frac {39 i \, \sqrt {2} a^{4} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{7 \, d}, x\right )}{35 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]