\[ \int \frac {\sec ^6(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {3003 \cos \left (d x +c \right )}{8192 d \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {1001 \sec \left (d x +c \right )}{5120 d \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {143 \left (\sec ^{3}\left (d x +c \right )\right )}{960 d \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {\sec ^{5}\left (d x +c \right )}{8 d \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {3003 \arctanh \left (\frac {\cos \left (d x +c \right ) \sqrt {a}\, \sqrt {2}}{2 \sqrt {a +a \sin \left (d x +c \right )}}\right ) \sqrt {2}}{16384 a^{\frac {3}{2}} d}+\frac {1001 \sec \left (d x +c \right )}{2048 a d \sqrt {a +a \sin \left (d x +c \right )}}+\frac {143 \left (\sec ^{3}\left (d x +c \right )\right )}{640 a d \sqrt {a +a \sin \left (d x +c \right )}}+\frac {13 \left (\sec ^{5}\left (d x +c \right )\right )}{80 a d \sqrt {a +a \sin \left (d x +c \right )}} \]
command
integrate(sec(d*x+c)^6/(a+a*sin(d*x+c))^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\sqrt {a} {\left (\frac {45045 \, \sqrt {2} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {45045 \, \sqrt {2} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {10 \, {\left (3249 \, \sqrt {2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 10633 \, \sqrt {2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 11767 \, \sqrt {2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4431 \, \sqrt {2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4} a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {256 \, \sqrt {2} {\left (225 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 25 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3\right )}}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}\right )}}{491520 \, d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________