\[ \int \frac {\sec ^6(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx \]
Optimal antiderivative \[ -\frac {231 a \cos \left (d x +c \right )}{512 d \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {77 a \sec \left (d x +c \right )}{320 d \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {11 a \left (\sec ^{3}\left (d x +c \right )\right )}{60 d \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {231 \arctanh \left (\frac {\cos \left (d x +c \right ) \sqrt {a}\, \sqrt {2}}{2 \sqrt {a +a \sin \left (d x +c \right )}}\right ) \sqrt {2}}{1024 d \sqrt {a}}+\frac {77 \sec \left (d x +c \right )}{128 d \sqrt {a +a \sin \left (d x +c \right )}}+\frac {11 \left (\sec ^{3}\left (d x +c \right )\right )}{40 d \sqrt {a +a \sin \left (d x +c \right )}}+\frac {\sec ^{5}\left (d x +c \right )}{5 d \sqrt {a +a \sin \left (d x +c \right )}} \]
command
integrate(sec(d*x+c)^6/(a+a*sin(d*x+c))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {3465 \, \sqrt {2} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {3465 \, \sqrt {2} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {10 \, \sqrt {2} {\left (213 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 472 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 267 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {32 \, \sqrt {2} {\left (150 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 20 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, \sqrt {a}\right )}}{a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{30720 \, d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________