\[ \int \sec ^6(c+d x) (a+b \sin (c+d x))^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {\left (\sec ^{3}\left (d x +c \right )\right ) \left (b -8 a \sin \left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{30 d}+\frac {\left (\sec ^{5}\left (d x +c \right )\right ) \left (b +a \sin \left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{5 d}-\frac {\sec \left (d x +c \right ) \left (b \left (8 a^{4}-13 a^{2} b^{2}+5 b^{4}\right )-a \left (32 a^{4}-61 a^{2} b^{2}+29 b^{4}\right ) \sin \left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{60 \left (a^{2}-b^{2}\right )^{2} d}+\frac {a \left (32 a^{2}-29 b^{2}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \sin \left (d x +c \right )}}{60 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \left (a^{2}-b^{2}\right ) d \sqrt {\frac {a +b \sin \left (d x +c \right )}{a +b}}}-\frac {\left (32 a^{2}-5 b^{2}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \sin \left (d x +c \right )}{a +b}}}{60 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) d \sqrt {a +b \sin \left (d x +c \right )}} \]
command
integrate(sec(d*x+c)^6*(a+b*sin(d*x+c))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\sqrt {2} {\left (64 \, a^{4} - 82 \, a^{2} b^{2} + 15 \, b^{4}\right )} \sqrt {i \, b} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right ) + \sqrt {2} {\left (64 \, a^{4} - 82 \, a^{2} b^{2} + 15 \, b^{4}\right )} \sqrt {-i \, b} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right ) + 3 \, \sqrt {2} {\left (32 i \, a^{3} b - 29 i \, a b^{3}\right )} \sqrt {i \, b} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right )\right ) + 3 \, \sqrt {2} {\left (-32 i \, a^{3} b + 29 i \, a b^{3}\right )} \sqrt {-i \, b} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right )\right ) - 6 \, {\left ({\left (8 \, a^{2} b^{2} - 5 \, b^{4}\right )} \cos \left (d x + c\right )^{4} - 12 \, a^{2} b^{2} + 12 \, b^{4} + 2 \, {\left (a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2} - {\left ({\left (32 \, a^{3} b - 29 \, a b^{3}\right )} \cos \left (d x + c\right )^{4} + 12 \, a^{3} b - 12 \, a b^{3} + 16 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}}{360 \, {\left (a^{2} b - b^{3}\right )} d \cos \left (d x + c\right )^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (b \sec \left (d x + c\right )^{6} \sin \left (d x + c\right ) + a \sec \left (d x + c\right )^{6}\right )} \sqrt {b \sin \left (d x + c\right ) + a}, x\right ) \]