\[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 b \left (\sec ^{3}\left (d x +c \right )\right )}{\left (a^{2}-b^{2}\right ) d \sqrt {a +b \sin \left (d x +c \right )}}-\frac {\left (\sec ^{3}\left (d x +c \right )\right ) \left (8 a b -\left (a^{2}+7 b^{2}\right ) \sin \left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{3 \left (a^{2}-b^{2}\right )^{2} d}-\frac {\sec \left (d x +c \right ) \left (a b \left (a^{2}-33 b^{2}\right )-\left (4 a^{4}-15 a^{2} b^{2}-21 b^{4}\right ) \sin \left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{6 \left (a^{2}-b^{2}\right )^{3} d}+\frac {\left (4 a^{4}-15 a^{2} b^{2}-21 b^{4}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \sin \left (d x +c \right )}}{6 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \left (a^{2}-b^{2}\right )^{3} d \sqrt {\frac {a +b \sin \left (d x +c \right )}{a +b}}}-\frac {2 a \left (a^{2}-3 b^{2}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \sin \left (d x +c \right )}{a +b}}}{3 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \left (a^{2}-b^{2}\right )^{2} d \sqrt {a +b \sin \left (d x +c \right )}} \]
command
integrate(sec(d*x+c)^4/(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 {{\left (\sqrt {2} {\left (8 \, a^{5} b - 33 \, a^{3} b^{3} + 57 \, a b^{5}\right )} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + \sqrt {2} {\left (8 \, a^{6} - 33 \, a^{4} b^{2} + 57 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt {i \, b} {\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 ) + {\left (\sqrt {2} {\left (8 \, a^{5} b - 33 \, a^{3} b^{3} + 57 \, a b^{5}\right )} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + \sqrt {2} {\left (8 \, a^{6} - 33 \, a^{4} b^{2} + 57 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt {-i \, b} {\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 \, {\left (\sqrt {2} {\left (4 i \, a^{4} b^{2} - 15 i \, a^{2} b^{4} - 21 i \, b^{6}\right )} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + \sqrt {2} {\left (4 i \, a^{5} b - 15 i \, a^{3} b^{3} - 21 i \, a b^{5}\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt {i \, b} {\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 \, {\left (\sqrt {2} {\left (-4 i \, a^{4} b^{2} + 15 i \, a^{2} b^{4} + 21 i \, b^{6}\right )} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + \sqrt {2} {\left (-4 i \, a^{5} b + 15 i \, a^{3} b^{3} + 21 i \, a b^{5}\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt {-i \, b} {\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 (2 \, a^{4} b^{2} - 4 \, a^{2} b^{4} + 2 \, b^{6} + {\left (4 \, a^{4} b^{2} - 15 \, a^{2} b^{4} - 21 \, b^{6}\right )} \cos \left (d x + c\right )^{4} - {\left (a^{4} b^{2} + 6 \, a^{2} b^{4} - 7 \, b^{6}\right )} \cos \left (d x + c\right )^{2} - 2 \, {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5} + 2 \, {\left (a^{5} b - 4 \, a^{3} b^{3} + 3 \, a b^{5}\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}}{36 \, {\left ({\left (a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right )} d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} d \cos \left (d x + c\right )^{3}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {b \sin \left (d x + c\right ) + a} \sec \left (d x + c\right )^{4}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}, x\right ) \]