\[ \int \frac {1}{(a+b \sin (c+d x))^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {2 b \cos \left (d x +c \right )}{5 \left (a^{2}-b^{2}\right ) d \left (a +b \sin \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {16 a b \cos \left (d x +c \right )}{15 \left (a^{2}-b^{2}\right )^{2} d \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {2 b \left (23 a^{2}+9 b^{2}\right ) \cos \left (d x +c \right )}{15 \left (a^{2}-b^{2}\right )^{3} d \sqrt {a +b \sin \left (d x +c \right )}}-\frac {2 \left (23 a^{2}+9 b^{2}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \sin \left (d x +c \right )}}{15 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \left (a^{2}-b^{2}\right )^{3} d \sqrt {\frac {a +b \sin \left (d x +c \right )}{a +b}}}+\frac {16 a \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \sin \left (d x +c \right )}{a +b}}}{15 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \left (a^{2}-b^{2}\right )^{2} d \sqrt {a +b \sin \left (d x +c \right )}} \]
command
integrate(1/(a+b*sin(d*x+c))^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {{\left (3 \, \sqrt {2} {\left (a^{4} b^{2} - 33 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (\sqrt {2} {\left (a^{3} b^{3} - 33 \, a b^{5}\right )} \cos \left (d x + c\right )^{2} - \sqrt {2} {\left (3 \, a^{5} b - 98 \, a^{3} b^{3} - 33 \, a b^{5}\right )}\right )} \sin \left (d x + c\right ) - \sqrt {2} {\left (a^{6} - 30 \, a^{4} b^{2} - 99 \, a^{2} b^{4}\right )}\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 (3 \, \sqrt {2} {\left (a^{4} b^{2} - 33 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (\sqrt {2} {\left (a^{3} b^{3} - 33 \, a b^{5}\right )} \cos \left (d x + c\right )^{2} - \sqrt {2} {\left (3 \, a^{5} b - 98 \, a^{3} b^{3} - 33 \, a b^{5}\right )}\right )} \sin \left (d x + c\right ) - \sqrt {2} {\left (a^{6} - 30 \, a^{4} b^{2} - 99 \, a^{2} b^{4}\right )}\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 (3 \, \sqrt {2} {\left (-23 i \, a^{3} b^{3} - 9 i \, a b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (\sqrt {2} {\left (-23 i \, a^{2} b^{4} - 9 i \, b^{6}\right )} \cos \left (d x + c\right )^{2} + \sqrt {2} {\left (69 i \, a^{4} b^{2} + 50 i \, a^{2} b^{4} + 9 i \, b^{6}\right )}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (23 i \, a^{5} b + 78 i \, a^{3} b^{3} + 27 i \, a b^{5}\right )}\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 (3 \, \sqrt {2} {\left (23 i \, a^{3} b^{3} + 9 i \, a b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (\sqrt {2} {\left (23 i \, a^{2} b^{4} + 9 i \, b^{6}\right )} \cos \left (d x + c\right )^{2} + \sqrt {2} {\left (-69 i \, a^{4} b^{2} - 50 i \, a^{2} b^{4} - 9 i \, b^{6}\right )}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (-23 i \, a^{5} b - 78 i \, a^{3} b^{3} - 27 i \, a b^{5}\right )}\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 ({\left (23 \, a^{2} b^{4} + 9 \, b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (27 \, a^{3} b^{3} + 5 \, a b^{5}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, {\left (17 \, a^{4} b^{2} + 9 \, a^{2} b^{4} + 6 \, b^{6}\right )} \cos \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}}{45 \, {\left (3 \, {\left (a^{7} b^{3} - 3 \, a^{5} b^{5} + 3 \, a^{3} b^{7} - a b^{9}\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{9} b - 6 \, a^{5} b^{5} + 8 \, a^{3} b^{7} - 3 \, a b^{9}\right )} d + {\left ({\left (a^{6} b^{4} - 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} - b^{10}\right )} d \cos \left (d x + c\right )^{2} - {\left (3 \, a^{8} b^{2} - 8 \, a^{6} b^{4} + 6 \, a^{4} b^{6} - b^{10}\right )} d\right )} \sin \left (d x + c\right )\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {b \sin \left (d x + c\right ) + a}}{b^{4} \cos \left (d x + c\right )^{4} + a^{4} + 6 \, a^{2} b^{2} + b^{4} - 2 \, {\left (3 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2} - 4 \, {\left (a b^{3} \cos \left (d x + c\right )^{2} - a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )}, x\right ) \]