\[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3} \, dx \]
Optimal antiderivative \[ \frac {9 b^{\frac {7}{2}} \left (11 a^{2}+2 b^{2}\right ) \arctan \! \left (\frac {\sqrt {b}\, \sqrt {e \cos \left (d x +c \right )}}{\left (-a^{2}+b^{2}\right )^{\frac {1}{4}} \sqrt {e}}\right )}{8 \left (-a^{2}+b^{2}\right )^{\frac {17}{4}} d \,e^{\frac {7}{2}}}-\frac {9 b^{\frac {7}{2}} \left (11 a^{2}+2 b^{2}\right ) \arctanh \! \left (\frac {\sqrt {b}\, \sqrt {e \cos \left (d x +c \right )}}{\left (-a^{2}+b^{2}\right )^{\frac {1}{4}} \sqrt {e}}\right )}{8 \left (-a^{2}+b^{2}\right )^{\frac {17}{4}} d \,e^{\frac {7}{2}}}+\frac {b}{2 \left (a^{2}-b^{2}\right ) d e \left (e \cos \! \left (d x +c \right )\right )^{\frac {5}{2}} \left (a +b \sin \! \left (d x +c \right )\right )^{2}}+\frac {13 a b}{4 \left (a^{2}-b^{2}\right )^{2} d e \left (e \cos \! \left (d x +c \right )\right )^{\frac {5}{2}} \left (a +b \sin \! \left (d x +c \right )\right )}+\frac {-9 b \left (11 a^{2}+2 b^{2}\right )+a \left (8 a^{2}+109 b^{2}\right ) \sin \! \left (d x +c \right )}{20 \left (a^{2}-b^{2}\right )^{3} d e \left (e \cos \! \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {\frac {9 b^{3} \left (11 a^{2}+2 b^{2}\right )}{4}+\frac {3 a \left (8 a^{4}-64 a^{2} b^{2}-139 b^{4}\right ) \sin \left (d x +c \right )}{20}}{\left (a^{2}-b^{2}\right )^{4} d \,e^{3} \sqrt {e \cos \! \left (d x +c \right )}}+\frac {9 a \,b^{3} \left (11 a^{2}+2 b^{2}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticPi \! \left (\sin \! \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 b}{b -\sqrt {-a^{2}+b^{2}}}, \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{8 \cos \! \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a^{2}-b^{2}\right )^{4} d \,e^{3} \left (b -\sqrt {-a^{2}+b^{2}}\right ) \sqrt {e \cos \! \left (d x +c \right )}}+\frac {9 a \,b^{3} \left (11 a^{2}+2 b^{2}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticPi \! \left (\sin \! \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 b}{b +\sqrt {-a^{2}+b^{2}}}, \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{8 \cos \! \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a^{2}-b^{2}\right )^{4} d \,e^{3} \left (b +\sqrt {-a^{2}+b^{2}}\right ) \sqrt {e \cos \! \left (d x +c \right )}}-\frac {3 a \left (8 a^{4}-64 a^{2} b^{2}-139 b^{4}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \! \left (\sin \! \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {e \cos \! \left (d x +c \right )}}{20 \cos \! \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a^{2}-b^{2}\right )^{4} d \,e^{4} \sqrt {\cos \! \left (d x +c \right )}} \]
command
int(1/(e*cos(d*x+c))^(7/2)/(a+b*sin(d*x+c))^3,x)
Maple 2022.1 output
\[\int \frac {1}{\left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}} \left (a +b \sin \left (d x +c \right )\right )^{3}}\, dx\]
Maple 2021.1 output
\[ \text {output too large to display} \]