\[ \int \frac {(e \cos (c+d x))^{3/2}}{(a+b \sin (c+d x))^4} \, dx \]
Optimal antiderivative \[ -\frac {a \left (a^{2}+6 b^{2}\right ) e^{\frac {3}{2}} \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 )}{16 b^{\frac {3}{2}} \left (-a^{2}+b^{2}\right )^{\frac {11}{4}} d}-\frac {a \left (a^{2}+6 b^{2}\right ) e^{\frac {3}{2}} \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 )}{16 b^{\frac {3}{2}} \left (-a^{2}+b^{2}\right )^{\frac {11}{4}} d}-\frac {\left (3 a^{2}+4 b^{2}\right ) e^{2} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \! \left (\sin \! \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{24 \cos \! \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2} \left (a^{2}-b^{2}\right )^{2} d \sqrt {e \cos \! \left (d x +c \right )}}+\frac {a^{2} \left (a^{2}+6 b^{2}\right ) e^{2} \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 )}{16 \cos \! \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2} \left (a^{2}-b^{2}\right )^{2} d \left (a^{2}-b \left (b -\sqrt {-a^{2}+b^{2}}\right )\right ) \sqrt {e \cos \! \left (d x +c \right )}}+\frac {a^{2} \left (a^{2}+6 b^{2}\right ) e^{2} \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 )}{16 \cos \! \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2} \left (a^{2}-b^{2}\right )^{2} d \left (a^{2}-b \left (b +\sqrt {-a^{2}+b^{2}}\right )\right ) \sqrt {e \cos \! \left (d x +c \right )}}-\frac {e \sqrt {e \cos \! \left (d x +c \right )}}{3 b d \left (a +b \sin \! \left (d x +c \right )\right )^{3}}+\frac {a e \sqrt {e \cos \! \left (d x +c \right )}}{12 b \left (a^{2}-b^{2}\right ) d \left (a +b \sin \! \left (d x +c \right )\right )^{2}}+\frac {\left (3 a^{2}+4 b^{2}\right ) e \sqrt {e \cos \! \left (d x +c \right )}}{24 b \left (a^{2}-b^{2}\right )^{2} d \left (a +b \sin \! \left (d x +c \right )\right )} \]
command
int((e*cos(d*x+c))^(3/2)/(a+b*sin(d*x+c))^4,x)
Maple 2022.1 output
\[\int \frac {\left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{\left (a +b \sin \left (d x +c \right )\right )^{4}}\, dx\]
Maple 2021.1 output
\[ \text {output too large to display} \]