\[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4} \, dx \]
Optimal antiderivative \[ -\frac {15 a \,b^{\frac {3}{2}} \left (7 a^{2}+6 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 )}{16 \left (-a^{2}+b^{2}\right )^{\frac {17}{4}} d \,e^{\frac {3}{2}}}+\frac {15 a \,b^{\frac {3}{2}} \left (7 a^{2}+6 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 )}{16 \left (-a^{2}+b^{2}\right )^{\frac {17}{4}} d \,e^{\frac {3}{2}}}+\frac {b}{3 \left (a^{2}-b^{2}\right ) d e \left (a +b \sin \! \left (d x +c \right )\right )^{3} \sqrt {e \cos \! \left (d x +c \right )}}+\frac {13 a b}{12 \left (a^{2}-b^{2}\right )^{2} d e \left (a +b \sin \! \left (d x +c \right )\right )^{2} \sqrt {e \cos \! \left (d x +c \right )}}+\frac {b \left (89 a^{2}+28 b^{2}\right )}{24 \left (a^{2}-b^{2}\right )^{3} d e \left (a +b \sin \! \left (d x +c \right )\right ) \sqrt {e \cos \! \left (d x +c \right )}}+\frac {-15 a b \left (7 a^{2}+6 b^{2}\right )+\left (16 a^{4}+151 a^{2} b^{2}+28 b^{4}\right ) \sin \! \left (d x +c \right )}{8 \left (a^{2}-b^{2}\right )^{4} d e \sqrt {e \cos \! \left (d x +c \right )}}-\frac {15 a^{2} b \left (7 a^{2}+6 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 )}{16 \cos \! \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a^{2}-b^{2}\right )^{4} d e \left (b -\sqrt {-a^{2}+b^{2}}\right ) \sqrt {e \cos \! \left (d x +c \right )}}-\frac {15 a^{2} b \left (7 a^{2}+6 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 )}{16 \cos \! \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a^{2}-b^{2}\right )^{4} d e \left (b +\sqrt {-a^{2}+b^{2}}\right ) \sqrt {e \cos \! \left (d x +c \right )}}-\frac {\left (16 a^{4}+151 a^{2} b^{2}+28 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 )}}{8 \cos \! \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a^{2}-b^{2}\right )^{4} d \,e^{2} \sqrt {\cos \! \left (d x +c \right )}} \]
command
int(1/(e*cos(d*x+c))^(3/2)/(a+b*sin(d*x+c))^4,x)
Maple 2022.1 output
\[\int \frac {1}{\left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}} \left (a +b \sin \left (d x +c \right )\right )^{4}}\, dx\]
Maple 2021.1 output
\[ \text {output too large to display} \]