\[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {179 a^{\frac {3}{2}} \arctanh \left (\frac {\cos \left (d x +c \right ) \sqrt {a}}{\sqrt {a +a \sin \left (d x +c \right )}}\right )}{512 d}-\frac {\cot \left (d x +c \right ) \left (\csc ^{5}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{6 d}-\frac {179 a^{2} \cot \left (d x +c \right )}{512 d \sqrt {a +a \sin \left (d x +c \right )}}+\frac {111 a^{2} \cot \left (d x +c \right ) \csc \left (d x +c \right )}{256 d \sqrt {a +a \sin \left (d x +c \right )}}+\frac {239 a^{2} \cot \left (d x +c \right ) \left (\csc ^{2}\left (d x +c \right )\right )}{320 d \sqrt {a +a \sin \left (d x +c \right )}}+\frac {137 a^{2} \cot \left (d x +c \right ) \left (\csc ^{3}\left (d x +c \right )\right )}{480 d \sqrt {a +a \sin \left (d x +c \right )}}-\frac {a \cot \left (d x +c \right ) \left (\csc ^{4}\left (d x +c \right )\right ) \sqrt {a +a \sin \left (d x +c \right )}}{20 d} \]
command
integrate(cos(d*x+c)^4*csc(d*x+c)^7*(a+a*sin(d*x+c))^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\sqrt {2} {\left (2685 \, \sqrt {2} a \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + \frac {4 \, {\left (85920 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 161520 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 62352 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 43464 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 30430 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2685 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{6}}\right )} \sqrt {a}}{30720 \, d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________