\[ \int \frac {\cot ^4(c+d x) \csc (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx \]
Optimal antiderivative \[ -\frac {11 \arctanh \! \left (\frac {\cos \left (d x +c \right ) \sqrt {a}}{\sqrt {a +a \sin \left (d x +c \right )}}\right )}{64 d \sqrt {a}}-\frac {11 \cot \! \left (d x +c \right )}{64 d \sqrt {a +a \sin \! \left (d x +c \right )}}+\frac {53 \cot \! \left (d x +c \right ) \csc \! \left (d x +c \right )}{96 d \sqrt {a +a \sin \! \left (d x +c \right )}}+\frac {\cot \! \left (d x +c \right ) \left (\csc ^{2}\left (d x +c \right )\right )}{24 d \sqrt {a +a \sin \! \left (d x +c \right )}}-\frac {\cot \! \left (d x +c \right ) \left (\csc ^{3}\left (d x +c \right )\right )}{4 d \sqrt {a +a \sin \! \left (d x +c \right )}} \]
command
integrate(cos(d*x+c)^4*csc(d*x+c)^5/(a+a*sin(d*x+c))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {Exception raised: TypeError} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {output too large to display} \]