\[ \int \csc ^7(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx \]
Optimal antiderivative \[ \frac {a^{2} c \arctanh \left (\cos \left (f x +e \right )\right )}{16 f}-\frac {a^{2} c \left (\cot ^{3}\left (f x +e \right )\right )}{3 f}-\frac {a^{2} c \left (\cot ^{5}\left (f x +e \right )\right )}{5 f}+\frac {a^{2} c \cot \left (f x +e \right ) \csc \left (f x +e \right )}{16 f}+\frac {a^{2} c \cot \left (f x +e \right ) \left (\csc ^{3}\left (f x +e \right )\right )}{24 f}-\frac {a^{2} c \cot \left (f x +e \right ) \left (\csc ^{5}\left (f x +e \right )\right )}{6 f} \]
command
integrate(csc(f*x+e)^7*(a+a*sin(f*x+e))^2*(c-c*sin(f*x+e)),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {5 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 12 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 15 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 20 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 15 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 120 \, a^{2} c \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}\right ) - 120 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + \frac {294 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 120 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 15 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 20 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 15 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 12 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 5 \, a^{2} c}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6}}}{1920 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________