43.25 Problem number 172

\[ \int \csc ^3(e+f x) (a+b \sin (e+f x))^3 \, dx \]

Optimal antiderivative \[ b^{3} x -\frac {a \left (a^{2}+6 b^{2}\right ) \arctanh \left (\cos \left (f x +e \right )\right )}{2 f}-\frac {5 a^{2} b \cot \left (f x +e \right )}{2 f}-\frac {a^{2} \cot \left (f x +e \right ) \csc \left (f x +e \right ) \left (a +b \sin \left (f x +e \right )\right )}{2 f} \]

command

integrate(csc(f*x+e)^3*(a+b*sin(f*x+e))^3,x, algorithm="giac")

Giac 1.9.0-11 via sagemath 9.6 output

\[ \frac {a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 8 \, {\left (f x + e\right )} b^{3} + 12 \, a^{2} b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 4 \, {\left (a^{3} + 6 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}\right ) - \frac {6 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 36 \, a b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 12 \, a^{2} b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a^{3}}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}}{8 \, f} \]

Giac 1.7.0 via sagemath 9.3 output

\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________