\[ \int \frac {\csc ^5(e+f x)}{a+b \sec ^2(e+f x)} \, dx \]
Optimal antiderivative \[ -\frac {\left (3 a^{2}-6 a b -b^{2}\right ) \arctanh \left (\cos \left (f x +e \right )\right )}{8 \left (a +b \right )^{3} f}-\frac {\left (3 a -b \right ) \cot \left (f x +e \right ) \csc \left (f x +e \right )}{8 \left (a +b \right )^{2} f}-\frac {\cot \left (f x +e \right ) \left (\csc ^{3}\left (f x +e \right )\right )}{4 \left (a +b \right ) f}+\frac {a^{\frac {3}{2}} \arctan \left (\frac {\cos \left (f x +e \right ) \sqrt {a}}{\sqrt {b}}\right ) \sqrt {b}}{\left (a +b \right )^{3} f} \]
command
integrate(csc(f*x+e)^5/(a+b*sec(f*x+e)^2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\frac {64 \, a^{2} b \arctan \left (-\frac {a \cos \left (f x + e\right ) - b}{\sqrt {a b} \cos \left (f x + e\right ) + \sqrt {a b}}\right )}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt {a b}} - \frac {4 \, {\left (3 \, a^{2} - 6 \, a b - b^{2}\right )} \log \left (\frac {{\left | -\cos \left (f x + e\right ) + 1 \right |}}{{\left | \cos \left (f x + e\right ) + 1 \right |}}\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac {\frac {8 \, a {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {a {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}}{a^{2} + 2 \, a b + b^{2}} + \frac {{\left (a^{2} + 2 \, a b + b^{2} - \frac {8 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {8 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {18 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {36 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {6 \, b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{2}}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}}{64 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________