\[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {3 \left (a^{2}-8 a b +8 b^{2}\right ) \arctanh \left (\cos \left (f x +e \right )\right )}{8 a^{4} f}-\frac {\left (5 a -6 b \right ) \cot \left (f x +e \right ) \csc \left (f x +e \right )}{8 a^{2} f \left (a -b +b \left (\sec ^{2}\left (f x +e \right )\right )\right )}-\frac {\left (\cot ^{3}\left (f x +e \right )\right ) \csc \left (f x +e \right )}{4 a f \left (a -b +b \left (\sec ^{2}\left (f x +e \right )\right )\right )}-\frac {3 \left (3 a -4 b \right ) b \sec \left (f x +e \right )}{8 a^{3} f \left (a -b +b \left (\sec ^{2}\left (f x +e \right )\right )\right )}-\frac {3 \left (a -2 b \right ) \arctan \left (\frac {\sec \left (f x +e \right ) \sqrt {b}}{\sqrt {a -b}}\right ) \sqrt {a -b}\, \sqrt {b}}{2 a^{4} f} \]
command
integrate(csc(f*x+e)^5/(a+b*tan(f*x+e)^2)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {12 \, {\left (a^{2} - 8 \, a b + 8 \, 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^{4}} - \frac {96 \, {\left (a^{2} b - 3 \, a b^{2} + 2 \, b^{3}\right )} \arctan \left (-\frac {a \cos \left (f x + e\right ) - b \cos \left (f x + e\right ) - b}{\sqrt {a b - b^{2}} \cos \left (f x + e\right ) + \sqrt {a b - b^{2}}}\right )}{\sqrt {a b - b^{2}} a^{4}} - \frac {\frac {8 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {16 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}}{a^{4}} - \frac {{\left (a^{2} - \frac {8 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {16 \, 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 {144 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {144 \, 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}}{a^{4} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}} - \frac {64 \, {\left (a^{2} b - a b^{2} + \frac {a^{2} b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {3 \, a b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {2 \, b^{3} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1}\right )}}{{\left (a + \frac {2 \, a {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {4 \, b {\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}}\right )} a^{4}}}{64 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________