\[ \int \frac {\csc (e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {\arctanh \left (\cos \left (f x +e \right )\right )}{a^{3} f}-\frac {b \sec \left (f x +e \right )}{4 a \left (a -b \right ) f \left (a -b +b \left (\sec ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {\left (7 a -4 b \right ) b \sec \left (f x +e \right )}{8 a^{2} \left (a -b \right )^{2} f \left (a -b +b \left (\sec ^{2}\left (f x +e \right )\right )\right )}-\frac {\left (15 a^{2}-20 a b +8 b^{2}\right ) \arctan \left (\frac {\sec \left (f x +e \right ) \sqrt {b}}{\sqrt {a -b}}\right ) \sqrt {b}}{8 a^{3} \left (a -b \right )^{\frac {5}{2}} f} \]
command
integrate(csc(f*x+e)/(a+b*tan(f*x+e)^2)^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\frac {{\left (15 \, a^{2} b - 20 \, a b^{2} + 8 \, 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 )}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} \sqrt {a b - b^{2}}} + \frac {2 \, {\left (9 \, a^{3} b - 6 \, a^{2} b^{2} + \frac {27 \, a^{3} b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {68 \, a^{2} b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {32 \, a b^{3} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {27 \, a^{3} b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {90 \, a^{2} b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {120 \, a b^{3} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {48 \, b^{4} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {9 \, a^{3} b {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {28 \, a^{2} b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {16 \, a b^{3} {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\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 )}^{2}} - \frac {4 \, \log \left (\frac {{\left | -\cos \left (f x + e\right ) + 1 \right |}}{{\left | \cos \left (f x + e\right ) + 1 \right |}}\right )}{a^{3}}}{8 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________