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