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