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