\[ \int \sin ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx \]
Optimal antiderivative \[ \frac {\left (\cos ^{3}\left (f x +e \right )\right ) \left (a -b +b \left (\sec ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}{3 \left (a -b \right ) f}+\frac {\arctanh \left (\frac {\sec \left (f x +e \right ) \sqrt {b}}{\sqrt {a -b +b \left (\sec ^{2}\left (f x +e \right )\right )}}\right ) \sqrt {b}}{f}-\frac {\cos \left (f x +e \right ) \sqrt {a -b +b \left (\sec ^{2}\left (f x +e \right )\right )}}{f} \]
command
integrate(sin(f*x+e)^3*(a+b*tan(f*x+e)^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{3} \, {\left (\frac {{\left (3 \, a b \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) - 3 \, b^{2} \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) + 3 \, a \sqrt {-b} \sqrt {b} - 4 \, \sqrt {-b} b^{\frac {3}{2}}\right )} \mathrm {sgn}\left (f\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{a \sqrt {-b} f^{2} - \sqrt {-b} b f^{2}} - \frac {3 \, b \arctan \left (\frac {\sqrt {a \cos \left (f x + e\right )^{2} - b \cos \left (f x + e\right )^{2} + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\left (f\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{\sqrt {-b} f^{2}} + \frac {{\left (a \cos \left (f x + e\right )^{2} - b \cos \left (f x + e\right )^{2} + b\right )}^{\frac {3}{2}} a^{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^{3} f^{4} \mathrm {sgn}\left (f\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - 2 \, {\left (a \cos \left (f x + e\right )^{2} - b \cos \left (f x + e\right )^{2} + b\right )}^{\frac {3}{2}} a b f^{4} \mathrm {sgn}\left (f\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 9 \, \sqrt {a \cos \left (f x + e\right )^{2} - b \cos \left (f x + e\right )^{2} + b} a^{2} b f^{4} \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} + b\right )}^{\frac {3}{2}} b^{2} f^{4} \mathrm {sgn}\left (f\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - 9 \, \sqrt {a \cos \left (f x + e\right )^{2} - b \cos \left (f x + e\right )^{2} + b} a b^{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} b^{3} f^{4} \mathrm {sgn}\left (f\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{a^{3} f^{6} - 3 \, a^{2} b f^{6} + 3 \, a b^{2} f^{6} - b^{3} f^{6}}\right )} {\left | f \right |} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________