\[ \int \frac {\sin ^3(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx \]
Optimal antiderivative \[ -\frac {\left (3 a -b \right ) \cos \left (f x +e \right ) \sqrt {a -b +b \left (\sec ^{2}\left (f x +e \right )\right )}}{3 \left (a -b \right )^{2} f}+\frac {\left (\cos ^{3}\left (f x +e \right )\right ) \sqrt {a -b +b \left (\sec ^{2}\left (f x +e \right )\right )}}{3 \left (a -b \right ) f} \]
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
\[ \frac {{\left (3 \, a \sqrt {b} - b^{\frac {3}{2}}\right )} \mathrm {sgn}\left (f\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{3 \, {\left (a^{2} {\left | f \right |} - 2 \, a b {\left | f \right |} + b^{2} {\left | f \right |}\right )}} + \frac {{\left (a \cos \left (f x + e\right )^{2} - b \cos \left (f x + e\right )^{2} + b\right )}^{\frac {3}{2}} f^{2}}{3 \, {\left (a {\left | f \right |} \mathrm {sgn}\left (f\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - b {\left | f \right |} \mathrm {sgn}\left (f\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )\right )} {\left (a f^{2} - b f^{2}\right )}} - \frac {\sqrt {a \cos \left (f x + e\right )^{2} - b \cos \left (f x + e\right )^{2} + b} a}{a^{2} {\left | f \right |} \mathrm {sgn}\left (f\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - 2 \, a b {\left | f \right |} \mathrm {sgn}\left (f\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + b^{2} {\left | f \right |} \mathrm {sgn}\left (f\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________