\[ \int (a+a \sin (e+f x))^{3/2} \tan ^4(e+f x) \, dx \]
Optimal antiderivative \[ \frac {2 a^{3} \left (\cos ^{3}\left (f x +e \right )\right )}{3 f \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {\left (\sec ^{3}\left (f x +e \right )\right ) \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{3 f}-\frac {a^{\frac {3}{2}} \arctanh \left (\frac {\cos \left (f x +e \right ) \sqrt {a}\, \sqrt {2}}{2 \sqrt {a +a \sin \left (f x +e \right )}}\right ) \sqrt {2}}{4 f}-\frac {4 a^{2} \cos \left (f x +e \right )}{f \sqrt {a +a \sin \left (f x +e \right )}}-\frac {7 a \sec \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}}{2 f} \]
command
integrate((a+a*sin(f*x+e))^(3/2)*tan(f*x+e)^4,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Timed out} \]_______________________________________________________