\[ \int (a+a \sin (e+f x))^{5/2} \tan ^4(e+f x) \, dx \]
Optimal antiderivative \[ -\frac {2 a^{5} \left (\cos ^{5}\left (f x +e \right )\right )}{5 f \left (a +a \sin \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {8 a^{4} \left (\cos ^{3}\left (f x +e \right )\right )}{3 f \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {2 a \left (\sec ^{3}\left (f x +e \right )\right ) \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{3 f}-\frac {12 a^{3} \cos \left (f x +e \right )}{f \sqrt {a +a \sin \left (f x +e \right )}}-\frac {8 a^{2} \sec \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}}{f} \]
command
integrate((a+a*sin(f*x+e))^(5/2)*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 \[ \text {Timed out} \]_______________________________________________________