\[ \int (d \sec (e+f x))^{7/2} (a+b \tan (e+f x)) \, dx \]
Optimal antiderivative \[ \frac {2 b \left (d \sec \left (f x +e \right )\right )^{\frac {7}{2}}}{7 f}+\frac {2 a d \left (d \sec \left (f x +e \right )\right )^{\frac {5}{2}} \sin \left (f x +e \right )}{5 f}-\frac {6 a \,d^{4} \sqrt {\frac {\cos \left (f x +e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {f x}{2}+\frac {e}{2}\right ), \sqrt {2}\right )}{5 \cos \left (\frac {f x}{2}+\frac {e}{2}\right ) f \sqrt {\cos \left (f x +e \right )}\, \sqrt {d \sec \left (f x +e \right )}}+\frac {6 a \,d^{3} \sin \left (f x +e \right ) \sqrt {d \sec \left (f x +e \right )}}{5 f} \]
command
integrate((d*sec(f*x+e))^(7/2)*(a+b*tan(f*x+e)),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {-21 i \, \sqrt {2} a d^{\frac {7}{2}} \cos \left (f x + e\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 21 i \, \sqrt {2} a d^{\frac {7}{2}} \cos \left (f x + e\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + 2 \, {\left (5 \, b d^{3} + 7 \, {\left (3 \, a d^{3} \cos \left (f x + e\right )^{3} + a d^{3} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{35 \, f \cos \left (f x + e\right )^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (b d^{3} \sec \left (f x + e\right )^{3} \tan \left (f x + e\right ) + a d^{3} \sec \left (f x + e\right )^{3}\right )} \sqrt {d \sec \left (f x + e\right )}, x\right ) \]