\[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx \]
Optimal antiderivative \[ \frac {2 \EllipticF \left (\sin \left (f x +e \right ), \sqrt {\frac {a}{a +b}}\right ) \sqrt {1-\frac {a \left (\sin ^{2}\left (f x +e \right )\right )}{a +b}}}{f \sqrt {2 \cos \left (2 f x +2 e \right )+2}\, \sqrt {\left (\sec ^{2}\left (f x +e \right )\right ) \left (a +b -a \left (\sin ^{2}\left (f x +e \right )\right )\right )}} \]
command
integrate(sec(f*x+e)/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {{\left (2 i \, a^{\frac {3}{2}} \sqrt {\frac {a b + b^{2}}{a^{2}}} + \sqrt {a} {\left (i \, a + 2 i \, b\right )}\right )} \sqrt {\frac {2 \, a \sqrt {\frac {a b + b^{2}}{a^{2}}} - a - 2 \, b}{a}} {\rm ellipticF}\left (\sqrt {\frac {2 \, a \sqrt {\frac {a b + b^{2}}{a^{2}}} - a - 2 \, b}{a}} {\left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )}, \frac {a^{2} + 8 \, a b + 8 \, b^{2} + 4 \, {\left (a^{2} + 2 \, a b\right )} \sqrt {\frac {a b + b^{2}}{a^{2}}}}{a^{2}}\right ) + {\left (-2 i \, a^{\frac {3}{2}} \sqrt {\frac {a b + b^{2}}{a^{2}}} + \sqrt {a} {\left (-i \, a - 2 i \, b\right )}\right )} \sqrt {\frac {2 \, a \sqrt {\frac {a b + b^{2}}{a^{2}}} - a - 2 \, b}{a}} {\rm ellipticF}\left (\sqrt {\frac {2 \, a \sqrt {\frac {a b + b^{2}}{a^{2}}} - a - 2 \, b}{a}} {\left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}, \frac {a^{2} + 8 \, a b + 8 \, b^{2} + 4 \, {\left (a^{2} + 2 \, a b\right )} \sqrt {\frac {a b + b^{2}}{a^{2}}}}{a^{2}}\right )}{a^{2} f} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sec \left (f x + e\right )}{\sqrt {b \sec \left (f x + e\right )^{2} + a}}, x\right ) \]