\[ \int \frac {1}{\sqrt {3-2 \sec (c+d x)} \sqrt {\sec (c+d x)}} \, dx \]
Optimal antiderivative \[ \frac {2 \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {6}\right ) \sqrt {3-2 \sec \left (d x +c \right )}}{3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {-2+3 \cos \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right )}}+\frac {4 \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {6}\right ) \sqrt {-2+3 \cos \left (d x +c \right )}\, \left (\sqrt {\sec }\left (d x +c \right )\right )}{3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {3-2 \sec \left (d x +c \right )}} \]
command
integrate(1/(3-2*sec(d*x+c))^(1/2)/sec(d*x+c)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {-4 i \, \sqrt {6} {\rm weierstrassPInverse}\left (-\frac {44}{27}, -\frac {784}{729}, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {4}{9}\right ) + 4 i \, \sqrt {6} {\rm weierstrassPInverse}\left (-\frac {44}{27}, -\frac {784}{729}, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {4}{9}\right ) + 9 i \, \sqrt {6} {\rm weierstrassZeta}\left (-\frac {44}{27}, -\frac {784}{729}, {\rm weierstrassPInverse}\left (-\frac {44}{27}, -\frac {784}{729}, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {4}{9}\right )\right ) - 9 i \, \sqrt {6} {\rm weierstrassZeta}\left (-\frac {44}{27}, -\frac {784}{729}, {\rm weierstrassPInverse}\left (-\frac {44}{27}, -\frac {784}{729}, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {4}{9}\right )\right )}{27 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {-2 \, \sec \left (d x + c\right ) + 3} \sqrt {\sec \left (d x + c\right )}}{2 \, \sec \left (d x + c\right )^{2} - 3 \, \sec \left (d x + c\right )}, x\right ) \]