\[ \int \frac {1}{\sqrt {2 \cos (c+d x)+3 \sin (c+d x)}} \, dx \]
Optimal antiderivative \[ \frac {2 \,13^{\frac {3}{4}} \sqrt {\frac {\cos \left (c +d x -\arctan \left (\frac {3}{2}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {c}{2}+\frac {d x}{2}-\frac {\arctan \left (\frac {3}{2}\right )}{2}\right ), \sqrt {2}\right )}{13 \cos \left (\frac {c}{2}+\frac {d x}{2}-\frac {\arctan \left (\frac {3}{2}\right )}{2}\right ) d} \]
command
integrate(1/(2*cos(d*x+c)+3*sin(d*x+c))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\left (2 i + 3\right ) \, \sqrt {3 i + 2} \sqrt {2} {\rm weierstrassPInverse}\left (\frac {48}{13} i + \frac {20}{13}, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - \left (2 i - 3\right ) \, \sqrt {2} \sqrt {-3 i + 2} {\rm weierstrassPInverse}\left (-\frac {48}{13} i + \frac {20}{13}, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )}{13 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {1}{\sqrt {2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )}}, x\right ) \]