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