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