\[ \int \sin (c+d x) \left (a+\frac {b}{\sqrt {\sin (c+d x)}}+c \sin (c+d x)\right )^2 \, dx \]
Optimal antiderivative \[ b^{2} x +a c x -\frac {a^{2} \cos \left (d x +c \right )}{d}-\frac {c^{2} \cos \left (d x +c \right )}{d}+\frac {c^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{3 d}-\frac {4 a b \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\right )}{\sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) d}-\frac {4 b c \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\right )}{3 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) d}-\frac {a c \cos \left (d x +c \right ) \sin \left (d x +c \right )}{d}-\frac {4 b c \cos \left (d x +c \right ) \left (\sqrt {\sin }\left (d x +c \right )\right )}{3 d} \]
command
integrate(sin(d*x+c)*(a+c*sin(d*x+c)+b/sin(d*x+c)^(1/2))^2,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {c^{2} \cos \left (d x + c\right )^{3} - 3 \, a c \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 2 \, \sqrt {2} \sqrt {-i} b c {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 2 \, \sqrt {2} \sqrt {i} b c {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 6 i \, \sqrt {2} \sqrt {-i} a b {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 6 i \, \sqrt {2} \sqrt {i} a b {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 4 \, b c \cos \left (d x + c\right ) \sqrt {\sin \left (d x + c\right )} + 3 \, {\left (b^{2} + a c\right )} \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right ) - 3 \, {\left (a^{2} + c^{2}\right )} \cos \left (d x + c\right )}{3 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-2 \, a c \cos \left (d x + c\right )^{2} + b^{2} + 2 \, a c - {\left (c^{2} \cos \left (d x + c\right )^{2} - a^{2} - c^{2}\right )} \sin \left (d x + c\right ) + 2 \, {\left (b c \sin \left (d x + c\right ) + a b\right )} \sqrt {\sin \left (d x + c\right )}, x\right ) \]