\[ \int \cos ^4(c+d x) \sqrt {a+b \sin (c+d x)} \, dx \]
Optimal antiderivative \[ \frac {2 \left (\cos ^{3}\left (d x +c \right )\right ) \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{9 b d}-\frac {4 a \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{21 b d}-\frac {4 \cos \left (d x +c \right ) \left (4 a \left (a^{2}-3 b^{2}\right )-3 b \left (a^{2}+7 b^{2}\right ) \sin \left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{315 b^{3} d}+\frac {8 \left (4 a^{4}-15 a^{2} b^{2}-21 b^{4}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \sin \left (d x +c \right )}}{315 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) b^{4} d \sqrt {\frac {a +b \sin \left (d x +c \right )}{a +b}}}-\frac {32 a \left (a^{4}-4 a^{2} b^{2}+3 b^{4}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \sin \left (d x +c \right )}{a +b}}}{315 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) b^{4} d \sqrt {a +b \sin \left (d x +c \right )}} \]
command
integrate(cos(d*x+c)^4*(a+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 {2 \, {\left (2 \, \sqrt {2} {\left (8 \, a^{5} - 33 \, a^{3} b^{2} + 57 \, a b^{4}\right )} \sqrt {i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right ) + 2 \, \sqrt {2} {\left (8 \, a^{5} - 33 \, a^{3} b^{2} + 57 \, a b^{4}\right )} \sqrt {-i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right ) - 6 \, \sqrt {2} {\left (-4 i \, a^{4} b + 15 i \, a^{2} b^{3} + 21 i \, b^{5}\right )} \sqrt {i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right )\right ) - 6 \, \sqrt {2} {\left (4 i \, a^{4} b - 15 i \, a^{2} b^{3} - 21 i \, b^{5}\right )} \sqrt {-i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right )\right ) + 3 \, {\left (5 \, a b^{4} \cos \left (d x + c\right )^{3} - 8 \, {\left (a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right ) + {\left (35 \, b^{5} \cos \left (d x + c\right )^{3} + 6 \, {\left (a^{2} b^{3} + 7 \, b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}\right )}}{945 \, b^{5} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {b \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4}, x\right ) \]