\[ \int (a+b \cos (c+d x))^3 \sqrt {\sec (c+d x)} \, dx \]
Optimal antiderivative \[ \frac {2 b^{2} \left (b +a \sec \left (d x +c \right )\right ) \sin \left (d x +c \right )}{5 d \sec \left (d x +c \right )^{\frac {3}{2}}}+\frac {8 a \,b^{2} \sin \left (d x +c \right )}{5 d \sqrt {\sec \left (d x +c \right )}}+\frac {6 b \left (5 a^{2}+b^{2}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 a \left (a^{2}+b^{2}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d} \]
command
integrate((a+b*cos(d*x+c))^3*sec(d*x+c)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {5 \, \sqrt {2} {\left (i \, a^{3} + i \, a b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-i \, a^{3} - i \, a b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 3 \, \sqrt {2} {\left (-5 i \, a^{2} b - i \, b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, \sqrt {2} {\left (5 i \, a^{2} b + i \, b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (b^{3} \cos \left (d x + c\right )^{2} + 5 \, a b^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{5 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (b^{3} \cos \left (d x + c\right )^{3} + 3 \, a b^{2} \cos \left (d x + c\right )^{2} + 3 \, a^{2} b \cos \left (d x + c\right ) + a^{3}\right )} \sqrt {\sec \left (d x + c\right )}, x\right ) \]