\[ \int \frac {A+B \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^3} \, dx \]
Optimal antiderivative \[ \frac {7 \left (17 A -7 B \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{10 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} d}+\frac {\left (33 A -13 B \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{6 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} d}+\frac {\left (33 A -13 B \right ) \sin \left (d x +c \right )}{6 a^{3} d \cos \left (d x +c \right )^{\frac {3}{2}}}-\frac {\left (A -B \right ) \sin \left (d x +c \right )}{5 d \cos \left (d x +c \right )^{\frac {3}{2}} \left (a +a \cos \left (d x +c \right )\right )^{3}}-\frac {\left (2 A -B \right ) \sin \left (d x +c \right )}{3 a d \cos \left (d x +c \right )^{\frac {3}{2}} \left (a +a \cos \left (d x +c \right )\right )^{2}}-\frac {7 \left (17 A -7 B \right ) \sin \left (d x +c \right )}{30 d \cos \left (d x +c \right )^{\frac {3}{2}} \left (a^{3}+a^{3} \cos \left (d x +c \right )\right )}-\frac {7 \left (17 A -7 B \right ) \sin \left (d x +c \right )}{10 a^{3} d \sqrt {\cos \left (d x +c \right )}} \]
command
integrate((A+B*cos(d*x+c))/cos(d*x+c)^(5/2)/(a+a*cos(d*x+c))^3,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (21 \, {\left (17 \, A - 7 \, B\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (453 \, A - 188 \, B\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left (139 \, A - 59 \, B\right )} \cos \left (d x + c\right )^{2} + 60 \, {\left (2 \, A - B\right )} \cos \left (d x + c\right ) - 20 \, A\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + 5 \, {\left (\sqrt {2} {\left (33 i \, A - 13 i \, B\right )} \cos \left (d x + c\right )^{5} + 3 \, \sqrt {2} {\left (33 i \, A - 13 i \, B\right )} \cos \left (d x + c\right )^{4} + 3 \, \sqrt {2} {\left (33 i \, A - 13 i \, B\right )} \cos \left (d x + c\right )^{3} + \sqrt {2} {\left (33 i \, A - 13 i \, B\right )} \cos \left (d x + c\right )^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, {\left (\sqrt {2} {\left (-33 i \, A + 13 i \, B\right )} \cos \left (d x + c\right )^{5} + 3 \, \sqrt {2} {\left (-33 i \, A + 13 i \, B\right )} \cos \left (d x + c\right )^{4} + 3 \, \sqrt {2} {\left (-33 i \, A + 13 i \, B\right )} \cos \left (d x + c\right )^{3} + \sqrt {2} {\left (-33 i \, A + 13 i \, B\right )} \cos \left (d x + c\right )^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, {\left (\sqrt {2} {\left (-17 i \, A + 7 i \, B\right )} \cos \left (d x + c\right )^{5} + 3 \, \sqrt {2} {\left (-17 i \, A + 7 i \, B\right )} \cos \left (d x + c\right )^{4} + 3 \, \sqrt {2} {\left (-17 i \, A + 7 i \, B\right )} \cos \left (d x + c\right )^{3} + \sqrt {2} {\left (-17 i \, A + 7 i \, B\right )} \cos \left (d x + c\right )^{2}\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 ) + 21 \, {\left (\sqrt {2} {\left (17 i \, A - 7 i \, B\right )} \cos \left (d x + c\right )^{5} + 3 \, \sqrt {2} {\left (17 i \, A - 7 i \, B\right )} \cos \left (d x + c\right )^{4} + 3 \, \sqrt {2} {\left (17 i \, A - 7 i \, B\right )} \cos \left (d x + c\right )^{3} + \sqrt {2} {\left (17 i \, A - 7 i \, B\right )} \cos \left (d x + c\right )^{2}\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 )}{60 \, {\left (a^{3} d \cos \left (d x + c\right )^{5} + 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B \cos \left (d x + c\right ) + A\right )} \sqrt {\cos \left (d x + c\right )}}{a^{3} \cos \left (d x + c\right )^{6} + 3 \, a^{3} \cos \left (d x + c\right )^{5} + 3 \, a^{3} \cos \left (d x + c\right )^{4} + a^{3} \cos \left (d x + c\right )^{3}}, x\right ) \]