\[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (\sec ^{\frac {3}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{2 d \left (a +a \cos \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {7 \left (\sec ^{\frac {3}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{6 a d \sqrt {a +a \cos \left (d x +c \right )}}+\frac {11 \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {a}\, \sqrt {2}}{2 \sqrt {\cos \left (d x +c \right )}\, \sqrt {a +a \cos \left (d x +c \right )}}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right ) \sqrt {2}}{4 a^{\frac {3}{2}} d}-\frac {19 \sin \left (d x +c \right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{6 a d \sqrt {a +a \cos \left (d x +c \right )}} \]
command
Integrate[Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(3/2),x]
Mathematica 13.1 output
\[ \frac {\cot ^3\left (\frac {c}{2}+\frac {d x}{2}\right ) \csc ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \left (\frac {1}{1-2 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}\right )^{7/2} \left (-80 \cos ^6\left (\frac {1}{2} (c+d x)\right ) \, _4F_3\left (2,2,2,\frac {7}{2};1,1,\frac {11}{2};\frac {\sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}{-1+2 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \sin ^{10}\left (\frac {c}{2}+\frac {d x}{2}\right )+120 \cos ^4\left (\frac {1}{2} (c+d x)\right ) \, _3F_2\left (2,2,\frac {7}{2};1,\frac {11}{2};\frac {\sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}{-1+2 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \sin ^{10}\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (-5+4 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )\right )+21 \left (1-2 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )\right )^3 \sqrt {\frac {\sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}{-1+2 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}} \left (-15 \tanh ^{-1}\left (\sqrt {\frac {\sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}{-1+2 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}}\right ) \left (-392+2347 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )-5391 \sin ^4\left (\frac {c}{2}+\frac {d x}{2}\right )+5972 \sin ^6\left (\frac {c}{2}+\frac {d x}{2}\right )-3232 \sin ^8\left (\frac {c}{2}+\frac {d x}{2}\right )+696 \sin ^{10}\left (\frac {c}{2}+\frac {d x}{2}\right )\right )+\sqrt {\frac {\sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}{-1+2 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}} \left (-5880+37165 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )-89856 \sin ^4\left (\frac {c}{2}+\frac {d x}{2}\right )+103992 \sin ^6\left (\frac {c}{2}+\frac {d x}{2}\right )-58336 \sin ^8\left (\frac {c}{2}+\frac {d x}{2}\right )+12960 \sin ^{10}\left (\frac {c}{2}+\frac {d x}{2}\right )\right )\right )\right )}{945 d (a (1+\cos (c+d x)))^{3/2}} \]
Mathematica 12.3 output
\[ \text {\$Aborted} \]________________________________________________________________________________________