\[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx \]
Optimal antiderivative \[ -\frac {b \left (12 A \,b^{6}-12 a^{5} b B +15 a^{3} b^{3} B -6 a \,b^{5} B -a^{2} b^{4} \left (29 A -2 C \right )+5 a^{4} b^{2} \left (4 A -C \right )+6 a^{6} C \right ) \arctan \left (\frac {\sqrt {a -b}\, \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {a +b}}\right )}{a^{5} \left (a -b \right )^{\frac {5}{2}} \left (a +b \right )^{\frac {5}{2}} d}+\frac {\left (12 A \,b^{2}-6 a b B +a^{2} \left (A +2 C \right )\right ) \arctanh \left (\sin \left (d x +c \right )\right )}{2 a^{5} d}-\frac {\left (12 A \,b^{5}-2 B \,a^{5}+11 B \,a^{3} b^{2}-6 B a \,b^{4}+a^{4} b \left (6 A -5 C \right )-a^{2} b^{3} \left (21 A -2 C \right )\right ) \tan \left (d x +c \right )}{2 a^{4} \left (a^{2}-b^{2}\right )^{2} d}+\frac {\left (6 A \,b^{4}+6 a^{3} b B -3 a \,b^{3} B +a^{4} \left (A -4 C \right )-a^{2} b^{2} \left (10 A -C \right )\right ) \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 a^{3} \left (a^{2}-b^{2}\right )^{2} d}+\frac {\left (A \,b^{2}-a \left (b B -a C \right )\right ) \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 a \left (a^{2}-b^{2}\right ) d \left (a +b \cos \left (d x +c \right )\right )^{2}}+\frac {\left (7 a^{2} A \,b^{2}-4 A \,b^{4}-5 a^{3} b B +2 a \,b^{3} B +3 a^{4} C \right ) \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 a^{2} \left (a^{2}-b^{2}\right )^{2} d \left (a +b \cos \left (d x +c \right )\right )} \]
command
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^3,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ \text {Timed out} \]