\[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx \]
Optimal antiderivative \[ -\frac {\left (35 a^{4} A \,b^{4}-28 a^{2} A \,b^{6}+8 A \,b^{8}+8 a^{7} b B -8 a^{5} b^{3} B +7 a^{3} b^{5} B -2 a \,b^{7} B -2 a^{8} C -a^{6} b^{2} \left (20 A +3 C \right )\right ) \arctan \left (\frac {\sqrt {a -b}\, \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {a +b}}\right )}{a^{5} \left (a -b \right )^{\frac {7}{2}} \left (a +b \right )^{\frac {7}{2}} d}-\frac {\left (4 A b -B a \right ) \arctanh \left (\sin \left (d x +c \right )\right )}{a^{5} d}+\frac {\left (68 a^{2} A \,b^{4}-24 A \,b^{6}+26 a^{5} b B -17 a^{3} b^{3} B +6 a \,b^{5} B +a^{6} \left (6 A -11 C \right )-a^{4} b^{2} \left (65 A +4 C \right )\right ) \tan \left (d x +c \right )}{6 a^{4} \left (a^{2}-b^{2}\right )^{3} d}+\frac {\left (A \,b^{2}-a \left (b B -a C \right )\right ) \tan \left (d x +c \right )}{3 a \left (a^{2}-b^{2}\right ) d \left (a +b \cos \left (d x +c \right )\right )^{3}}-\frac {\left (4 A \,b^{4}+6 a^{3} b B -a \,b^{3} B -3 a^{4} C -a^{2} b^{2} \left (9 A +2 C \right )\right ) \tan \left (d x +c \right )}{6 a^{2} \left (a^{2}-b^{2}\right )^{2} d \left (a +b \cos \left (d x +c \right )\right )^{2}}-\frac {\left (11 a^{2} A \,b^{4}-4 A \,b^{6}+6 a^{5} b B -2 a^{3} b^{3} B +a \,b^{5} B -2 a^{6} C -3 a^{4} b^{2} \left (4 A +C \right )\right ) \tan \left (d x +c \right )}{2 a^{3} \left (a^{2}-b^{2}\right )^{3} d \left (a +b \cos \left (d x +c \right )\right )} \]
command
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^2/(a+b*cos(d*x+c))^4,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 \[ \text {Timed out} \]