\[ \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx \]
Optimal antiderivative \[ -\frac {b^{2} \left (40 A \,a^{6} b -84 A \,a^{4} b^{3}+69 A \,a^{2} b^{5}-20 A \,b^{7}-20 B \,a^{7}+35 B \,a^{5} b^{2}-28 B \,a^{3} b^{4}+8 B a \,b^{6}\right ) \arctan \left (\frac {\sqrt {a -b}\, \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {a +b}}\right )}{a^{6} \left (a -b \right )^{\frac {7}{2}} \left (a +b \right )^{\frac {7}{2}} d}+\frac {\left (a^{2} A +20 A \,b^{2}-8 a b B \right ) \arctanh \left (\sin \left (d x +c \right )\right )}{2 a^{6} d}-\frac {\left (24 A \,a^{6} b -146 A \,a^{4} b^{3}+167 A \,a^{2} b^{5}-60 A \,b^{7}-6 B \,a^{7}+65 B \,a^{5} b^{2}-68 B \,a^{3} b^{4}+24 B a \,b^{6}\right ) \tan \left (d x +c \right )}{6 a^{5} \left (a^{2}-b^{2}\right )^{3} d}+\frac {\left (A \,a^{6}-23 A \,a^{4} b^{2}+27 A \,a^{2} b^{4}-10 A \,b^{6}+12 B \,a^{5} b -11 B \,a^{3} b^{3}+4 B a \,b^{5}\right ) \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 a^{4} \left (a^{2}-b^{2}\right )^{3} d}+\frac {b \left (A b -B a \right ) \sec \left (d x +c \right ) \tan \left (d x +c \right )}{3 a \left (a^{2}-b^{2}\right ) d \left (a +b \cos \left (d x +c \right )\right )^{3}}+\frac {b \left (10 A \,a^{2} b -5 A \,b^{3}-7 a^{3} B +2 B a \,b^{2}\right ) \sec \left (d x +c \right ) \tan \left (d x +c \right )}{6 a^{2} \left (a^{2}-b^{2}\right )^{2} d \left (a +b \cos \left (d x +c \right )\right )^{2}}+\frac {b \left (48 A \,a^{4} b -53 A \,a^{2} b^{3}+20 A \,b^{5}-27 B \,a^{5}+20 B \,a^{3} b^{2}-8 B a \,b^{4}\right ) \sec \left (d x +c \right ) \tan \left (d x +c \right )}{6 a^{3} \left (a^{2}-b^{2}\right )^{3} d \left (a +b \cos \left (d x +c \right )\right )} \]
command
integrate((A+B*cos(d*x+c))*sec(d*x+c)^3/(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 \[ \text {Timed out} \]