\[ \int \frac {(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx \]
Optimal antiderivative \[ \frac {b \left (20 A \,a^{6} b -35 A \,a^{4} b^{3}+28 A \,a^{2} b^{5}-8 A \,b^{7}-8 B \,a^{7}+8 B \,a^{5} b^{2}-7 B \,a^{3} b^{4}+2 B a \,b^{6}\right ) \arctan \left (\frac {\sqrt {a -b}\, \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {a +b}}\right )}{a^{5} \left (a -b \right )^{\frac {7}{2}} \left (a +b \right )^{\frac {7}{2}} d}-\frac {\left (4 A b -B a \right ) \arctanh \left (\sin \left (d x +c \right )\right )}{a^{5} d}+\frac {\left (6 A \,a^{6}-65 A \,a^{4} b^{2}+68 A \,a^{2} b^{4}-24 A \,b^{6}+26 B \,a^{5} b -17 B \,a^{3} b^{3}+6 B a \,b^{5}\right ) \tan \left (d x +c \right )}{6 a^{4} \left (a^{2}-b^{2}\right )^{3} d}+\frac {b \left (A b -B a \right ) \tan \left (d x +c \right )}{3 a \left (a^{2}-b^{2}\right ) d \left (a +b \cos \left (d x +c \right )\right )^{3}}+\frac {b \left (9 A \,a^{2} b -4 A \,b^{3}-6 a^{3} B +B a \,b^{2}\right ) \tan \left (d x +c \right )}{6 a^{2} \left (a^{2}-b^{2}\right )^{2} d \left (a +b \cos \left (d x +c \right )\right )^{2}}+\frac {b \left (12 A \,a^{4} b -11 A \,a^{2} b^{3}+4 A \,b^{5}-6 B \,a^{5}+2 B \,a^{3} b^{2}-B a \,b^{4}\right ) \tan \left (d x +c \right )}{2 a^{3} \left (a^{2}-b^{2}\right )^{3} d \left (a +b \cos \left (d x +c \right )\right )} \]
command
integrate((A+B*cos(d*x+c))*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 \[ \text {Timed out} \]