Integrand size = 41, antiderivative size = 320 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{16} \left (18 a^2 b B+8 b^3 B+6 a b^2 (3 A+4 C)+a^3 (5 A+6 C)\right ) x+\frac {\left (12 a^3 B+42 a b^2 B+9 a^2 b (4 A+5 C)+b^3 (11 A+15 C)\right ) \sin (c+d x)}{15 d}+\frac {\left (18 a^2 b B+8 b^3 B+6 a b^2 (3 A+4 C)+a^3 (5 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \left (6 A b^2+42 a b B+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {(A b+2 a B) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}-\frac {\left (A b^3+4 a^3 B+12 a b^2 B+3 a^2 b (4 A+5 C)\right ) \sin ^3(c+d x)}{15 d} \]
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Time = 0.99 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {4179, 4159, 4132, 2715, 8, 4129, 3092} \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a \sin (c+d x) \cos ^3(c+d x) \left (5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right )}{120 d}-\frac {\sin ^3(c+d x) \left (4 a^3 B+3 a^2 b (4 A+5 C)+12 a b^2 B+A b^3\right )}{15 d}+\frac {\sin (c+d x) \left (12 a^3 B+9 a^2 b (4 A+5 C)+42 a b^2 B+b^3 (11 A+15 C)\right )}{15 d}+\frac {\sin (c+d x) \cos (c+d x) \left (a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right )}{16 d}+\frac {1}{16} x \left (a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right )+\frac {(2 a B+A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{10 d}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d} \]
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Rule 8
Rule 2715
Rule 3092
Rule 4129
Rule 4132
Rule 4159
Rule 4179
Rubi steps \begin{align*} \text {integral}& = \frac {A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {1}{6} \int \cos ^5(c+d x) (a+b \sec (c+d x))^2 \left (3 (A b+2 a B)+(5 a A+6 b B+6 a C) \sec (c+d x)+2 b (A+3 C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {(A b+2 a B) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {1}{30} \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (6 A b^2+42 a b B+5 a^2 (5 A+6 C)+\left (24 a^2 B+30 b^2 B+a b (47 A+60 C)\right ) \sec (c+d x)+2 b (8 A b+6 a B+15 b C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {a \left (6 A b^2+42 a b B+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {(A b+2 a B) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}-\frac {1}{120} \int \cos ^3(c+d x) \left (-24 \left (A b^3+4 a^3 B+12 a b^2 B+3 a^2 b (4 A+5 C)\right )-15 \left (18 a^2 b B+8 b^3 B+6 a b^2 (3 A+4 C)+a^3 (5 A+6 C)\right ) \sec (c+d x)-8 b^2 (8 A b+6 a B+15 b C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {a \left (6 A b^2+42 a b B+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {(A b+2 a B) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}-\frac {1}{120} \int \cos ^3(c+d x) \left (-24 \left (A b^3+4 a^3 B+12 a b^2 B+3 a^2 b (4 A+5 C)\right )-8 b^2 (8 A b+6 a B+15 b C) \sec ^2(c+d x)\right ) \, dx-\frac {1}{8} \left (-18 a^2 b B-8 b^3 B-6 a b^2 (3 A+4 C)-a^3 (5 A+6 C)\right ) \int \cos ^2(c+d x) \, dx \\ & = \frac {\left (18 a^2 b B+8 b^3 B+6 a b^2 (3 A+4 C)+a^3 (5 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \left (6 A b^2+42 a b B+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {(A b+2 a B) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}-\frac {1}{120} \int \cos (c+d x) \left (-8 b^2 (8 A b+6 a B+15 b C)-24 \left (A b^3+4 a^3 B+12 a b^2 B+3 a^2 b (4 A+5 C)\right ) \cos ^2(c+d x)\right ) \, dx-\frac {1}{16} \left (-18 a^2 b B-8 b^3 B-6 a b^2 (3 A+4 C)-a^3 (5 A+6 C)\right ) \int 1 \, dx \\ & = \frac {1}{16} \left (18 a^2 b B+8 b^3 B+6 a b^2 (3 A+4 C)+a^3 (5 A+6 C)\right ) x+\frac {\left (18 a^2 b B+8 b^3 B+6 a b^2 (3 A+4 C)+a^3 (5 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \left (6 A b^2+42 a b B+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {(A b+2 a B) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {\text {Subst}\left (\int \left (-8 b^2 (8 A b+6 a B+15 b C)-24 \left (A b^3+4 a^3 B+12 a b^2 B+3 a^2 b (4 A+5 C)\right )+24 \left (A b^3+4 a^3 B+12 a b^2 B+3 a^2 b (4 A+5 C)\right ) x^2\right ) \, dx,x,-\sin (c+d x)\right )}{120 d} \\ & = \frac {1}{16} \left (18 a^2 b B+8 b^3 B+6 a b^2 (3 A+4 C)+a^3 (5 A+6 C)\right ) x+\frac {\left (12 a^3 B+42 a b^2 B+9 a^2 b (4 A+5 C)+b^3 (11 A+15 C)\right ) \sin (c+d x)}{15 d}+\frac {\left (18 a^2 b B+8 b^3 B+6 a b^2 (3 A+4 C)+a^3 (5 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \left (6 A b^2+42 a b B+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {(A b+2 a B) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}-\frac {\left (A b^3+4 a^3 B+12 a b^2 B+3 a^2 b (4 A+5 C)\right ) \sin ^3(c+d x)}{15 d} \\ \end{align*}
Time = 1.99 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.15 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {300 a^3 A c+1080 a A b^2 c+1080 a^2 b B c+480 b^3 B c+360 a^3 c C+1440 a b^2 c C+300 a^3 A d x+1080 a A b^2 d x+1080 a^2 b B d x+480 b^3 B d x+360 a^3 C d x+1440 a b^2 C d x+120 \left (5 a^3 B+18 a b^2 B+2 b^3 (3 A+4 C)+3 a^2 b (5 A+6 C)\right ) \sin (c+d x)+15 \left (48 a^2 b B+16 b^3 B+48 a b^2 (A+C)+a^3 (15 A+16 C)\right ) \sin (2 (c+d x))+300 a^2 A b \sin (3 (c+d x))+80 A b^3 \sin (3 (c+d x))+100 a^3 B \sin (3 (c+d x))+240 a b^2 B \sin (3 (c+d x))+240 a^2 b C \sin (3 (c+d x))+45 a^3 A \sin (4 (c+d x))+90 a A b^2 \sin (4 (c+d x))+90 a^2 b B \sin (4 (c+d x))+30 a^3 C \sin (4 (c+d x))+36 a^2 A b \sin (5 (c+d x))+12 a^3 B \sin (5 (c+d x))+5 a^3 A \sin (6 (c+d x))}{960 d} \]
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Time = 0.99 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.76
method | result | size |
parallelrisch | \(\frac {\left (\left (225 A +240 C \right ) a^{3}+720 B \,a^{2} b +720 b^{2} \left (A +C \right ) a +240 B \,b^{3}\right ) \sin \left (2 d x +2 c \right )+\left (100 B \,a^{3}+300 b \left (A +\frac {4 C}{5}\right ) a^{2}+240 B a \,b^{2}+80 A \,b^{3}\right ) \sin \left (3 d x +3 c \right )+45 \left (a^{2} \left (A +\frac {2 C}{3}\right )+2 B a b +2 A \,b^{2}\right ) a \sin \left (4 d x +4 c \right )+\left (36 A \,a^{2} b +12 B \,a^{3}\right ) \sin \left (5 d x +5 c \right )+5 a^{3} A \sin \left (6 d x +6 c \right )+\left (600 B \,a^{3}+1800 b \left (A +\frac {6 C}{5}\right ) a^{2}+2160 B a \,b^{2}+720 b^{3} \left (A +\frac {4 C}{3}\right )\right ) \sin \left (d x +c \right )+300 d x \left (a^{3} \left (A +\frac {6 C}{5}\right )+\frac {18 B \,a^{2} b}{5}+\frac {18 \left (A +\frac {4 C}{3}\right ) a \,b^{2}}{5}+\frac {8 B \,b^{3}}{5}\right )}{960 d}\) | \(244\) |
derivativedivides | \(\frac {a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {3 A \,a^{2} b \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {B \,a^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+3 a A \,b^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 B \,a^{2} b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{3} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {A \,b^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B a \,b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+a^{2} b C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+B \,b^{3} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 C a \,b^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \sin \left (d x +c \right ) b^{3}}{d}\) | \(370\) |
default | \(\frac {a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {3 A \,a^{2} b \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {B \,a^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+3 a A \,b^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 B \,a^{2} b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{3} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {A \,b^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B a \,b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+a^{2} b C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+B \,b^{3} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 C a \,b^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \sin \left (d x +c \right ) b^{3}}{d}\) | \(370\) |
risch | \(\frac {9 A a \,b^{2} x}{8}+\frac {15 \sin \left (2 d x +2 c \right ) a^{3} A}{64 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} C}{4 d}+\frac {\sin \left (3 d x +3 c \right ) A \,b^{3}}{12 d}+\frac {\sin \left (2 d x +2 c \right ) B \,b^{3}}{4 d}+\frac {5 a^{3} A x}{16}+\frac {5 \sin \left (3 d x +3 c \right ) A \,a^{2} b}{16 d}+\frac {3 \sin \left (2 d x +2 c \right ) a A \,b^{2}}{4 d}+\frac {3 a^{3} x C}{8}+\frac {3 \sin \left (2 d x +2 c \right ) B \,a^{2} b}{4 d}+\frac {\sin \left (4 d x +4 c \right ) a^{3} C}{32 d}+\frac {a^{3} A \sin \left (6 d x +6 c \right )}{192 d}+\frac {9 B \,a^{2} b x}{8}+\frac {5 B \,a^{3} \sin \left (3 d x +3 c \right )}{48 d}+\frac {3 x C a \,b^{2}}{2}+\frac {15 \sin \left (d x +c \right ) A \,a^{2} b}{8 d}+\frac {9 \sin \left (d x +c \right ) B a \,b^{2}}{4 d}+\frac {9 \sin \left (d x +c \right ) a^{2} b C}{4 d}+\frac {3 \sin \left (5 d x +5 c \right ) A \,a^{2} b}{80 d}+\frac {3 \sin \left (4 d x +4 c \right ) a A \,b^{2}}{32 d}+\frac {3 \sin \left (4 d x +4 c \right ) B \,a^{2} b}{32 d}+\frac {\sin \left (3 d x +3 c \right ) B a \,b^{2}}{4 d}+\frac {\sin \left (3 d x +3 c \right ) a^{2} b C}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) C a \,b^{2}}{4 d}+\frac {3 a^{3} A \sin \left (4 d x +4 c \right )}{64 d}+\frac {x B \,b^{3}}{2}+\frac {3 \sin \left (d x +c \right ) A \,b^{3}}{4 d}+\frac {\sin \left (d x +c \right ) C \,b^{3}}{d}+\frac {\sin \left (5 d x +5 c \right ) B \,a^{3}}{80 d}+\frac {5 a^{3} B \sin \left (d x +c \right )}{8 d}\) | \(472\) |
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Time = 0.30 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.80 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} + 18 \, B a^{2} b + 6 \, {\left (3 \, A + 4 \, C\right )} a b^{2} + 8 \, B b^{3}\right )} d x + {\left (40 \, A a^{3} \cos \left (d x + c\right )^{5} + 48 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )^{4} + 128 \, B a^{3} + 96 \, {\left (4 \, A + 5 \, C\right )} a^{2} b + 480 \, B a b^{2} + 80 \, {\left (2 \, A + 3 \, C\right )} b^{3} + 10 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} + 18 \, B a^{2} b + 18 \, A a b^{2}\right )} \cos \left (d x + c\right )^{3} + 16 \, {\left (4 \, B a^{3} + 3 \, {\left (4 \, A + 5 \, C\right )} a^{2} b + 15 \, B a b^{2} + 5 \, A b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} + 18 \, B a^{2} b + 6 \, {\left (3 \, A + 4 \, C\right )} a b^{2} + 8 \, B b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \]
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Timed out. \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Time = 0.25 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.12 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 64 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{3} - 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 192 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{2} b - 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} b + 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} b - 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a b^{2} + 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a b^{2} - 720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a b^{2} + 320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A b^{3} - 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B b^{3} - 960 \, C b^{3} \sin \left (d x + c\right )}{960 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1307 vs. \(2 (306) = 612\).
Time = 0.37 (sec) , antiderivative size = 1307, normalized size of antiderivative = 4.08 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
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Time = 18.72 (sec) , antiderivative size = 471, normalized size of antiderivative = 1.47 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {5\,A\,a^3\,x}{16}+\frac {B\,b^3\,x}{2}+\frac {3\,C\,a^3\,x}{8}+\frac {9\,A\,a\,b^2\,x}{8}+\frac {9\,B\,a^2\,b\,x}{8}+\frac {3\,C\,a\,b^2\,x}{2}+\frac {3\,A\,b^3\,\sin \left (c+d\,x\right )}{4\,d}+\frac {5\,B\,a^3\,\sin \left (c+d\,x\right )}{8\,d}+\frac {C\,b^3\,\sin \left (c+d\,x\right )}{d}+\frac {15\,A\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{64\,d}+\frac {3\,A\,a^3\,\sin \left (4\,c+4\,d\,x\right )}{64\,d}+\frac {A\,a^3\,\sin \left (6\,c+6\,d\,x\right )}{192\,d}+\frac {A\,b^3\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {5\,B\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{48\,d}+\frac {B\,a^3\,\sin \left (5\,c+5\,d\,x\right )}{80\,d}+\frac {B\,b^3\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,a^3\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {3\,A\,a\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {5\,A\,a^2\,b\,\sin \left (3\,c+3\,d\,x\right )}{16\,d}+\frac {3\,A\,a\,b^2\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {3\,A\,a^2\,b\,\sin \left (5\,c+5\,d\,x\right )}{80\,d}+\frac {3\,B\,a^2\,b\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,a\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{4\,d}+\frac {3\,B\,a^2\,b\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {3\,C\,a\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,a^2\,b\,\sin \left (3\,c+3\,d\,x\right )}{4\,d}+\frac {15\,A\,a^2\,b\,\sin \left (c+d\,x\right )}{8\,d}+\frac {9\,B\,a\,b^2\,\sin \left (c+d\,x\right )}{4\,d}+\frac {9\,C\,a^2\,b\,\sin \left (c+d\,x\right )}{4\,d} \]
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