Integrand size = 31, antiderivative size = 241 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {1}{16} a^4 (49 A+44 B) x+\frac {a^4 (252 A+227 B) \sin (c+d x)}{35 d}+\frac {a^4 (49 A+44 B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^4 (301 A+276 B) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {a B \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(7 A+10 B) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac {7 (A+B) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{15 d}-\frac {a^4 (252 A+227 B) \sin ^3(c+d x)}{105 d} \]
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Time = 0.64 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {3055, 3047, 3102, 2827, 2715, 8, 2713} \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=-\frac {a^4 (252 A+227 B) \sin ^3(c+d x)}{105 d}+\frac {a^4 (252 A+227 B) \sin (c+d x)}{35 d}+\frac {a^4 (301 A+276 B) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac {7 (A+B) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{15 d}+\frac {a^4 (49 A+44 B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} a^4 x (49 A+44 B)+\frac {(7 A+10 B) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{42 d}+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d} \]
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Rule 8
Rule 2713
Rule 2715
Rule 2827
Rule 3047
Rule 3055
Rule 3102
Rubi steps \begin{align*} \text {integral}& = \frac {a B \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {1}{7} \int \cos ^2(c+d x) (a+a \cos (c+d x))^3 (a (7 A+3 B)+a (7 A+10 B) \cos (c+d x)) \, dx \\ & = \frac {a B \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(7 A+10 B) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac {1}{42} \int \cos ^2(c+d x) (a+a \cos (c+d x))^2 \left (3 a^2 (21 A+16 B)+98 a^2 (A+B) \cos (c+d x)\right ) \, dx \\ & = \frac {a B \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(7 A+10 B) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac {7 (A+B) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{210} \int \cos ^2(c+d x) (a+a \cos (c+d x)) \left (3 a^3 (203 A+178 B)+3 a^3 (301 A+276 B) \cos (c+d x)\right ) \, dx \\ & = \frac {a B \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(7 A+10 B) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac {7 (A+B) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{210} \int \cos ^2(c+d x) \left (3 a^4 (203 A+178 B)+\left (3 a^4 (203 A+178 B)+3 a^4 (301 A+276 B)\right ) \cos (c+d x)+3 a^4 (301 A+276 B) \cos ^2(c+d x)\right ) \, dx \\ & = \frac {a^4 (301 A+276 B) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {a B \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(7 A+10 B) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac {7 (A+B) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{840} \int \cos ^2(c+d x) \left (105 a^4 (49 A+44 B)+24 a^4 (252 A+227 B) \cos (c+d x)\right ) \, dx \\ & = \frac {a^4 (301 A+276 B) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {a B \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(7 A+10 B) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac {7 (A+B) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{8} \left (a^4 (49 A+44 B)\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{35} \left (a^4 (252 A+227 B)\right ) \int \cos ^3(c+d x) \, dx \\ & = \frac {a^4 (49 A+44 B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^4 (301 A+276 B) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {a B \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(7 A+10 B) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac {7 (A+B) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{16} \left (a^4 (49 A+44 B)\right ) \int 1 \, dx-\frac {\left (a^4 (252 A+227 B)\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{35 d} \\ & = \frac {1}{16} a^4 (49 A+44 B) x+\frac {a^4 (252 A+227 B) \sin (c+d x)}{35 d}+\frac {a^4 (49 A+44 B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^4 (301 A+276 B) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {a B \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(7 A+10 B) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac {7 (A+B) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{15 d}-\frac {a^4 (252 A+227 B) \sin ^3(c+d x)}{105 d} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.65 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {a^4 (18480 B c+20580 A d x+18480 B d x+105 (352 A+323 B) \sin (c+d x)+105 (127 A+124 B) \sin (2 (c+d x))+5040 A \sin (3 (c+d x))+5495 B \sin (3 (c+d x))+1575 A \sin (4 (c+d x))+2100 B \sin (4 (c+d x))+336 A \sin (5 (c+d x))+651 B \sin (5 (c+d x))+35 A \sin (6 (c+d x))+140 B \sin (6 (c+d x))+15 B \sin (7 (c+d x)))}{6720 d} \]
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Time = 5.70 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.54
method | result | size |
parallelrisch | \(\frac {15 \left (\frac {\left (127 A +124 B \right ) \sin \left (2 d x +2 c \right )}{15}+\frac {\left (16 A +\frac {157 B}{9}\right ) \sin \left (3 d x +3 c \right )}{5}+\left (A +\frac {4 B}{3}\right ) \sin \left (4 d x +4 c \right )+\frac {\left (16 A +31 B \right ) \sin \left (5 d x +5 c \right )}{75}+\frac {\left (A +4 B \right ) \sin \left (6 d x +6 c \right )}{45}+\frac {B \sin \left (7 d x +7 c \right )}{105}+\frac {\left (352 A +323 B \right ) \sin \left (d x +c \right )}{15}+\frac {196 \left (A +\frac {44 B}{49}\right ) x d}{15}\right ) a^{4}}{64 d}\) | \(131\) |
risch | \(\frac {49 a^{4} x A}{16}+\frac {11 a^{4} B x}{4}+\frac {11 \sin \left (d x +c \right ) a^{4} A}{2 d}+\frac {323 \sin \left (d x +c \right ) B \,a^{4}}{64 d}+\frac {B \,a^{4} \sin \left (7 d x +7 c \right )}{448 d}+\frac {\sin \left (6 d x +6 c \right ) a^{4} A}{192 d}+\frac {\sin \left (6 d x +6 c \right ) B \,a^{4}}{48 d}+\frac {\sin \left (5 d x +5 c \right ) a^{4} A}{20 d}+\frac {31 \sin \left (5 d x +5 c \right ) B \,a^{4}}{320 d}+\frac {15 \sin \left (4 d x +4 c \right ) a^{4} A}{64 d}+\frac {5 \sin \left (4 d x +4 c \right ) B \,a^{4}}{16 d}+\frac {3 \sin \left (3 d x +3 c \right ) a^{4} A}{4 d}+\frac {157 \sin \left (3 d x +3 c \right ) B \,a^{4}}{192 d}+\frac {127 \sin \left (2 d x +2 c \right ) a^{4} A}{64 d}+\frac {31 \sin \left (2 d x +2 c \right ) B \,a^{4}}{16 d}\) | \(244\) |
parts | \(\frac {\left (a^{4} A +4 B \,a^{4}\right ) \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {\left (4 a^{4} A +B \,a^{4}\right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (4 a^{4} A +6 B \,a^{4}\right ) \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}+\frac {\left (6 a^{4} A +4 B \,a^{4}\right ) \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {B \,a^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7 d}+\frac {a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(263\) |
derivativedivides | \(\frac {a^{4} A \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {B \,a^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}+\frac {4 a^{4} A \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+4 B \,a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+6 a^{4} A \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {6 B \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {4 a^{4} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 B \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {B \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(358\) |
default | \(\frac {a^{4} A \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {B \,a^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}+\frac {4 a^{4} A \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+4 B \,a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+6 a^{4} A \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {6 B \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {4 a^{4} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 B \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {B \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(358\) |
norman | \(\frac {\frac {a^{4} \left (49 A +44 B \right ) x}{16}+\frac {128 a^{4} \left (49 A +44 B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {283 a^{4} \left (49 A +44 B \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{120 d}+\frac {5 a^{4} \left (49 A +44 B \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {a^{4} \left (49 A +44 B \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {7 a^{4} \left (49 A +44 B \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {21 a^{4} \left (49 A +44 B \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {35 a^{4} \left (49 A +44 B \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {35 a^{4} \left (49 A +44 B \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {21 a^{4} \left (49 A +44 B \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {7 a^{4} \left (49 A +44 B \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {a^{4} \left (49 A +44 B \right ) x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {a^{4} \left (207 A +212 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {a^{4} \left (523 A +420 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {a^{4} \left (19157 A +18012 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{120 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}\) | \(379\) |
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Time = 0.31 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.62 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {105 \, {\left (49 \, A + 44 \, B\right )} a^{4} d x + {\left (240 \, B a^{4} \cos \left (d x + c\right )^{6} + 280 \, {\left (A + 4 \, B\right )} a^{4} \cos \left (d x + c\right )^{5} + 192 \, {\left (7 \, A + 12 \, B\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \, {\left (41 \, A + 44 \, B\right )} a^{4} \cos \left (d x + c\right )^{3} + 16 \, {\left (252 \, A + 227 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 105 \, {\left (49 \, A + 44 \, B\right )} a^{4} \cos \left (d x + c\right ) + 32 \, {\left (252 \, A + 227 \, B\right )} a^{4}\right )} \sin \left (d x + c\right )}{1680 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 960 vs. \(2 (226) = 452\).
Time = 0.59 (sec) , antiderivative size = 960, normalized size of antiderivative = 3.98 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\text {Too large to display} \]
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Time = 0.22 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.48 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {1792 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{4} - 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 8960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} + 1260 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 1680 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 192 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} B a^{4} + 2688 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{4} - 140 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 2240 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} + 840 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4}}{6720 \, d} \]
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Time = 0.34 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.80 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {B a^{4} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {1}{16} \, {\left (49 \, A a^{4} + 44 \, B a^{4}\right )} x + \frac {{\left (A a^{4} + 4 \, B a^{4}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {{\left (16 \, A a^{4} + 31 \, B a^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {5 \, {\left (3 \, A a^{4} + 4 \, B a^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (144 \, A a^{4} + 157 \, B a^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {{\left (127 \, A a^{4} + 124 \, B a^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (352 \, A a^{4} + 323 \, B a^{4}\right )} \sin \left (d x + c\right )}{64 \, d} \]
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Time = 1.79 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.46 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {\left (\frac {49\,A\,a^4}{8}+\frac {11\,B\,a^4}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (\frac {245\,A\,a^4}{6}+\frac {110\,B\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {13867\,A\,a^4}{120}+\frac {3113\,B\,a^4}{30}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {896\,A\,a^4}{5}+\frac {5632\,B\,a^4}{35}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {19157\,A\,a^4}{120}+\frac {1501\,B\,a^4}{10}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {523\,A\,a^4}{6}+70\,B\,a^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {207\,A\,a^4}{8}+\frac {53\,B\,a^4}{2}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {a^4\,\left (49\,A+44\,B\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{8\,d}+\frac {a^4\,\mathrm {atan}\left (\frac {a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (49\,A+44\,B\right )}{8\,\left (\frac {49\,A\,a^4}{8}+\frac {11\,B\,a^4}{2}\right )}\right )\,\left (49\,A+44\,B\right )}{8\,d} \]
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