Integrand size = 29, antiderivative size = 129 \[ \int \cos (c+d x) (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\frac {1}{8} a^2 (8 A+7 B) x+\frac {a^2 (8 A+7 B) \sin (c+d x)}{6 d}+\frac {a^2 (8 A+7 B) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {(4 A-B) (a+a \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac {B (a+a \cos (c+d x))^3 \sin (c+d x)}{4 a d} \]
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Time = 0.20 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {3047, 3102, 2830, 2723} \[ \int \cos (c+d x) (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\frac {a^2 (8 A+7 B) \sin (c+d x)}{6 d}+\frac {a^2 (8 A+7 B) \sin (c+d x) \cos (c+d x)}{24 d}+\frac {1}{8} a^2 x (8 A+7 B)+\frac {(4 A-B) \sin (c+d x) (a \cos (c+d x)+a)^2}{12 d}+\frac {B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 a d} \]
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Rule 2723
Rule 2830
Rule 3047
Rule 3102
Rubi steps \begin{align*} \text {integral}& = \int (a+a \cos (c+d x))^2 \left (A \cos (c+d x)+B \cos ^2(c+d x)\right ) \, dx \\ & = \frac {B (a+a \cos (c+d x))^3 \sin (c+d x)}{4 a d}+\frac {\int (a+a \cos (c+d x))^2 (3 a B+a (4 A-B) \cos (c+d x)) \, dx}{4 a} \\ & = \frac {(4 A-B) (a+a \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac {B (a+a \cos (c+d x))^3 \sin (c+d x)}{4 a d}+\frac {1}{12} (8 A+7 B) \int (a+a \cos (c+d x))^2 \, dx \\ & = \frac {1}{8} a^2 (8 A+7 B) x+\frac {a^2 (8 A+7 B) \sin (c+d x)}{6 d}+\frac {a^2 (8 A+7 B) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {(4 A-B) (a+a \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac {B (a+a \cos (c+d x))^3 \sin (c+d x)}{4 a d} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.67 \[ \int \cos (c+d x) (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\frac {a^2 (84 B c+96 A d x+84 B d x+24 (7 A+6 B) \sin (c+d x)+48 (A+B) \sin (2 (c+d x))+8 A \sin (3 (c+d x))+16 B \sin (3 (c+d x))+3 B \sin (4 (c+d x)))}{96 d} \]
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Time = 2.84 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.57
method | result | size |
parallelrisch | \(\frac {\left (\frac {\left (A +B \right ) \sin \left (2 d x +2 c \right )}{2}+\frac {\left (\frac {A}{2}+B \right ) \sin \left (3 d x +3 c \right )}{6}+\frac {\sin \left (4 d x +4 c \right ) B}{32}+\frac {\left (\frac {7 A}{2}+3 B \right ) \sin \left (d x +c \right )}{2}+d x \left (A +\frac {7 B}{8}\right )\right ) a^{2}}{d}\) | \(74\) |
parts | \(\frac {\left (A \,a^{2}+2 B \,a^{2}\right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (2 A \,a^{2}+B \,a^{2}\right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\sin \left (d x +c \right ) A \,a^{2}}{d}+\frac {B \,a^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(128\) |
risch | \(a^{2} x A +\frac {7 a^{2} B x}{8}+\frac {7 \sin \left (d x +c \right ) A \,a^{2}}{4 d}+\frac {3 \sin \left (d x +c \right ) B \,a^{2}}{2 d}+\frac {\sin \left (4 d x +4 c \right ) B \,a^{2}}{32 d}+\frac {\sin \left (3 d x +3 c \right ) A \,a^{2}}{12 d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{2}}{6 d}+\frac {\sin \left (2 d x +2 c \right ) A \,a^{2}}{2 d}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{2}}{2 d}\) | \(135\) |
derivativedivides | \(\frac {\frac {A \,a^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+B \,a^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 A \,a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {2 B \,a^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+A \,a^{2} \sin \left (d x +c \right )+B \,a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(154\) |
default | \(\frac {\frac {A \,a^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+B \,a^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 A \,a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {2 B \,a^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+A \,a^{2} \sin \left (d x +c \right )+B \,a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(154\) |
norman | \(\frac {\frac {a^{2} \left (8 A +7 B \right ) x}{8}+\frac {11 a^{2} \left (8 A +7 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {a^{2} \left (8 A +7 B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {a^{2} \left (8 A +7 B \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {3 a^{2} \left (8 A +7 B \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {a^{2} \left (8 A +7 B \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {a^{2} \left (8 A +7 B \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {a^{2} \left (24 A +25 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {a^{2} \left (136 A +83 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}\) | \(229\) |
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Time = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.70 \[ \int \cos (c+d x) (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\frac {3 \, {\left (8 \, A + 7 \, B\right )} a^{2} d x + {\left (6 \, B a^{2} \cos \left (d x + c\right )^{3} + 8 \, {\left (A + 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} + 3 \, {\left (8 \, A + 7 \, B\right )} a^{2} \cos \left (d x + c\right ) + 8 \, {\left (5 \, A + 4 \, B\right )} a^{2}\right )} \sin \left (d x + c\right )}{24 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (112) = 224\).
Time = 0.19 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.62 \[ \int \cos (c+d x) (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\begin {cases} A a^{2} x \sin ^{2}{\left (c + d x \right )} + A a^{2} x \cos ^{2}{\left (c + d x \right )} + \frac {2 A a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {A a^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {A a^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {A a^{2} \sin {\left (c + d x \right )}}{d} + \frac {3 B a^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 B a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {B a^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {3 B a^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {B a^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {3 B a^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {4 B a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {5 B a^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {2 B a^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {B a^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (A + B \cos {\left (c \right )}\right ) \left (a \cos {\left (c \right )} + a\right )^{2} \cos {\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.25 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.12 \[ \int \cos (c+d x) (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=-\frac {32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} - 48 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} + 64 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{2} - 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} - 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} - 96 \, A a^{2} \sin \left (d x + c\right )}{96 \, d} \]
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Time = 0.32 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.85 \[ \int \cos (c+d x) (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\frac {B a^{2} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {1}{8} \, {\left (8 \, A a^{2} + 7 \, B a^{2}\right )} x + \frac {{\left (A a^{2} + 2 \, B a^{2}\right )} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {{\left (A a^{2} + B a^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{2 \, d} + \frac {{\left (7 \, A a^{2} + 6 \, B a^{2}\right )} \sin \left (d x + c\right )}{4 \, d} \]
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Time = 0.19 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.04 \[ \int \cos (c+d x) (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=A\,a^2\,x+\frac {7\,B\,a^2\,x}{8}+\frac {7\,A\,a^2\,\sin \left (c+d\,x\right )}{4\,d}+\frac {3\,B\,a^2\,\sin \left (c+d\,x\right )}{2\,d}+\frac {A\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {A\,a^2\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {B\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {B\,a^2\,\sin \left (3\,c+3\,d\,x\right )}{6\,d}+\frac {B\,a^2\,\sin \left (4\,c+4\,d\,x\right )}{32\,d} \]
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