Integrand size = 21, antiderivative size = 47 \[ \int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx=\frac {1}{2} a (2 A+B) x+\frac {a (A+B) \sin (c+d x)}{d}+\frac {a B \cos (c+d x) \sin (c+d x)}{2 d} \]
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Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2813} \[ \int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx=\frac {a (A+B) \sin (c+d x)}{d}+\frac {1}{2} a x (2 A+B)+\frac {a B \sin (c+d x) \cos (c+d x)}{2 d} \]
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Rule 2813
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} a (2 A+B) x+\frac {a (A+B) \sin (c+d x)}{d}+\frac {a B \cos (c+d x) \sin (c+d x)}{2 d} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.94 \[ \int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx=\frac {a (2 B c+4 A d x+2 B d x+4 (A+B) \sin (c+d x)+B \sin (2 (c+d x)))}{4 d} \]
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Time = 0.95 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.79
method | result | size |
parallelrisch | \(\frac {\left (\frac {\sin \left (2 d x +2 c \right ) B}{4}+\left (A +B \right ) \sin \left (d x +c \right )+d x \left (A +\frac {B}{2}\right )\right ) a}{d}\) | \(37\) |
risch | \(a x A +\frac {a B x}{2}+\frac {\sin \left (d x +c \right ) a A}{d}+\frac {a B \sin \left (d x +c \right )}{d}+\frac {\sin \left (2 d x +2 c \right ) B a}{4 d}\) | \(51\) |
parts | \(a x A +\frac {\left (a A +B a \right ) \sin \left (d x +c \right )}{d}+\frac {B a \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(51\) |
derivativedivides | \(\frac {B a \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a A \sin \left (d x +c \right )+B a \sin \left (d x +c \right )+a A \left (d x +c \right )}{d}\) | \(57\) |
default | \(\frac {B a \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a A \sin \left (d x +c \right )+B a \sin \left (d x +c \right )+a A \left (d x +c \right )}{d}\) | \(57\) |
norman | \(\frac {\frac {a \left (2 A +B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+a \left (2 A +B \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {a \left (2 A +3 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {a \left (2 A +B \right ) x}{2}+\frac {a \left (2 A +B \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}\) | \(108\) |
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Time = 0.37 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.81 \[ \int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx=\frac {{\left (2 \, A + B\right )} a d x + {\left (B a \cos \left (d x + c\right ) + 2 \, {\left (A + B\right )} a\right )} \sin \left (d x + c\right )}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (42) = 84\).
Time = 0.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.00 \[ \int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx=\begin {cases} A a x + \frac {A a \sin {\left (c + d x \right )}}{d} + \frac {B a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {B a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {B a \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {B a \sin {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A + B \cos {\left (c \right )}\right ) \left (a \cos {\left (c \right )} + a\right ) & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.17 \[ \int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx=\frac {4 \, {\left (d x + c\right )} A a + {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a + 4 \, A a \sin \left (d x + c\right ) + 4 \, B a \sin \left (d x + c\right )}{4 \, d} \]
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Time = 0.29 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.96 \[ \int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx=\frac {1}{2} \, {\left (2 \, A a + B a\right )} x + \frac {B a \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {{\left (A a + B a\right )} \sin \left (d x + c\right )}{d} \]
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Time = 0.12 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.06 \[ \int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx=A\,a\,x+\frac {B\,a\,x}{2}+\frac {A\,a\,\sin \left (c+d\,x\right )}{d}+\frac {B\,a\,\sin \left (c+d\,x\right )}{d}+\frac {B\,a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d} \]
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