Integrand size = 19, antiderivative size = 71 \[ \int \cos (c+d x) (a+b \cos (c+d x))^2 \, dx=a b x+\frac {2 \left (a^2+b^2\right ) \sin (c+d x)}{3 d}+\frac {a b \cos (c+d x) \sin (c+d x)}{3 d}+\frac {(a+b \cos (c+d x))^2 \sin (c+d x)}{3 d} \]
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Time = 0.06 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2832, 2813} \[ \int \cos (c+d x) (a+b \cos (c+d x))^2 \, dx=\frac {2 \left (a^2+b^2\right ) \sin (c+d x)}{3 d}+\frac {\sin (c+d x) (a+b \cos (c+d x))^2}{3 d}+\frac {a b \sin (c+d x) \cos (c+d x)}{3 d}+a b x \]
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Rule 2813
Rule 2832
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{3} \int (2 b+2 a \cos (c+d x)) (a+b \cos (c+d x)) \, dx \\ & = a b x+\frac {2 \left (a^2+b^2\right ) \sin (c+d x)}{3 d}+\frac {a b \cos (c+d x) \sin (c+d x)}{3 d}+\frac {(a+b \cos (c+d x))^2 \sin (c+d x)}{3 d} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.83 \[ \int \cos (c+d x) (a+b \cos (c+d x))^2 \, dx=\frac {3 \left (4 a^2+3 b^2\right ) \sin (c+d x)+b (12 a (c+d x)+6 a \sin (2 (c+d x))+b \sin (3 (c+d x)))}{12 d} \]
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Time = 1.94 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.86
method | result | size |
parallelrisch | \(\frac {12 a b x d +b^{2} \sin \left (3 d x +3 c \right )+6 a b \sin \left (2 d x +2 c \right )+12 a^{2} \sin \left (d x +c \right )+9 \sin \left (d x +c \right ) b^{2}}{12 d}\) | \(61\) |
derivativedivides | \(\frac {a^{2} \sin \left (d x +c \right )+2 a b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {b^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(63\) |
default | \(\frac {a^{2} \sin \left (d x +c \right )+2 a b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {b^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(63\) |
risch | \(a b x +\frac {a^{2} \sin \left (d x +c \right )}{d}+\frac {3 b^{2} \sin \left (d x +c \right )}{4 d}+\frac {\sin \left (3 d x +3 c \right ) b^{2}}{12 d}+\frac {a b \sin \left (2 d x +2 c \right )}{2 d}\) | \(66\) |
parts | \(\frac {b^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {a^{2} \sin \left (d x +c \right )}{d}+\frac {2 a b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(68\) |
norman | \(\frac {a b x +a b x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4 \left (3 a^{2}+b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {2 \left (a^{2}-a b +b^{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 \left (a^{2}+a b +b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+3 a b x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a b x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}\) | \(145\) |
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Time = 0.26 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.73 \[ \int \cos (c+d x) (a+b \cos (c+d x))^2 \, dx=\frac {3 \, a b d x + {\left (b^{2} \cos \left (d x + c\right )^{2} + 3 \, a b \cos \left (d x + c\right ) + 3 \, a^{2} + 2 \, b^{2}\right )} \sin \left (d x + c\right )}{3 \, d} \]
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Time = 0.12 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.51 \[ \int \cos (c+d x) (a+b \cos (c+d x))^2 \, dx=\begin {cases} \frac {a^{2} \sin {\left (c + d x \right )}}{d} + a b x \sin ^{2}{\left (c + d x \right )} + a b x \cos ^{2}{\left (c + d x \right )} + \frac {a b \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {2 b^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {b^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \cos {\left (c \right )}\right )^{2} \cos {\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.85 \[ \int \cos (c+d x) (a+b \cos (c+d x))^2 \, dx=\frac {3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a b - 2 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} b^{2} + 6 \, a^{2} \sin \left (d x + c\right )}{6 \, d} \]
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Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.85 \[ \int \cos (c+d x) (a+b \cos (c+d x))^2 \, dx=a b x + \frac {b^{2} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {a b \sin \left (2 \, d x + 2 \, c\right )}{2 \, d} + \frac {{\left (4 \, a^{2} + 3 \, b^{2}\right )} \sin \left (d x + c\right )}{4 \, d} \]
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Time = 14.69 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.01 \[ \int \cos (c+d x) (a+b \cos (c+d x))^2 \, dx=\frac {a^2\,\sin \left (c+d\,x\right )}{d}+\frac {2\,b^2\,\sin \left (c+d\,x\right )}{3\,d}+a\,b\,x+\frac {b^2\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d}+\frac {a\,b\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{d} \]
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