Integrand size = 20, antiderivative size = 31 \[ \int \cos ^3(a+b x) \sin ^3(2 a+2 b x) \, dx=-\frac {8 \cos ^7(a+b x)}{7 b}+\frac {8 \cos ^9(a+b x)}{9 b} \]
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Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4372, 2645, 14} \[ \int \cos ^3(a+b x) \sin ^3(2 a+2 b x) \, dx=\frac {8 \cos ^9(a+b x)}{9 b}-\frac {8 \cos ^7(a+b x)}{7 b} \]
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Rule 14
Rule 2645
Rule 4372
Rubi steps \begin{align*} \text {integral}& = 8 \int \cos ^6(a+b x) \sin ^3(a+b x) \, dx \\ & = -\frac {8 \text {Subst}\left (\int x^6 \left (1-x^2\right ) \, dx,x,\cos (a+b x)\right )}{b} \\ & = -\frac {8 \text {Subst}\left (\int \left (x^6-x^8\right ) \, dx,x,\cos (a+b x)\right )}{b} \\ & = -\frac {8 \cos ^7(a+b x)}{7 b}+\frac {8 \cos ^9(a+b x)}{9 b} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \cos ^3(a+b x) \sin ^3(2 a+2 b x) \, dx=\frac {4 \cos ^7(a+b x) (-11+7 \cos (2 (a+b x)))}{63 b} \]
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Time = 1.61 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.58
method | result | size |
parallelrisch | \(\frac {4864-1890 \cos \left (x b +a \right )+135 \cos \left (7 x b +7 a \right )+35 \cos \left (9 x b +9 a \right )-840 \cos \left (3 x b +3 a \right )}{10080 b}\) | \(49\) |
default | \(-\frac {3 \cos \left (x b +a \right )}{16 b}-\frac {\cos \left (3 x b +3 a \right )}{12 b}+\frac {3 \cos \left (7 x b +7 a \right )}{224 b}+\frac {\cos \left (9 x b +9 a \right )}{288 b}\) | \(55\) |
risch | \(-\frac {3 \cos \left (x b +a \right )}{16 b}-\frac {\cos \left (3 x b +3 a \right )}{12 b}+\frac {3 \cos \left (7 x b +7 a \right )}{224 b}+\frac {\cos \left (9 x b +9 a \right )}{288 b}\) | \(55\) |
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Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \cos ^3(a+b x) \sin ^3(2 a+2 b x) \, dx=\frac {8 \, {\left (7 \, \cos \left (b x + a\right )^{9} - 9 \, \cos \left (b x + a\right )^{7}\right )}}{63 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (26) = 52\).
Time = 4.92 (sec) , antiderivative size = 284, normalized size of antiderivative = 9.16 \[ \int \cos ^3(a+b x) \sin ^3(2 a+2 b x) \, dx=\begin {cases} - \frac {94 \sin ^{3}{\left (a + b x \right )} \sin ^{3}{\left (2 a + 2 b x \right )}}{315 b} - \frac {32 \sin ^{3}{\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{105 b} - \frac {4 \sin ^{2}{\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{7 b} - \frac {64 \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{105 b} + \frac {13 \sin {\left (a + b x \right )} \sin ^{3}{\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )}}{105 b} + \frac {8 \sin {\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{35 b} - \frac {46 \sin ^{2}{\left (2 a + 2 b x \right )} \cos ^{3}{\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{105 b} - \frac {16 \cos ^{3}{\left (a + b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{63 b} & \text {for}\: b \neq 0 \\x \sin ^{3}{\left (2 a \right )} \cos ^{3}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.52 \[ \int \cos ^3(a+b x) \sin ^3(2 a+2 b x) \, dx=\frac {7 \, \cos \left (9 \, b x + 9 \, a\right ) + 27 \, \cos \left (7 \, b x + 7 \, a\right ) - 168 \, \cos \left (3 \, b x + 3 \, a\right ) - 378 \, \cos \left (b x + a\right )}{2016 \, b} \]
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Time = 0.53 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \cos ^3(a+b x) \sin ^3(2 a+2 b x) \, dx=\frac {8 \, {\left (7 \, \cos \left (b x + a\right )^{9} - 9 \, \cos \left (b x + a\right )^{7}\right )}}{63 \, b} \]
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Time = 19.94 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \cos ^3(a+b x) \sin ^3(2 a+2 b x) \, dx=-\frac {8\,\left (9\,{\cos \left (a+b\,x\right )}^7-7\,{\cos \left (a+b\,x\right )}^9\right )}{63\,b} \]
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