Integrand size = 9, antiderivative size = 17 \[ \int \cos (x) \sin ^3(2 x) \, dx=-\frac {8}{5} \cos ^5(x)+\frac {8 \cos ^7(x)}{7} \]
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Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4372, 2645, 14} \[ \int \cos (x) \sin ^3(2 x) \, dx=\frac {8 \cos ^7(x)}{7}-\frac {8 \cos ^5(x)}{5} \]
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Rule 14
Rule 2645
Rule 4372
Rubi steps \begin{align*} \text {integral}& = 8 \int \cos ^4(x) \sin ^3(x) \, dx \\ & = -\left (8 \text {Subst}\left (\int x^4 \left (1-x^2\right ) \, dx,x,\cos (x)\right )\right ) \\ & = -\left (8 \text {Subst}\left (\int \left (x^4-x^6\right ) \, dx,x,\cos (x)\right )\right ) \\ & = -\frac {8}{5} \cos ^5(x)+\frac {8 \cos ^7(x)}{7} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.82 \[ \int \cos (x) \sin ^3(2 x) \, dx=-\frac {3 \cos (x)}{8}-\frac {1}{8} \cos (3 x)+\frac {1}{40} \cos (5 x)+\frac {1}{56} \cos (7 x) \]
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Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.41
method | result | size |
default | \(-\frac {3 \cos \left (x \right )}{8}-\frac {\cos \left (3 x \right )}{8}+\frac {\cos \left (5 x \right )}{40}+\frac {\cos \left (7 x \right )}{56}\) | \(24\) |
risch | \(-\frac {3 \cos \left (x \right )}{8}-\frac {\cos \left (3 x \right )}{8}+\frac {\cos \left (5 x \right )}{40}+\frac {\cos \left (7 x \right )}{56}\) | \(24\) |
parallelrisch | \(\frac {8}{21}-\frac {3 \cos \left (x \right )}{8}-\frac {\cos \left (3 x \right )}{8}+\frac {\cos \left (5 x \right )}{40}+\frac {\cos \left (7 x \right )}{56}\) | \(25\) |
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Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \cos (x) \sin ^3(2 x) \, dx=\frac {8}{7} \, \cos \left (x\right )^{7} - \frac {8}{5} \, \cos \left (x\right )^{5} \]
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (15) = 30\).
Time = 0.59 (sec) , antiderivative size = 65, normalized size of antiderivative = 3.82 \[ \int \cos (x) \sin ^3(2 x) \, dx=- \frac {9 \sin {\left (x \right )} \sin ^{3}{\left (2 x \right )}}{35} - \frac {8 \sin {\left (x \right )} \sin {\left (2 x \right )} \cos ^{2}{\left (2 x \right )}}{35} - \frac {22 \sin ^{2}{\left (2 x \right )} \cos {\left (x \right )} \cos {\left (2 x \right )}}{35} - \frac {16 \cos {\left (x \right )} \cos ^{3}{\left (2 x \right )}}{35} \]
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Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.35 \[ \int \cos (x) \sin ^3(2 x) \, dx=\frac {1}{56} \, \cos \left (7 \, x\right ) + \frac {1}{40} \, \cos \left (5 \, x\right ) - \frac {1}{8} \, \cos \left (3 \, x\right ) - \frac {3}{8} \, \cos \left (x\right ) \]
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Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \cos (x) \sin ^3(2 x) \, dx=\frac {8}{7} \, \cos \left (x\right )^{7} - \frac {8}{5} \, \cos \left (x\right )^{5} \]
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Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \cos (x) \sin ^3(2 x) \, dx=\frac {8\,{\cos \left (x\right )}^5\,\left (5\,{\cos \left (x\right )}^2-7\right )}{35} \]
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