Integrand size = 13, antiderivative size = 21 \[ \int \cos ^3(1+x) \sin ^3(1+x) \, dx=\frac {1}{4} \sin ^4(1+x)-\frac {1}{6} \sin ^6(1+x) \]
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Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2644, 14} \[ \int \cos ^3(1+x) \sin ^3(1+x) \, dx=\frac {1}{4} \sin ^4(x+1)-\frac {1}{6} \sin ^6(x+1) \]
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Rule 14
Rule 2644
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int x^3 \left (1-x^2\right ) \, dx,x,\sin (1+x)\right ) \\ & = \text {Subst}\left (\int \left (x^3-x^5\right ) \, dx,x,\sin (1+x)\right ) \\ & = \frac {1}{4} \sin ^4(1+x)-\frac {1}{6} \sin ^6(1+x) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \cos ^3(1+x) \sin ^3(1+x) \, dx=\frac {1}{8} \left (-\frac {3}{8} \cos (2 (1+x))+\frac {1}{24} \cos (6 (1+x))\right ) \]
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Time = 0.36 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(\frac {\sin \left (x +1\right )^{4}}{4}-\frac {\sin \left (x +1\right )^{6}}{6}\) | \(18\) |
default | \(\frac {\sin \left (x +1\right )^{4}}{4}-\frac {\sin \left (x +1\right )^{6}}{6}\) | \(18\) |
risch | \(\frac {\cos \left (6 x +6\right )}{192}-\frac {3 \cos \left (2 x +2\right )}{64}\) | \(18\) |
parallelrisch | \(\frac {7}{40}+\frac {\cos \left (6 x +6\right )}{192}-\frac {3 \cos \left (2 x +2\right )}{64}\) | \(19\) |
norman | \(\frac {4 \tan \left (\frac {x}{2}+\frac {1}{2}\right )^{4}+4 \tan \left (\frac {x}{2}+\frac {1}{2}\right )^{8}-\frac {8 \tan \left (\frac {x}{2}+\frac {1}{2}\right )^{6}}{3}}{\left (1+\tan \left (\frac {x}{2}+\frac {1}{2}\right )^{2}\right )^{6}}\) | \(45\) |
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Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \cos ^3(1+x) \sin ^3(1+x) \, dx=\frac {1}{6} \, \cos \left (x + 1\right )^{6} - \frac {1}{4} \, \cos \left (x + 1\right )^{4} \]
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Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \cos ^3(1+x) \sin ^3(1+x) \, dx=- \frac {\sin ^{2}{\left (x + 1 \right )} \cos ^{4}{\left (x + 1 \right )}}{4} - \frac {\cos ^{6}{\left (x + 1 \right )}}{12} \]
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Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \cos ^3(1+x) \sin ^3(1+x) \, dx=-\frac {1}{6} \, \sin \left (x + 1\right )^{6} + \frac {1}{4} \, \sin \left (x + 1\right )^{4} \]
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Time = 0.29 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \cos ^3(1+x) \sin ^3(1+x) \, dx=-\frac {1}{6} \, \sin \left (x + 1\right )^{6} + \frac {1}{4} \, \sin \left (x + 1\right )^{4} \]
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Time = 0.09 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int \cos ^3(1+x) \sin ^3(1+x) \, dx=-\frac {{\sin \left (x+1\right )}^4\,\left (2\,{\sin \left (x+1\right )}^2-3\right )}{12} \]
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