Integrand size = 10, antiderivative size = 44 \[ \int x^2 \cos (x) \sin ^2(x) \, dx=\frac {4}{9} x \cos (x)-\frac {4 \sin (x)}{9}+\frac {2}{9} x \cos (x) \sin ^2(x)-\frac {2 \sin ^3(x)}{27}+\frac {1}{3} x^2 \sin ^3(x) \]
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Time = 0.03 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3524, 3391, 3377, 2717} \[ \int x^2 \cos (x) \sin ^2(x) \, dx=\frac {1}{3} x^2 \sin ^3(x)-\frac {2 \sin ^3(x)}{27}-\frac {4 \sin (x)}{9}+\frac {4}{9} x \cos (x)+\frac {2}{9} x \sin ^2(x) \cos (x) \]
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Rule 2717
Rule 3377
Rule 3391
Rule 3524
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^2 \sin ^3(x)-\frac {2}{3} \int x \sin ^3(x) \, dx \\ & = \frac {2}{9} x \cos (x) \sin ^2(x)-\frac {2 \sin ^3(x)}{27}+\frac {1}{3} x^2 \sin ^3(x)-\frac {4}{9} \int x \sin (x) \, dx \\ & = \frac {4}{9} x \cos (x)+\frac {2}{9} x \cos (x) \sin ^2(x)-\frac {2 \sin ^3(x)}{27}+\frac {1}{3} x^2 \sin ^3(x)-\frac {4}{9} \int \cos (x) \, dx \\ & = \frac {4}{9} x \cos (x)-\frac {4 \sin (x)}{9}+\frac {2}{9} x \cos (x) \sin ^2(x)-\frac {2 \sin ^3(x)}{27}+\frac {1}{3} x^2 \sin ^3(x) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.89 \[ \int x^2 \cos (x) \sin ^2(x) \, dx=\frac {1}{54} \left (27 x \cos (x)-3 x \cos (3 x)+\left (-26+9 x^2+\left (2-9 x^2\right ) \cos (2 x)\right ) \sin (x)\right ) \]
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Time = 0.45 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.73
method | result | size |
default | \(\frac {x^{2} \left (\sin ^{3}\left (x \right )\right )}{3}+\frac {2 x \left (2+\sin ^{2}\left (x \right )\right ) \cos \left (x \right )}{9}-\frac {2 \left (\sin ^{3}\left (x \right )\right )}{27}-\frac {4 \sin \left (x \right )}{9}\) | \(32\) |
risch | \(\frac {x \cos \left (x \right )}{2}+\frac {\left (x^{2}-2\right ) \sin \left (x \right )}{4}-\frac {x \cos \left (3 x \right )}{18}-\frac {\left (9 x^{2}-2\right ) \sin \left (3 x \right )}{108}\) | \(36\) |
parallelrisch | \(\frac {x \cos \left (x \right )}{2}+\frac {x^{2} \sin \left (x \right )}{4}-\frac {\sin \left (x \right )}{2}-\frac {x \cos \left (3 x \right )}{18}-\frac {x^{2} \sin \left (3 x \right )}{12}+\frac {\sin \left (3 x \right )}{54}\) | \(40\) |
norman | \(\frac {\frac {4 x}{9}-\frac {64 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{27}-\frac {8 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{9}+\frac {4 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{3}-\frac {4 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{3}-\frac {4 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{9}+\frac {8 \left (\tan ^{3}\left (\frac {x}{2}\right )\right ) x^{2}}{3}-\frac {8 \tan \left (\frac {x}{2}\right )}{9}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{3}}\) | \(76\) |
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Time = 0.24 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.82 \[ \int x^2 \cos (x) \sin ^2(x) \, dx=-\frac {2}{9} \, x \cos \left (x\right )^{3} + \frac {2}{3} \, x \cos \left (x\right ) - \frac {1}{27} \, {\left ({\left (9 \, x^{2} - 2\right )} \cos \left (x\right )^{2} - 9 \, x^{2} + 14\right )} \sin \left (x\right ) \]
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Time = 0.18 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.20 \[ \int x^2 \cos (x) \sin ^2(x) \, dx=\frac {x^{2} \sin ^{3}{\left (x \right )}}{3} + \frac {2 x \sin ^{2}{\left (x \right )} \cos {\left (x \right )}}{3} + \frac {4 x \cos ^{3}{\left (x \right )}}{9} - \frac {14 \sin ^{3}{\left (x \right )}}{27} - \frac {4 \sin {\left (x \right )} \cos ^{2}{\left (x \right )}}{9} \]
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Time = 0.20 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.80 \[ \int x^2 \cos (x) \sin ^2(x) \, dx=-\frac {1}{18} \, x \cos \left (3 \, x\right ) + \frac {1}{2} \, x \cos \left (x\right ) - \frac {1}{108} \, {\left (9 \, x^{2} - 2\right )} \sin \left (3 \, x\right ) + \frac {1}{4} \, {\left (x^{2} - 2\right )} \sin \left (x\right ) \]
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Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.80 \[ \int x^2 \cos (x) \sin ^2(x) \, dx=-\frac {1}{18} \, x \cos \left (3 \, x\right ) + \frac {1}{2} \, x \cos \left (x\right ) - \frac {1}{108} \, {\left (9 \, x^{2} - 2\right )} \sin \left (3 \, x\right ) + \frac {1}{4} \, {\left (x^{2} - 2\right )} \sin \left (x\right ) \]
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Time = 0.10 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.91 \[ \int x^2 \cos (x) \sin ^2(x) \, dx=\frac {x^2\,{\sin \left (x\right )}^3}{3}+\frac {4\,x\,{\cos \left (x\right )}^3}{9}+\frac {2\,x\,\cos \left (x\right )\,{\sin \left (x\right )}^2}{3}-\frac {4\,{\cos \left (x\right )}^2\,\sin \left (x\right )}{9}-\frac {14\,{\sin \left (x\right )}^3}{27} \]
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