Optimal. Leaf size=44 \[ \frac{1}{3} x^2 \sin ^3(x)-\frac{2 \sin ^3(x)}{27}-\frac{4 \sin (x)}{9}+\frac{4}{9} x \cos (x)+\frac{2}{9} x \sin ^2(x) \cos (x) \]
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Rubi [A] time = 0.0420206, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3443, 3310, 3296, 2637} \[ \frac{1}{3} x^2 \sin ^3(x)-\frac{2 \sin ^3(x)}{27}-\frac{4 \sin (x)}{9}+\frac{4}{9} x \cos (x)+\frac{2}{9} x \sin ^2(x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 3443
Rule 3310
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int x^2 \cos (x) \sin ^2(x) \, dx &=\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*}
Mathematica [A] time = 0.121809, size = 39, normalized size = 0.89 \[ \frac{1}{54} \left (\sin (x) \left (9 x^2+\left (2-9 x^2\right ) \cos (2 x)-26\right )+27 x \cos (x)-3 x \cos (3 x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 32, normalized size = 0.7 \begin{align*}{\frac{{x}^{2} \left ( \sin \left ( x \right ) \right ) ^{3}}{3}}+{\frac{2\,x \left ( 2+ \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \cos \left ( x \right ) }{9}}-{\frac{2\, \left ( \sin \left ( x \right ) \right ) ^{3}}{27}}-{\frac{4\,\sin \left ( x \right ) }{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.95696, size = 47, normalized size = 1.07 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.25054, size = 111, normalized size = 2.52 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.20993, size = 53, normalized size = 1.2 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08224, size = 47, normalized size = 1.07 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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