Optimal. Leaf size=33 \[ \frac{\sin ^3(x)}{9}+\frac{2 \sin (x)}{3}-\frac{2}{3} x \cos (x)-\frac{1}{3} x \sin ^2(x) \cos (x) \]
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Rubi [A] time = 0.0216979, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3310, 3296, 2637} \[ \frac{\sin ^3(x)}{9}+\frac{2 \sin (x)}{3}-\frac{2}{3} x \cos (x)-\frac{1}{3} x \sin ^2(x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 3310
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int x \sin ^3(x) \, dx &=-\frac{1}{3} x \cos (x) \sin ^2(x)+\frac{\sin ^3(x)}{9}+\frac{2}{3} \int x \sin (x) \, dx\\ &=-\frac{2}{3} x \cos (x)-\frac{1}{3} x \cos (x) \sin ^2(x)+\frac{\sin ^3(x)}{9}+\frac{2}{3} \int \cos (x) \, dx\\ &=-\frac{2}{3} x \cos (x)+\frac{2 \sin (x)}{3}-\frac{1}{3} x \cos (x) \sin ^2(x)+\frac{\sin ^3(x)}{9}\\ \end{align*}
Mathematica [A] time = 0.0085734, size = 31, normalized size = 0.94 \[ \frac{3 \sin (x)}{4}-\frac{1}{36} \sin (3 x)-\frac{3}{4} x \cos (x)+\frac{1}{12} x \cos (3 x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 23, normalized size = 0.7 \begin{align*} -{\frac{x \left ( 2+ \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \cos \left ( x \right ) }{3}}+{\frac{ \left ( \sin \left ( x \right ) \right ) ^{3}}{9}}+{\frac{2\,\sin \left ( x \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.962111, size = 31, normalized size = 0.94 \begin{align*} \frac{1}{12} \, x \cos \left (3 \, x\right ) - \frac{3}{4} \, x \cos \left (x\right ) - \frac{1}{36} \, \sin \left (3 \, x\right ) + \frac{3}{4} \, \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82112, size = 74, normalized size = 2.24 \begin{align*} \frac{1}{3} \, x \cos \left (x\right )^{3} - x \cos \left (x\right ) - \frac{1}{9} \,{\left (\cos \left (x\right )^{2} - 7\right )} \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.603218, size = 39, normalized size = 1.18 \begin{align*} - x \sin ^{2}{\left (x \right )} \cos{\left (x \right )} - \frac{2 x \cos ^{3}{\left (x \right )}}{3} + \frac{7 \sin ^{3}{\left (x \right )}}{9} + \frac{2 \sin{\left (x \right )} \cos ^{2}{\left (x \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08484, size = 31, normalized size = 0.94 \begin{align*} \frac{1}{12} \, x \cos \left (3 \, x\right ) - \frac{3}{4} \, x \cos \left (x\right ) - \frac{1}{36} \, \sin \left (3 \, x\right ) + \frac{3}{4} \, \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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