Optimal. Leaf size=33 \[ \frac{2}{3} x \sin (x)+\frac{\cos ^3(x)}{9}+\frac{2 \cos (x)}{3}+\frac{1}{3} x \sin (x) \cos ^2(x) \]
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Rubi [A] time = 0.0230636, 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, 2638} \[ \frac{2}{3} x \sin (x)+\frac{\cos ^3(x)}{9}+\frac{2 \cos (x)}{3}+\frac{1}{3} x \sin (x) \cos ^2(x) \]
Antiderivative was successfully verified.
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Rule 3310
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int x \cos ^3(x) \, dx &=\frac{\cos ^3(x)}{9}+\frac{1}{3} x \cos ^2(x) \sin (x)+\frac{2}{3} \int x \cos (x) \, dx\\ &=\frac{\cos ^3(x)}{9}+\frac{2}{3} x \sin (x)+\frac{1}{3} x \cos ^2(x) \sin (x)-\frac{2}{3} \int \sin (x) \, dx\\ &=\frac{2 \cos (x)}{3}+\frac{\cos ^3(x)}{9}+\frac{2}{3} x \sin (x)+\frac{1}{3} x \cos ^2(x) \sin (x)\\ \end{align*}
Mathematica [A] time = 0.00813, size = 31, normalized size = 0.94 \[ \frac{3}{4} x \sin (x)+\frac{1}{12} x \sin (3 x)+\frac{3 \cos (x)}{4}+\frac{1}{36} \cos (3 x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 23, normalized size = 0.7 \begin{align*}{\frac{x \left ( 2+ \left ( \cos \left ( x \right ) \right ) ^{2} \right ) \sin \left ( x \right ) }{3}}+{\frac{ \left ( \cos \left ( x \right ) \right ) ^{3}}{9}}+{\frac{2\,\cos \left ( x \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.951987, size = 31, normalized size = 0.94 \begin{align*} \frac{1}{12} \, x \sin \left (3 \, x\right ) + \frac{3}{4} \, x \sin \left (x\right ) + \frac{1}{36} \, \cos \left (3 \, x\right ) + \frac{3}{4} \, \cos \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77494, size = 80, normalized size = 2.42 \begin{align*} \frac{1}{9} \, \cos \left (x\right )^{3} + \frac{1}{3} \,{\left (x \cos \left (x\right )^{2} + 2 \, x\right )} \sin \left (x\right ) + \frac{2}{3} \, \cos \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.60164, size = 39, normalized size = 1.18 \begin{align*} \frac{2 x \sin ^{3}{\left (x \right )}}{3} + x \sin{\left (x \right )} \cos ^{2}{\left (x \right )} + \frac{2 \sin ^{2}{\left (x \right )} \cos{\left (x \right )}}{3} + \frac{7 \cos ^{3}{\left (x \right )}}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06452, size = 31, normalized size = 0.94 \begin{align*} \frac{1}{12} \, x \sin \left (3 \, x\right ) + \frac{3}{4} \, x \sin \left (x\right ) + \frac{1}{36} \, \cos \left (3 \, x\right ) + \frac{3}{4} \, \cos \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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