Optimal. Leaf size=23 \[ x^3 \sin (x)+3 x^2 \cos (x)-6 x \sin (x)-6 \cos (x) \]
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Rubi [A] time = 0.0355037, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3296, 2638} \[ x^3 \sin (x)+3 x^2 \cos (x)-6 x \sin (x)-6 \cos (x) \]
Antiderivative was successfully verified.
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Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int x^3 \cos (x) \, dx &=x^3 \sin (x)-3 \int x^2 \sin (x) \, dx\\ &=3 x^2 \cos (x)+x^3 \sin (x)-6 \int x \cos (x) \, dx\\ &=3 x^2 \cos (x)-6 x \sin (x)+x^3 \sin (x)+6 \int \sin (x) \, dx\\ &=-6 \cos (x)+3 x^2 \cos (x)-6 x \sin (x)+x^3 \sin (x)\\ \end{align*}
Mathematica [A] time = 0.0129928, size = 19, normalized size = 0.83 \[ x \left (x^2-6\right ) \sin (x)+3 \left (x^2-2\right ) \cos (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 24, normalized size = 1. \begin{align*} -6\,\cos \left ( x \right ) +3\,{x}^{2}\cos \left ( x \right ) -6\,x\sin \left ( x \right ) +{x}^{3}\sin \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.952307, size = 27, normalized size = 1.17 \begin{align*} 3 \,{\left (x^{2} - 2\right )} \cos \left (x\right ) +{\left (x^{3} - 6 \, x\right )} \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.450515, size = 55, normalized size = 2.39 \begin{align*} 3 \,{\left (x^{2} - 2\right )} \cos \left (x\right ) +{\left (x^{3} - 6 \, x\right )} \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.553506, size = 26, normalized size = 1.13 \begin{align*} x^{3} \sin{\left (x \right )} + 3 x^{2} \cos{\left (x \right )} - 6 x \sin{\left (x \right )} - 6 \cos{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06311, size = 27, normalized size = 1.17 \begin{align*} 3 \,{\left (x^{2} - 2\right )} \cos \left (x\right ) +{\left (x^{3} - 6 \, x\right )} \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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