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 = 0.04, antiderivative size = 23, normalized size of antiderivative = 1.00, 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 2638
Rule 3296
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.40, size = 20, normalized size = 0.87 \[ 3 \, {\left (x^{2} - 2\right )} \cos \relax (x) + {\left (x^{3} - 6 \, x\right )} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.01, size = 20, normalized size = 0.87 \[ 3 \, {\left (x^{2} - 2\right )} \cos \relax (x) + {\left (x^{3} - 6 \, x\right )} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 24, normalized size = 1.04 \[ x^{3} \sin \relax (x )+3 x^{2} \cos \relax (x )-6 x \sin \relax (x )-6 \cos \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 20, normalized size = 0.87 \[ 3 \, {\left (x^{2} - 2\right )} \cos \relax (x) + {\left (x^{3} - 6 \, x\right )} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 24, normalized size = 1.04 \[ \cos \relax (x)\,\left (3\,x^2-6\right )-\sin \relax (x)\,\left (6\,x-x^3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.64, size = 26, normalized size = 1.13 \[ x^{3} \sin {\relax (x )} + 3 x^{2} \cos {\relax (x )} - 6 x \sin {\relax (x )} - 6 \cos {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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