Optimal. Leaf size=24 \[ 6 x \cos (x)-x^3 \cos (x)-6 \sin (x)+3 x^2 \sin (x) \]
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Rubi [A]
time = 0.02, antiderivative size = 24, 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 = {3377, 2717}
\begin {gather*} x^3 (-\cos (x))+3 x^2 \sin (x)-6 \sin (x)+6 x \cos (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 2717
Rule 3377
Rubi steps
\begin {align*} \int x^3 \sin (x) \, dx &=-x^3 \cos (x)+3 \int x^2 \cos (x) \, dx\\ &=-x^3 \cos (x)+3 x^2 \sin (x)-6 \int x \sin (x) \, dx\\ &=6 x \cos (x)-x^3 \cos (x)+3 x^2 \sin (x)-6 \int \cos (x) \, dx\\ &=6 x \cos (x)-x^3 \cos (x)-6 \sin (x)+3 x^2 \sin (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 20, normalized size = 0.83 \begin {gather*} -x \left (-6+x^2\right ) \cos (x)+3 \left (-2+x^2\right ) \sin (x) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.87, size = 24, normalized size = 1.00 \begin {gather*} 6 x \text {Cos}\left [x\right ]+3 x^2 \text {Sin}\left [x\right ]-x^3 \text {Cos}\left [x\right ]-6 \text {Sin}\left [x\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 25, normalized size = 1.04
method | result | size |
risch | \(\left (-x^{3}+6 x \right ) \cos \left (x \right )+3 \left (x^{2}-2\right ) \sin \left (x \right )\) | \(23\) |
default | \(6 x \cos \left (x \right )-x^{3} \cos \left (x \right )-6 \sin \left (x \right )+3 x^{2} \sin \left (x \right )\) | \(25\) |
meijerg | \(8 \sqrt {\pi }\, \left (\frac {x \left (-\frac {5 x^{2}}{2}+15\right ) \cos \left (x \right )}{20 \sqrt {\pi }}-\frac {\left (-\frac {15 x^{2}}{2}+15\right ) \sin \left (x \right )}{20 \sqrt {\pi }}\right )\) | \(36\) |
norman | \(\frac {x^{3} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+6 x -x^{3}-6 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+6 x^{2} \tan \left (\frac {x}{2}\right )-12 \tan \left (\frac {x}{2}\right )}{1+\tan ^{2}\left (\frac {x}{2}\right )}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 21, normalized size = 0.88 \begin {gather*} -{\left (x^{3} - 6 \, x\right )} \cos \left (x\right ) + 3 \, {\left (x^{2} - 2\right )} \sin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 21, normalized size = 0.88 \begin {gather*} -{\left (x^{3} - 6 \, x\right )} \cos \left (x\right ) + 3 \, {\left (x^{2} - 2\right )} \sin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 26, normalized size = 1.08 \begin {gather*} - x^{3} \cos {\left (x \right )} + 3 x^{2} \sin {\left (x \right )} + 6 x \cos {\left (x \right )} - 6 \sin {\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 22, normalized size = 0.92 \begin {gather*} \left (-x^{3}+6 x\right ) \cos x+\left (3 x^{2}-6\right ) \sin x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 23, normalized size = 0.96 \begin {gather*} \cos \left (x\right )\,\left (6\,x-x^3\right )+\sin \left (x\right )\,\left (3\,x^2-6\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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