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 = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 2637
Rule 3296
Rule 3310
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.42, size = 23, normalized size = 0.70 \[ \frac {1}{3} \, x \cos \relax (x)^{3} - x \cos \relax (x) - \frac {1}{9} \, {\left (\cos \relax (x)^{2} - 7\right )} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.01, size = 23, normalized size = 0.70 \[ \frac {1}{12} \, x \cos \left (3 \, x\right ) - \frac {3}{4} \, x \cos \relax (x) - \frac {1}{36} \, \sin \left (3 \, x\right ) + \frac {3}{4} \, \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 23, normalized size = 0.70 \[ \frac {\left (\sin ^{3}\relax (x )\right )}{9}-\frac {\left (\sin ^{2}\relax (x )+2\right ) x \cos \relax (x )}{3}+\frac {2 \sin \relax (x )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 23, normalized size = 0.70 \[ \frac {1}{12} \, x \cos \left (3 \, x\right ) - \frac {3}{4} \, x \cos \relax (x) - \frac {1}{36} \, \sin \left (3 \, x\right ) + \frac {3}{4} \, \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 25, normalized size = 0.76 \[ \frac {x\,{\cos \relax (x)}^3}{3}-\frac {\sin \relax (x)\,{\cos \relax (x)}^2}{9}-x\,\cos \relax (x)+\frac {7\,\sin \relax (x)}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.65, size = 39, normalized size = 1.18 \[ - x \sin ^{2}{\relax (x )} \cos {\relax (x )} - \frac {2 x \cos ^{3}{\relax (x )}}{3} + \frac {7 \sin ^{3}{\relax (x )}}{9} + \frac {2 \sin {\relax (x )} \cos ^{2}{\relax (x )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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