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 = 0.02, 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, 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 2638
Rule 3296
Rule 3310
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.44, size = 25, normalized size = 0.76 \[ \frac {1}{9} \, \cos \relax (x)^{3} + \frac {1}{3} \, {\left (x \cos \relax (x)^{2} + 2 \, x\right )} \sin \relax (x) + \frac {2}{3} \, \cos \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.38, size = 23, normalized size = 0.70 \[ \frac {1}{12} \, x \sin \left (3 \, x\right ) + \frac {3}{4} \, x \sin \relax (x) + \frac {1}{36} \, \cos \left (3 \, x\right ) + \frac {3}{4} \, \cos \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 23, normalized size = 0.70 \[ \frac {\left (\cos ^{3}\relax (x )\right )}{9}+\frac {\left (\cos ^{2}\relax (x )+2\right ) x \sin \relax (x )}{3}+\frac {2 \cos \relax (x )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 23, normalized size = 0.70 \[ \frac {1}{12} \, x \sin \left (3 \, x\right ) + \frac {3}{4} \, x \sin \relax (x) + \frac {1}{36} \, \cos \left (3 \, x\right ) + \frac {3}{4} \, \cos \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 25, normalized size = 0.76 \[ \frac {{\cos \relax (x)}^3}{9}+\frac {x\,\sin \relax (x)\,{\cos \relax (x)}^2}{3}+\frac {2\,\cos \relax (x)}{3}+\frac {2\,x\,\sin \relax (x)}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.61, size = 39, normalized size = 1.18 \[ \frac {2 x \sin ^{3}{\relax (x )}}{3} + x \sin {\relax (x )} \cos ^{2}{\relax (x )} + \frac {2 \sin ^{2}{\relax (x )} \cos {\relax (x )}}{3} + \frac {7 \cos ^{3}{\relax (x )}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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