Optimal. Leaf size=23 \[ -\frac {x}{4}+\frac {1}{2} x \sin ^2(x)+\frac {1}{4} \sin (x) \cos (x) \]
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Rubi [A] time = 0.01, antiderivative size = 23, 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 = {3443, 2635, 8} \[ -\frac {x}{4}+\frac {1}{2} x \sin ^2(x)+\frac {1}{4} \sin (x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 3443
Rubi steps
\begin {align*} \int x \cos (x) \sin (x) \, dx &=\frac {1}{2} x \sin ^2(x)-\frac {1}{2} \int \sin ^2(x) \, dx\\ &=\frac {1}{4} \cos (x) \sin (x)+\frac {1}{2} x \sin ^2(x)-\frac {\int 1 \, dx}{4}\\ &=-\frac {x}{4}+\frac {1}{4} \cos (x) \sin (x)+\frac {1}{2} x \sin ^2(x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 18, normalized size = 0.78 \[ \frac {1}{8} \sin (2 x)-\frac {1}{4} x \cos (2 x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 17, normalized size = 0.74 \[ -\frac {1}{2} \, x \cos \relax (x)^{2} + \frac {1}{4} \, \cos \relax (x) \sin \relax (x) + \frac {1}{4} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.01, size = 14, normalized size = 0.61 \[ -\frac {1}{4} \, x \cos \left (2 \, x\right ) + \frac {1}{8} \, \sin \left (2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 18, normalized size = 0.78 \[ -\frac {x \left (\cos ^{2}\relax (x )\right )}{2}+\frac {\cos \relax (x ) \sin \relax (x )}{4}+\frac {x}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 14, normalized size = 0.61 \[ -\frac {1}{4} \, x \cos \left (2 \, x\right ) + \frac {1}{8} \, \sin \left (2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 18, normalized size = 0.78 \[ \frac {\sin \left (2\,x\right )}{8}+\frac {x\,\left (2\,{\sin \relax (x)}^2-1\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.34, size = 24, normalized size = 1.04 \[ \frac {x \sin ^{2}{\relax (x )}}{4} - \frac {x \cos ^{2}{\relax (x )}}{4} + \frac {\sin {\relax (x )} \cos {\relax (x )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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