Optimal. Leaf size=18 \[ \frac{1}{2} x \sin (2 x)+\frac{1}{4} \cos (2 x) \]
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Rubi [A] time = 0.0122093, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3296, 2638} \[ \frac{1}{2} x \sin (2 x)+\frac{1}{4} \cos (2 x) \]
Antiderivative was successfully verified.
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Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int x \cos (2 x) \, dx &=\frac{1}{2} x \sin (2 x)-\frac{1}{2} \int \sin (2 x) \, dx\\ &=\frac{1}{4} \cos (2 x)+\frac{1}{2} x \sin (2 x)\\ \end{align*}
Mathematica [A] time = 0.0090039, size = 18, normalized size = 1. \[ \frac{1}{2} x \sin (2 x)+\frac{1}{4} \cos (2 x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 15, normalized size = 0.8 \begin{align*}{\frac{\cos \left ( 2\,x \right ) }{4}}+{\frac{x\sin \left ( 2\,x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.935605, size = 19, normalized size = 1.06 \begin{align*} \frac{1}{2} \, x \sin \left (2 \, x\right ) + \frac{1}{4} \, \cos \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.37485, size = 42, normalized size = 2.33 \begin{align*} \frac{1}{2} \, x \sin \left (2 \, x\right ) + \frac{1}{4} \, \cos \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.162415, size = 14, normalized size = 0.78 \begin{align*} \frac{x \sin{\left (2 x \right )}}{2} + \frac{\cos{\left (2 x \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04112, size = 19, normalized size = 1.06 \begin{align*} \frac{1}{2} \, x \sin \left (2 \, x\right ) + \frac{1}{4} \, \cos \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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