Optimal. Leaf size=29 \[ \frac{1}{3} x^2 \sin (3 x)-\frac{2}{27} \sin (3 x)+\frac{2}{9} x \cos (3 x) \]
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Rubi [A] time = 0.025313, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3296, 2637} \[ \frac{1}{3} x^2 \sin (3 x)-\frac{2}{27} \sin (3 x)+\frac{2}{9} x \cos (3 x) \]
Antiderivative was successfully verified.
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Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int x^2 \cos (3 x) \, dx &=\frac{1}{3} x^2 \sin (3 x)-\frac{2}{3} \int x \sin (3 x) \, dx\\ &=\frac{2}{9} x \cos (3 x)+\frac{1}{3} x^2 \sin (3 x)-\frac{2}{9} \int \cos (3 x) \, dx\\ &=\frac{2}{9} x \cos (3 x)-\frac{2}{27} \sin (3 x)+\frac{1}{3} x^2 \sin (3 x)\\ \end{align*}
Mathematica [A] time = 0.0267297, size = 25, normalized size = 0.86 \[ \frac{1}{27} \left (9 x^2-2\right ) \sin (3 x)+\frac{2}{9} x \cos (3 x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 24, normalized size = 0.8 \begin{align*}{\frac{2\,x\cos \left ( 3\,x \right ) }{9}}-{\frac{2\,\sin \left ( 3\,x \right ) }{27}}+{\frac{{x}^{2}\sin \left ( 3\,x \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.941103, size = 28, normalized size = 0.97 \begin{align*} \frac{2}{9} \, x \cos \left (3 \, x\right ) + \frac{1}{27} \,{\left (9 \, x^{2} - 2\right )} \sin \left (3 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02833, size = 59, normalized size = 2.03 \begin{align*} \frac{2}{9} \, x \cos \left (3 \, x\right ) + \frac{1}{27} \,{\left (9 \, x^{2} - 2\right )} \sin \left (3 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.303979, size = 27, normalized size = 0.93 \begin{align*} \frac{x^{2} \sin{\left (3 x \right )}}{3} + \frac{2 x \cos{\left (3 x \right )}}{9} - \frac{2 \sin{\left (3 x \right )}}{27} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04041, size = 28, normalized size = 0.97 \begin{align*} \frac{2}{9} \, x \cos \left (3 \, x\right ) + \frac{1}{27} \,{\left (9 \, x^{2} - 2\right )} \sin \left (3 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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