Optimal. Leaf size=29 \[ -\frac {1}{2} x^2 \cos (2 x)+\frac {1}{2} x \sin (2 x)+\frac {1}{4} \cos (2 x) \]
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Rubi [A] time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3296, 2638} \[ -\frac {1}{2} x^2 \cos (2 x)+\frac {1}{2} x \sin (2 x)+\frac {1}{4} \cos (2 x) \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rubi steps
\begin {align*} \int x^2 \sin (2 x) \, dx &=-\frac {1}{2} x^2 \cos (2 x)+\int x \cos (2 x) \, dx\\ &=-\frac {1}{2} x^2 \cos (2 x)+\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^2 \cos (2 x)+\frac {1}{2} x \sin (2 x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 25, normalized size = 0.86 \[ \frac {1}{2} x \sin (2 x)-\frac {1}{4} \left (2 x^2-1\right ) \cos (2 x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 21, normalized size = 0.72 \[ -\frac {1}{4} \, {\left (2 \, x^{2} - 1\right )} \cos \left (2 \, x\right ) + \frac {1}{2} \, x \sin \left (2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.95, size = 21, normalized size = 0.72 \[ -\frac {1}{4} \, {\left (2 \, x^{2} - 1\right )} \cos \left (2 \, x\right ) + \frac {1}{2} \, x \sin \left (2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 24, normalized size = 0.83 \[ -\frac {x^{2} \cos \left (2 x \right )}{2}+\frac {x \sin \left (2 x \right )}{2}+\frac {\cos \left (2 x \right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 21, normalized size = 0.72 \[ -\frac {1}{4} \, {\left (2 \, x^{2} - 1\right )} \cos \left (2 \, x\right ) + \frac {1}{2} \, x \sin \left (2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 24, normalized size = 0.83 \[ \frac {x\,\sin \left (2\,x\right )}{2}+\left (2\,{\sin \relax (x)}^2-1\right )\,\left (\frac {x^2}{2}-\frac {1}{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.33, size = 24, normalized size = 0.83 \[ - \frac {x^{2} \cos {\left (2 x \right )}}{2} + \frac {x \sin {\left (2 x \right )}}{2} + \frac {\cos {\left (2 x \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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