Optimal. Leaf size=28 \[ \frac{x^2}{4}+\frac{1}{2} x^2 \log ^2(x)-\frac{1}{2} x^2 \log (x) \]
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Rubi [A] time = 0.0097481, antiderivative size = 28, 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 = {2305, 2304} \[ \frac{x^2}{4}+\frac{1}{2} x^2 \log ^2(x)-\frac{1}{2} x^2 \log (x) \]
Antiderivative was successfully verified.
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Rule 2305
Rule 2304
Rubi steps
\begin{align*} \int x \log ^2(x) \, dx &=\frac{1}{2} x^2 \log ^2(x)-\int x \log (x) \, dx\\ &=\frac{x^2}{4}-\frac{1}{2} x^2 \log (x)+\frac{1}{2} x^2 \log ^2(x)\\ \end{align*}
Mathematica [A] time = 0.0008745, size = 28, normalized size = 1. \[ \frac{x^2}{4}+\frac{1}{2} x^2 \log ^2(x)-\frac{1}{2} x^2 \log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 23, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{4}}-{\frac{{x}^{2}\ln \left ( x \right ) }{2}}+{\frac{{x}^{2} \left ( \ln \left ( x \right ) \right ) ^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.925699, size = 23, normalized size = 0.82 \begin{align*} \frac{1}{4} \,{\left (2 \, \log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 1\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81658, size = 61, normalized size = 2.18 \begin{align*} \frac{1}{2} \, x^{2} \log \left (x\right )^{2} - \frac{1}{2} \, x^{2} \log \left (x\right ) + \frac{1}{4} \, x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.094922, size = 22, normalized size = 0.79 \begin{align*} \frac{x^{2} \log{\left (x \right )}^{2}}{2} - \frac{x^{2} \log{\left (x \right )}}{2} + \frac{x^{2}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05854, size = 30, normalized size = 1.07 \begin{align*} \frac{1}{2} \, x^{2} \log \left (x\right )^{2} - \frac{1}{2} \, x^{2} \log \left (x\right ) + \frac{1}{4} \, x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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