Optimal. Leaf size=32 \[ \frac {1}{2} x^2 \log ^2(c x)-\frac {1}{2} x^2 \log (c x)+\frac {x^2}{4} \]
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Rubi [A] time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2305, 2304} \[ \frac {1}{2} x^2 \log ^2(c x)-\frac {1}{2} x^2 \log (c x)+\frac {x^2}{4} \]
Antiderivative was successfully verified.
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Rule 2304
Rule 2305
Rubi steps
\begin {align*} \int x \log ^2(c x) \, dx &=\frac {1}{2} x^2 \log ^2(c x)-\int x \log (c x) \, dx\\ &=\frac {x^2}{4}-\frac {1}{2} x^2 \log (c x)+\frac {1}{2} x^2 \log ^2(c x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 32, normalized size = 1.00 \[ \frac {1}{2} x^2 \log ^2(c x)-\frac {1}{2} x^2 \log (c x)+\frac {x^2}{4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 26, normalized size = 0.81 \[ \frac {1}{2} \, x^{2} \log \left (c x\right )^{2} - \frac {1}{2} \, x^{2} \log \left (c x\right ) + \frac {1}{4} \, x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 26, normalized size = 0.81 \[ \frac {1}{2} \, x^{2} \log \left (c x\right )^{2} - \frac {1}{2} \, x^{2} \log \left (c x\right ) + \frac {1}{4} \, x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 27, normalized size = 0.84 \[ \frac {x^{2} \ln \left (c x \right )^{2}}{2}-\frac {x^{2} \ln \left (c x \right )}{2}+\frac {x^{2}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 21, normalized size = 0.66 \[ \frac {1}{4} \, {\left (2 \, \log \left (c x\right )^{2} - 2 \, \log \left (c x\right ) + 1\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.49, size = 21, normalized size = 0.66 \[ \frac {x^2\,\left (2\,{\ln \left (c\,x\right )}^2-2\,\ln \left (c\,x\right )+1\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 26, normalized size = 0.81 \[ \frac {x^{2} \log {\left (c x \right )}^{2}}{2} - \frac {x^{2} \log {\left (c x \right )}}{2} + \frac {x^{2}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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