Optimal. Leaf size=45 \[ \frac{1}{2} x^2 \log ^3(c x)-\frac{3}{4} x^2 \log ^2(c x)+\frac{3}{4} x^2 \log (c x)-\frac{3 x^2}{8} \]
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Rubi [A] time = 0.0181902, antiderivative size = 45, 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 = {2305, 2304} \[ \frac{1}{2} x^2 \log ^3(c x)-\frac{3}{4} x^2 \log ^2(c x)+\frac{3}{4} x^2 \log (c x)-\frac{3 x^2}{8} \]
Antiderivative was successfully verified.
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Rule 2305
Rule 2304
Rubi steps
\begin{align*} \int x \log ^3(c x) \, dx &=\frac{1}{2} x^2 \log ^3(c x)-\frac{3}{2} \int x \log ^2(c x) \, dx\\ &=-\frac{3}{4} x^2 \log ^2(c x)+\frac{1}{2} x^2 \log ^3(c x)+\frac{3}{2} \int x \log (c x) \, dx\\ &=-\frac{3 x^2}{8}+\frac{3}{4} x^2 \log (c x)-\frac{3}{4} x^2 \log ^2(c x)+\frac{1}{2} x^2 \log ^3(c x)\\ \end{align*}
Mathematica [A] time = 0.0011034, size = 45, normalized size = 1. \[ \frac{1}{2} x^2 \log ^3(c x)-\frac{3}{4} x^2 \log ^2(c x)+\frac{3}{4} x^2 \log (c x)-\frac{3 x^2}{8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 38, normalized size = 0.8 \begin{align*} -{\frac{3\,{x}^{2}}{8}}+{\frac{3\,{x}^{2}\ln \left ( cx \right ) }{4}}-{\frac{3\,{x}^{2} \left ( \ln \left ( cx \right ) \right ) ^{2}}{4}}+{\frac{{x}^{2} \left ( \ln \left ( cx \right ) \right ) ^{3}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0054, size = 39, normalized size = 0.87 \begin{align*} \frac{1}{8} \,{\left (4 \, \log \left (c x\right )^{3} - 6 \, \log \left (c x\right )^{2} + 6 \, \log \left (c x\right ) - 3\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.823187, size = 95, normalized size = 2.11 \begin{align*} \frac{1}{2} \, x^{2} \log \left (c x\right )^{3} - \frac{3}{4} \, x^{2} \log \left (c x\right )^{2} + \frac{3}{4} \, x^{2} \log \left (c x\right ) - \frac{3}{8} \, x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.122258, size = 42, normalized size = 0.93 \begin{align*} \frac{x^{2} \log{\left (c x \right )}^{3}}{2} - \frac{3 x^{2} \log{\left (c x \right )}^{2}}{4} + \frac{3 x^{2} \log{\left (c x \right )}}{4} - \frac{3 x^{2}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09851, size = 50, normalized size = 1.11 \begin{align*} \frac{1}{2} \, x^{2} \log \left (c x\right )^{3} - \frac{3}{4} \, x^{2} \log \left (c x\right )^{2} + \frac{3}{4} \, x^{2} \log \left (c x\right ) - \frac{3}{8} \, x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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