Optimal. Leaf size=28 \[ x \log ^3(c x)-3 x \log ^2(c x)+6 x \log (c x)-6 x \]
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Rubi [A] time = 0.0079641, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2296, 2295} \[ x \log ^3(c x)-3 x \log ^2(c x)+6 x \log (c x)-6 x \]
Antiderivative was successfully verified.
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Rule 2296
Rule 2295
Rubi steps
\begin{align*} \int \log ^3(c x) \, dx &=x \log ^3(c x)-3 \int \log ^2(c x) \, dx\\ &=-3 x \log ^2(c x)+x \log ^3(c x)+6 \int \log (c x) \, dx\\ &=-6 x+6 x \log (c x)-3 x \log ^2(c x)+x \log ^3(c x)\\ \end{align*}
Mathematica [A] time = 0.0009562, size = 28, normalized size = 1. \[ x \log ^3(c x)-3 x \log ^2(c x)+6 x \log (c x)-6 x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 29, normalized size = 1. \begin{align*} -6\,x+6\,x\ln \left ( cx \right ) -3\,x \left ( \ln \left ( cx \right ) \right ) ^{2}+x \left ( \ln \left ( cx \right ) \right ) ^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02186, size = 32, normalized size = 1.14 \begin{align*}{\left (\log \left (c x\right )^{3} - 3 \, \log \left (c x\right )^{2} + 6 \, \log \left (c x\right ) - 6\right )} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.747944, size = 70, normalized size = 2.5 \begin{align*} x \log \left (c x\right )^{3} - 3 \, x \log \left (c x\right )^{2} + 6 \, x \log \left (c x\right ) - 6 \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.110192, size = 29, normalized size = 1.04 \begin{align*} x \log{\left (c x \right )}^{3} - 3 x \log{\left (c x \right )}^{2} + 6 x \log{\left (c x \right )} - 6 x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13001, size = 38, normalized size = 1.36 \begin{align*} x \log \left (c x\right )^{3} - 3 \, x \log \left (c x\right )^{2} + 6 \, x \log \left (c x\right ) - 6 \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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