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 = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, 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 2295
Rule 2296
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*}
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Mathematica [A] time = 0.00, size = 28, normalized size = 1.00 \[ x \log ^3(c x)-3 x \log ^2(c x)+6 x \log (c x)-6 x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 28, normalized size = 1.00 \[ x \log \left (c x\right )^{3} - 3 \, x \log \left (c x\right )^{2} + 6 \, x \log \left (c x\right ) - 6 \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 28, normalized size = 1.00 \[ x \log \left (c x\right )^{3} - 3 \, x \log \left (c x\right )^{2} + 6 \, x \log \left (c x\right ) - 6 \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 29, normalized size = 1.04 \[ x \ln \left (c x \right )^{3}-3 x \ln \left (c x \right )^{2}+6 x \ln \left (c x \right )-6 x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 24, normalized size = 0.86 \[ {\left (\log \left (c x\right )^{3} - 3 \, \log \left (c x\right )^{2} + 6 \, \log \left (c x\right ) - 6\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.56, size = 24, normalized size = 0.86 \[ x\,\left ({\ln \left (c\,x\right )}^3-3\,{\ln \left (c\,x\right )}^2+6\,\ln \left (c\,x\right )-6\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 29, normalized size = 1.04 \[ x \log {\left (c x \right )}^{3} - 3 x \log {\left (c x \right )}^{2} + 6 x \log {\left (c x \right )} - 6 x \]
Verification of antiderivative is not currently implemented for this CAS.
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