Optimal. Leaf size=45 \[ -\frac{\log ^3(c x)}{2 x^2}-\frac{3 \log ^2(c x)}{4 x^2}-\frac{3 \log (c x)}{4 x^2}-\frac{3}{8 x^2} \]
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Rubi [A] time = 0.0331861, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2305, 2304} \[ -\frac{\log ^3(c x)}{2 x^2}-\frac{3 \log ^2(c x)}{4 x^2}-\frac{3 \log (c x)}{4 x^2}-\frac{3}{8 x^2} \]
Antiderivative was successfully verified.
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Rule 2305
Rule 2304
Rubi steps
\begin{align*} \int \frac{\log ^3(c x)}{x^3} \, dx &=-\frac{\log ^3(c x)}{2 x^2}+\frac{3}{2} \int \frac{\log ^2(c x)}{x^3} \, dx\\ &=-\frac{3 \log ^2(c x)}{4 x^2}-\frac{\log ^3(c x)}{2 x^2}+\frac{3}{2} \int \frac{\log (c x)}{x^3} \, dx\\ &=-\frac{3}{8 x^2}-\frac{3 \log (c x)}{4 x^2}-\frac{3 \log ^2(c x)}{4 x^2}-\frac{\log ^3(c x)}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0014731, size = 45, normalized size = 1. \[ -\frac{\log ^3(c x)}{2 x^2}-\frac{3 \log ^2(c x)}{4 x^2}-\frac{3 \log (c x)}{4 x^2}-\frac{3}{8 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 38, normalized size = 0.8 \begin{align*} -{\frac{3}{8\,{x}^{2}}}-{\frac{3\,\ln \left ( cx \right ) }{4\,{x}^{2}}}-{\frac{3\, \left ( \ln \left ( cx \right ) \right ) ^{2}}{4\,{x}^{2}}}-{\frac{ \left ( \ln \left ( cx \right ) \right ) ^{3}}{2\,{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06371, size = 39, normalized size = 0.87 \begin{align*} -\frac{4 \, \log \left (c x\right )^{3} + 6 \, \log \left (c x\right )^{2} + 6 \, \log \left (c x\right ) + 3}{8 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.885273, size = 77, normalized size = 1.71 \begin{align*} -\frac{4 \, \log \left (c x\right )^{3} + 6 \, \log \left (c x\right )^{2} + 6 \, \log \left (c x\right ) + 3}{8 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.150497, size = 44, normalized size = 0.98 \begin{align*} - \frac{\log{\left (c x \right )}^{3}}{2 x^{2}} - \frac{3 \log{\left (c x \right )}^{2}}{4 x^{2}} - \frac{3 \log{\left (c x \right )}}{4 x^{2}} - \frac{3}{8 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13499, size = 50, normalized size = 1.11 \begin{align*} -\frac{\log \left (c x\right )^{3}}{2 \, x^{2}} - \frac{3 \, \log \left (c x\right )^{2}}{4 \, x^{2}} - \frac{3 \, \log \left (c x\right )}{4 \, x^{2}} - \frac{3}{8 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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