Optimal. Leaf size=32 \[ -\frac{\log ^2(c x)}{2 x^2}-\frac{\log (c x)}{2 x^2}-\frac{1}{4 x^2} \]
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Rubi [A] time = 0.0189641, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2305, 2304} \[ -\frac{\log ^2(c x)}{2 x^2}-\frac{\log (c x)}{2 x^2}-\frac{1}{4 x^2} \]
Antiderivative was successfully verified.
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Rule 2305
Rule 2304
Rubi steps
\begin{align*} \int \frac{\log ^2(c x)}{x^3} \, dx &=-\frac{\log ^2(c x)}{2 x^2}+\int \frac{\log (c x)}{x^3} \, dx\\ &=-\frac{1}{4 x^2}-\frac{\log (c x)}{2 x^2}-\frac{\log ^2(c x)}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0012215, size = 32, normalized size = 1. \[ -\frac{\log ^2(c x)}{2 x^2}-\frac{\log (c x)}{2 x^2}-\frac{1}{4 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 27, normalized size = 0.8 \begin{align*} -{\frac{1}{4\,{x}^{2}}}-{\frac{\ln \left ( cx \right ) }{2\,{x}^{2}}}-{\frac{ \left ( \ln \left ( cx \right ) \right ) ^{2}}{2\,{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.985426, size = 28, normalized size = 0.88 \begin{align*} -\frac{2 \, \log \left (c x\right )^{2} + 2 \, \log \left (c x\right ) + 1}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.805439, size = 57, normalized size = 1.78 \begin{align*} -\frac{2 \, \log \left (c x\right )^{2} + 2 \, \log \left (c x\right ) + 1}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.128722, size = 29, normalized size = 0.91 \begin{align*} - \frac{\log{\left (c x \right )}^{2}}{2 x^{2}} - \frac{\log{\left (c x \right )}}{2 x^{2}} - \frac{1}{4 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1034, size = 35, normalized size = 1.09 \begin{align*} -\frac{\log \left (c x\right )^{2}}{2 \, x^{2}} - \frac{\log \left (c x\right )}{2 \, x^{2}} - \frac{1}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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