Optimal. Leaf size=36 \[ 2 c^2 \text{Ei}(-2 \log (c x))-\frac{1}{2 x^2 \log ^2(c x)}+\frac{1}{x^2 \log (c x)} \]
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Rubi [A] time = 0.049051, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2306, 2309, 2178} \[ 2 c^2 \text{Ei}(-2 \log (c x))-\frac{1}{2 x^2 \log ^2(c x)}+\frac{1}{x^2 \log (c x)} \]
Antiderivative was successfully verified.
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Rule 2306
Rule 2309
Rule 2178
Rubi steps
\begin{align*} \int \frac{1}{x^3 \log ^3(c x)} \, dx &=-\frac{1}{2 x^2 \log ^2(c x)}-\int \frac{1}{x^3 \log ^2(c x)} \, dx\\ &=-\frac{1}{2 x^2 \log ^2(c x)}+\frac{1}{x^2 \log (c x)}+2 \int \frac{1}{x^3 \log (c x)} \, dx\\ &=-\frac{1}{2 x^2 \log ^2(c x)}+\frac{1}{x^2 \log (c x)}+\left (2 c^2\right ) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{x} \, dx,x,\log (c x)\right )\\ &=2 c^2 \text{Ei}(-2 \log (c x))-\frac{1}{2 x^2 \log ^2(c x)}+\frac{1}{x^2 \log (c x)}\\ \end{align*}
Mathematica [A] time = 0.0159106, size = 36, normalized size = 1. \[ 2 c^2 \text{Ei}(-2 \log (c x))-\frac{1}{2 x^2 \log ^2(c x)}+\frac{1}{x^2 \log (c x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 36, normalized size = 1. \begin{align*} -{\frac{1}{2\,{x}^{2} \left ( \ln \left ( cx \right ) \right ) ^{2}}}+{\frac{1}{{x}^{2}\ln \left ( cx \right ) }}-2\,{c}^{2}{\it Ei} \left ( 1,2\,\ln \left ( cx \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.24094, size = 18, normalized size = 0.5 \begin{align*} -4 \, c^{2} \Gamma \left (-2, 2 \, \log \left (c x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.717287, size = 119, normalized size = 3.31 \begin{align*} \frac{4 \, c^{2} x^{2} \log \left (c x\right )^{2} \logintegral \left (\frac{1}{c^{2} x^{2}}\right ) + 2 \, \log \left (c x\right ) - 1}{2 \, x^{2} \log \left (c x\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} 2 \int \frac{1}{x^{3} \log{\left (c x \right )}}\, dx + \frac{2 \log{\left (c x \right )} - 1}{2 x^{2} \log{\left (c x \right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{3} \log \left (c x\right )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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