Optimal. Leaf size=39 \[ \frac{1}{2} c \text{Ei}(-\log (c x))-\frac{1}{2 x \log ^2(c x)}+\frac{1}{2 x \log (c x)} \]
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Rubi [A] time = 0.0495498, antiderivative size = 39, 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} \[ \frac{1}{2} c \text{Ei}(-\log (c x))-\frac{1}{2 x \log ^2(c x)}+\frac{1}{2 x \log (c x)} \]
Antiderivative was successfully verified.
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Rule 2306
Rule 2309
Rule 2178
Rubi steps
\begin{align*} \int \frac{1}{x^2 \log ^3(c x)} \, dx &=-\frac{1}{2 x \log ^2(c x)}-\frac{1}{2} \int \frac{1}{x^2 \log ^2(c x)} \, dx\\ &=-\frac{1}{2 x \log ^2(c x)}+\frac{1}{2 x \log (c x)}+\frac{1}{2} \int \frac{1}{x^2 \log (c x)} \, dx\\ &=-\frac{1}{2 x \log ^2(c x)}+\frac{1}{2 x \log (c x)}+\frac{1}{2} c \operatorname{Subst}\left (\int \frac{e^{-x}}{x} \, dx,x,\log (c x)\right )\\ &=\frac{1}{2} c \text{Ei}(-\log (c x))-\frac{1}{2 x \log ^2(c x)}+\frac{1}{2 x \log (c x)}\\ \end{align*}
Mathematica [A] time = 0.0145465, size = 39, normalized size = 1. \[ \frac{1}{2} c \text{Ei}(-\log (c x))-\frac{1}{2 x \log ^2(c x)}+\frac{1}{2 x \log (c x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 33, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,x \left ( \ln \left ( cx \right ) \right ) ^{2}}}+{\frac{1}{2\,x\ln \left ( cx \right ) }}-{\frac{c{\it Ei} \left ( 1,\ln \left ( cx \right ) \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16639, size = 12, normalized size = 0.31 \begin{align*} -c \Gamma \left (-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 = 0.846329, size = 100, normalized size = 2.56 \begin{align*} \frac{c x \log \left (c x\right )^{2} \logintegral \left (\frac{1}{c x}\right ) + \log \left (c x\right ) - 1}{2 \, x \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*} \frac{\int \frac{1}{x^{2} \log{\left (c x \right )}}\, dx}{2} + \frac{\log{\left (c x \right )} - 1}{2 x \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^{2} \log \left (c x\right )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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