Optimal. Leaf size=27 \[ \frac{1}{(\log (x)-1)^2}+\frac{1}{1-\log (x)}-\frac{2}{3 (1-\log (x))^3} \]
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Rubi [A] time = 0.0494096, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {698} \[ \frac{1}{(\log (x)-1)^2}+\frac{1}{1-\log (x)}-\frac{2}{3 (1-\log (x))^3} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \frac{1-4 \log (x)+\log ^2(x)}{x (-1+\log (x))^4} \, dx &=\operatorname{Subst}\left (\int \frac{1-4 x+x^2}{(-1+x)^4} \, dx,x,\log (x)\right )\\ &=\operatorname{Subst}\left (\int \left (-\frac{2}{(-1+x)^4}-\frac{2}{(-1+x)^3}+\frac{1}{(-1+x)^2}\right ) \, dx,x,\log (x)\right )\\ &=-\frac{2}{3 (1-\log (x))^3}+\frac{1}{1-\log (x)}+\frac{1}{(-1+\log (x))^2}\\ \end{align*}
Mathematica [A] time = 0.0414772, size = 22, normalized size = 0.81 \[ \frac{-3 \log ^2(x)+9 \log (x)-4}{3 (\log (x)-1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 24, normalized size = 0.9 \begin{align*} - \left ( -1+\ln \left ( x \right ) \right ) ^{-1}+{\frac{2}{3\, \left ( -1+\ln \left ( x \right ) \right ) ^{3}}}+ \left ( -1+\ln \left ( x \right ) \right ) ^{-2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04224, size = 43, normalized size = 1.59 \begin{align*} -\frac{3 \, \log \left (x\right )^{2} - 9 \, \log \left (x\right ) + 4}{3 \,{\left (\log \left (x\right )^{3} - 3 \, \log \left (x\right )^{2} + 3 \, \log \left (x\right ) - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01171, size = 99, normalized size = 3.67 \begin{align*} -\frac{3 \, \log \left (x\right )^{2} - 9 \, \log \left (x\right ) + 4}{3 \,{\left (\log \left (x\right )^{3} - 3 \, \log \left (x\right )^{2} + 3 \, \log \left (x\right ) - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.120792, size = 32, normalized size = 1.19 \begin{align*} \frac{- 3 \log{\left (x \right )}^{2} + 9 \log{\left (x \right )} - 4}{3 \log{\left (x \right )}^{3} - 9 \log{\left (x \right )}^{2} + 9 \log{\left (x \right )} - 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.35821, size = 27, normalized size = 1. \begin{align*} -\frac{3 \, \log \left (x\right )^{2} - 9 \, \log \left (x\right ) + 4}{3 \,{\left (\log \left (x\right ) - 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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