Optimal. Leaf size=41 \[ \frac{1}{2} \log \left (\log ^2(3 x)-\log (3 x)+1\right )+\frac{\tan ^{-1}\left (\frac{1-2 \log (3 x)}{\sqrt{3}}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0575292, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {634, 618, 204, 628} \[ \frac{1}{2} \log \left (\log ^2(3 x)-\log (3 x)+1\right )+\frac{\tan ^{-1}\left (\frac{1-2 \log (3 x)}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{-1+\log (3 x)}{x \left (1-\log (3 x)+\log ^2(3 x)\right )} \, dx &=\operatorname{Subst}\left (\int \frac{-1+x}{1-x+x^2} \, dx,x,\log (3 x)\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1-x+x^2} \, dx,x,\log (3 x)\right )\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{-1+2 x}{1-x+x^2} \, dx,x,\log (3 x)\right )\\ &=\frac{1}{2} \log \left (1-\log (3 x)+\log ^2(3 x)\right )+\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 \log (3 x)\right )\\ &=-\frac{\tan ^{-1}\left (\frac{-1+2 \log (3 x)}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{1}{2} \log \left (1-\log (3 x)+\log ^2(3 x)\right )\\ \end{align*}
Mathematica [A] time = 0.0890602, size = 42, normalized size = 1.02 \[ \frac{1}{2} \log \left (\log ^2(3 x)-\log (3 x)+1\right )-\frac{\tan ^{-1}\left (\frac{2 \log (3 x)-1}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 38, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( 1-\ln \left ( 3\,x \right ) + \left ( \ln \left ( 3\,x \right ) \right ) ^{2} \right ) }{2}}-{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( -1+2\,\ln \left ( 3\,x \right ) \right ) \sqrt{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52494, size = 50, normalized size = 1.22 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \log \left (3 \, x\right ) - 1\right )}\right ) + \frac{1}{2} \, \log \left (\log \left (3 \, x\right )^{2} - \log \left (3 \, x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84896, size = 127, normalized size = 3.1 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{2}{3} \, \sqrt{3} \log \left (3 \, x\right ) - \frac{1}{3} \, \sqrt{3}\right ) + \frac{1}{2} \, \log \left (\log \left (3 \, x\right )^{2} - \log \left (3 \, x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.178734, size = 22, normalized size = 0.54 \begin{align*} \operatorname{RootSum}{\left (3 z^{2} - 3 z + 1, \left ( i \mapsto i \log{\left (- 3 i + \log{\left (3 x \right )} + 1 \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23167, size = 50, normalized size = 1.22 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \log \left (3 \, x\right ) - 1\right )}\right ) + \frac{1}{2} \, \log \left (\log \left (3 \, x\right )^{2} - \log \left (3 \, x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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