Optimal. Leaf size=41 \[ \frac {\tan ^{-1}\left (\frac {1-2 \log (3 x)}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (1-\log (3 x)+\log ^2(3 x)\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 41, normalized size of antiderivative = 1.00, 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 = {648, 632, 210,
642} \begin {gather*} \frac {\text {ArcTan}\left (\frac {1-2 \log (3 x)}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (\log ^2(3 x)-\log (3 x)+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {-1+\log (3 x)}{x \left (1-\log (3 x)+\log ^2(3 x)\right )} \, dx &=\text {Subst}\left (\int \frac {-1+x}{1-x+x^2} \, dx,x,\log (3 x)\right )\\ &=-\left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\log (3 x)\right )\right )+\frac {1}{2} \text {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 )+\text {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*}
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Mathematica [A]
time = 0.04, size = 42, normalized size = 1.02 \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 38, normalized size = 0.93
method | result | size |
derivativedivides | \(\frac {\ln \left (1-\ln \left (3 x \right )+\ln \left (3 x \right )^{2}\right )}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (-1+2 \ln \left (3 x \right )\right ) \sqrt {3}}{3}\right )}{3}\) | \(38\) |
default | \(\frac {\ln \left (1-\ln \left (3 x \right )+\ln \left (3 x \right )^{2}\right )}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (-1+2 \ln \left (3 x \right )\right ) \sqrt {3}}{3}\right )}{3}\) | \(38\) |
risch | \(\frac {\ln \left (\ln \left (3 x \right )-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{2}+\frac {i \ln \left (\ln \left (3 x \right )-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{6}+\frac {\ln \left (\ln \left (3 x \right )-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{2}-\frac {i \ln \left (\ln \left (3 x \right )-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{6}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, size = 37, normalized size = 0.90 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 39, normalized size = 0.95 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.18, size = 22, normalized size = 0.54 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.57, size = 37, normalized size = 0.90 \begin {gather*} \frac {\ln \left ({\ln \left (3\,x\right )}^2-\ln \left (3\,x\right )+1\right )}{2}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,\left (2\,\ln \left (3\,x\right )-1\right )}{3}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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