Optimal. Leaf size=42 \[ -\sqrt {3} \tan ^{-1}\left (\frac {1+2 \log (3 x)}{\sqrt {3}}\right )+\log (x)-\frac {1}{2} \log \left (1+\log (3 x)+\log ^2(3 x)\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1671, 648, 632,
210, 642} \begin {gather*} -\sqrt {3} \text {ArcTan}\left (\frac {2 \log (3 x)+1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (\log ^2(3 x)+\log (3 x)+1\right )+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1671
Rubi steps
\begin {align*} \int \frac {-1+\log ^2(3 x)}{x+x \log (3 x)+x \log ^2(3 x)} \, dx &=\text {Subst}\left (\int \frac {-1+x^2}{1+x+x^2} \, dx,x,\log (3 x)\right )\\ &=\text {Subst}\left (\int \left (1-\frac {2+x}{1+x+x^2}\right ) \, dx,x,\log (3 x)\right )\\ &=\log (x)-\text {Subst}\left (\int \frac {2+x}{1+x+x^2} \, dx,x,\log (3 x)\right )\\ &=\log (x)-\frac {1}{2} \text {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\log (3 x)\right )-\frac {3}{2} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\log (3 x)\right )\\ &=\log (x)-\frac {1}{2} \log \left (1+\log (3 x)+\log ^2(3 x)\right )+3 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \log (3 x)\right )\\ &=-\sqrt {3} \tan ^{-1}\left (\frac {1+2 \log (3 x)}{\sqrt {3}}\right )+\log (x)-\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 = 44, normalized size = 1.05 \begin {gather*} -\sqrt {3} \tan ^{-1}\left (\frac {1+2 \log (3 x)}{\sqrt {3}}\right )+\log (3 x)-\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 = 40, normalized size = 0.95
method | result | size |
derivativedivides | \(\ln \left (3 x \right )-\frac {\ln \left (1+\ln \left (3 x \right )+\ln \left (3 x \right )^{2}\right )}{2}-\arctan \left (\frac {\left (1+2 \ln \left (3 x \right )\right ) \sqrt {3}}{3}\right ) \sqrt {3}\) | \(40\) |
default | \(\ln \left (3 x \right )-\frac {\ln \left (1+\ln \left (3 x \right )+\ln \left (3 x \right )^{2}\right )}{2}-\arctan \left (\frac {\left (1+2 \ln \left (3 x \right )\right ) \sqrt {3}}{3}\right ) \sqrt {3}\) | \(40\) |
risch | \(\ln \left (x \right )-\frac {\ln \left (\ln \left (3 x \right )+\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{2}+\frac {i \ln \left (\ln \left (3 x \right )+\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{2}-\frac {\ln \left (\ln \left (3 x \right )+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{2}-\frac {i \ln \left (\ln \left (3 x \right )+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{2}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 41, normalized size = 0.98 \begin {gather*} -\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 ) + \log \left (3 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.17, size = 19, normalized size = 0.45 \begin {gather*} \log {\left (x \right )} + \operatorname {RootSum} {\left (z^{2} + z + 1, \left ( i \mapsto i \log {\left (- i + \log {\left (3 x \right )} \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 = 0.60, size = 37, normalized size = 0.88 \begin {gather*} \ln \left (x\right )-\frac {\ln \left ({\ln \left (3\,x\right )}^2+\ln \left (3\,x\right )+1\right )}{2}-\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,\left (2\,\ln \left (3\,x\right )+1\right )}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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