Optimal. Leaf size=42 \[ -\frac {1}{2} \log \left (\log ^2(3 x)+\log (3 x)+1\right )+\log (x)-\sqrt {3} \tan ^{-1}\left (\frac {2 \log (3 x)+1}{\sqrt {3}}\right ) \]
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Rubi [A] time = 0.05, 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 = {1657, 634, 618, 204, 628} \[ -\frac {1}{2} \log \left (\log ^2(3 x)+\log (3 x)+1\right )+\log (x)-\sqrt {3} \tan ^{-1}\left (\frac {2 \log (3 x)+1}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1657
Rubi steps
\begin {align*} \int \frac {-1+\log ^2(3 x)}{x+x \log (3 x)+x \log ^2(3 x)} \, dx &=\operatorname {Subst}\left (\int \frac {-1+x^2}{1+x+x^2} \, dx,x,\log (3 x)\right )\\ &=\operatorname {Subst}\left (\int \left (1-\frac {2+x}{1+x+x^2}\right ) \, dx,x,\log (3 x)\right )\\ &=\log (x)-\operatorname {Subst}\left (\int \frac {2+x}{1+x+x^2} \, dx,x,\log (3 x)\right )\\ &=\log (x)-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\log (3 x)\right )-\frac {3}{2} \operatorname {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 \operatorname {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 = 0.09, size = 44, normalized size = 1.05 \[ -\frac {1}{2} \log \left (\log ^2(3 x)+\log (3 x)+1\right )+\log (3 x)-\sqrt {3} \tan ^{-1}\left (\frac {2 \log (3 x)+1}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 41, normalized size = 0.98 \[ -\sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \log \left (3 \, x\right ) + \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{2} \, \log \left (\log \left (3 \, x\right )^{2} + \log \left (3 \, x\right ) + 1\right ) + \log \left (3 \, x\right ) \]
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 \[ \int \frac {\log \left (3 \, x\right )^{2} - 1}{x \log \left (3 \, x\right )^{2} + x \log \left (3 \, x\right ) + x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 40, normalized size = 0.95 \[ -\sqrt {3}\, \arctan \left (\frac {\left (2 \ln \left (3 x \right )+1\right ) \sqrt {3}}{3}\right )+\ln \left (3 x \right )-\frac {\ln \left (\ln \left (3 x \right )^{2}+\ln \left (3 x \right )+1\right )}{2} \]
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 \[ -\int \frac {\log \relax (3) + \log \relax (x) + 2}{x {\left (2 \, \log \relax (3) + 1\right )} \log \relax (x) + x \log \relax (x)^{2} + {\left (\log \relax (3)^{2} + \log \relax (3) + 1\right )} x}\,{d x} + \log \relax (x) \]
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 \[ \ln \relax (x)-\frac {\ln \left ({\ln \left (3\,x\right )}^2+\ln \left (3\,x\right )+1\right )}{2}-\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,\left (2\,\ln \left (3\,x\right )+1\right )}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 19, normalized size = 0.45 \[ \log {\relax (x )} + \operatorname {RootSum} {\left (z^{2} + z + 1, \left (i \mapsto i \log {\left (- i + \log {\left (3 x \right )} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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