Integrand size = 27, antiderivative size = 42 \[ \int \frac {-1+\log ^2(3 x)}{x+x \log (3 x)+x \log ^2(3 x)} \, dx=-\sqrt {3} \arctan \left (\frac {1+2 \log (3 x)}{\sqrt {3}}\right )+\log (x)-\frac {1}{2} \log \left (1+\log (3 x)+\log ^2(3 x)\right ) \]
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Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1671, 648, 632, 210, 642} \[ \int \frac {-1+\log ^2(3 x)}{x+x \log (3 x)+x \log ^2(3 x)} \, dx=-\sqrt {3} \arctan \left (\frac {2 \log (3 x)+1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (\log ^2(3 x)+\log (3 x)+1\right )+\log (x) \]
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1671
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.06 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.05 \[ \int \frac {-1+\log ^2(3 x)}{x+x \log (3 x)+x \log ^2(3 x)} \, dx=-\sqrt {3} \arctan \left (\frac {1+2 \log (3 x)}{\sqrt {3}}\right )+\log (3 x)-\frac {1}{2} \log \left (1+\log (3 x)+\log ^2(3 x)\right ) \]
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Time = 0.45 (sec) , antiderivative size = 40, normalized size of antiderivative = 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 | \(\ln \left (x \right )-\frac {\ln \left (\ln \left (3 x \right )+\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{2}+\frac {i \ln \left (\ln \left (3 x \right )+\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{2}-\frac {\ln \left (\ln \left (3 x \right )+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{2}-\frac {i \ln \left (\ln \left (3 x \right )+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{2}\) | \(72\) |
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Time = 0.29 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.98 \[ \int \frac {-1+\log ^2(3 x)}{x+x \log (3 x)+x \log ^2(3 x)} \, dx=-\sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \log \left (3 \, x\right ) + \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{2} \, \log \left (\log \left (3 \, x\right )^{2} + \log \left (3 \, x\right ) + 1\right ) + \log \left (3 \, x\right ) \]
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Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.45 \[ \int \frac {-1+\log ^2(3 x)}{x+x \log (3 x)+x \log ^2(3 x)} \, dx=\log {\left (x \right )} + \operatorname {RootSum} {\left (z^{2} + z + 1, \left ( i \mapsto i \log {\left (- i + \log {\left (3 x \right )} \right )} \right )\right )} \]
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\[ \int \frac {-1+\log ^2(3 x)}{x+x \log (3 x)+x \log ^2(3 x)} \, dx=\int { \frac {\log \left (3 \, x\right )^{2} - 1}{x \log \left (3 \, x\right )^{2} + x \log \left (3 \, x\right ) + x} \,d x } \]
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\[ \int \frac {-1+\log ^2(3 x)}{x+x \log (3 x)+x \log ^2(3 x)} \, dx=\int { \frac {\log \left (3 \, x\right )^{2} - 1}{x \log \left (3 \, x\right )^{2} + x \log \left (3 \, x\right ) + x} \,d x } \]
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Time = 1.74 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.88 \[ \int \frac {-1+\log ^2(3 x)}{x+x \log (3 x)+x \log ^2(3 x)} \, dx=\ln \left (x\right )-\frac {\ln \left ({\ln \left (3\,x\right )}^2+\ln \left (3\,x\right )+1\right )}{2}-\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,\left (2\,\ln \left (3\,x\right )+1\right )}{3}\right ) \]
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