Integrand size = 68, antiderivative size = 20 \[ \int \frac {-2+5 x+7 x^2+e \left (x+x^2\right )+\left (3 x+21 x^2+e \left (x+3 x^2\right )\right ) \log (x)}{\left (-2 x+5 x^2+7 x^3+e \left (x^2+x^3\right )\right ) \log (x)} \, dx=\log \left ((1+x)^2\right )+\log \left (\left (7+e-\frac {2}{x}\right ) x \log (x)\right ) \]
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Time = 0.12 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {6820, 78, 2339, 29} \[ \int \frac {-2+5 x+7 x^2+e \left (x+x^2\right )+\left (3 x+21 x^2+e \left (x+3 x^2\right )\right ) \log (x)}{\left (-2 x+5 x^2+7 x^3+e \left (x^2+x^3\right )\right ) \log (x)} \, dx=2 \log (x+1)+\log (2-(7+e) x)+\log (\log (x)) \]
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Rule 29
Rule 78
Rule 2339
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3+e+3 (7+e) x}{(1+x) (-2+(7+e) x)}+\frac {1}{x \log (x)}\right ) \, dx \\ & = \int \frac {3+e+3 (7+e) x}{(1+x) (-2+(7+e) x)} \, dx+\int \frac {1}{x \log (x)} \, dx \\ & = \int \left (\frac {2}{1+x}+\frac {-7-e}{2-(7+e) x}\right ) \, dx+\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right ) \\ & = 2 \log (1+x)+\log (2-(7+e) x)+\log (\log (x)) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25 \[ \int \frac {-2+5 x+7 x^2+e \left (x+x^2\right )+\left (3 x+21 x^2+e \left (x+3 x^2\right )\right ) \log (x)}{\left (-2 x+5 x^2+7 x^3+e \left (x^2+x^3\right )\right ) \log (x)} \, dx=2 \log (1+x)+\log (9+e-7 (1+x)-e (1+x))+\log (\log (x)) \]
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Time = 0.19 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05
method | result | size |
default | \(\ln \left (\ln \left (x \right )\right )+\ln \left (x \,{\mathrm e}+7 x -2\right )+2 \ln \left (1+x \right )\) | \(21\) |
norman | \(\ln \left (\ln \left (x \right )\right )+\ln \left (x \,{\mathrm e}+7 x -2\right )+2 \ln \left (1+x \right )\) | \(21\) |
parts | \(\ln \left (\ln \left (x \right )\right )+\ln \left (x \,{\mathrm e}+7 x -2\right )+2 \ln \left (1+x \right )\) | \(21\) |
risch | \(\ln \left (x \left (7+{\mathrm e}\right )-2\right )+2 \ln \left (-1-x \right )+\ln \left (\ln \left (x \right )\right )\) | \(22\) |
parallelrisch | \(\ln \left (\ln \left (x \right )\right )+\ln \left (\frac {x \,{\mathrm e}+7 x -2}{7+{\mathrm e}}\right )+2 \ln \left (1+x \right )\) | \(28\) |
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Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {-2+5 x+7 x^2+e \left (x+x^2\right )+\left (3 x+21 x^2+e \left (x+3 x^2\right )\right ) \log (x)}{\left (-2 x+5 x^2+7 x^3+e \left (x^2+x^3\right )\right ) \log (x)} \, dx=\log \left (x e + 7 \, x - 2\right ) + 2 \, \log \left (x + 1\right ) + \log \left (\log \left (x\right )\right ) \]
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Time = 0.46 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {-2+5 x+7 x^2+e \left (x+x^2\right )+\left (3 x+21 x^2+e \left (x+3 x^2\right )\right ) \log (x)}{\left (-2 x+5 x^2+7 x^3+e \left (x^2+x^3\right )\right ) \log (x)} \, dx=2 \log {\left (x + 1 \right )} + \log {\left (x - \frac {2}{e + 7} \right )} + \log {\left (\log {\left (x \right )} \right )} \]
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Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {-2+5 x+7 x^2+e \left (x+x^2\right )+\left (3 x+21 x^2+e \left (x+3 x^2\right )\right ) \log (x)}{\left (-2 x+5 x^2+7 x^3+e \left (x^2+x^3\right )\right ) \log (x)} \, dx=\log \left (x {\left (e + 7\right )} - 2\right ) + 2 \, \log \left (x + 1\right ) + \log \left (\log \left (x\right )\right ) \]
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Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {-2+5 x+7 x^2+e \left (x+x^2\right )+\left (3 x+21 x^2+e \left (x+3 x^2\right )\right ) \log (x)}{\left (-2 x+5 x^2+7 x^3+e \left (x^2+x^3\right )\right ) \log (x)} \, dx=\log \left (x e + 7 \, x - 2\right ) + 2 \, \log \left (x + 1\right ) + \log \left (\log \left (x\right )\right ) \]
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Time = 15.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {-2+5 x+7 x^2+e \left (x+x^2\right )+\left (3 x+21 x^2+e \left (x+3 x^2\right )\right ) \log (x)}{\left (-2 x+5 x^2+7 x^3+e \left (x^2+x^3\right )\right ) \log (x)} \, dx=2\,\ln \left (x+1\right )+\ln \left (\ln \left (x\right )\right )+\ln \left (7\,x+x\,\mathrm {e}-2\right ) \]
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