Integrand size = 31, antiderivative size = 20 \[ \int \frac {-3 \log (3)-3 \log (3) \log (x)+\left (-x+x^2\right ) \log ^2(x)}{x^2 \log ^2(x)} \, dx=e^3+x+\frac {3 \log (3)}{x \log (x)}-\log (x) \]
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Time = 0.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {6874, 45, 2343, 2346, 2209} \[ \int \frac {-3 \log (3)-3 \log (3) \log (x)+\left (-x+x^2\right ) \log ^2(x)}{x^2 \log ^2(x)} \, dx=x-\log (x)+\frac {3 \log (3)}{x \log (x)} \]
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Rule 45
Rule 2209
Rule 2343
Rule 2346
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {-1+x}{x}-\frac {3 \log (3)}{x^2 \log ^2(x)}-\frac {3 \log (3)}{x^2 \log (x)}\right ) \, dx \\ & = -\left ((3 \log (3)) \int \frac {1}{x^2 \log ^2(x)} \, dx\right )-(3 \log (3)) \int \frac {1}{x^2 \log (x)} \, dx+\int \frac {-1+x}{x} \, dx \\ & = \frac {3 \log (3)}{x \log (x)}+(3 \log (3)) \int \frac {1}{x^2 \log (x)} \, dx-(3 \log (3)) \text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right )+\int \left (1-\frac {1}{x}\right ) \, dx \\ & = x-3 \operatorname {ExpIntegralEi}(-\log (x)) \log (3)+\frac {3 \log (3)}{x \log (x)}-\log (x)+(3 \log (3)) \text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right ) \\ & = x+\frac {3 \log (3)}{x \log (x)}-\log (x) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {-3 \log (3)-3 \log (3) \log (x)+\left (-x+x^2\right ) \log ^2(x)}{x^2 \log ^2(x)} \, dx=x+\frac {3 \log (3)}{x \log (x)}-\log (x) \]
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Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90
method | result | size |
risch | \(x -\ln \left (x \right )+\frac {3 \ln \left (3\right )}{\ln \left (x \right ) x}\) | \(18\) |
norman | \(\frac {x^{2} \ln \left (x \right )-x \ln \left (x \right )^{2}+3 \ln \left (3\right )}{x \ln \left (x \right )}\) | \(27\) |
parallelrisch | \(\frac {x^{2} \ln \left (x \right )-x \ln \left (x \right )^{2}+3 \ln \left (3\right )}{x \ln \left (x \right )}\) | \(27\) |
default | \(x -\ln \left (x \right )+3 \ln \left (3\right ) \operatorname {Ei}_{1}\left (\ln \left (x \right )\right )-3 \ln \left (3\right ) \left (-\frac {1}{x \ln \left (x \right )}+\operatorname {Ei}_{1}\left (\ln \left (x \right )\right )\right )\) | \(33\) |
parts | \(x -\ln \left (x \right )+3 \ln \left (3\right ) \operatorname {Ei}_{1}\left (\ln \left (x \right )\right )-3 \ln \left (3\right ) \left (-\frac {1}{x \ln \left (x \right )}+\operatorname {Ei}_{1}\left (\ln \left (x \right )\right )\right )\) | \(33\) |
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none
Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30 \[ \int \frac {-3 \log (3)-3 \log (3) \log (x)+\left (-x+x^2\right ) \log ^2(x)}{x^2 \log ^2(x)} \, dx=\frac {x^{2} \log \left (x\right ) - x \log \left (x\right )^{2} + 3 \, \log \left (3\right )}{x \log \left (x\right )} \]
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Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70 \[ \int \frac {-3 \log (3)-3 \log (3) \log (x)+\left (-x+x^2\right ) \log ^2(x)}{x^2 \log ^2(x)} \, dx=x - \log {\left (x \right )} + \frac {3 \log {\left (3 \right )}}{x \log {\left (x \right )}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {-3 \log (3)-3 \log (3) \log (x)+\left (-x+x^2\right ) \log ^2(x)}{x^2 \log ^2(x)} \, dx=-3 \, {\rm Ei}\left (-\log \left (x\right )\right ) \log \left (3\right ) + 3 \, \Gamma \left (-1, \log \left (x\right )\right ) \log \left (3\right ) + x - \log \left (x\right ) \]
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none
Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {-3 \log (3)-3 \log (3) \log (x)+\left (-x+x^2\right ) \log ^2(x)}{x^2 \log ^2(x)} \, dx=x + \frac {3 \, \log \left (3\right )}{x \log \left (x\right )} - \log \left (x\right ) \]
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Time = 7.78 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {-3 \log (3)-3 \log (3) \log (x)+\left (-x+x^2\right ) \log ^2(x)}{x^2 \log ^2(x)} \, dx=x-\ln \left (x\right )+\frac {3\,\ln \left (3\right )}{x\,\ln \left (x\right )} \]
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