Integrand size = 29, antiderivative size = 27 \[ \int \frac {-5+\left (-1+e^{4+x} x-8 x^2\right ) \log ^2(x)}{x \log ^2(x)} \, dx=e^{4+x}-4 x^2-\frac {\log (10)}{3}+\frac {5}{\log (x)}-\log (x) \]
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Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {6874, 2225, 6820, 14, 2339, 30} \[ \int \frac {-5+\left (-1+e^{4+x} x-8 x^2\right ) \log ^2(x)}{x \log ^2(x)} \, dx=-4 x^2+e^{x+4}-\log (x)+\frac {5}{\log (x)} \]
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Rule 14
Rule 30
Rule 2225
Rule 2339
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (e^{4+x}+\frac {-5-\log ^2(x)-8 x^2 \log ^2(x)}{x \log ^2(x)}\right ) \, dx \\ & = \int e^{4+x} \, dx+\int \frac {-5-\log ^2(x)-8 x^2 \log ^2(x)}{x \log ^2(x)} \, dx \\ & = e^{4+x}+\int \frac {-1-8 x^2-\frac {5}{\log ^2(x)}}{x} \, dx \\ & = e^{4+x}+\int \left (\frac {-1-8 x^2}{x}-\frac {5}{x \log ^2(x)}\right ) \, dx \\ & = e^{4+x}-5 \int \frac {1}{x \log ^2(x)} \, dx+\int \frac {-1-8 x^2}{x} \, dx \\ & = e^{4+x}-5 \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (x)\right )+\int \left (-\frac {1}{x}-8 x\right ) \, dx \\ & = e^{4+x}-4 x^2+\frac {5}{\log (x)}-\log (x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {-5+\left (-1+e^{4+x} x-8 x^2\right ) \log ^2(x)}{x \log ^2(x)} \, dx=e^{4+x}-4 x^2+\frac {5}{\log (x)}-\log (x) \]
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Time = 0.04 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78
method | result | size |
default | \(\frac {5}{\ln \left (x \right )}-4 x^{2}-\ln \left (x \right )+{\mathrm e}^{4+x}\) | \(21\) |
risch | \(\frac {5}{\ln \left (x \right )}-4 x^{2}-\ln \left (x \right )+{\mathrm e}^{4+x}\) | \(21\) |
parts | \(\frac {5}{\ln \left (x \right )}-4 x^{2}-\ln \left (x \right )+{\mathrm e}^{4+x}\) | \(21\) |
norman | \(\frac {5-\ln \left (x \right )^{2}+{\mathrm e}^{4+x} \ln \left (x \right )-4 x^{2} \ln \left (x \right )}{\ln \left (x \right )}\) | \(28\) |
parallelrisch | \(\frac {5-\ln \left (x \right )^{2}+{\mathrm e}^{4+x} \ln \left (x \right )-4 x^{2} \ln \left (x \right )}{\ln \left (x \right )}\) | \(28\) |
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Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {-5+\left (-1+e^{4+x} x-8 x^2\right ) \log ^2(x)}{x \log ^2(x)} \, dx=-\frac {{\left (4 \, x^{2} - e^{\left (x + 4\right )}\right )} \log \left (x\right ) + \log \left (x\right )^{2} - 5}{\log \left (x\right )} \]
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Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.63 \[ \int \frac {-5+\left (-1+e^{4+x} x-8 x^2\right ) \log ^2(x)}{x \log ^2(x)} \, dx=- 4 x^{2} + e^{x + 4} - \log {\left (x \right )} + \frac {5}{\log {\left (x \right )}} \]
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none
Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {-5+\left (-1+e^{4+x} x-8 x^2\right ) \log ^2(x)}{x \log ^2(x)} \, dx=-4 \, x^{2} + \frac {5}{\log \left (x\right )} + e^{\left (x + 4\right )} - \log \left (x\right ) \]
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Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {-5+\left (-1+e^{4+x} x-8 x^2\right ) \log ^2(x)}{x \log ^2(x)} \, dx=-\frac {4 \, x^{2} \log \left (x\right ) - e^{\left (x + 4\right )} \log \left (x\right ) + \log \left (x\right )^{2} - 5}{\log \left (x\right )} \]
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Time = 9.17 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {-5+\left (-1+e^{4+x} x-8 x^2\right ) \log ^2(x)}{x \log ^2(x)} \, dx={\mathrm {e}}^{x+4}-\ln \left (x\right )+\frac {5}{\ln \left (x\right )}-4\,x^2 \]
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