Integrand size = 30, antiderivative size = 25 \[ \int \frac {1+2 x+2 x^2-4 e^{2 x^2} x^2-2 \log (x)}{x} \, dx=-5-e^{2 x^2}+2 x+x^2+\log (x)-\log ^2(x) \]
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Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {14, 2240, 2338} \[ \int \frac {1+2 x+2 x^2-4 e^{2 x^2} x^2-2 \log (x)}{x} \, dx=x^2-e^{2 x^2}+2 x-\log ^2(x)+\log (x) \]
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Rule 14
Rule 2240
Rule 2338
Rubi steps \begin{align*} \text {integral}& = \int \left (-4 e^{2 x^2} x+\frac {1+2 x+2 x^2-2 \log (x)}{x}\right ) \, dx \\ & = -\left (4 \int e^{2 x^2} x \, dx\right )+\int \frac {1+2 x+2 x^2-2 \log (x)}{x} \, dx \\ & = -e^{2 x^2}+\int \left (\frac {1+2 x+2 x^2}{x}-\frac {2 \log (x)}{x}\right ) \, dx \\ & = -e^{2 x^2}-2 \int \frac {\log (x)}{x} \, dx+\int \frac {1+2 x+2 x^2}{x} \, dx \\ & = -e^{2 x^2}-\log ^2(x)+\int \left (2+\frac {1}{x}+2 x\right ) \, dx \\ & = -e^{2 x^2}+2 x+x^2+\log (x)-\log ^2(x) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {1+2 x+2 x^2-4 e^{2 x^2} x^2-2 \log (x)}{x} \, dx=-e^{2 x^2}+2 x+x^2+\log (x)-\log ^2(x) \]
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Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96
method | result | size |
default | \(2 x +\ln \left (x \right )+x^{2}-{\mathrm e}^{2 x^{2}}-\ln \left (x \right )^{2}\) | \(24\) |
norman | \(2 x +\ln \left (x \right )+x^{2}-{\mathrm e}^{2 x^{2}}-\ln \left (x \right )^{2}\) | \(24\) |
risch | \(2 x +\ln \left (x \right )+x^{2}-{\mathrm e}^{2 x^{2}}-\ln \left (x \right )^{2}\) | \(24\) |
parallelrisch | \(2 x +\ln \left (x \right )+x^{2}-{\mathrm e}^{2 x^{2}}-\ln \left (x \right )^{2}\) | \(24\) |
parts | \(2 x +\ln \left (x \right )+x^{2}-{\mathrm e}^{2 x^{2}}-\ln \left (x \right )^{2}\) | \(24\) |
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Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {1+2 x+2 x^2-4 e^{2 x^2} x^2-2 \log (x)}{x} \, dx=x^{2} - \log \left (x\right )^{2} + 2 \, x - e^{\left (2 \, x^{2}\right )} + \log \left (x\right ) \]
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Time = 0.12 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {1+2 x+2 x^2-4 e^{2 x^2} x^2-2 \log (x)}{x} \, dx=x^{2} + 2 x - e^{2 x^{2}} - \log {\left (x \right )}^{2} + \log {\left (x \right )} \]
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Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {1+2 x+2 x^2-4 e^{2 x^2} x^2-2 \log (x)}{x} \, dx=x^{2} - \log \left (x\right )^{2} + 2 \, x - e^{\left (2 \, x^{2}\right )} + \log \left (x\right ) \]
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Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {1+2 x+2 x^2-4 e^{2 x^2} x^2-2 \log (x)}{x} \, dx=x^{2} - \log \left (x\right )^{2} + 2 \, x - e^{\left (2 \, x^{2}\right )} + \log \left (x\right ) \]
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Time = 8.91 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {1+2 x+2 x^2-4 e^{2 x^2} x^2-2 \log (x)}{x} \, dx=2\,x-{\mathrm {e}}^{2\,x^2}+\ln \left (x\right )-{\ln \left (x\right )}^2+x^2 \]
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