Integrand size = 23, antiderivative size = 20 \[ \int \frac {2-x-2 e^x x-4 x^2}{2 x} \, dx=-x \left (\frac {1}{2}+x\right )+\log \left (3 e^{-e^x} x\right ) \]
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Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {12, 14, 2225} \[ \int \frac {2-x-2 e^x x-4 x^2}{2 x} \, dx=-x^2-\frac {x}{2}-e^x+\log (x) \]
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Rule 12
Rule 14
Rule 2225
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \frac {2-x-2 e^x x-4 x^2}{x} \, dx \\ & = \frac {1}{2} \int \left (-2 e^x+\frac {2-x-4 x^2}{x}\right ) \, dx \\ & = \frac {1}{2} \int \frac {2-x-4 x^2}{x} \, dx-\int e^x \, dx \\ & = -e^x+\frac {1}{2} \int \left (-1+\frac {2}{x}-4 x\right ) \, dx \\ & = -e^x-\frac {x}{2}-x^2+\log (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {2-x-2 e^x x-4 x^2}{2 x} \, dx=-e^x-\frac {x}{2}-x^2+\log (x) \]
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Time = 0.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80
method | result | size |
default | \(-x^{2}-\frac {x}{2}+\ln \left (x \right )-{\mathrm e}^{x}\) | \(16\) |
norman | \(-x^{2}-\frac {x}{2}+\ln \left (x \right )-{\mathrm e}^{x}\) | \(16\) |
risch | \(-x^{2}-\frac {x}{2}+\ln \left (x \right )-{\mathrm e}^{x}\) | \(16\) |
parallelrisch | \(-x^{2}-\frac {x}{2}+\ln \left (x \right )-{\mathrm e}^{x}\) | \(16\) |
parts | \(-x^{2}-\frac {x}{2}+\ln \left (x \right )-{\mathrm e}^{x}\) | \(16\) |
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Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \frac {2-x-2 e^x x-4 x^2}{2 x} \, dx=-x^{2} - \frac {1}{2} \, x - e^{x} + \log \left (x\right ) \]
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Time = 0.06 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.60 \[ \int \frac {2-x-2 e^x x-4 x^2}{2 x} \, dx=- x^{2} - \frac {x}{2} - e^{x} + \log {\left (x \right )} \]
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Time = 0.18 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \frac {2-x-2 e^x x-4 x^2}{2 x} \, dx=-x^{2} - \frac {1}{2} \, x - e^{x} + \log \left (x\right ) \]
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Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \frac {2-x-2 e^x x-4 x^2}{2 x} \, dx=-x^{2} - \frac {1}{2} \, x - e^{x} + \log \left (x\right ) \]
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Time = 12.07 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \frac {2-x-2 e^x x-4 x^2}{2 x} \, dx=\ln \left (x\right )-{\mathrm {e}}^x-\frac {x}{2}-x^2 \]
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