Integrand size = 24, antiderivative size = 17 \[ \int \frac {-4+e^x+e (-1+x)}{4-e x+e^x x} \, dx=1-x+\log \left (4-e x+e^x x\right ) \]
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\[ \int \frac {-4+e^x+e (-1+x)}{4-e x+e^x x} \, dx=\int \frac {-4+e^x+e (-1+x)}{4-e x+e^x x} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{x}+\frac {-4-4 x+e x^2}{x \left (4-e x+e^x x\right )}\right ) \, dx \\ & = \log (x)+\int \frac {-4-4 x+e x^2}{x \left (4-e x+e^x x\right )} \, dx \\ & = \log (x)+\int \left (-\frac {e x}{-4+e x-e^x x}-\frac {4}{4-e x+e^x x}-\frac {4}{x \left (4-e x+e^x x\right )}\right ) \, dx \\ & = \log (x)-4 \int \frac {1}{4-e x+e^x x} \, dx-4 \int \frac {1}{x \left (4-e x+e^x x\right )} \, dx-e \int \frac {x}{-4+e x-e^x x} \, dx \\ \end{align*}
Time = 0.43 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {-4+e^x+e (-1+x)}{4-e x+e^x x} \, dx=-x+\log \left (4-e x+e^x x\right ) \]
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Time = 0.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00
method | result | size |
norman | \(-x +\ln \left (x \,{\mathrm e}-{\mathrm e}^{x} x -4\right )\) | \(17\) |
parallelrisch | \(-x +\ln \left ({\mathrm e}^{x} x -x \,{\mathrm e}+4\right )\) | \(17\) |
risch | \(\ln \left (x \right )-x +\ln \left ({\mathrm e}^{x}-\frac {x \,{\mathrm e}-4}{x}\right )\) | \(22\) |
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Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.35 \[ \int \frac {-4+e^x+e (-1+x)}{4-e x+e^x x} \, dx=-x + \log \left (x\right ) + \log \left (-\frac {x e - x e^{x} - 4}{x}\right ) \]
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Time = 0.11 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {-4+e^x+e (-1+x)}{4-e x+e^x x} \, dx=- x + \log {\left (x \right )} + \log {\left (e^{x} + \frac {- e x + 4}{x} \right )} \]
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Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.35 \[ \int \frac {-4+e^x+e (-1+x)}{4-e x+e^x x} \, dx=-x + \log \left (x\right ) + \log \left (-\frac {x e - x e^{x} - 4}{x}\right ) \]
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Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {-4+e^x+e (-1+x)}{4-e x+e^x x} \, dx=-x + \log \left (x e - x e^{x} - 4\right ) \]
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Time = 0.10 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {-4+e^x+e (-1+x)}{4-e x+e^x x} \, dx=\ln \left (x\,{\mathrm {e}}^x-x\,\mathrm {e}+4\right )-x \]
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