Integrand size = 39, antiderivative size = 20 \[ \int \frac {-1+7 x-x^2-x \log \left (4 e^{-x} x\right )}{4 x-8 x^2+4 x^3} \, dx=\frac {-5+\log \left (4 e^{-x} x\right )}{4 (-1+x)} \]
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Time = 0.13 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.60, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1608, 27, 12, 6874, 907, 2631, 29} \[ \int \frac {-1+7 x-x^2-x \log \left (4 e^{-x} x\right )}{4 x-8 x^2+4 x^3} \, dx=\frac {5}{4 (1-x)}-\frac {\log \left (4 e^{-x} x\right )}{4 (1-x)} \]
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Rule 12
Rule 27
Rule 29
Rule 907
Rule 1608
Rule 2631
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {-1+7 x-x^2-x \log \left (4 e^{-x} x\right )}{x \left (4-8 x+4 x^2\right )} \, dx \\ & = \int \frac {-1+7 x-x^2-x \log \left (4 e^{-x} x\right )}{4 (-1+x)^2 x} \, dx \\ & = \frac {1}{4} \int \frac {-1+7 x-x^2-x \log \left (4 e^{-x} x\right )}{(-1+x)^2 x} \, dx \\ & = \frac {1}{4} \int \left (\frac {-1+7 x-x^2}{(-1+x)^2 x}-\frac {\log \left (4 e^{-x} x\right )}{(-1+x)^2}\right ) \, dx \\ & = \frac {1}{4} \int \frac {-1+7 x-x^2}{(-1+x)^2 x} \, dx-\frac {1}{4} \int \frac {\log \left (4 e^{-x} x\right )}{(-1+x)^2} \, dx \\ & = -\frac {\log \left (4 e^{-x} x\right )}{4 (1-x)}+\frac {1}{4} \int \left (\frac {5}{(-1+x)^2}-\frac {1}{x}\right ) \, dx+\frac {1}{4} \int \frac {1}{x} \, dx \\ & = \frac {5}{4 (1-x)}-\frac {\log \left (4 e^{-x} x\right )}{4 (1-x)} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {-1+7 x-x^2-x \log \left (4 e^{-x} x\right )}{4 x-8 x^2+4 x^3} \, dx=-\frac {5-\log \left (4 e^{-x} x\right )}{4 (-1+x)} \]
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Time = 0.14 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90
method | result | size |
parallelrisch | \(\frac {-5+\ln \left (4 x \,{\mathrm e}^{-x}\right )}{-4+4 x}\) | \(18\) |
norman | \(\frac {\frac {\ln \left (4 x \,{\mathrm e}^{-x}\right )}{4}-\frac {5}{4}}{-1+x}\) | \(19\) |
default | \(\frac {\ln \left (4 x \,{\mathrm e}^{-x}\right )}{-4+4 x}-\frac {5}{4 \left (-1+x \right )}\) | \(24\) |
parts | \(\frac {\ln \left (4 x \,{\mathrm e}^{-x}\right )}{-4+4 x}-\frac {5}{4 \left (-1+x \right )}\) | \(24\) |
risch | \(-\frac {\ln \left ({\mathrm e}^{x}\right )}{4 \left (-1+x \right )}+\frac {-10-i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i x \,{\mathrm e}^{-x}\right )+i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{2}+i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{2}-i \pi \operatorname {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{3}+4 \ln \left (2\right )+2 \ln \left (x \right )}{8 x -8}\) | \(113\) |
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Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {-1+7 x-x^2-x \log \left (4 e^{-x} x\right )}{4 x-8 x^2+4 x^3} \, dx=\frac {\log \left (4 \, x e^{\left (-x\right )}\right ) - 5}{4 \, {\left (x - 1\right )}} \]
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Time = 0.11 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {-1+7 x-x^2-x \log \left (4 e^{-x} x\right )}{4 x-8 x^2+4 x^3} \, dx=\frac {\log {\left (4 x e^{- x} \right )}}{4 x - 4} - \frac {5}{4 x - 4} \]
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Time = 0.21 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {-1+7 x-x^2-x \log \left (4 e^{-x} x\right )}{4 x-8 x^2+4 x^3} \, dx=\frac {\log \left (4 \, x e^{\left (-x\right )}\right )}{4 \, {\left (x - 1\right )}} - \frac {5}{4 \, {\left (x - 1\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {-1+7 x-x^2-x \log \left (4 e^{-x} x\right )}{4 x-8 x^2+4 x^3} \, dx=\frac {\log \left (2\right ) - 3}{2 \, {\left (x - 1\right )}} + \frac {\log \left (x\right )}{4 \, {\left (x - 1\right )}} \]
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Time = 11.97 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {-1+7 x-x^2-x \log \left (4 e^{-x} x\right )}{4 x-8 x^2+4 x^3} \, dx=\frac {2\,\ln \left (2\right )-6\,x+\ln \left (x\right )}{4\,\left (x-1\right )} \]
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