Integrand size = 51, antiderivative size = 19 \[ \int \frac {e^{-x} \left (-12 x^2+7 x^3-x^4+x^5+\left (-20 x^4+9 x^5-x^6\right ) \log (-4+x)\right )}{-4+x} \, dx=e^{-x} x^2 \left (x+x^3 \log (-4+x)\right ) \]
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Time = 0.50 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21, number of steps used = 39, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.137, Rules used = {6874, 2230, 2225, 2209, 2207, 2227, 2634} \[ \int \frac {e^{-x} \left (-12 x^2+7 x^3-x^4+x^5+\left (-20 x^4+9 x^5-x^6\right ) \log (-4+x)\right )}{-4+x} \, dx=e^{-x} x^5 \log (x-4)+e^{-x} x^3 \]
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Rule 2207
Rule 2209
Rule 2225
Rule 2227
Rule 2230
Rule 2634
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {e^{-x} x^2 \left (-12+7 x-x^2+x^3\right )}{-4+x}-e^{-x} (-5+x) x^4 \log (-4+x)\right ) \, dx \\ & = \int \frac {e^{-x} x^2 \left (-12+7 x-x^2+x^3\right )}{-4+x} \, dx-\int e^{-x} (-5+x) x^4 \log (-4+x) \, dx \\ & = e^{-x} x^5 \log (-4+x)-\int \frac {e^{-x} x^5}{-4+x} \, dx+\int \left (256 e^{-x}+\frac {1024 e^{-x}}{-4+x}+64 e^{-x} x+19 e^{-x} x^2+3 e^{-x} x^3+e^{-x} x^4\right ) \, dx \\ & = e^{-x} x^5 \log (-4+x)+3 \int e^{-x} x^3 \, dx+19 \int e^{-x} x^2 \, dx+64 \int e^{-x} x \, dx+256 \int e^{-x} \, dx+1024 \int \frac {e^{-x}}{-4+x} \, dx+\int e^{-x} x^4 \, dx-\int \left (256 e^{-x}+\frac {1024 e^{-x}}{-4+x}+64 e^{-x} x+16 e^{-x} x^2+4 e^{-x} x^3+e^{-x} x^4\right ) \, dx \\ & = -256 e^{-x}-64 e^{-x} x-19 e^{-x} x^2-3 e^{-x} x^3-e^{-x} x^4+\frac {1024 \text {Ei}(4-x)}{e^4}+e^{-x} x^5 \log (-4+x)+9 \int e^{-x} x^2 \, dx-16 \int e^{-x} x^2 \, dx+38 \int e^{-x} x \, dx+64 \int e^{-x} \, dx-64 \int e^{-x} x \, dx-256 \int e^{-x} \, dx-1024 \int \frac {e^{-x}}{-4+x} \, dx-\int e^{-x} x^4 \, dx \\ & = -64 e^{-x}-38 e^{-x} x-12 e^{-x} x^2-3 e^{-x} x^3+e^{-x} x^5 \log (-4+x)-4 \int e^{-x} x^3 \, dx+18 \int e^{-x} x \, dx-32 \int e^{-x} x \, dx+38 \int e^{-x} \, dx-64 \int e^{-x} \, dx \\ & = -38 e^{-x}-24 e^{-x} x-12 e^{-x} x^2+e^{-x} x^3+e^{-x} x^5 \log (-4+x)-12 \int e^{-x} x^2 \, dx+18 \int e^{-x} \, dx-32 \int e^{-x} \, dx \\ & = -24 e^{-x}-24 e^{-x} x+e^{-x} x^3+e^{-x} x^5 \log (-4+x)-24 \int e^{-x} x \, dx \\ & = -24 e^{-x}+e^{-x} x^3+e^{-x} x^5 \log (-4+x)-24 \int e^{-x} \, dx \\ & = e^{-x} x^3+e^{-x} x^5 \log (-4+x) \\ \end{align*}
Time = 1.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-x} \left (-12 x^2+7 x^3-x^4+x^5+\left (-20 x^4+9 x^5-x^6\right ) \log (-4+x)\right )}{-4+x} \, dx=e^{-x} x^3 \left (1+x^2 \log (-4+x)\right ) \]
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Time = 0.74 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16
method | result | size |
risch | \(x^{5} {\mathrm e}^{-x} \ln \left (x -4\right )+x^{3} {\mathrm e}^{-x}\) | \(22\) |
parallelrisch | \(-\frac {\left (-8 \ln \left (x -4\right ) x^{5}-8 x^{3}\right ) {\mathrm e}^{-x}}{8}\) | \(22\) |
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Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {e^{-x} \left (-12 x^2+7 x^3-x^4+x^5+\left (-20 x^4+9 x^5-x^6\right ) \log (-4+x)\right )}{-4+x} \, dx=x^{5} e^{\left (-x\right )} \log \left (x - 4\right ) + x^{3} e^{\left (-x\right )} \]
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Time = 0.13 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int \frac {e^{-x} \left (-12 x^2+7 x^3-x^4+x^5+\left (-20 x^4+9 x^5-x^6\right ) \log (-4+x)\right )}{-4+x} \, dx=\left (x^{5} \log {\left (x - 4 \right )} + x^{3}\right ) e^{- x} \]
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Time = 0.23 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {e^{-x} \left (-12 x^2+7 x^3-x^4+x^5+\left (-20 x^4+9 x^5-x^6\right ) \log (-4+x)\right )}{-4+x} \, dx=x^{5} e^{\left (-x\right )} \log \left (x - 4\right ) + x^{3} e^{\left (-x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (18) = 36\).
Time = 0.28 (sec) , antiderivative size = 142, normalized size of antiderivative = 7.47 \[ \int \frac {e^{-x} \left (-12 x^2+7 x^3-x^4+x^5+\left (-20 x^4+9 x^5-x^6\right ) \log (-4+x)\right )}{-4+x} \, dx={\left ({\left (x - 4\right )}^{5} e^{\left (-x + 4\right )} \log \left (x - 4\right ) + 20 \, {\left (x - 4\right )}^{4} e^{\left (-x + 4\right )} \log \left (x - 4\right ) + 160 \, {\left (x - 4\right )}^{3} e^{\left (-x + 4\right )} \log \left (x - 4\right ) + {\left (x - 4\right )}^{3} e^{\left (-x + 4\right )} + 640 \, {\left (x - 4\right )}^{2} e^{\left (-x + 4\right )} \log \left (x - 4\right ) + 12 \, {\left (x - 4\right )}^{2} e^{\left (-x + 4\right )} + 1280 \, {\left (x - 4\right )} e^{\left (-x + 4\right )} \log \left (x - 4\right ) + 48 \, {\left (x - 4\right )} e^{\left (-x + 4\right )} + 1024 \, e^{\left (-x + 4\right )} \log \left (x - 4\right ) + 64 \, e^{\left (-x + 4\right )}\right )} e^{\left (-4\right )} \]
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Time = 12.68 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {e^{-x} \left (-12 x^2+7 x^3-x^4+x^5+\left (-20 x^4+9 x^5-x^6\right ) \log (-4+x)\right )}{-4+x} \, dx=x^3\,{\mathrm {e}}^{-x}\,\left (x^2\,\ln \left (x-4\right )+1\right ) \]
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