Integrand size = 68, antiderivative size = 19 \[ \int \frac {e^{-8+4 x} \left (4 x^3-4 x^4\right )}{e^{-16+8 x}+x^8-2 x^8 \log (3)+x^8 \log ^2(3)+e^{-8+4 x} \left (2 x^4-2 x^4 \log (3)\right )} \, dx=\frac {1}{1+\frac {e^{-8+4 x}}{x^4}-\log (3)} \]
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Time = 0.80 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.37, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {6, 1607, 6820, 12, 6843, 32} \[ \int \frac {e^{-8+4 x} \left (4 x^3-4 x^4\right )}{e^{-16+8 x}+x^8-2 x^8 \log (3)+x^8 \log ^2(3)+e^{-8+4 x} \left (2 x^4-2 x^4 \log (3)\right )} \, dx=\frac {e^8}{\frac {e^{4 x}}{x^4}+e^8 (1-\log (3))} \]
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Rule 6
Rule 12
Rule 32
Rule 1607
Rule 6820
Rule 6843
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{-8+4 x} \left (4 x^3-4 x^4\right )}{e^{-16+8 x}+x^8 (1-2 \log (3))+x^8 \log ^2(3)+e^{-8+4 x} \left (2 x^4-2 x^4 \log (3)\right )} \, dx \\ & = \int \frac {e^{-8+4 x} \left (4 x^3-4 x^4\right )}{e^{-16+8 x}+e^{-8+4 x} \left (2 x^4-2 x^4 \log (3)\right )+x^8 \left (1-2 \log (3)+\log ^2(3)\right )} \, dx \\ & = \int \frac {e^{-8+4 x} (4-4 x) x^3}{e^{-16+8 x}+e^{-8+4 x} \left (2 x^4-2 x^4 \log (3)\right )+x^8 \left (1-2 \log (3)+\log ^2(3)\right )} \, dx \\ & = \int \frac {4 e^{8+4 x} (1-x) x^3}{\left (e^{4 x}-e^8 x^4 (-1+\log (3))\right )^2} \, dx \\ & = 4 \int \frac {e^{8+4 x} (1-x) x^3}{\left (e^{4 x}-e^8 x^4 (-1+\log (3))\right )^2} \, dx \\ & = -\left (e^8 \text {Subst}\left (\int \frac {1}{\left (x-e^8 (-1+\log (3))\right )^2} \, dx,x,\frac {e^{4 x}}{x^4}\right )\right ) \\ & = \frac {e^8}{\frac {e^{4 x}}{x^4}+e^8 (1-\log (3))} \\ \end{align*}
Time = 0.52 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.58 \[ \int \frac {e^{-8+4 x} \left (4 x^3-4 x^4\right )}{e^{-16+8 x}+x^8-2 x^8 \log (3)+x^8 \log ^2(3)+e^{-8+4 x} \left (2 x^4-2 x^4 \log (3)\right )} \, dx=-\frac {4 e^8 x^4}{-4 e^{4 x}+4 e^8 x^4 (-1+\log (3))} \]
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Time = 0.20 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.47
method | result | size |
risch | \(-\frac {x^{4}}{x^{4} \ln \left (3\right )-{\mathrm e}^{4 x -8}-x^{4}}\) | \(28\) |
parallelrisch | \(-\frac {x^{4}}{x^{4} \ln \left (3\right )-{\mathrm e}^{4 x -8}-x^{4}}\) | \(28\) |
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Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.42 \[ \int \frac {e^{-8+4 x} \left (4 x^3-4 x^4\right )}{e^{-16+8 x}+x^8-2 x^8 \log (3)+x^8 \log ^2(3)+e^{-8+4 x} \left (2 x^4-2 x^4 \log (3)\right )} \, dx=-\frac {x^{4}}{x^{4} \log \left (3\right ) - x^{4} - e^{\left (4 \, x - 8\right )}} \]
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Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-8+4 x} \left (4 x^3-4 x^4\right )}{e^{-16+8 x}+x^8-2 x^8 \log (3)+x^8 \log ^2(3)+e^{-8+4 x} \left (2 x^4-2 x^4 \log (3)\right )} \, dx=\frac {x^{4}}{- x^{4} \log {\left (3 \right )} + x^{4} + e^{4 x - 8}} \]
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Time = 0.35 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.37 \[ \int \frac {e^{-8+4 x} \left (4 x^3-4 x^4\right )}{e^{-16+8 x}+x^8-2 x^8 \log (3)+x^8 \log ^2(3)+e^{-8+4 x} \left (2 x^4-2 x^4 \log (3)\right )} \, dx=-\frac {x^{4} e^{8}}{x^{4} {\left (\log \left (3\right ) - 1\right )} e^{8} - e^{\left (4 \, x\right )}} \]
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Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.63 \[ \int \frac {e^{-8+4 x} \left (4 x^3-4 x^4\right )}{e^{-16+8 x}+x^8-2 x^8 \log (3)+x^8 \log ^2(3)+e^{-8+4 x} \left (2 x^4-2 x^4 \log (3)\right )} \, dx=-\frac {x^{4} e^{8}}{x^{4} e^{8} \log \left (3\right ) - x^{4} e^{8} - e^{\left (4 \, x\right )}} \]
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Timed out. \[ \int \frac {e^{-8+4 x} \left (4 x^3-4 x^4\right )}{e^{-16+8 x}+x^8-2 x^8 \log (3)+x^8 \log ^2(3)+e^{-8+4 x} \left (2 x^4-2 x^4 \log (3)\right )} \, dx=\int \frac {{\mathrm {e}}^{4\,x-8}\,\left (4\,x^3-4\,x^4\right )}{{\mathrm {e}}^{8\,x-16}+x^8\,{\ln \left (3\right )}^2-{\mathrm {e}}^{4\,x-8}\,\left (2\,x^4\,\ln \left (3\right )-2\,x^4\right )-2\,x^8\,\ln \left (3\right )+x^8} \,d x \]
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