Optimal. Leaf size=19 \[ \frac {1}{1+\frac {e^{-8+4 x}}{x^4}-\log (3)} \]
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Rubi [A] time = 1.51, antiderivative size = 26, normalized size of antiderivative = 1.37, number of steps used = 7, number of rules used = 6, integrand size = 68, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {6, 1593, 6688, 12, 6711, 32} \begin {gather*} \frac {e^8}{\frac {e^{4 x}}{x^4}+e^8 (1-\log (3))} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 32
Rule 1593
Rule 6688
Rule 6711
Rubi steps
\begin {gather*} \begin {aligned} \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 \operatorname {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 {aligned} \end {gather*}
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Mathematica [A] time = 0.27, size = 30, normalized size = 1.58 \begin {gather*} -\frac {4 e^8 x^4}{-4 e^{4 x}+4 e^8 x^4 (-1+\log (3))} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 27, normalized size = 1.42 \begin {gather*} -\frac {x^{4}}{x^{4} \log \relax (3) - x^{4} - e^{\left (4 \, x - 8\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 31, normalized size = 1.63 \begin {gather*} -\frac {x^{4} e^{8}}{x^{4} e^{8} \log \relax (3) - x^{4} e^{8} - e^{\left (4 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 28, normalized size = 1.47
method | result | size |
risch | \(-\frac {x^{4}}{x^{4} \ln \relax (3)-{\mathrm e}^{4 x -8}-x^{4}}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.26, size = 26, normalized size = 1.37 \begin {gather*} -\frac {x^{4} e^{8}}{x^{4} {\left (\log \relax (3) - 1\right )} e^{8} - e^{\left (4 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {{\mathrm {e}}^{4\,x-8}\,\left (4\,x^3-4\,x^4\right )}{{\mathrm {e}}^{8\,x-16}+x^8\,{\ln \relax (3)}^2-{\mathrm {e}}^{4\,x-8}\,\left (2\,x^4\,\ln \relax (3)-2\,x^4\right )-2\,x^8\,\ln \relax (3)+x^8} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 19, normalized size = 1.00 \begin {gather*} \frac {x^{4}}{- x^{4} \log {\relax (3 )} + x^{4} + e^{4 x - 8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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