Optimal. Leaf size=32 \[ \frac {3 e^{-x} x}{1+\frac {e^x}{-4+3 \left (x-e^{-x} x\right )}} \]
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Rubi [F] time = 9.67, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {3 e^{-x} x \left (9 x^2-9 x^3+e^{3 x} \left (-4+14 x-6 x^2\right )+e^{2 x} \left (16-46 x+42 x^2-9 x^3\right )+e^x \left (24 x-42 x^2+18 x^3\right )\right )}{e^{4 x} x+9 x^3+e^{3 x} \left (-8 x+6 x^2\right )+e^x \left (24 x^2-18 x^3\right )+e^{2 x} \left (16 x-30 x^2+9 x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=3 \int \frac {e^{-x} x \left (9 x^2-9 x^3+e^{3 x} \left (-4+14 x-6 x^2\right )+e^{2 x} \left (16-46 x+42 x^2-9 x^3\right )+e^x \left (24 x-42 x^2+18 x^3\right )\right )}{e^{4 x} x+9 x^3+e^{3 x} \left (-8 x+6 x^2\right )+e^x \left (24 x^2-18 x^3\right )+e^{2 x} \left (16 x-30 x^2+9 x^3\right )} \, dx\\ &=3 \int \frac {e^{-x} \left (-9 (-1+x) x^2-2 e^{3 x} \left (2-7 x+3 x^2\right )+6 e^x x \left (4-7 x+3 x^2\right )+e^{2 x} \left (16-46 x+42 x^2-9 x^3\right )\right )}{\left (e^{2 x}-3 x+e^x (-4+3 x)\right )^2} \, dx\\ &=3 \int \left (-\frac {e^{-x} \left (4 e^x-22 x-14 e^x x+24 x^2+6 e^x x^2-9 x^3\right )}{-4 e^x+e^{2 x}-3 x+3 e^x x}-\frac {e^{-x} x \left (-100 e^x-75 x+162 e^x x+81 x^2-108 e^x x^2-27 x^3+27 e^x x^3\right )}{\left (-4 e^x+e^{2 x}-3 x+3 e^x x\right )^2}\right ) \, dx\\ &=-\left (3 \int \frac {e^{-x} \left (4 e^x-22 x-14 e^x x+24 x^2+6 e^x x^2-9 x^3\right )}{-4 e^x+e^{2 x}-3 x+3 e^x x} \, dx\right )-3 \int \frac {e^{-x} x \left (-100 e^x-75 x+162 e^x x+81 x^2-108 e^x x^2-27 x^3+27 e^x x^3\right )}{\left (-4 e^x+e^{2 x}-3 x+3 e^x x\right )^2} \, dx\\ &=-\left (3 \int \left (-\frac {100 x}{\left (-4 e^x+e^{2 x}-3 x+3 e^x x\right )^2}+\frac {162 x^2}{\left (-4 e^x+e^{2 x}-3 x+3 e^x x\right )^2}-\frac {75 e^{-x} x^2}{\left (-4 e^x+e^{2 x}-3 x+3 e^x x\right )^2}-\frac {108 x^3}{\left (-4 e^x+e^{2 x}-3 x+3 e^x x\right )^2}+\frac {81 e^{-x} x^3}{\left (-4 e^x+e^{2 x}-3 x+3 e^x x\right )^2}+\frac {27 x^4}{\left (-4 e^x+e^{2 x}-3 x+3 e^x x\right )^2}-\frac {27 e^{-x} x^4}{\left (-4 e^x+e^{2 x}-3 x+3 e^x x\right )^2}\right ) \, dx\right )-3 \int \left (\frac {4}{-4 e^x+e^{2 x}-3 x+3 e^x x}-\frac {14 x}{-4 e^x+e^{2 x}-3 x+3 e^x x}-\frac {22 e^{-x} x}{-4 e^x+e^{2 x}-3 x+3 e^x x}+\frac {6 x^2}{-4 e^x+e^{2 x}-3 x+3 e^x x}+\frac {24 e^{-x} x^2}{-4 e^x+e^{2 x}-3 x+3 e^x x}-\frac {9 e^{-x} x^3}{-4 e^x+e^{2 x}-3 x+3 e^x x}\right ) \, dx\\ &=-\left (12 \int \frac {1}{-4 e^x+e^{2 x}-3 x+3 e^x x} \, dx\right )-18 \int \frac {x^2}{-4 e^x+e^{2 x}-3 x+3 e^x x} \, dx+27 \int \frac {e^{-x} x^3}{-4 e^x+e^{2 x}-3 x+3 e^x x} \, dx+42 \int \frac {x}{-4 e^x+e^{2 x}-3 x+3 e^x x} \, dx+66 \int \frac {e^{-x} x}{-4 e^x+e^{2 x}-3 x+3 e^x x} \, dx-72 \int \frac {e^{-x} x^2}{-4 e^x+e^{2 x}-3 x+3 e^x x} \, dx-81 \int \frac {x^4}{\left (-4 e^x+e^{2 x}-3 x+3 e^x x\right )^2} \, dx+81 \int \frac {e^{-x} x^4}{\left (-4 e^x+e^{2 x}-3 x+3 e^x x\right )^2} \, dx+225 \int \frac {e^{-x} x^2}{\left (-4 e^x+e^{2 x}-3 x+3 e^x x\right )^2} \, dx-243 \int \frac {e^{-x} x^3}{\left (-4 e^x+e^{2 x}-3 x+3 e^x x\right )^2} \, dx+300 \int \frac {x}{\left (-4 e^x+e^{2 x}-3 x+3 e^x x\right )^2} \, dx+324 \int \frac {x^3}{\left (-4 e^x+e^{2 x}-3 x+3 e^x x\right )^2} \, dx-486 \int \frac {x^2}{\left (-4 e^x+e^{2 x}-3 x+3 e^x x\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.07, size = 41, normalized size = 1.28 \begin {gather*} \frac {3 e^{-x} x \left (-3 x+e^x (-4+3 x)\right )}{e^{2 x}-3 x+e^x (-4+3 x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 72, normalized size = 2.25 \begin {gather*} \frac {{\left (3 \, x - 4\right )} e^{\left (-2 \, x + 2 \, \log \relax (3) + 2 \, \log \relax (x)\right )} - e^{\left (-3 \, x + 3 \, \log \relax (3) + 3 \, \log \relax (x)\right )}}{{\left (3 \, x - 4\right )} e^{\left (-x + \log \relax (3) + \log \relax (x)\right )} + 3 \, x - e^{\left (-2 \, x + 2 \, \log \relax (3) + 2 \, \log \relax (x)\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.57, size = 52, normalized size = 1.62 \begin {gather*} \frac {3 \, {\left (3 \, x^{2} e^{x} - 3 \, x^{2} - x e^{\left (2 \, x\right )} - 4 \, x e^{x}\right )}}{3 \, x e^{\left (2 \, x\right )} - 3 \, x e^{x} + e^{\left (3 \, x\right )} - 4 \, e^{\left (2 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 33, normalized size = 1.03
method | result | size |
risch | \(3 x \,{\mathrm e}^{-x}-\frac {3 x \,{\mathrm e}^{x}}{{\mathrm e}^{2 x}+3 \,{\mathrm e}^{x} x -4 \,{\mathrm e}^{x}-3 x}\) | \(33\) |
norman | \(\frac {\left (-12 \,{\mathrm e}^{x} x -9 x^{2}+9 \,{\mathrm e}^{x} x^{2}\right ) {\mathrm e}^{-x}}{{\mathrm e}^{2 x}+3 \,{\mathrm e}^{x} x -4 \,{\mathrm e}^{x}-3 x}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 29, normalized size = 0.91 \begin {gather*} \frac {3 \, {\left (3 \, x^{2} - 4 \, x\right )}}{{\left (3 \, x - 4\right )} e^{x} - 3 \, x + e^{\left (2 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.13, size = 46, normalized size = 1.44 \begin {gather*} \frac {12\,x\,{\mathrm {e}}^x-9\,x^2\,{\mathrm {e}}^x+9\,x^2}{4\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{3\,x}-3\,x\,{\mathrm {e}}^{2\,x}+3\,x\,{\mathrm {e}}^x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.28, size = 29, normalized size = 0.91 \begin {gather*} 3 x e^{- x} - \frac {3 x e^{x}}{- 3 x + \left (3 x - 4\right ) e^{x} + e^{2 x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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