Optimal. Leaf size=18 \[ e^{\frac {6}{-\frac {e^x}{x}+x}} x \]
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Rubi [F] time = 3.52, 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 {e^{-\frac {6 x}{e^x-x^2}} \left (e^{2 x}-6 x^3+x^4+e^x \left (-6 x+4 x^2\right )\right )}{e^{2 x}-2 e^x x^2+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-\frac {6 x}{e^x-x^2}} \left (e^{2 x}-6 x^3+x^4+e^x \left (-6 x+4 x^2\right )\right )}{\left (e^x-x^2\right )^2} \, dx\\ &=\int \left (e^{-\frac {6 x}{e^x-x^2}}+\frac {6 e^{-\frac {6 x}{e^x-x^2}} (-2+x) x^3}{\left (-e^x+x^2\right )^2}-\frac {6 e^{-\frac {6 x}{e^x-x^2}} (-1+x) x}{-e^x+x^2}\right ) \, dx\\ &=6 \int \frac {e^{-\frac {6 x}{e^x-x^2}} (-2+x) x^3}{\left (-e^x+x^2\right )^2} \, dx-6 \int \frac {e^{-\frac {6 x}{e^x-x^2}} (-1+x) x}{-e^x+x^2} \, dx+\int e^{-\frac {6 x}{e^x-x^2}} \, dx\\ &=6 \int \left (-\frac {2 e^{-\frac {6 x}{e^x-x^2}} x^3}{\left (-e^x+x^2\right )^2}+\frac {e^{-\frac {6 x}{e^x-x^2}} x^4}{\left (-e^x+x^2\right )^2}\right ) \, dx-6 \int \left (-\frac {e^{-\frac {6 x}{e^x-x^2}} x}{-e^x+x^2}+\frac {e^{-\frac {6 x}{e^x-x^2}} x^2}{-e^x+x^2}\right ) \, dx+\int e^{-\frac {6 x}{e^x-x^2}} \, dx\\ &=6 \int \frac {e^{-\frac {6 x}{e^x-x^2}} x^4}{\left (-e^x+x^2\right )^2} \, dx+6 \int \frac {e^{-\frac {6 x}{e^x-x^2}} x}{-e^x+x^2} \, dx-6 \int \frac {e^{-\frac {6 x}{e^x-x^2}} x^2}{-e^x+x^2} \, dx-12 \int \frac {e^{-\frac {6 x}{e^x-x^2}} x^3}{\left (-e^x+x^2\right )^2} \, dx+\int e^{-\frac {6 x}{e^x-x^2}} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.55, size = 18, normalized size = 1.00 \begin {gather*} e^{-\frac {6 x}{e^x-x^2}} x \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 16, normalized size = 0.89 \begin {gather*} x e^{\left (\frac {6 \, x}{x^{2} - e^{x}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.88, size = 30, normalized size = 1.67 \begin {gather*} x e^{\left (-x + \frac {x^{3} - x e^{x} + 6 \, x}{x^{2} - e^{x}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 17, normalized size = 0.94
method | result | size |
risch | \({\mathrm e}^{-\frac {6 x}{{\mathrm e}^{x}-x^{2}}} x\) | \(17\) |
norman | \(\frac {x^{3} {\mathrm e}^{-\frac {6 x}{{\mathrm e}^{x}-x^{2}}}-{\mathrm e}^{x} x \,{\mathrm e}^{-\frac {6 x}{{\mathrm e}^{x}-x^{2}}}}{-{\mathrm e}^{x}+x^{2}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} - 6 \, x^{3} + 2 \, {\left (2 \, x^{2} - 3 \, x\right )} e^{x} + e^{\left (2 \, x\right )}\right )} e^{\left (\frac {6 \, x}{x^{2} - e^{x}}\right )}}{x^{4} - 2 \, x^{2} e^{x} + e^{\left (2 \, x\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.33, size = 16, normalized size = 0.89 \begin {gather*} x\,{\mathrm {e}}^{-\frac {6\,x}{{\mathrm {e}}^x-x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.52, size = 12, normalized size = 0.67 \begin {gather*} x e^{- \frac {6 x}{- x^{2} + e^{x}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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