Optimal. Leaf size=30 \[ \frac {4}{3}-x+(1+x)^2+4 x \left (-x+\frac {x}{e^x-x}\right ) \]
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Rubi [F] time = 0.63, 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^{2 x} (1-6 x)-3 x^2-6 x^3+e^x \left (6 x+8 x^2\right )}{e^{2 x}-2 e^x x+x^2} \, 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^{2 x} (1-6 x)-3 x^2-6 x^3+e^x \left (6 x+8 x^2\right )}{\left (e^x-x\right )^2} \, dx\\ &=\int \left (1-6 x-\frac {4 (-2+x) x}{e^x-x}-\frac {4 (-1+x) x^2}{\left (e^x-x\right )^2}\right ) \, dx\\ &=x-3 x^2-4 \int \frac {(-2+x) x}{e^x-x} \, dx-4 \int \frac {(-1+x) x^2}{\left (e^x-x\right )^2} \, dx\\ &=x-3 x^2-4 \int \left (-\frac {2 x}{e^x-x}+\frac {x^2}{e^x-x}\right ) \, dx-4 \int \left (-\frac {x^2}{\left (e^x-x\right )^2}+\frac {x^3}{\left (e^x-x\right )^2}\right ) \, dx\\ &=x-3 x^2+4 \int \frac {x^2}{\left (e^x-x\right )^2} \, dx-4 \int \frac {x^2}{e^x-x} \, dx-4 \int \frac {x^3}{\left (e^x-x\right )^2} \, dx+8 \int \frac {x}{e^x-x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.17, size = 21, normalized size = 0.70 \begin {gather*} x-3 x^2+\frac {4 x^2}{e^x-x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 34, normalized size = 1.13 \begin {gather*} -\frac {3 \, x^{3} + 3 \, x^{2} - {\left (3 \, x^{2} - x\right )} e^{x}}{x - e^{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 32, normalized size = 1.07 \begin {gather*} -\frac {3 \, x^{3} - 3 \, x^{2} e^{x} + 3 \, x^{2} + x e^{x}}{x - e^{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 21, normalized size = 0.70
method | result | size |
risch | \(-3 x^{2}+x -\frac {4 x^{2}}{x -{\mathrm e}^{x}}\) | \(21\) |
norman | \(\frac {-3 x^{2}-3 x^{3}-{\mathrm e}^{x} x +3 \,{\mathrm e}^{x} x^{2}}{x -{\mathrm e}^{x}}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 34, normalized size = 1.13 \begin {gather*} -\frac {3 \, x^{3} + 3 \, x^{2} - {\left (3 \, x^{2} - x\right )} e^{x}}{x - e^{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.64, size = 20, normalized size = 0.67 \begin {gather*} x-\frac {4\,x^2}{x-{\mathrm {e}}^x}-3\,x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 15, normalized size = 0.50 \begin {gather*} - 3 x^{2} + \frac {4 x^{2}}{- x + e^{x}} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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