Optimal. Leaf size=20 \[ \frac {x}{2}+\frac {x}{4-e^{-20+x} x} \]
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Rubi [F] time = 0.77, 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 {24 e^{40-2 x}+x^2+e^{20-x} \left (-8 x+2 x^2\right )}{32 e^{40-2 x}-16 e^{20-x} x+2 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} \left (24 e^{40-2 x}+x^2+e^{20-x} \left (-8 x+2 x^2\right )\right )}{2 \left (4 e^{20}-e^x x\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {e^{2 x} \left (24 e^{40-2 x}+x^2+e^{20-x} \left (-8 x+2 x^2\right )\right )}{\left (4 e^{20}-e^x x\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {24 e^{40}+2 e^{20+x} (-4+x) x+e^{2 x} x^2}{\left (4 e^{20}-e^x x\right )^2} \, dx\\ &=\frac {1}{2} \int \left (1+\frac {8 e^{40} (1+x)}{\left (4 e^{20}-e^x x\right )^2}-\frac {2 e^{20} x}{4 e^{20}-e^x x}\right ) \, dx\\ &=\frac {x}{2}-e^{20} \int \frac {x}{4 e^{20}-e^x x} \, dx+\left (4 e^{40}\right ) \int \frac {1+x}{\left (4 e^{20}-e^x x\right )^2} \, dx\\ &=\frac {x}{2}-e^{20} \int \frac {x}{4 e^{20}-e^x x} \, dx+\left (4 e^{40}\right ) \int \left (\frac {1}{\left (4 e^{20}-e^x x\right )^2}+\frac {x}{\left (-4 e^{20}+e^x x\right )^2}\right ) \, dx\\ &=\frac {x}{2}-e^{20} \int \frac {x}{4 e^{20}-e^x x} \, dx+\left (4 e^{40}\right ) \int \frac {1}{\left (4 e^{20}-e^x x\right )^2} \, dx+\left (4 e^{40}\right ) \int \frac {x}{\left (-4 e^{20}+e^x x\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.14, size = 25, normalized size = 1.25 \begin {gather*} \frac {1}{2} \left (x-\frac {2 e^{20} x}{-4 e^{20}+e^x x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 27, normalized size = 1.35 \begin {gather*} \frac {x^{2} - 6 \, x e^{\left (-x + 20\right )}}{2 \, {\left (x - 4 \, e^{\left (-x + 20\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.45, size = 107, normalized size = 5.35 \begin {gather*} \frac {x^{4} e^{x} - 10 \, x^{3} e^{20} - 2 \, x^{2} e^{20} - x^{2} e^{x} + 24 \, x^{2} e^{\left (-x + 40\right )} + 8 \, x e^{20} + 8 \, x e^{\left (-x + 40\right )} - 16 \, e^{\left (-x + 40\right )}}{2 \, {\left (x^{3} e^{x} - 8 \, x^{2} e^{20} + x^{2} e^{x} - 8 \, x e^{20} + 16 \, x e^{\left (-x + 40\right )} + 16 \, e^{\left (-x + 40\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 22, normalized size = 1.10
method | result | size |
risch | \(\frac {3 x}{4}-\frac {x^{2}}{4 \left (x -4 \,{\mathrm e}^{-x +20}\right )}\) | \(22\) |
norman | \(\frac {\frac {x^{2}}{2}-3 x \,{\mathrm e}^{-x +20}}{x -4 \,{\mathrm e}^{-x +20}}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.85, size = 25, normalized size = 1.25 \begin {gather*} \frac {x^{2} e^{x} - 6 \, x e^{20}}{2 \, {\left (x e^{x} - 4 \, e^{20}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 23, normalized size = 1.15 \begin {gather*} \frac {3\,x}{4}-\frac {x^2}{2\,\left (2\,x-8\,{\mathrm {e}}^{20-x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 17, normalized size = 0.85 \begin {gather*} \frac {x^{2}}{- 4 x + 16 e^{20 - x}} + \frac {3 x}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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