Optimal. Leaf size=19 \[ e^{\frac {2 x}{-4+2 x}}+4 x+x^4 \]
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Rubi [A] time = 0.14, antiderivative size = 20, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {27, 6688, 2230, 2209} \begin {gather*} x^4+4 x+e^{1-\frac {2}{2-x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 2209
Rule 2230
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {16-2 e^{\frac {2 x}{-4+2 x}}-16 x+4 x^2+16 x^3-16 x^4+4 x^5}{(-2+x)^2} \, dx\\ &=\int \left (4-\frac {2 e^{\frac {x}{-2+x}}}{(-2+x)^2}+4 x^3\right ) \, dx\\ &=4 x+x^4-2 \int \frac {e^{\frac {x}{-2+x}}}{(-2+x)^2} \, dx\\ &=4 x+x^4-2 \int \frac {e^{1+\frac {2}{-2+x}}}{(-2+x)^2} \, dx\\ &=e^{1-\frac {2}{2-x}}+4 x+x^4\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 16, normalized size = 0.84 \begin {gather*} e^{\frac {x}{-2+x}}+4 x+x^4 \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 15, normalized size = 0.79 \begin {gather*} x^{4} + 4 \, x + e^{\left (\frac {x}{x - 2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 153, normalized size = 8.05 \begin {gather*} \frac {\frac {4 \, x e^{\left (\frac {x}{x - 2}\right )}}{x - 2} - \frac {6 \, x^{2} e^{\left (\frac {x}{x - 2}\right )}}{{\left (x - 2\right )}^{2}} + \frac {4 \, x^{3} e^{\left (\frac {x}{x - 2}\right )}}{{\left (x - 2\right )}^{3}} - \frac {x^{4} e^{\left (\frac {x}{x - 2}\right )}}{{\left (x - 2\right )}^{4}} - \frac {88 \, x}{x - 2} + \frac {120 \, x^{2}}{{\left (x - 2\right )}^{2}} - \frac {72 \, x^{3}}{{\left (x - 2\right )}^{3}} - e^{\left (\frac {x}{x - 2}\right )} + 24}{\frac {4 \, x}{x - 2} - \frac {6 \, x^{2}}{{\left (x - 2\right )}^{2}} + \frac {4 \, x^{3}}{{\left (x - 2\right )}^{3}} - \frac {x^{4}}{{\left (x - 2\right )}^{4}} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 16, normalized size = 0.84
method | result | size |
risch | \(4 x +x^{4}+{\mathrm e}^{\frac {x}{x -2}}\) | \(16\) |
derivativedivides | \(36 x -72+\left (x -2\right )^{4}+8 \left (x -2\right )^{3}+24 \left (x -2\right )^{2}+{\mathrm e}^{1+\frac {2}{x -2}}\) | \(35\) |
default | \(36 x -72+\left (x -2\right )^{4}+8 \left (x -2\right )^{3}+24 \left (x -2\right )^{2}+{\mathrm e}^{1+\frac {2}{x -2}}\) | \(35\) |
norman | \(\frac {x^{5}+x \,{\mathrm e}^{\frac {2 x}{2 x -4}}+4 x^{2}-2 x^{4}-2 \,{\mathrm e}^{\frac {2 x}{2 x -4}}-16}{x -2}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 17, normalized size = 0.89 \begin {gather*} x^{4} + 4 \, x + e^{\left (\frac {2}{x - 2} + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.80, size = 15, normalized size = 0.79 \begin {gather*} 4\,x+{\mathrm {e}}^{\frac {x}{x-2}}+x^4 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 15, normalized size = 0.79 \begin {gather*} x^{4} + 4 x + e^{\frac {2 x}{2 x - 4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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