Optimal. Leaf size=20 \[ e^{-4+(2+x)^2+\frac {2}{1-x^2}} \]
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Rubi [F] time = 1.52, 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 {-2-4 x-x^2+4 x^3+x^4}{-1+x^2}} \left (4+6 x-8 x^2-4 x^3+4 x^4+2 x^5\right )}{1-2 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 {-2-4 x-x^2+4 x^3+x^4}{-1+x^2}} \left (4+6 x-8 x^2-4 x^3+4 x^4+2 x^5\right )}{\left (-1+x^2\right )^2} \, dx\\ &=\int \frac {2 e^{\frac {-2-4 x-x^2+4 x^3+x^4}{-1+x^2}} \left (2+3 x-4 x^2-2 x^3+2 x^4+x^5\right )}{\left (1-x^2\right )^2} \, dx\\ &=2 \int \frac {e^{\frac {-2-4 x-x^2+4 x^3+x^4}{-1+x^2}} \left (2+3 x-4 x^2-2 x^3+2 x^4+x^5\right )}{\left (1-x^2\right )^2} \, dx\\ &=2 \int \left (2 e^{\frac {-2-4 x-x^2+4 x^3+x^4}{-1+x^2}}+\frac {e^{\frac {-2-4 x-x^2+4 x^3+x^4}{-1+x^2}}}{2 (-1+x)^2}+e^{\frac {-2-4 x-x^2+4 x^3+x^4}{-1+x^2}} x-\frac {e^{\frac {-2-4 x-x^2+4 x^3+x^4}{-1+x^2}}}{2 (1+x)^2}\right ) \, dx\\ &=2 \int e^{\frac {-2-4 x-x^2+4 x^3+x^4}{-1+x^2}} x \, dx+4 \int e^{\frac {-2-4 x-x^2+4 x^3+x^4}{-1+x^2}} \, dx+\int \frac {e^{\frac {-2-4 x-x^2+4 x^3+x^4}{-1+x^2}}}{(-1+x)^2} \, dx-\int \frac {e^{\frac {-2-4 x-x^2+4 x^3+x^4}{-1+x^2}}}{(1+x)^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.29, size = 18, normalized size = 0.90 \begin {gather*} e^{4 x+x^2-\frac {2}{-1+x^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 27, normalized size = 1.35 \begin {gather*} e^{\left (\frac {x^{4} + 4 \, x^{3} - x^{2} - 4 \, x - 2}{x^{2} - 1}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 56, normalized size = 2.80 \begin {gather*} e^{\left (\frac {x^{4}}{x^{2} - 1} + \frac {4 \, x^{3}}{x^{2} - 1} - \frac {x^{2}}{x^{2} - 1} - \frac {4 \, x}{x^{2} - 1} - \frac {2}{x^{2} - 1}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 28, normalized size = 1.40
method | result | size |
gosper | \({\mathrm e}^{\frac {x^{4}+4 x^{3}-x^{2}-4 x -2}{x^{2}-1}}\) | \(28\) |
risch | \({\mathrm e}^{\frac {x^{4}+4 x^{3}-x^{2}-4 x -2}{\left (x -1\right ) \left (x +1\right )}}\) | \(31\) |
norman | \(\frac {x^{2} {\mathrm e}^{\frac {x^{4}+4 x^{3}-x^{2}-4 x -2}{x^{2}-1}}-{\mathrm e}^{\frac {x^{4}+4 x^{3}-x^{2}-4 x -2}{x^{2}-1}}}{x^{2}-1}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 20, normalized size = 1.00 \begin {gather*} e^{\left (x^{2} + 4 \, x + \frac {1}{x + 1} - \frac {1}{x - 1}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.74, size = 60, normalized size = 3.00 \begin {gather*} {\mathrm {e}}^{-\frac {x^2}{x^2-1}}\,{\mathrm {e}}^{\frac {x^4}{x^2-1}}\,{\mathrm {e}}^{\frac {4\,x^3}{x^2-1}}\,{\mathrm {e}}^{-\frac {2}{x^2-1}}\,{\mathrm {e}}^{-\frac {4\,x}{x^2-1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 22, normalized size = 1.10 \begin {gather*} e^{\frac {x^{4} + 4 x^{3} - x^{2} - 4 x - 2}{x^{2} - 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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