Optimal. Leaf size=13 \[ e^{\frac {1}{-1+x^2}} (1+x) \]
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Rubi [F]
time = 0.25, 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 {1}{-1+x^2}} \left (1-3 x-x^2+x^3\right )}{1-x-x^2+x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {e^{\frac {1}{-1+x^2}} \left (1-3 x-x^2+x^3\right )}{1-x-x^2+x^3} \, dx &=\int \left (e^{\frac {1}{-1+x^2}}-\frac {2 e^{\frac {1}{-1+x^2}} x}{1-x-x^2+x^3}\right ) \, dx\\ &=-\left (2 \int \frac {e^{\frac {1}{-1+x^2}} x}{1-x-x^2+x^3} \, dx\right )+\int e^{\frac {1}{-1+x^2}} \, dx\\ &=-\left (2 \int \left (\frac {e^{\frac {1}{-1+x^2}}}{2 (-1+x)^2}+\frac {e^{\frac {1}{-1+x^2}}}{2 \left (-1+x^2\right )}\right ) \, dx\right )+\int e^{\frac {1}{-1+x^2}} \, dx\\ &=\int e^{\frac {1}{-1+x^2}} \, dx-\int \frac {e^{\frac {1}{-1+x^2}}}{(-1+x)^2} \, dx-\int \frac {e^{\frac {1}{-1+x^2}}}{-1+x^2} \, dx\\ &=\int e^{\frac {1}{-1+x^2}} \, dx-\int \frac {e^{\frac {1}{-1+x^2}}}{(-1+x)^2} \, dx-\int \left (-\frac {e^{\frac {1}{-1+x^2}}}{2 (1-x)}-\frac {e^{\frac {1}{-1+x^2}}}{2 (1+x)}\right ) \, dx\\ &=\frac {1}{2} \int \frac {e^{\frac {1}{-1+x^2}}}{1-x} \, dx+\frac {1}{2} \int \frac {e^{\frac {1}{-1+x^2}}}{1+x} \, dx+\int e^{\frac {1}{-1+x^2}} \, dx-\int \frac {e^{\frac {1}{-1+x^2}}}{(-1+x)^2} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 13, normalized size = 1.00 \begin {gather*} e^{\frac {1}{-1+x^2}} (1+x) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.75, size = 13, normalized size = 1.00 \begin {gather*} \left (1+x\right ) E^{\frac {1}{-1+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 13, normalized size = 1.00
method | result | size |
gosper | \({\mathrm e}^{\frac {1}{x^{2}-1}} \left (1+x \right )\) | \(13\) |
risch | \({\mathrm e}^{\frac {1}{\left (-1+x \right ) \left (1+x \right )}} \left (1+x \right )\) | \(17\) |
norman | \(\frac {x^{2} {\mathrm e}^{\frac {1}{x^{2}-1}}-{\mathrm e}^{\frac {1}{x^{2}-1}}}{-1+x}\) | \(30\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 12, normalized size = 0.92 \begin {gather*} {\left (x + 1\right )} e^{\left (\frac {1}{x^{2} - 1}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 10, normalized size = 0.77 \begin {gather*} \left (x + 1\right ) e^{\frac {1}{x^{2} - 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 33 vs.
\(2 (12) = 24\).
time = 0.00, size = 29, normalized size = 2.23 \begin {gather*} \frac {x \mathrm {e}^{\frac {x^{2}}{x^{2}-1}}+\mathrm {e}^{\frac {x^{2}}{x^{2}-1}}}{\mathrm {e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.24, size = 12, normalized size = 0.92 \begin {gather*} {\mathrm {e}}^{\frac {1}{x^2-1}}\,\left (x+1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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