Optimal. Leaf size=29 \[ e^{\frac {1}{2} \left (-e^4-\frac {-e+\frac {1}{x}}{x}\right ) \left (x+x^3\right )} \]
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Rubi [A] time = 0.71, antiderivative size = 33, normalized size of antiderivative = 1.14, number of steps used = 2, number of rules used = 2, integrand size = 73, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {12, 6706} \begin {gather*} \exp \left (-\frac {-e \left (x^3+x\right )+x^2+e^4 \left (x^4+x^2\right )+1}{2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 6706
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {\exp \left (\frac {-1-x^2+e \left (x+x^3\right )+e^4 \left (-x^2-x^4\right )}{2 x}\right ) \left (1-x^2+2 e x^3+e^4 \left (-x^2-3 x^4\right )\right )}{x^2} \, dx\\ &=\exp \left (-\frac {1+x^2-e \left (x+x^3\right )+e^4 \left (x^2+x^4\right )}{2 x}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.19, size = 27, normalized size = 0.93 \begin {gather*} e^{-\frac {\left (1+x^2\right ) \left (1-e x+e^4 x^2\right )}{2 x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 30, normalized size = 1.03 \begin {gather*} e^{\left (-\frac {x^{2} + {\left (x^{4} + x^{2}\right )} e^{4} - {\left (x^{3} + x\right )} e + 1}{2 \, x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 33, normalized size = 1.14 \begin {gather*} e^{\left (-\frac {1}{2} \, x^{3} e^{4} + \frac {1}{2} \, x^{2} e - \frac {1}{2} \, x e^{4} - \frac {1}{2} \, x - \frac {1}{2 \, x} + \frac {1}{2} \, e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 25, normalized size = 0.86
method | result | size |
risch | \({\mathrm e}^{\frac {\left (x^{2}+1\right ) \left (-x^{2} {\mathrm e}^{4}+x \,{\mathrm e}-1\right )}{2 x}}\) | \(25\) |
norman | \({\mathrm e}^{\frac {\left (-x^{4}-x^{2}\right ) {\mathrm e}^{4}+\left (x^{3}+x \right ) {\mathrm e}-x^{2}-1}{2 x}}\) | \(36\) |
gosper | \({\mathrm e}^{\frac {-x^{4} {\mathrm e}^{4}+x^{3} {\mathrm e}-x^{2} {\mathrm e}^{4}+x \,{\mathrm e}-x^{2}-1}{2 x}}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 33, normalized size = 1.14 \begin {gather*} e^{\left (-\frac {1}{2} \, x^{3} e^{4} + \frac {1}{2} \, x^{2} e - \frac {1}{2} \, x e^{4} - \frac {1}{2} \, x - \frac {1}{2 \, x} + \frac {1}{2} \, e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.31, size = 38, normalized size = 1.31 \begin {gather*} {\mathrm {e}}^{\frac {x^2\,\mathrm {e}}{2}}\,{\mathrm {e}}^{-\frac {x^3\,{\mathrm {e}}^4}{2}}\,{\mathrm {e}}^{\frac {\mathrm {e}}{2}}\,{\mathrm {e}}^{-\frac {x}{2}}\,{\mathrm {e}}^{-\frac {1}{2\,x}}\,{\mathrm {e}}^{-\frac {x\,{\mathrm {e}}^4}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.23, size = 34, normalized size = 1.17 \begin {gather*} e^{\frac {- \frac {x^{2}}{2} + \frac {e \left (x^{3} + x\right )}{2} + \frac {\left (- x^{4} - x^{2}\right ) e^{4}}{2} - \frac {1}{2}}{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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