Optimal. Leaf size=27 \[ \frac {-14-e^2+e^{-2 (1-x)+\frac {1}{x}}-x}{e^2} \]
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Rubi [A] time = 0.14, antiderivative size = 17, normalized size of antiderivative = 0.63, number of steps used = 4, number of rules used = 3, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {12, 14, 6706} \begin {gather*} e^{2 x+\frac {1}{x}-4}-\frac {x}{e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 6706
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-x^2+e^{\frac {1-2 x+2 x^2}{x}} \left (-1+2 x^2\right )}{x^2} \, dx}{e^2}\\ &=\frac {\int \left (-1+\frac {e^{-2+\frac {1}{x}+2 x} \left (-1+2 x^2\right )}{x^2}\right ) \, dx}{e^2}\\ &=-\frac {x}{e^2}+\frac {\int \frac {e^{-2+\frac {1}{x}+2 x} \left (-1+2 x^2\right )}{x^2} \, dx}{e^2}\\ &=e^{-4+\frac {1}{x}+2 x}-\frac {x}{e^2}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.06, size = 17, normalized size = 0.63 \begin {gather*} e^{-4+\frac {1}{x}+2 x}-\frac {x}{e^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 23, normalized size = 0.85 \begin {gather*} -{\left (x - e^{\left (\frac {2 \, x^{2} - 2 \, x + 1}{x}\right )}\right )} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 23, normalized size = 0.85 \begin {gather*} -{\left (x - e^{\left (\frac {2 \, x^{2} - 2 \, x + 1}{x}\right )}\right )} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 22, normalized size = 0.81
method | result | size |
risch | \(-x \,{\mathrm e}^{-2}+{\mathrm e}^{\frac {2 x^{2}-4 x +1}{x}}\) | \(22\) |
norman | \(\frac {x \,{\mathrm e}^{-2} {\mathrm e}^{\frac {2 x^{2}-2 x +1}{x}}-x^{2} {\mathrm e}^{-2}}{x}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.82, size = 17, normalized size = 0.63 \begin {gather*} -{\left (x - e^{\left (2 \, x + \frac {1}{x} - 2\right )}\right )} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.12, size = 17, normalized size = 0.63 \begin {gather*} {\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{1/x}\,{\mathrm {e}}^{-4}-x\,{\mathrm {e}}^{-2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 20, normalized size = 0.74 \begin {gather*} - \frac {x}{e^{2}} + \frac {e^{\frac {2 x^{2} - 2 x + 1}{x}}}{e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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