Optimal. Leaf size=27 \[ 18+\frac {e^x}{3}+e^{\frac {x-(2+x)^2}{4 x^2}} \]
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Rubi [A] time = 0.20, antiderivative size = 26, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {12, 14, 2194, 6706} \begin {gather*} e^{-\frac {1}{x^2}-\frac {3}{4 x}-\frac {1}{4}}+\frac {e^x}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2194
Rule 6706
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{12} \int \frac {4 e^x x^3+e^{\frac {-4-3 x-x^2}{4 x^2}} (24+9 x)}{x^3} \, dx\\ &=\frac {1}{12} \int \left (4 e^x+\frac {3 e^{-\frac {1}{4}-\frac {1}{x^2}-\frac {3}{4 x}} (8+3 x)}{x^3}\right ) \, dx\\ &=\frac {1}{4} \int \frac {e^{-\frac {1}{4}-\frac {1}{x^2}-\frac {3}{4 x}} (8+3 x)}{x^3} \, dx+\frac {\int e^x \, dx}{3}\\ &=e^{-\frac {1}{4}-\frac {1}{x^2}-\frac {3}{4 x}}+\frac {e^x}{3}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.13, size = 26, normalized size = 0.96 \begin {gather*} e^{-\frac {1}{4}-\frac {1}{x^2}-\frac {3}{4 x}}+\frac {e^x}{3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 19, normalized size = 0.70 \begin {gather*} \frac {1}{3} \, e^{x} + e^{\left (-\frac {x^{2} + 3 \, x + 4}{4 \, x^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 18, normalized size = 0.67 \begin {gather*} \frac {1}{3} \, e^{x} + e^{\left (-\frac {3}{4 \, x} - \frac {1}{x^{2}} - \frac {1}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 19, normalized size = 0.70
method | result | size |
default | \({\mathrm e}^{-\frac {1}{4}-\frac {3}{4 x}-\frac {1}{x^{2}}}+\frac {{\mathrm e}^{x}}{3}\) | \(19\) |
risch | \(\frac {{\mathrm e}^{x}}{3}+{\mathrm e}^{-\frac {x^{2}+3 x +4}{4 x^{2}}}\) | \(20\) |
norman | \(\frac {x^{2} {\mathrm e}^{\frac {-x^{2}-3 x -4}{4 x^{2}}}+\frac {{\mathrm e}^{x} x^{2}}{3}}{x^{2}}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 26, normalized size = 0.96 \begin {gather*} \frac {1}{3} \, {\left (e^{\left (x + \frac {1}{x^{2}} + \frac {1}{4}\right )} + 3 \, e^{\left (-\frac {3}{4 \, x}\right )}\right )} e^{\left (-\frac {1}{x^{2}} - \frac {1}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.24, size = 18, normalized size = 0.67 \begin {gather*} {\mathrm {e}}^{-\frac {3}{4\,x}-\frac {1}{x^2}-\frac {1}{4}}+\frac {{\mathrm {e}}^x}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 22, normalized size = 0.81 \begin {gather*} \frac {e^{x}}{3} + e^{\frac {- \frac {x^{2}}{4} - \frac {3 x}{4} - 1}{x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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