Optimal. Leaf size=28 \[ \frac {1}{3} \left (-1+e^{\frac {\left (3+\frac {-3+x}{x}\right )^2}{4 e}}\right )^2 \]
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Rubi [A] time = 0.49, antiderivative size = 47, normalized size of antiderivative = 1.68, number of steps used = 5, number of rules used = 3, integrand size = 64, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {12, 14, 6706} \begin {gather*} \frac {1}{3} e^{\frac {(3-4 x)^2}{2 e x^2}}-\frac {2}{3} e^{\frac {(3-4 x)^2}{4 e x^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 {e^{\frac {9-24 x+16 x^2}{4 e x^2}} (3-4 x)+e^{\frac {9-24 x+16 x^2}{2 e x^2}} (-3+4 x)}{x^3} \, dx}{e}\\ &=\frac {\int \left (-\frac {e^{\frac {(3-4 x)^2}{4 e x^2}} (-3+4 x)}{x^3}+\frac {e^{\frac {(3-4 x)^2}{2 e x^2}} (-3+4 x)}{x^3}\right ) \, dx}{e}\\ &=-\frac {\int \frac {e^{\frac {(3-4 x)^2}{4 e x^2}} (-3+4 x)}{x^3} \, dx}{e}+\frac {\int \frac {e^{\frac {(3-4 x)^2}{2 e x^2}} (-3+4 x)}{x^3} \, dx}{e}\\ &=-\frac {2}{3} e^{\frac {(3-4 x)^2}{4 e x^2}}+\frac {1}{3} e^{\frac {(3-4 x)^2}{2 e x^2}}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.45, size = 44, normalized size = 1.57 \begin {gather*} \frac {1}{3} e^{\frac {(3-4 x)^2}{4 e x^2}} \left (-2+e^{\frac {(3-4 x)^2}{4 e x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 41, normalized size = 1.46 \begin {gather*} \frac {1}{3} \, e^{\left (\frac {{\left (16 \, x^{2} - 24 \, x + 9\right )} e^{\left (-1\right )}}{2 \, x^{2}}\right )} - \frac {2}{3} \, e^{\left (\frac {{\left (16 \, x^{2} - 24 \, x + 9\right )} e^{\left (-1\right )}}{4 \, x^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left ({\left (4 \, x - 3\right )} e^{\left (\frac {{\left (16 \, x^{2} - 24 \, x + 9\right )} e^{\left (-1\right )}}{2 \, x^{2}}\right )} - {\left (4 \, x - 3\right )} e^{\left (\frac {{\left (16 \, x^{2} - 24 \, x + 9\right )} e^{\left (-1\right )}}{4 \, x^{2}}\right )}\right )} e^{\left (-1\right )}}{x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 36, normalized size = 1.29
method | result | size |
risch | \(\frac {{\mathrm e}^{\frac {\left (4 x -3\right )^{2} {\mathrm e}^{-1}}{2 x^{2}}}}{3}-\frac {2 \,{\mathrm e}^{\frac {\left (4 x -3\right )^{2} {\mathrm e}^{-1}}{4 x^{2}}}}{3}\) | \(36\) |
norman | \(\frac {\frac {x^{2} {\mathrm e}^{\frac {\left (16 x^{2}-24 x +9\right ) {\mathrm e}^{-1}}{2 x^{2}}}}{3}-\frac {2 \,{\mathrm e}^{\frac {\left (16 x^{2}-24 x +9\right ) {\mathrm e}^{-1}}{4 x^{2}}} x^{2}}{3}}{x^{2}}\) | \(58\) |
default | \({\mathrm e}^{-1} \left (-\frac {2 \,{\mathrm e} \,{\mathrm e}^{4 \,{\mathrm e}^{-1}-\frac {6 \,{\mathrm e}^{-1}}{x}+\frac {9 \,{\mathrm e}^{-1}}{4 x^{2}}}}{3}+\frac {{\mathrm e} \,{\mathrm e}^{\frac {9 \,{\mathrm e}^{-1}}{2 x^{2}}-\frac {12 \,{\mathrm e}^{-1}}{x}+8 \,{\mathrm e}^{-1}}}{3}\right )\) | \(59\) |
derivativedivides | \(-{\mathrm e}^{-1} \left (\frac {2 \,{\mathrm e} \,{\mathrm e}^{4 \,{\mathrm e}^{-1}-\frac {6 \,{\mathrm e}^{-1}}{x}+\frac {9 \,{\mathrm e}^{-1}}{4 x^{2}}}}{3}-\frac {{\mathrm e} \,{\mathrm e}^{\frac {9 \,{\mathrm e}^{-1}}{2 x^{2}}-\frac {12 \,{\mathrm e}^{-1}}{x}+8 \,{\mathrm e}^{-1}}}{3}\right )\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.50, size = 52, normalized size = 1.86 \begin {gather*} -\frac {1}{3} \, {\left (2 \, e^{\left (\frac {6 \, e^{\left (-1\right )}}{x} + \frac {9 \, e^{\left (-1\right )}}{4 \, x^{2}} + 4 \, e^{\left (-1\right )} + 1\right )} - e^{\left (\frac {9 \, e^{\left (-1\right )}}{2 \, x^{2}} + 8 \, e^{\left (-1\right )} + 1\right )}\right )} e^{\left (-\frac {12 \, e^{\left (-1\right )}}{x} - 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.26, size = 45, normalized size = 1.61 \begin {gather*} {\mathrm {e}}^{4\,{\mathrm {e}}^{-1}-\frac {6\,{\mathrm {e}}^{-1}}{x}+\frac {9\,{\mathrm {e}}^{-1}}{4\,x^2}}\,\left (\frac {{\mathrm {e}}^{4\,{\mathrm {e}}^{-1}-\frac {6\,{\mathrm {e}}^{-1}}{x}+\frac {9\,{\mathrm {e}}^{-1}}{4\,x^2}}}{3}-\frac {2}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.20, size = 44, normalized size = 1.57 \begin {gather*} \frac {e^{\frac {2 \left (4 x^{2} - 6 x + \frac {9}{4}\right )}{e x^{2}}}}{3} - \frac {2 e^{\frac {4 x^{2} - 6 x + \frac {9}{4}}{e x^{2}}}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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