Optimal. Leaf size=19 \[ \frac {1}{16} e^{-3+\frac {9}{x^2}} \left (4+e^x\right ) x \]
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Rubi [F] time = 0.19, 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 {9-3 x^2-x^2 \log (16)}{x^2}} \left (-72+4 x^2\right )+e^{x+\frac {9-3 x^2-x^2 \log (16)}{x^2}} \left (-18+x^2+x^3\right )}{x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {1}{4} e^{-3+\frac {9}{x^2}}+\frac {1}{16} e^{-3+\frac {9}{x^2}+x}-\frac {9 e^{-3+\frac {9}{x^2}}}{2 x^2}-\frac {9 e^{-3+\frac {9}{x^2}+x}}{8 x^2}+\frac {1}{16} e^{-3+\frac {9}{x^2}+x} x\right ) \, dx\\ &=\frac {1}{16} \int e^{-3+\frac {9}{x^2}+x} \, dx+\frac {1}{16} \int e^{-3+\frac {9}{x^2}+x} x \, dx+\frac {1}{4} \int e^{-3+\frac {9}{x^2}} \, dx-\frac {9}{8} \int \frac {e^{-3+\frac {9}{x^2}+x}}{x^2} \, dx-\frac {9}{2} \int \frac {e^{-3+\frac {9}{x^2}}}{x^2} \, dx\\ &=\frac {1}{4} e^{-3+\frac {9}{x^2}} x+\frac {1}{16} \int e^{-3+\frac {9}{x^2}+x} \, dx+\frac {1}{16} \int e^{-3+\frac {9}{x^2}+x} x \, dx-\frac {9}{8} \int \frac {e^{-3+\frac {9}{x^2}+x}}{x^2} \, dx+\frac {9}{2} \int \frac {e^{-3+\frac {9}{x^2}}}{x^2} \, dx+\frac {9}{2} \operatorname {Subst}\left (\int e^{-3+9 x^2} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{4} e^{-3+\frac {9}{x^2}} x+\frac {3 \sqrt {\pi } \text {erfi}\left (\frac {3}{x}\right )}{4 e^3}+\frac {1}{16} \int e^{-3+\frac {9}{x^2}+x} \, dx+\frac {1}{16} \int e^{-3+\frac {9}{x^2}+x} x \, dx-\frac {9}{8} \int \frac {e^{-3+\frac {9}{x^2}+x}}{x^2} \, dx-\frac {9}{2} \operatorname {Subst}\left (\int e^{-3+9 x^2} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{4} e^{-3+\frac {9}{x^2}} x+\frac {1}{16} \int e^{-3+\frac {9}{x^2}+x} \, dx+\frac {1}{16} \int e^{-3+\frac {9}{x^2}+x} x \, dx-\frac {9}{8} \int \frac {e^{-3+\frac {9}{x^2}+x}}{x^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.06, size = 19, normalized size = 1.00 \begin {gather*} \frac {1}{16} e^{-3+\frac {9}{x^2}} \left (4+e^x\right ) x \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.22, size = 48, normalized size = 2.53 \begin {gather*} x e^{\left (\frac {x^{3} - 4 \, x^{2} \log \relax (2) - 3 \, x^{2} + 9}{x^{2}}\right )} + 4 \, x e^{\left (-\frac {4 \, x^{2} \log \relax (2) + 3 \, x^{2} - 9}{x^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 33, normalized size = 1.74 \begin {gather*} \frac {1}{16} \, x e^{\left (\frac {x^{3} - 3 \, x^{2} + 9}{x^{2}}\right )} + \frac {1}{4} \, x e^{\left (-\frac {3 \, {\left (x^{2} - 3\right )}}{x^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 22, normalized size = 1.16
method | result | size |
risch | \(\frac {\left ({\mathrm e}^{x} x +4 x \right ) {\mathrm e}^{-\frac {3 \left (x^{2}-3\right )}{x^{2}}}}{16}\) | \(22\) |
norman | \(\frac {{\mathrm e}^{x} x^{2} {\mathrm e}^{\frac {-4 x^{2} \ln \relax (2)-3 x^{2}+9}{x^{2}}}+4 x^{2} {\mathrm e}^{\frac {-4 x^{2} \ln \relax (2)-3 x^{2}+9}{x^{2}}}}{x}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.70, size = 61, normalized size = 3.21 \begin {gather*} \frac {3}{8} \, x \sqrt {-\frac {1}{x^{2}}} e^{\left (-3\right )} \Gamma \left (-\frac {1}{2}, -\frac {9}{x^{2}}\right ) + \frac {1}{16} \, x e^{\left (x + \frac {9}{x^{2}} - 3\right )} + \frac {3 \, \sqrt {\pi } {\left (\operatorname {erf}\left (3 \, \sqrt {-\frac {1}{x^{2}}}\right ) - 1\right )} e^{\left (-3\right )}}{4 \, x \sqrt {-\frac {1}{x^{2}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.89, size = 15, normalized size = 0.79 \begin {gather*} \frac {x\,{\mathrm {e}}^{-3}\,{\mathrm {e}}^{\frac {9}{x^2}}\,\left ({\mathrm {e}}^x+4\right )}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.48, size = 27, normalized size = 1.42 \begin {gather*} \left (x e^{x} + 4 x\right ) e^{\frac {- 3 x^{2} - 4 x^{2} \log {\relax (2 )} + 9}{x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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