Optimal. Leaf size=31 \[ -3-e^3+e^{\frac {1}{2} e^{\frac {1}{x^2}} \left (3-e^{e^x x}+x\right )}+x \]
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Rubi [F]
time = 12.13, 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 {2 x^3+\exp \left (\frac {1}{2} \left (-e^{\frac {1}{x^2}+e^x x}+e^{\frac {1}{x^2}} (3+x)\right )\right ) \left (e^{\frac {1}{x^2}} \left (-6-2 x+x^3\right )+e^{e^x x} \left (2 e^{\frac {1}{x^2}}+e^{\frac {1}{x^2}+x} \left (-x^3-x^4\right )\right )\right )}{2 x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {2 x^3+\exp \left (\frac {1}{2} \left (-e^{\frac {1}{x^2}+e^x x}+e^{\frac {1}{x^2}} (3+x)\right )\right ) \left (e^{\frac {1}{x^2}} \left (-6-2 x+x^3\right )+e^{e^x x} \left (2 e^{\frac {1}{x^2}}+e^{\frac {1}{x^2}+x} \left (-x^3-x^4\right )\right )\right )}{x^3} \, dx\\ &=\frac {1}{2} \int \left (2-\frac {\exp \left (-\frac {1}{2} e^{\frac {1}{x^2}+e^x x}+\frac {1}{x^2}+\frac {1}{2} e^{\frac {1}{x^2}} (3+x)\right ) \left (6-2 e^{e^x x}+2 x-x^3+e^{x+e^x x} x^3+e^{x+e^x x} x^4\right )}{x^3}\right ) \, dx\\ &=x-\frac {1}{2} \int \frac {\exp \left (-\frac {1}{2} e^{\frac {1}{x^2}+e^x x}+\frac {1}{x^2}+\frac {1}{2} e^{\frac {1}{x^2}} (3+x)\right ) \left (6-2 e^{e^x x}+2 x-x^3+e^{x+e^x x} x^3+e^{x+e^x x} x^4\right )}{x^3} \, dx\\ &=x-\frac {1}{2} \int \left (\frac {\exp \left (-\frac {1}{2} e^{\frac {1}{x^2}+e^x x}+\frac {1}{x^2}+\frac {1}{2} e^{\frac {1}{x^2}} (3+x)\right ) \left (6+2 x-x^3\right )}{x^3}+\frac {\exp \left (-\frac {1}{2} e^{\frac {1}{x^2}+e^x x}+\frac {1}{x^2}+e^x x+\frac {1}{2} e^{\frac {1}{x^2}} (3+x)\right ) \left (-2+e^x x^3+e^x x^4\right )}{x^3}\right ) \, dx\\ &=x-\frac {1}{2} \int \frac {\exp \left (-\frac {1}{2} e^{\frac {1}{x^2}+e^x x}+\frac {1}{x^2}+\frac {1}{2} e^{\frac {1}{x^2}} (3+x)\right ) \left (6+2 x-x^3\right )}{x^3} \, dx-\frac {1}{2} \int \frac {\exp \left (-\frac {1}{2} e^{\frac {1}{x^2}+e^x x}+\frac {1}{x^2}+e^x x+\frac {1}{2} e^{\frac {1}{x^2}} (3+x)\right ) \left (-2+e^x x^3+e^x x^4\right )}{x^3} \, dx\\ &=x-\frac {1}{2} \int \left (-\exp \left (-\frac {1}{2} e^{\frac {1}{x^2}+e^x x}+\frac {1}{x^2}+\frac {1}{2} e^{\frac {1}{x^2}} (3+x)\right )+\frac {6 \exp \left (-\frac {1}{2} e^{\frac {1}{x^2}+e^x x}+\frac {1}{x^2}+\frac {1}{2} e^{\frac {1}{x^2}} (3+x)\right )}{x^3}+\frac {2 \exp \left (-\frac {1}{2} e^{\frac {1}{x^2}+e^x x}+\frac {1}{x^2}+\frac {1}{2} e^{\frac {1}{x^2}} (3+x)\right )}{x^2}\right ) \, dx-\frac {1}{2} \int \left (-\frac {2 \exp \left (-\frac {1}{2} e^{\frac {1}{x^2}+e^x x}+\frac {1}{x^2}+e^x x+\frac {1}{2} e^{\frac {1}{x^2}} (3+x)\right )}{x^3}+\exp \left (-\frac {1}{2} e^{\frac {1}{x^2}+e^x x}+\frac {1}{x^2}+x+e^x x+\frac {1}{2} e^{\frac {1}{x^2}} (3+x)\right ) (1+x)\right ) \, dx\\ &=x+\frac {1}{2} \int \exp \left (-\frac {1}{2} e^{\frac {1}{x^2}+e^x x}+\frac {1}{x^2}+\frac {1}{2} e^{\frac {1}{x^2}} (3+x)\right ) \, dx-\frac {1}{2} \int \exp \left (-\frac {1}{2} e^{\frac {1}{x^2}+e^x x}+\frac {1}{x^2}+x+e^x x+\frac {1}{2} e^{\frac {1}{x^2}} (3+x)\right ) (1+x) \, dx-3 \int \frac {\exp \left (-\frac {1}{2} e^{\frac {1}{x^2}+e^x x}+\frac {1}{x^2}+\frac {1}{2} e^{\frac {1}{x^2}} (3+x)\right )}{x^3} \, dx+\int \frac {\exp \left (-\frac {1}{2} e^{\frac {1}{x^2}+e^x x}+\frac {1}{x^2}+e^x x+\frac {1}{2} e^{\frac {1}{x^2}} (3+x)\right )}{x^3} \, dx-\int \frac {\exp \left (-\frac {1}{2} e^{\frac {1}{x^2}+e^x x}+\frac {1}{x^2}+\frac {1}{2} e^{\frac {1}{x^2}} (3+x)\right )}{x^2} \, dx\\ &=x+\frac {1}{2} \int \exp \left (-\frac {1}{2} e^{\frac {1}{x^2}+e^x x}+\frac {1}{x^2}+\frac {1}{2} e^{\frac {1}{x^2}} (3+x)\right ) \, dx-\frac {1}{2} \int \left (\exp \left (-\frac {1}{2} e^{\frac {1}{x^2}+e^x x}+\frac {1}{x^2}+x+e^x x+\frac {1}{2} e^{\frac {1}{x^2}} (3+x)\right )+\exp \left (-\frac {1}{2} e^{\frac {1}{x^2}+e^x x}+\frac {1}{x^2}+x+e^x x+\frac {1}{2} e^{\frac {1}{x^2}} (3+x)\right ) x\right ) \, dx-3 \int \frac {\exp \left (-\frac {1}{2} e^{\frac {1}{x^2}+e^x x}+\frac {1}{x^2}+\frac {1}{2} e^{\frac {1}{x^2}} (3+x)\right )}{x^3} \, dx+\int \frac {\exp \left (-\frac {1}{2} e^{\frac {1}{x^2}+e^x x}+\frac {1}{x^2}+e^x x+\frac {1}{2} e^{\frac {1}{x^2}} (3+x)\right )}{x^3} \, dx-\int \frac {\exp \left (-\frac {1}{2} e^{\frac {1}{x^2}+e^x x}+\frac {1}{x^2}+\frac {1}{2} e^{\frac {1}{x^2}} (3+x)\right )}{x^2} \, dx\\ &=x+\frac {1}{2} \int \exp \left (-\frac {1}{2} e^{\frac {1}{x^2}+e^x x}+\frac {1}{x^2}+\frac {1}{2} e^{\frac {1}{x^2}} (3+x)\right ) \, dx-\frac {1}{2} \int \exp \left (-\frac {1}{2} e^{\frac {1}{x^2}+e^x x}+\frac {1}{x^2}+x+e^x x+\frac {1}{2} e^{\frac {1}{x^2}} (3+x)\right ) \, dx-\frac {1}{2} \int \exp \left (-\frac {1}{2} e^{\frac {1}{x^2}+e^x x}+\frac {1}{x^2}+x+e^x x+\frac {1}{2} e^{\frac {1}{x^2}} (3+x)\right ) x \, dx-3 \int \frac {\exp \left (-\frac {1}{2} e^{\frac {1}{x^2}+e^x x}+\frac {1}{x^2}+\frac {1}{2} e^{\frac {1}{x^2}} (3+x)\right )}{x^3} \, dx+\int \frac {\exp \left (-\frac {1}{2} e^{\frac {1}{x^2}+e^x x}+\frac {1}{x^2}+e^x x+\frac {1}{2} e^{\frac {1}{x^2}} (3+x)\right )}{x^3} \, dx-\int \frac {\exp \left (-\frac {1}{2} e^{\frac {1}{x^2}+e^x x}+\frac {1}{x^2}+\frac {1}{2} e^{\frac {1}{x^2}} (3+x)\right )}{x^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 2.05, size = 32, normalized size = 1.03 \begin {gather*} e^{-\frac {1}{2} e^{\frac {1}{x^2}+e^x x}+\frac {1}{2} e^{\frac {1}{x^2}} (3+x)}+x \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 33, normalized size = 1.06
method | result | size |
risch | \(x +{\mathrm e}^{-\frac {{\mathrm e}^{\frac {1+{\mathrm e}^{x} x^{3}}{x^{2}}}}{2}+\frac {x \,{\mathrm e}^{\frac {1}{x^{2}}}}{2}+\frac {3 \,{\mathrm e}^{\frac {1}{x^{2}}}}{2}}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.38, size = 28, normalized size = 0.90 \begin {gather*} x + e^{\left (\frac {1}{2} \, x e^{\left (\frac {1}{x^{2}}\right )} - \frac {1}{2} \, e^{\left (x e^{x} + \frac {1}{x^{2}}\right )} + \frac {3}{2} \, e^{\left (\frac {1}{x^{2}}\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 41, normalized size = 1.32 \begin {gather*} x + e^{\left (\frac {1}{2} \, {\left ({\left (x + 3\right )} e^{\left (\frac {x^{3} + 1}{x^{2}}\right )} - e^{\left (x + \frac {x^{3} e^{x} + 1}{x^{2}}\right )}\right )} e^{\left (-x\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 6.69, size = 31, normalized size = 1.00 \begin {gather*} x + e^{\left (\frac {x}{2} + \frac {3}{2}\right ) e^{\frac {1}{x^{2}}} - \frac {e^{\frac {1}{x^{2}}} e^{x e^{x}}}{2}} \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*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.76, size = 28, normalized size = 0.90 \begin {gather*} x+{\mathrm {e}}^{\frac {3\,{\mathrm {e}}^{\frac {1}{x^2}}}{2}+\frac {x\,{\mathrm {e}}^{\frac {1}{x^2}}}{2}-\frac {{\mathrm {e}}^{x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{\frac {1}{x^2}}}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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