Optimal. Leaf size=31 \[ -e^{\frac {1}{2} \left (\left (3+\frac {2}{x^2}\right ) x-x^2\right ) (x+\log (x))}+x^2 \]
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Rubi [F] time = 4.67, 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 {4 x^3+\exp \left (\frac {2 x+3 x^3-x^4+\left (2+3 x^2-x^3\right ) \log (x)}{2 x}\right ) \left (-2-3 x^2-5 x^3+3 x^4+\left (2-3 x^2+2 x^3\right ) \log (x)\right )}{2 x^2} \, 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 {4 x^3+\exp \left (\frac {2 x+3 x^3-x^4+\left (2+3 x^2-x^3\right ) \log (x)}{2 x}\right ) \left (-2-3 x^2-5 x^3+3 x^4+\left (2-3 x^2+2 x^3\right ) \log (x)\right )}{x^2} \, dx\\ &=\frac {1}{2} \int \left (4 x+e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{-2+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \left (-2-3 x^2-5 x^3+3 x^4+2 \log (x)-3 x^2 \log (x)+2 x^3 \log (x)\right )\right ) \, dx\\ &=x^2+\frac {1}{2} \int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{-2+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \left (-2-3 x^2-5 x^3+3 x^4+2 \log (x)-3 x^2 \log (x)+2 x^3 \log (x)\right ) \, dx\\ &=x^2+\frac {1}{2} \int \left (-2 e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{-2+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}}-3 e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}}-5 e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{1+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}}+3 e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{2+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}}+2 e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{-2+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \log (x)-3 e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \log (x)+2 e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{1+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \log (x)\right ) \, dx\\ &=x^2-\frac {3}{2} \int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \, dx+\frac {3}{2} \int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{2+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \, dx-\frac {3}{2} \int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \log (x) \, dx-\frac {5}{2} \int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{1+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \, dx-\int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{-2+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \, dx+\int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{-2+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \log (x) \, dx+\int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{1+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \log (x) \, dx\\ &=x^2+\frac {3}{2} \int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{2+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \, dx-\frac {3}{2} \int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{\frac {1+\frac {3 x^2}{2}-\frac {x^3}{2}}{x}} \, dx+\frac {3}{2} \int \frac {\int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{\frac {1+\frac {3 x^2}{2}-\frac {x^3}{2}}{x}} \, dx}{x} \, dx-\frac {5}{2} \int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{1+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \, dx+\log (x) \int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{-2+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \, dx+\log (x) \int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{1+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \, dx-\frac {1}{2} (3 \log (x)) \int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{\frac {1+\frac {3 x^2}{2}-\frac {x^3}{2}}{x}} \, dx-\int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{-2+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \, dx-\int \frac {\int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{-2+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \, dx}{x} \, dx-\int \frac {\int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{1+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \, dx}{x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 1.36, size = 42, normalized size = 1.35 \begin {gather*} x^2-e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 37, normalized size = 1.19 \begin {gather*} x^{2} - e^{\left (-\frac {x^{4} - 3 \, x^{3} + {\left (x^{3} - 3 \, x^{2} - 2\right )} \log \relax (x) - 2 \, x}{2 \, x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.68, size = 37, normalized size = 1.19 \begin {gather*} x^{2} - e^{\left (-\frac {1}{2} \, x^{3} - \frac {1}{2} \, x^{2} \log \relax (x) + \frac {3}{2} \, x^{2} + \frac {3}{2} \, x \log \relax (x) + \frac {\log \relax (x)}{x} + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 27, normalized size = 0.87
method | result | size |
risch | \(-{\mathrm e}^{-\frac {\left (x^{3}-3 x^{2}-2\right ) \left (x +\ln \relax (x )\right )}{2 x}}+x^{2}\) | \(27\) |
default | \(-{\mathrm e}^{\frac {\left (-x^{3}+3 x^{2}+2\right ) \ln \relax (x )-x^{4}+3 x^{3}+2 x}{2 x}}+x^{2}\) | \(42\) |
norman | \(\frac {x^{3}-x \,{\mathrm e}^{\frac {\left (-x^{3}+3 x^{2}+2\right ) \ln \relax (x )-x^{4}+3 x^{3}+2 x}{2 x}}}{x}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 37, normalized size = 1.19 \begin {gather*} x^{2} - e^{\left (-\frac {1}{2} \, x^{3} - \frac {1}{2} \, x^{2} \log \relax (x) + \frac {3}{2} \, x^{2} + \frac {3}{2} \, x \log \relax (x) + \frac {\log \relax (x)}{x} + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.10, size = 39, normalized size = 1.26 \begin {gather*} x^2-\frac {x^{1/x}\,{\left ({\mathrm {e}}^{x^2}\right )}^{3/2}\,{\mathrm {e}}^{\frac {3\,x\,\ln \relax (x)}{2}}\,\mathrm {e}\,{\mathrm {e}}^{-\frac {x^2\,\ln \relax (x)}{2}}}{\sqrt {{\mathrm {e}}^{x^3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.42, size = 34, normalized size = 1.10 \begin {gather*} x^{2} - e^{\frac {- \frac {x^{4}}{2} + \frac {3 x^{3}}{2} + x + \frac {\left (- x^{3} + 3 x^{2} + 2\right ) \log {\relax (x )}}{2}}{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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