Optimal. Leaf size=26 \[ 6+\left (e^{\frac {\left (e^2+x\right ) \left (4+x^2\right )}{x}}-x\right )^2+\log (x) \]
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Rubi [F] time = 0.37, 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 {x+2 x^3+e^{\frac {2 \left (4 x+x^3+e^2 \left (4+x^2\right )\right )}{x}} \left (4 x^3+e^2 \left (-8+2 x^2\right )\right )+e^{\frac {4 x+x^3+e^2 \left (4+x^2\right )}{x}} \left (-2 x^2-4 x^4+e^2 \left (8 x-2 x^3\right )\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+2 x^2}{x}+\frac {2 e^{\frac {2 \left (e^2+x\right ) \left (4+x^2\right )}{x}} \left (-4 e^2+e^2 x^2+2 x^3\right )}{x^2}-\frac {2 e^{\frac {\left (e^2+x\right ) \left (4+x^2\right )}{x}} \left (-4 e^2+x+e^2 x^2+2 x^3\right )}{x}\right ) \, dx\\ &=2 \int \frac {e^{\frac {2 \left (e^2+x\right ) \left (4+x^2\right )}{x}} \left (-4 e^2+e^2 x^2+2 x^3\right )}{x^2} \, dx-2 \int \frac {e^{\frac {\left (e^2+x\right ) \left (4+x^2\right )}{x}} \left (-4 e^2+x+e^2 x^2+2 x^3\right )}{x} \, dx+\int \frac {1+2 x^2}{x} \, dx\\ &=\frac {2 e^{\frac {\left (e^2+x\right ) \left (4+x^2\right )}{x}} \left (4 e^2-e^2 x^2-2 x^3\right )}{x \left (2 \left (e^2+x\right )+\frac {4+x^2}{x}-\frac {\left (e^2+x\right ) \left (4+x^2\right )}{x^2}\right )}+2 \int \left (e^{2+\frac {2 \left (e^2+x\right ) \left (4+x^2\right )}{x}}-\frac {4 e^{2+\frac {2 \left (e^2+x\right ) \left (4+x^2\right )}{x}}}{x^2}+2 e^{\frac {2 \left (e^2+x\right ) \left (4+x^2\right )}{x}} x\right ) \, dx+\int \left (\frac {1}{x}+2 x\right ) \, dx\\ &=x^2+\frac {2 e^{\frac {\left (e^2+x\right ) \left (4+x^2\right )}{x}} \left (4 e^2-e^2 x^2-2 x^3\right )}{x \left (2 \left (e^2+x\right )+\frac {4+x^2}{x}-\frac {\left (e^2+x\right ) \left (4+x^2\right )}{x^2}\right )}+\log (x)+2 \int e^{2+\frac {2 \left (e^2+x\right ) \left (4+x^2\right )}{x}} \, dx+4 \int e^{\frac {2 \left (e^2+x\right ) \left (4+x^2\right )}{x}} x \, dx-8 \int \frac {e^{2+\frac {2 \left (e^2+x\right ) \left (4+x^2\right )}{x}}}{x^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.12, size = 25, normalized size = 0.96 \begin {gather*} \left (e^{\frac {\left (e^2+x\right ) \left (4+x^2\right )}{x}}-x\right )^2+\log (x) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.07, size = 50, normalized size = 1.92 \begin {gather*} x^{2} - 2 \, x e^{\left (\frac {x^{3} + {\left (x^{2} + 4\right )} e^{2} + 4 \, x}{x}\right )} + e^{\left (\frac {2 \, {\left (x^{3} + {\left (x^{2} + 4\right )} e^{2} + 4 \, x\right )}}{x}\right )} + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 54, normalized size = 2.08 \begin {gather*} x^{2} - 2 \, x e^{\left (\frac {x^{3} + x^{2} e^{2} + 4 \, x + 4 \, e^{2}}{x}\right )} + e^{\left (\frac {2 \, {\left (x^{3} + x^{2} e^{2} + 4 \, x + 4 \, e^{2}\right )}}{x}\right )} + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 39, normalized size = 1.50
method | result | size |
risch | \(\ln \relax (x )+x^{2}+{\mathrm e}^{\frac {2 \left (x^{2}+4\right ) \left (x +{\mathrm e}^{2}\right )}{x}}-2 \,{\mathrm e}^{\frac {\left (x^{2}+4\right ) \left (x +{\mathrm e}^{2}\right )}{x}} x\) | \(39\) |
norman | \(\frac {x^{3}+x \,{\mathrm e}^{\frac {2 \left (x^{2}+4\right ) {\mathrm e}^{2}+2 x^{3}+8 x}{x}}-2 x^{2} {\mathrm e}^{\frac {\left (x^{2}+4\right ) {\mathrm e}^{2}+x^{3}+4 x}{x}}}{x}+\ln \relax (x )\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 46, normalized size = 1.77 \begin {gather*} x^{2} - 2 \, x e^{\left (x^{2} + x e^{2} + \frac {4 \, e^{2}}{x} + 4\right )} + e^{\left (2 \, x^{2} + 2 \, x e^{2} + \frac {8 \, e^{2}}{x} + 8\right )} + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.25, size = 46, normalized size = 1.77 \begin {gather*} {\mathrm {e}}^{2\,x\,{\mathrm {e}}^2+\frac {8\,{\mathrm {e}}^2}{x}+2\,x^2+8}+\ln \relax (x)-2\,x\,{\mathrm {e}}^{x\,{\mathrm {e}}^2+\frac {4\,{\mathrm {e}}^2}{x}+x^2+4}+x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.30, size = 49, normalized size = 1.88 \begin {gather*} x^{2} - 2 x e^{\frac {x^{3} + 4 x + \left (x^{2} + 4\right ) e^{2}}{x}} + e^{\frac {2 \left (x^{3} + 4 x + \left (x^{2} + 4\right ) e^{2}\right )}{x}} + \log {\relax (x )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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