Optimal. Leaf size=22 \[ \left (e^{1+x+\frac {x}{\log (x-\log (x))}}-x\right )^2 \]
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Rubi [B] time = 2.95, antiderivative size = 155, normalized size of antiderivative = 7.05, number of steps used = 4, number of rules used = 3, integrand size = 187, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {6742, 6706, 2288} \begin {gather*} x^2-\frac {2 e^{x+\frac {x}{\log (x-\log (x))}+1} \left (-x^2+x^2 \log ^2(x-\log (x))+x^2 \log (x-\log (x))+x-x \log (x) \log ^2(x-\log (x))-x \log (x) \log (x-\log (x))\right )}{(x-\log (x)) \left (-\frac {\left (1-\frac {1}{x}\right ) x}{(x-\log (x)) \log ^2(x-\log (x))}+\frac {1}{\log (x-\log (x))}+1\right ) \log ^2(x-\log (x))}+e^{2 x+\frac {2 x}{\log (x-\log (x))}+2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2288
Rule 6706
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2 x+\frac {2 e^{2+2 x+\frac {2 x}{\log (x-\log (x))}} \left (1-x+x \log (x-\log (x))-\log (x) \log (x-\log (x))+x \log ^2(x-\log (x))-\log (x) \log ^2(x-\log (x))\right )}{(x-\log (x)) \log ^2(x-\log (x))}-\frac {2 e^{1+x+\frac {x}{\log (x-\log (x))}} \left (x-x^2+x^2 \log (x-\log (x))-x \log (x) \log (x-\log (x))+x \log ^2(x-\log (x))+x^2 \log ^2(x-\log (x))-\log (x) \log ^2(x-\log (x))-x \log (x) \log ^2(x-\log (x))\right )}{(x-\log (x)) \log ^2(x-\log (x))}\right ) \, dx\\ &=x^2+2 \int \frac {e^{2+2 x+\frac {2 x}{\log (x-\log (x))}} \left (1-x+x \log (x-\log (x))-\log (x) \log (x-\log (x))+x \log ^2(x-\log (x))-\log (x) \log ^2(x-\log (x))\right )}{(x-\log (x)) \log ^2(x-\log (x))} \, dx-2 \int \frac {e^{1+x+\frac {x}{\log (x-\log (x))}} \left (x-x^2+x^2 \log (x-\log (x))-x \log (x) \log (x-\log (x))+x \log ^2(x-\log (x))+x^2 \log ^2(x-\log (x))-\log (x) \log ^2(x-\log (x))-x \log (x) \log ^2(x-\log (x))\right )}{(x-\log (x)) \log ^2(x-\log (x))} \, dx\\ &=e^{2+2 x+\frac {2 x}{\log (x-\log (x))}}+x^2-\frac {2 e^{1+x+\frac {x}{\log (x-\log (x))}} \left (x-x^2+x^2 \log (x-\log (x))-x \log (x) \log (x-\log (x))+x^2 \log ^2(x-\log (x))-x \log (x) \log ^2(x-\log (x))\right )}{(x-\log (x)) \left (1-\frac {\left (1-\frac {1}{x}\right ) x}{(x-\log (x)) \log ^2(x-\log (x))}+\frac {1}{\log (x-\log (x))}\right ) \log ^2(x-\log (x))}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.30, size = 22, normalized size = 1.00 \begin {gather*} \left (e^{1+x+\frac {x}{\log (x-\log (x))}}-x\right )^2 \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.66, size = 56, normalized size = 2.55 \begin {gather*} x^{2} - 2 \, x e^{\left (\frac {{\left (x + 1\right )} \log \left (x - \log \relax (x)\right ) + x}{\log \left (x - \log \relax (x)\right )}\right )} + e^{\left (\frac {2 \, {\left ({\left (x + 1\right )} \log \left (x - \log \relax (x)\right ) + x\right )}}{\log \left (x - \log \relax (x)\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.13, size = 54, normalized size = 2.45 \begin {gather*} x^{2} - 2 \, x e^{\left (\frac {x \log \left (x - \log \relax (x)\right ) + x + \log \left (x - \log \relax (x)\right )}{\log \left (x - \log \relax (x)\right )}\right )} + e^{\left (2 \, x + \frac {2 \, x}{\log \left (x - \log \relax (x)\right )} + 2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 67, normalized size = 3.05
method | result | size |
risch | \(x^{2}-2 x \,{\mathrm e}^{\frac {x \ln \left (x -\ln \relax (x )\right )+\ln \left (x -\ln \relax (x )\right )+x}{\ln \left (x -\ln \relax (x )\right )}}+{\mathrm e}^{\frac {2 x \ln \left (x -\ln \relax (x )\right )+2 \ln \left (x -\ln \relax (x )\right )+2 x}{\ln \left (x -\ln \relax (x )\right )}}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.82, size = 43, normalized size = 1.95 \begin {gather*} x^2+{\mathrm {e}}^{\frac {2\,x}{\ln \left (x-\ln \relax (x)\right )}}\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^2-2\,x\,{\mathrm {e}}^{\frac {x}{\ln \left (x-\ln \relax (x)\right )}}\,\mathrm {e}\,{\mathrm {e}}^x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.03, size = 49, normalized size = 2.23 \begin {gather*} x^{2} - 2 x e^{\frac {x + \left (x + 1\right ) \log {\left (x - \log {\relax (x )} \right )}}{\log {\left (x - \log {\relax (x )} \right )}}} + e^{\frac {2 \left (x + \left (x + 1\right ) \log {\left (x - \log {\relax (x )} \right )}\right )}{\log {\left (x - \log {\relax (x )} \right )}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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