Optimal. Leaf size=17 \[ -4+e^x-e (x+\log (2 x+\log (x))) \]
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Rubi [A] time = 0.57, antiderivative size = 18, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 4, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.085, Rules used = {2561, 6742, 2194, 6684} \begin {gather*} -e x+e^x-e \log (2 x+\log (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 2194
Rule 2561
Rule 6684
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^x x^2+e \left (-1-2 x-2 x^2\right )+\left (-e x+e^x x\right ) \log (x)}{x (2 x+\log (x))} \, dx\\ &=\int \left (e^x-\frac {e \left (1+2 x+2 x^2+x \log (x)\right )}{x (2 x+\log (x))}\right ) \, dx\\ &=-\left (e \int \frac {1+2 x+2 x^2+x \log (x)}{x (2 x+\log (x))} \, dx\right )+\int e^x \, dx\\ &=e^x-e \int \left (1+\frac {1+2 x}{x (2 x+\log (x))}\right ) \, dx\\ &=e^x-e x-e \int \frac {1+2 x}{x (2 x+\log (x))} \, dx\\ &=e^x-e x-e \log (2 x+\log (x))\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.17, size = 18, normalized size = 1.06 \begin {gather*} e^x-e x-e \log (2 x+\log (x)) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 19, normalized size = 1.12 \begin {gather*} -x e - e \log \left (2 \, x + \log \relax (x)\right ) + e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 19, normalized size = 1.12 \begin {gather*} -x e - e \log \left (2 \, x + \log \relax (x)\right ) + e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 20, normalized size = 1.18
method | result | size |
default | \(-x \,{\mathrm e}-{\mathrm e} \ln \left (2 x +\ln \relax (x )\right )+{\mathrm e}^{x}\) | \(20\) |
norman | \(-x \,{\mathrm e}-{\mathrm e} \ln \left (2 x +\ln \relax (x )\right )+{\mathrm e}^{x}\) | \(20\) |
risch | \(-x \,{\mathrm e}-{\mathrm e} \ln \left (2 x +\ln \relax (x )\right )+{\mathrm e}^{x}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 19, normalized size = 1.12 \begin {gather*} -x e - e \log \left (2 \, x + \log \relax (x)\right ) + e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.47, size = 19, normalized size = 1.12 \begin {gather*} {\mathrm {e}}^x-x\,\mathrm {e}-\mathrm {e}\,\ln \left (2\,x+\ln \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.37, size = 19, normalized size = 1.12 \begin {gather*} - e x + e^{x} - e \log {\left (2 x + \log {\relax (x )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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