Optimal. Leaf size=22 \[ -\log \left (2 x \left (\log (x)-e^{-2-x} \log (x)\right )^2\right ) \]
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Rubi [A] time = 1.61, antiderivative size = 25, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 9, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.170, Rules used = {6741, 6688, 6742, 2282, 36, 31, 29, 2365, 43} \begin {gather*} 2 x-2 \log \left (1-e^{x+2}\right )-\log (x)-2 \log (\log (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 43
Rule 2282
Rule 2365
Rule 6688
Rule 6741
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{2+x} \left (2 \log (x)+\log ^2(x)+e^{-2-x} \log (x) (-2+(-1+2 x) \log (x))\right )}{\left (1-e^{2+x}\right ) x \log ^2(x)} \, dx\\ &=\int \frac {-2+2 e^{2+x}+\left (-1+e^{2+x}+2 x\right ) \log (x)}{\left (1-e^{2+x}\right ) x \log (x)} \, dx\\ &=\int \left (-\frac {2}{-1+e^{2+x}}+\frac {-2-\log (x)}{x \log (x)}\right ) \, dx\\ &=-\left (2 \int \frac {1}{-1+e^{2+x}} \, dx\right )+\int \frac {-2-\log (x)}{x \log (x)} \, dx\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{(-1+x) x} \, dx,x,e^{2+x}\right )\right )+\operatorname {Subst}\left (\int \frac {-2-x}{x} \, dx,x,\log (x)\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{-1+x} \, dx,x,e^{2+x}\right )\right )+2 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^{2+x}\right )+\operatorname {Subst}\left (\int \left (-1-\frac {2}{x}\right ) \, dx,x,\log (x)\right )\\ &=2 x-2 \log \left (1-e^{2+x}\right )-\log (x)-2 \log (\log (x))\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.13, size = 25, normalized size = 1.14 \begin {gather*} 2 x-2 \log \left (1-e^{2+x}\right )-\log (x)-2 \log (\log (x)) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 22, normalized size = 1.00 \begin {gather*} -\log \relax (x) - 2 \, \log \left (e^{\left (-x + \log \left (\log \relax (x)\right ) - 2\right )} - \log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.49, size = 24, normalized size = 1.09 \begin {gather*} 2 \, x - \log \relax (x) - 2 \, \log \left (-e^{\left (x + 2\right )} + 1\right ) - 2 \, \log \left (\log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 24, normalized size = 1.09
method | result | size |
risch | \(-\ln \relax (x )-4-2 \ln \left (-\ln \relax (x )+\ln \relax (x ) {\mathrm e}^{-x -2}\right )\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 25, normalized size = 1.14 \begin {gather*} 2 \, x - 2 \, \log \left ({\left (e^{\left (x + 2\right )} - 1\right )} e^{\left (-2\right )}\right ) - \log \relax (x) - 2 \, \log \left (\log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.64, size = 21, normalized size = 0.95 \begin {gather*} -2\,\ln \left (\ln \relax (x)-{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-2}\,\ln \relax (x)\right )-\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.28, size = 22, normalized size = 1.00 \begin {gather*} - \log {\relax (x )} - 2 \log {\left (e^{- x - 2} - 1 \right )} - 2 \log {\left (\log {\relax (x )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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