Optimal. Leaf size=25 \[ \log \left (\frac {\left (4+e^4\right ) \left (e^{2-x}+\log (5)\right )}{6 (1+x)}\right ) \]
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Rubi [A] time = 0.84, antiderivative size = 21, normalized size of antiderivative = 0.84, number of steps used = 8, number of rules used = 7, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.180, Rules used = {6741, 6688, 6742, 2282, 36, 29, 31} \begin {gather*} -x-\log (x+1)+\log \left (e^x \log (5)+e^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2282
Rule 6688
Rule 6741
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x \left (e^{2-x} (-2-x)-\log (5)\right )}{(1+x) \left (e^2+e^x \log (5)\right )} \, dx\\ &=\int \frac {-e^2 (2+x)-e^x \log (5)}{(1+x) \left (e^2+e^x \log (5)\right )} \, dx\\ &=\int \left (\frac {1}{-1-x}-\frac {e^2}{e^2+e^x \log (5)}\right ) \, dx\\ &=-\log (1+x)-e^2 \int \frac {1}{e^2+e^x \log (5)} \, dx\\ &=-\log (1+x)-e^2 \operatorname {Subst}\left (\int \frac {1}{x \left (e^2+x \log (5)\right )} \, dx,x,e^x\right )\\ &=-\log (1+x)+\log (5) \operatorname {Subst}\left (\int \frac {1}{e^2+x \log (5)} \, dx,x,e^x\right )-\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )\\ &=-x-\log (1+x)+\log \left (e^2+e^x \log (5)\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.07, size = 21, normalized size = 0.84 \begin {gather*} -x-\log (1+x)+\log \left (e^2+e^x \log (5)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 17, normalized size = 0.68 \begin {gather*} -\log \left (x + 1\right ) + \log \left (e^{\left (-x + 2\right )} + \log \relax (5)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 21, normalized size = 0.84 \begin {gather*} -\log \left (x + 1\right ) + \log \left (-e^{\left (-x + 2\right )} - \log \relax (5)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 18, normalized size = 0.72
method | result | size |
norman | \(-\ln \left (x +1\right )+\ln \left ({\mathrm e}^{2-x}+\ln \relax (5)\right )\) | \(18\) |
risch | \(-\ln \left (x +1\right )-2+\ln \left ({\mathrm e}^{2-x}+\ln \relax (5)\right )\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.82, size = 24, normalized size = 0.96 \begin {gather*} -x - \log \left (x + 1\right ) + \log \left (\frac {e^{x} \log \relax (5) + e^{2}}{\log \relax (5)}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 18, normalized size = 0.72 \begin {gather*} \ln \left (\ln \relax (5)+{\mathrm {e}}^{-x}\,{\mathrm {e}}^2\right )-\ln \left (x+1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 14, normalized size = 0.56 \begin {gather*} - \log {\left (x + 1 \right )} + \log {\left (e^{2 - x} + \log {\relax (5 )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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