Optimal. Leaf size=16 \[ \log \left (\frac {4 e^4 x^2}{-5+e^x}\right ) \]
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Rubi [A] time = 0.14, antiderivative size = 15, normalized size of antiderivative = 0.94, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6742, 2282, 36, 31, 29, 43} \begin {gather*} 2 \log (x)-\log \left (5-e^x\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 43
Rule 2282
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {5}{-5+e^x}+\frac {2-x}{x}\right ) \, dx\\ &=-\left (5 \int \frac {1}{-5+e^x} \, dx\right )+\int \frac {2-x}{x} \, dx\\ &=-\left (5 \operatorname {Subst}\left (\int \frac {1}{(-5+x) x} \, dx,x,e^x\right )\right )+\int \left (-1+\frac {2}{x}\right ) \, dx\\ &=-x+2 \log (x)-\operatorname {Subst}\left (\int \frac {1}{-5+x} \, dx,x,e^x\right )+\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )\\ &=-\log \left (5-e^x\right )+2 \log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.03, size = 15, normalized size = 0.94 \begin {gather*} -\log \left (5-e^x\right )+2 \log (x) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 12, normalized size = 0.75 \begin {gather*} 2 \, \log \relax (x) - \log \left (e^{x} - 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 12, normalized size = 0.75 \begin {gather*} 2 \, \log \relax (x) - \log \left (e^{x} - 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 13, normalized size = 0.81
method | result | size |
norman | \(2 \ln \relax (x )-\ln \left ({\mathrm e}^{x}-5\right )\) | \(13\) |
risch | \(2 \ln \relax (x )-\ln \left ({\mathrm e}^{x}-5\right )\) | \(13\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 12, normalized size = 0.75 \begin {gather*} 2 \, \log \relax (x) - \log \left (e^{x} - 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 12, normalized size = 0.75 \begin {gather*} 2\,\ln \relax (x)-\ln \left ({\mathrm {e}}^x-5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 10, normalized size = 0.62 \begin {gather*} 2 \log {\relax (x )} - \log {\left (e^{x} - 5 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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