Optimal. Leaf size=34 \[ \log \left (\frac {\left (2-2 e^x\right )^2}{4+x+\frac {3-\frac {x}{4}+\frac {x^2}{16}}{x}}\right ) \]
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Rubi [A] time = 0.73, antiderivative size = 26, normalized size of antiderivative = 0.76, number of steps used = 10, number of rules used = 8, integrand size = 61, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.131, Rules used = {6741, 6728, 2282, 36, 31, 29, 1628, 628} \begin {gather*} -\log \left (17 x^2+60 x+48\right )+2 \log \left (1-e^x\right )+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 628
Rule 1628
Rule 2282
Rule 6728
Rule 6741
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {48-17 x^2-e^x \left (48+96 x+103 x^2+34 x^3\right )}{\left (1-e^x\right ) x \left (48+60 x+17 x^2\right )} \, dx\\ &=\int \left (\frac {2}{-1+e^x}+\frac {48+96 x+103 x^2+34 x^3}{x \left (48+60 x+17 x^2\right )}\right ) \, dx\\ &=2 \int \frac {1}{-1+e^x} \, dx+\int \frac {48+96 x+103 x^2+34 x^3}{x \left (48+60 x+17 x^2\right )} \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {1}{(-1+x) x} \, dx,x,e^x\right )+\int \left (2+\frac {1}{x}-\frac {2 (30+17 x)}{48+60 x+17 x^2}\right ) \, dx\\ &=2 x+\log (x)-2 \int \frac {30+17 x}{48+60 x+17 x^2} \, dx+2 \operatorname {Subst}\left (\int \frac {1}{-1+x} \, dx,x,e^x\right )-2 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )\\ &=2 \log \left (1-e^x\right )+\log (x)-\log \left (48+60 x+17 x^2\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.08, size = 26, normalized size = 0.76 \begin {gather*} 2 \log \left (1-e^x\right )+\log (x)-\log \left (48+60 x+17 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 23, normalized size = 0.68 \begin {gather*} -\log \left (17 \, x^{2} + 60 \, x + 48\right ) + \log \relax (x) + 2 \, \log \left (e^{x} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 23, normalized size = 0.68 \begin {gather*} -\log \left (17 \, x^{2} + 60 \, x + 48\right ) + \log \relax (x) + 2 \, \log \left (e^{x} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 24, normalized size = 0.71
method | result | size |
norman | \(2 \ln \left ({\mathrm e}^{x}-1\right )-\ln \left (17 x^{2}+60 x +48\right )+\ln \relax (x )\) | \(24\) |
risch | \(2 \ln \left ({\mathrm e}^{x}-1\right )-\ln \left (17 x^{2}+60 x +48\right )+\ln \relax (x )\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 23, normalized size = 0.68 \begin {gather*} -\log \left (17 \, x^{2} + 60 \, x + 48\right ) + \log \relax (x) + 2 \, \log \left (e^{x} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 23, normalized size = 0.68 \begin {gather*} 2\,\ln \left ({\mathrm {e}}^x-1\right )-\ln \left (17\,x^2+60\,x+48\right )+\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 22, normalized size = 0.65 \begin {gather*} \log {\relax (x )} + 2 \log {\left (e^{x} - 1 \right )} - \log {\left (17 x^{2} + 60 x + 48 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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