Optimal. Leaf size=26 \[ \frac {1}{3} \log \left (\frac {2 \left (2+3 e^{-x/2}\right )}{\log (1-x)}\right ) \]
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Rubi [A] time = 0.91, antiderivative size = 33, normalized size of antiderivative = 1.27, number of steps used = 11, number of rules used = 9, integrand size = 80, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.112, Rules used = {6688, 12, 6742, 2282, 36, 29, 31, 2390, 2302} \begin {gather*} -\frac {x}{6}+\frac {1}{3} \log \left (2 e^{x/2}+3\right )-\frac {1}{3} \log (\log (1-x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 29
Rule 31
Rule 36
Rule 2282
Rule 2302
Rule 2390
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {6+4 e^{x/2}+3 (-1+x) \log (1-x)}{6 \left (3+2 e^{x/2}\right ) (1-x) \log (1-x)} \, dx\\ &=\frac {1}{6} \int \frac {6+4 e^{x/2}+3 (-1+x) \log (1-x)}{\left (3+2 e^{x/2}\right ) (1-x) \log (1-x)} \, dx\\ &=\frac {1}{6} \int \left (-\frac {3}{3+2 e^{x/2}}-\frac {2}{(-1+x) \log (1-x)}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {1}{(-1+x) \log (1-x)} \, dx\right )-\frac {1}{2} \int \frac {1}{3+2 e^{x/2}} \, dx\\ &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,1-x\right )\right )-\operatorname {Subst}\left (\int \frac {1}{x (3+2 x)} \, dx,x,e^{x/2}\right )\\ &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^{x/2}\right )\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (1-x)\right )+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{3+2 x} \, dx,x,e^{x/2}\right )\\ &=-\frac {x}{6}+\frac {1}{3} \log \left (3+2 e^{x/2}\right )-\frac {1}{3} \log (\log (1-x))\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.09, size = 31, normalized size = 1.19 \begin {gather*} \frac {1}{6} \left (-x+2 \log \left (3+2 e^{x/2}\right )-2 \log (\log (1-x))\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 22, normalized size = 0.85 \begin {gather*} \frac {1}{3} \, \log \left (e^{\left (-\frac {1}{2} \, x + \log \relax (3)\right )} + 2\right ) - \frac {1}{3} \, \log \left (\log \left (-x + 1\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 21, normalized size = 0.81 \begin {gather*} \frac {1}{3} \, \log \left (3 \, e^{\left (-\frac {1}{2} \, x\right )} + 2\right ) - \frac {1}{3} \, \log \left (\log \left (-x + 1\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 23, normalized size = 0.88
method | result | size |
norman | \(-\frac {\ln \left (\ln \left (1-x \right )\right )}{3}+\frac {\ln \left ({\mathrm e}^{\ln \relax (3)-\frac {x}{2}}+2\right )}{3}\) | \(23\) |
risch | \(-\frac {\ln \relax (3)}{3}+\frac {\ln \left (3 \,{\mathrm e}^{-\frac {x}{2}}+2\right )}{3}-\frac {\ln \left (\ln \left (1-x \right )\right )}{3}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.23, size = 22, normalized size = 0.85 \begin {gather*} -\frac {1}{6} \, x + \frac {1}{3} \, \log \left (e^{\left (\frac {1}{2} \, x\right )} + \frac {3}{2}\right ) - \frac {1}{3} \, \log \left (\log \left (-x + 1\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.54, size = 21, normalized size = 0.81 \begin {gather*} \frac {\ln \left (\frac {3}{\sqrt {{\mathrm {e}}^x}}+2\right )}{3}-\frac {\ln \left (\ln \left (1-x\right )\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.39, size = 20, normalized size = 0.77 \begin {gather*} \frac {\log {\left (\frac {2}{3} + e^{- \frac {x}{2}} \right )}}{3} - \frac {\log {\left (\log {\left (1 - x \right )} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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