Optimal. Leaf size=20 \[ \log \left (\frac {5}{3} \left (-x+\frac {2 x}{5 (x+\log (x))}\right )\right ) \]
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Rubi [A]
time = 0.22, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {6874, 6816}
\begin {gather*} \log (x)+\log (-5 x-5 \log (x)+2)-\log (x+\log (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 6816
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {1}{x}+\frac {-1-x}{x (x+\log (x))}+\frac {5 (1+x)}{x (-2+5 x+5 \log (x))}\right ) \, dx\\ &=\log (x)+5 \int \frac {1+x}{x (-2+5 x+5 \log (x))} \, dx+\int \frac {-1-x}{x (x+\log (x))} \, dx\\ &=\log (x)+\log (2-5 x-5 \log (x))-\log (x+\log (x))\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.05, size = 20, normalized size = 1.00 \begin {gather*} \log (x)+\log (2-5 x-5 \log (x))-\log (x+\log (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.01, size = 21, normalized size = 1.05
method | result | size |
risch | \(\ln \left (x \right )+\ln \left (-\frac {2}{5}+x +\ln \left (x \right )\right )-\ln \left (x +\ln \left (x \right )\right )\) | \(17\) |
default | \(\ln \left (x \right )-\ln \left (x +\ln \left (x \right )\right )+\ln \left (5 \ln \left (x \right )+5 x -2\right )\) | \(21\) |
norman | \(\ln \left (x \right )-\ln \left (x +\ln \left (x \right )\right )+\ln \left (5 \ln \left (x \right )+5 x -2\right )\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 16, normalized size = 0.80 \begin {gather*} -\log \left (x + \log \left (x\right )\right ) + \log \left (x + \log \left (x\right ) - \frac {2}{5}\right ) + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 20, normalized size = 1.00 \begin {gather*} \log \left (5 \, x + 5 \, \log \left (x\right ) - 2\right ) - \log \left (x + \log \left (x\right )\right ) + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 19, normalized size = 0.95 \begin {gather*} \log {\left (x \right )} - \log {\left (x + \log {\left (x \right )} \right )} + \log {\left (x + \log {\left (x \right )} - \frac {2}{5} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 20, normalized size = 1.00 \begin {gather*} \log \left (5 \, x + 5 \, \log \left (x\right ) - 2\right ) - \log \left (x + \log \left (x\right )\right ) + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.60, size = 16, normalized size = 0.80 \begin {gather*} \ln \left (x+\ln \left (x\right )-\frac {2}{5}\right )-\ln \left (x+\ln \left (x\right )\right )+\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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