Optimal. Leaf size=19 \[ 4+\log ^2\left (\frac {x}{5-x}\right )-\log (\log (x)) \]
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Rubi [A]
time = 0.15, antiderivative size = 18, normalized size of antiderivative = 0.95, number of steps
used = 7, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1607, 6874,
2339, 29, 2562, 2338} \begin {gather*} \log ^2\left (\frac {x}{5-x}\right )-\log (\log (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 1607
Rule 2338
Rule 2339
Rule 2562
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5-x-10 \log (x) \log \left (-\frac {x}{-5+x}\right )}{(-5+x) x \log (x)} \, dx\\ &=\int \left (-\frac {1}{x \log (x)}-\frac {10 \log \left (-\frac {x}{-5+x}\right )}{(-5+x) x}\right ) \, dx\\ &=-\left (10 \int \frac {\log \left (-\frac {x}{-5+x}\right )}{(-5+x) x} \, dx\right )-\int \frac {1}{x \log (x)} \, dx\\ &=\log ^2\left (\frac {x}{5-x}\right )-\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=\log ^2\left (\frac {x}{5-x}\right )-\log (\log (x))\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 17, normalized size = 0.89 \begin {gather*} \log ^2\left (-\frac {x}{-5+x}\right )-\log (\log (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.02, size = 19, normalized size = 1.00
method | result | size |
norman | \(\ln \left (-\frac {x}{x -5}\right )^{2}-\ln \left (\ln \left (x \right )\right )\) | \(18\) |
default | \(-\ln \left (\ln \left (x \right )\right )+\ln \left (-1-\frac {5}{x -5}\right )^{2}\) | \(19\) |
risch | \(\ln \left (x -5\right )^{2}-2 \ln \left (x -5\right ) \ln \left (x \right )+i \pi \ln \left (x \right ) \mathrm {csgn}\left (\frac {i}{x -5}\right ) \mathrm {csgn}\left (\frac {i x}{x -5}\right )^{2}-2 i \pi \ln \left (x \right ) \mathrm {csgn}\left (\frac {i x}{x -5}\right )^{2}-i \pi \ln \left (x -5\right ) \mathrm {csgn}\left (\frac {i x}{x -5}\right )^{3}+i \pi \ln \left (x -5\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i}{x -5}\right ) \mathrm {csgn}\left (\frac {i x}{x -5}\right )-2 i \pi \ln \left (x -5\right )+2 i \pi \ln \left (x \right )+i \pi \ln \left (x \right ) \mathrm {csgn}\left (\frac {i x}{x -5}\right )^{3}+2 i \pi \ln \left (x -5\right ) \mathrm {csgn}\left (\frac {i x}{x -5}\right )^{2}+i \pi \ln \left (x \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x}{x -5}\right )^{2}-i \pi \ln \left (x -5\right ) \mathrm {csgn}\left (\frac {i}{x -5}\right ) \mathrm {csgn}\left (\frac {i x}{x -5}\right )^{2}-i \pi \ln \left (x \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i}{x -5}\right ) \mathrm {csgn}\left (\frac {i x}{x -5}\right )-i \pi \ln \left (x -5\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x}{x -5}\right )^{2}+\ln \left (x \right )^{2}-\ln \left (\ln \left (x \right )\right )\) | \(281\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.43, size = 28, normalized size = 1.47 \begin {gather*} \log \left (x\right )^{2} - 2 \, \log \left (x\right ) \log \left (-x + 5\right ) + \log \left (-x + 5\right )^{2} - \log \left (\log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 17, normalized size = 0.89 \begin {gather*} \log \left (-\frac {x}{x - 5}\right )^{2} - \log \left (\log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 14, normalized size = 0.74 \begin {gather*} \log {\left (- \frac {x}{x - 5} \right )}^{2} - \log {\left (\log {\left (x \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 0.39, size = 45, normalized size = 2.37 \begin {gather*} -2 i \, \pi \log \left (x - 5\right ) + 2 \, {\left (\log \left (x - 5\right ) - \log \left (x\right )\right )} \log \left (x - 5\right ) - \log \left (x - 5\right )^{2} + 2 i \, \pi \log \left (x\right ) + \log \left (x\right )^{2} - \log \left (\log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.86, size = 17, normalized size = 0.89 \begin {gather*} {\ln \left (-\frac {x}{x-5}\right )}^2-\ln \left (\ln \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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