Optimal. Leaf size=19 \[ \log \left (x^4 \left (x+\log \left (-\frac {x^2}{\log (5)}\right )\right )^2\right ) \]
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Rubi [A]
time = 0.19, antiderivative size = 20, normalized size of antiderivative = 1.05, number of steps
used = 6, number of rules used = 5, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {2641, 6873, 12,
6874, 6816} \begin {gather*} 2 \log \left (\log \left (-\frac {x^2}{\log (5)}\right )+x\right )+4 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2641
Rule 6816
Rule 6873
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4+6 x+4 \log \left (-\frac {x^2}{\log (5)}\right )}{x \left (x+\log \left (-\frac {x^2}{\log (5)}\right )\right )} \, dx\\ &=\int \frac {2 \left (2+3 x+2 \log \left (-\frac {x^2}{\log (5)}\right )\right )}{x \left (x+\log \left (-\frac {x^2}{\log (5)}\right )\right )} \, dx\\ &=2 \int \frac {2+3 x+2 \log \left (-\frac {x^2}{\log (5)}\right )}{x \left (x+\log \left (-\frac {x^2}{\log (5)}\right )\right )} \, dx\\ &=2 \int \left (\frac {2}{x}+\frac {2+x}{x \left (x+\log \left (-\frac {x^2}{\log (5)}\right )\right )}\right ) \, dx\\ &=4 \log (x)+2 \int \frac {2+x}{x \left (x+\log \left (-\frac {x^2}{\log (5)}\right )\right )} \, dx\\ &=4 \log (x)+2 \log \left (x+\log \left (-\frac {x^2}{\log (5)}\right )\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.03, size = 20, normalized size = 1.05 \begin {gather*} 4 \log (x)+2 \log \left (x+\log \left (-\frac {x^2}{\log (5)}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.37, size = 21, normalized size = 1.11
method | result | size |
norman | \(4 \ln \left (x \right )+2 \ln \left (\ln \left (-\frac {x^{2}}{\ln \left (5\right )}\right )+x \right )\) | \(21\) |
risch | \(4 \ln \left (x \right )+2 \ln \left (\ln \left (-\frac {x^{2}}{\ln \left (5\right )}\right )+x \right )\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 21, normalized size = 1.11 \begin {gather*} 4 \, \log \left (x\right ) + 2 \, \log \left (\frac {1}{2} \, x + \log \left (x\right ) - \frac {1}{2} \, \log \left (-\log \left (5\right )\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 28, normalized size = 1.47 \begin {gather*} 2 \, \log \left (x + \log \left (-\frac {x^{2}}{\log \left (5\right )}\right )\right ) + 2 \, \log \left (-\frac {x^{2}}{\log \left (5\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 19, normalized size = 1.00 \begin {gather*} 4 \log {\left (x \right )} + 2 \log {\left (x + \log {\left (- \frac {x^{2}}{\log {\left (5 \right )}} \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 21, normalized size = 1.11 \begin {gather*} 2 \, \log \left (x + \log \left (-x^{2}\right ) - \log \left (\log \left (5\right )\right )\right ) + 4 \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.76, size = 23, normalized size = 1.21 \begin {gather*} 2\,\ln \left (x+\ln \left (-x^2\right )-\ln \left (\ln \left (5\right )\right )\right )+2\,\ln \left (x^2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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