Optimal. Leaf size=20 \[ 4+\log \left (\frac {4}{x \left (1+x+x^2\right ) \log (x)}\right ) \]
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Rubi [A]
time = 0.30, antiderivative size = 19, normalized size of antiderivative = 0.95, number of steps
used = 8, number of rules used = 6, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1608, 6860,
1642, 642, 2339, 29} \begin {gather*} -\log \left (x^2+x+1\right )-\log (x)-\log (\log (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 642
Rule 1608
Rule 1642
Rule 2339
Rule 6860
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1-x-x^2+\left (-1-2 x-3 x^2\right ) \log (x)}{x \left (1+x+x^2\right ) \log (x)} \, dx\\ &=\int \left (\frac {-1-2 x-3 x^2}{x \left (1+x+x^2\right )}-\frac {1}{x \log (x)}\right ) \, dx\\ &=\int \frac {-1-2 x-3 x^2}{x \left (1+x+x^2\right )} \, dx-\int \frac {1}{x \log (x)} \, dx\\ &=\int \left (-\frac {1}{x}+\frac {-1-2 x}{1+x+x^2}\right ) \, dx-\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=-\log (x)-\log (\log (x))+\int \frac {-1-2 x}{1+x+x^2} \, dx\\ &=-\log (x)-\log \left (1+x+x^2\right )-\log (\log (x))\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.03, size = 19, normalized size = 0.95 \begin {gather*} -\log (x)-\log \left (1+x+x^2\right )-\log (\log (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 18, normalized size = 0.90
method | result | size |
default | \(-\ln \left (\ln \left (x \right )\right )-\ln \left (x \left (x^{2}+x +1\right )\right )\) | \(18\) |
risch | \(-\ln \left (x^{3}+x^{2}+x \right )-\ln \left (\ln \left (x \right )\right )\) | \(18\) |
norman | \(-\ln \left (x \right )-\ln \left (\ln \left (x \right )\right )-\ln \left (x^{2}+x +1\right )\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 19, normalized size = 0.95 \begin {gather*} -\log \left (x^{2} + x + 1\right ) - \log \left (x\right ) - \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 = 0.40, size = 17, normalized size = 0.85 \begin {gather*} -\log \left (x^{3} + x^{2} + x\right ) - \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 = 0.07, size = 15, normalized size = 0.75 \begin {gather*} - \log {\left (x^{3} + x^{2} + x \right )} - \log {\left (\log {\left (x \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.38, size = 19, normalized size = 0.95 \begin {gather*} -\log \left (x^{2} + x + 1\right ) - \log \left (x\right ) - \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 = 1.85, size = 17, normalized size = 0.85 \begin {gather*} -\ln \left (\ln \left (x\right )\,\left (x^2+x+1\right )\right )-\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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