Optimal. Leaf size=26 \[ \log \left (\frac {7 \left (4-x^2\right ) \log (4) (-x+\log (4 \log (x)))}{3 x}\right ) \]
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Rubi [A]
time = 1.21, antiderivative size = 23, normalized size of antiderivative = 0.88, number of steps
used = 7, number of rules used = 5, integrand size = 58, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {6873, 6857,
457, 78, 6816} \begin {gather*} \log \left (4-x^2\right )-\log (x)+\log (x-\log (4 \log (x))) \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rule 6816
Rule 6857
Rule 6873
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4+x^2-2 x^3 \log (x)+\left (4+x^2\right ) \log (x) \log (4 \log (x))}{x \left (4-x^2\right ) \log (x) (x-\log (4 \log (x)))} \, dx\\ &=\int \left (\frac {4+x^2}{x \left (-4+x^2\right )}+\frac {-1+x \log (x)}{x \log (x) (x-\log (4 \log (x)))}\right ) \, dx\\ &=\int \frac {4+x^2}{x \left (-4+x^2\right )} \, dx+\int \frac {-1+x \log (x)}{x \log (x) (x-\log (4 \log (x)))} \, dx\\ &=\log (x-\log (4 \log (x)))+\frac {1}{2} \text {Subst}\left (\int \frac {4+x}{(-4+x) x} \, dx,x,x^2\right )\\ &=\log (x-\log (4 \log (x)))+\frac {1}{2} \text {Subst}\left (\int \left (\frac {2}{-4+x}-\frac {1}{x}\right ) \, dx,x,x^2\right )\\ &=-\log (x)+\log \left (4-x^2\right )+\log (x-\log (4 \log (x)))\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.15, size = 23, normalized size = 0.88 \begin {gather*} -\log (x)+\log \left (4-x^2\right )+\log (x-\log (4 \log (x))) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.75, size = 22, normalized size = 0.85
method | result | size |
risch | \(-\ln \left (x \right )+\ln \left (x^{2}-4\right )+\ln \left (\ln \left (4 \ln \left (x \right )\right )-x \right )\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 25, normalized size = 0.96 \begin {gather*} \log \left (x + 2\right ) + \log \left (x - 2\right ) - \log \left (x\right ) + \log \left (-x + 2 \, \log \left (2\right ) + \log \left (\log \left (x\right )\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 21, normalized size = 0.81 \begin {gather*} \log \left (x^{2} - 4\right ) - \log \left (x\right ) + \log \left (-x + \log \left (4 \, \log \left (x\right )\right )\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 = 0.73 \begin {gather*} - \log {\left (x \right )} + \log {\left (- x + \log {\left (4 \log {\left (x \right )} \right )} \right )} + \log {\left (x^{2} - 4 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 21, normalized size = 0.81 \begin {gather*} \log \left (x^{2} - 4\right ) - \log \left (x\right ) + \log \left (-x + \log \left (4 \, \log \left (x\right )\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.89, size = 21, normalized size = 0.81 \begin {gather*} \ln \left (\ln \left (4\,\ln \left (x\right )\right )-x\right )+\ln \left (x^2-4\right )-\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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