Optimal. Leaf size=21 \[ \log \left (\left (-\frac {32}{x}+x\right )^2 (-1+\log (-1+x+\log (x)))^2\right ) \]
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Rubi [A] time = 3.42, antiderivative size = 28, normalized size of antiderivative = 1.33, number of steps used = 8, number of rules used = 6, integrand size = 117, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {6688, 12, 6725, 446, 72, 6684} \begin {gather*} 2 \log \left (32-x^2\right )-2 \log (x)+2 \log (1-\log (x+\log (x)-1)) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 72
Rule 446
Rule 6684
Rule 6688
Rule 6725
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (-2 (-32+x) x-\left (32+x^2\right ) \log (x) (-1+\log (-1+x+\log (x)))-\left (-32+32 x-x^2+x^3\right ) \log (-1+x+\log (x))\right )}{x \left (32-x^2\right ) (1-x-\log (x)) (1-\log (-1+x+\log (x)))} \, dx\\ &=2 \int \frac {-2 (-32+x) x-\left (32+x^2\right ) \log (x) (-1+\log (-1+x+\log (x)))-\left (-32+32 x-x^2+x^3\right ) \log (-1+x+\log (x))}{x \left (32-x^2\right ) (1-x-\log (x)) (1-\log (-1+x+\log (x)))} \, dx\\ &=2 \int \left (\frac {32+x^2}{x \left (-32+x^2\right )}+\frac {1+x}{x (-1+x+\log (x)) (-1+\log (-1+x+\log (x)))}\right ) \, dx\\ &=2 \int \frac {32+x^2}{x \left (-32+x^2\right )} \, dx+2 \int \frac {1+x}{x (-1+x+\log (x)) (-1+\log (-1+x+\log (x)))} \, dx\\ &=2 \log (1-\log (-1+x+\log (x)))+\operatorname {Subst}\left (\int \frac {32+x}{(-32+x) x} \, dx,x,x^2\right )\\ &=2 \log (1-\log (-1+x+\log (x)))+\operatorname {Subst}\left (\int \left (\frac {2}{-32+x}-\frac {1}{x}\right ) \, dx,x,x^2\right )\\ &=-2 \log (x)+2 \log \left (32-x^2\right )+2 \log (1-\log (-1+x+\log (x)))\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.24, size = 26, normalized size = 1.24 \begin {gather*} 2 \left (-\log (x)+\log \left (32-x^2\right )+\log (1-\log (-1+x+\log (x)))\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 24, normalized size = 1.14 \begin {gather*} 2 \, \log \left (x^{2} - 32\right ) - 2 \, \log \relax (x) + 2 \, \log \left (\log \left (x + \log \relax (x) - 1\right ) - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 24, normalized size = 1.14 \begin {gather*} 2 \, \log \left (x^{2} - 32\right ) - 2 \, \log \relax (x) + 2 \, \log \left (\log \left (x + \log \relax (x) - 1\right ) - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 25, normalized size = 1.19
method | result | size |
risch | \(-2 \ln \relax (x )+2 \ln \left (x^{2}-32\right )+2 \ln \left (\ln \left (-1+\ln \relax (x )+x \right )-1\right )\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 24, normalized size = 1.14 \begin {gather*} 2 \, \log \left (x^{2} - 32\right ) - 2 \, \log \relax (x) + 2 \, \log \left (\log \left (x + \log \relax (x) - 1\right ) - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.42, size = 24, normalized size = 1.14 \begin {gather*} 2\,\ln \left (x^2-32\right )-2\,\ln \relax (x)+2\,\ln \left (\ln \left (x+\ln \relax (x)-1\right )-1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.49, size = 26, normalized size = 1.24 \begin {gather*} - 2 \log {\relax (x )} + 2 \log {\left (x^{2} - 32 \right )} + 2 \log {\left (\log {\left (x + \log {\relax (x )} - 1 \right )} - 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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