Optimal. Leaf size=21 \[ \log ((2-2 x) x)+\frac {1}{\left (3-\log ^2\left (x^2\right )\right )^2} \]
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Rubi [A] time = 0.58, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 109, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.055, Rules used = {6688, 6742, 72, 199, 207, 261} \begin {gather*} \frac {1}{\left (3-\log ^2\left (x^2\right )\right )^2}+\log (1-x)+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 72
Rule 199
Rule 207
Rule 261
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {27-54 x-8 (-1+x) \log \left (x^2\right )+27 (-1+2 x) \log ^2\left (x^2\right )+(9-18 x) \log ^4\left (x^2\right )+(-1+2 x) \log ^6\left (x^2\right )}{(1-x) x \left (3-\log ^2\left (x^2\right )\right )^3} \, dx\\ &=\int \left (\frac {-1+2 x}{(-1+x) x}-\frac {8 \log \left (x^2\right )}{x \left (-3+\log ^2\left (x^2\right )\right )^3}\right ) \, dx\\ &=-\left (8 \int \frac {\log \left (x^2\right )}{x \left (-3+\log ^2\left (x^2\right )\right )^3} \, dx\right )+\int \frac {-1+2 x}{(-1+x) x} \, dx\\ &=-\left (4 \operatorname {Subst}\left (\int \frac {x}{\left (-3+x^2\right )^3} \, dx,x,\log \left (x^2\right )\right )\right )+\int \left (\frac {1}{-1+x}+\frac {1}{x}\right ) \, dx\\ &=\log (1-x)+\log (x)+\frac {1}{\left (3-\log ^2\left (x^2\right )\right )^2}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.19, size = 19, normalized size = 0.90 \begin {gather*} \log (1-x)+\log (x)+\frac {1}{\left (-3+\log ^2\left (x^2\right )\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 62, normalized size = 2.95 \begin {gather*} \frac {\log \left (x^{2} - x\right ) \log \left (x^{2}\right )^{4} - 6 \, \log \left (x^{2} - x\right ) \log \left (x^{2}\right )^{2} + 9 \, \log \left (x^{2} - x\right ) + 1}{\log \left (x^{2}\right )^{4} - 6 \, \log \left (x^{2}\right )^{2} + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.67, size = 25, normalized size = 1.19 \begin {gather*} \frac {1}{\log \left (x^{2}\right )^{4} - 6 \, \log \left (x^{2}\right )^{2} + 9} + \log \left (x - 1\right ) + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 20, normalized size = 0.95
method | result | size |
risch | \(\ln \left (x^{2}-x \right )+\frac {1}{\left (\ln \left (x^{2}\right )^{2}-3\right )^{2}}\) | \(20\) |
norman | \(\frac {-3 \ln \left (x^{2}\right )^{3}+\frac {\ln \left (x^{2}\right )^{5}}{2}+\frac {9 \ln \left (x^{2}\right )}{2}+1}{\left (\ln \left (x^{2}\right )^{2}-3\right )^{2}}+\ln \left (x -1\right )\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 23, normalized size = 1.10 \begin {gather*} \frac {1}{16 \, \log \relax (x)^{4} - 24 \, \log \relax (x)^{2} + 9} + \log \left (x - 1\right ) + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.58, size = 17, normalized size = 0.81 \begin {gather*} \ln \left (x\,\left (x-1\right )\right )+\frac {1}{{\left ({\ln \left (x^2\right )}^2-3\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 24, normalized size = 1.14 \begin {gather*} \log {\left (x^{2} - x \right )} + \frac {1}{\log {\left (x^{2} \right )}^{4} - 6 \log {\left (x^{2} \right )}^{2} + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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