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 = 0.38, 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 = {6820, 6874,
78, 205, 213, 267} \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 78
Rule 205
Rule 213
Rule 267
Rule 6820
Rule 6874
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 \text {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 = 0.09, 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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Maple [A]
time = 0.72, 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 = 0.31, size = 23, normalized size = 1.10 \begin {gather*} \frac {1}{16 \, \log \left (x\right )^{4} - 24 \, \log \left (x\right )^{2} + 9} + \log \left (x - 1\right ) + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 62 vs.
\(2 (18) = 36\).
time = 0.36, 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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Sympy [A]
time = 0.07, 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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Giac [A]
time = 0.53, 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 \left (x\right ) \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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