Optimal. Leaf size=22 \[ \frac {x}{\log (-1+x) \log \left (4+4 \left (x-x^2\right )\right )} \]
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Rubi [F] time = 2.52, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (x+x^2-x^3\right ) \log \left (4+4 x-4 x^2\right )+\log (-1+x) \left (-x+3 x^2-2 x^3+\left (1-2 x^2+x^3\right ) \log \left (4+4 x-4 x^2\right )\right )}{\left (1-2 x^2+x^3\right ) \log ^2(-1+x) \log ^2\left (4+4 x-4 x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {x (-1+2 x)}{\left (1+x-x^2\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )}+\frac {-x-\log (-1+x)+x \log (-1+x)}{(-1+x) \log ^2(-1+x) \log \left (4+4 x-4 x^2\right )}\right ) \, dx\\ &=\int \frac {x (-1+2 x)}{\left (1+x-x^2\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )} \, dx+\int \frac {-x-\log (-1+x)+x \log (-1+x)}{(-1+x) \log ^2(-1+x) \log \left (4+4 x-4 x^2\right )} \, dx\\ &=\int \left (-\frac {2}{\log (-1+x) \log ^2\left (4+4 x-4 x^2\right )}+\frac {-2-x}{\left (-1-x+x^2\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )}\right ) \, dx+\int \left (-\frac {x}{(-1+x) \log ^2(-1+x) \log \left (4+4 x-4 x^2\right )}-\frac {1}{(-1+x) \log (-1+x) \log \left (4+4 x-4 x^2\right )}+\frac {x}{(-1+x) \log (-1+x) \log \left (4+4 x-4 x^2\right )}\right ) \, dx\\ &=-\left (2 \int \frac {1}{\log (-1+x) \log ^2\left (4+4 x-4 x^2\right )} \, dx\right )+\int \frac {-2-x}{\left (-1-x+x^2\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )} \, dx-\int \frac {x}{(-1+x) \log ^2(-1+x) \log \left (4+4 x-4 x^2\right )} \, dx-\int \frac {1}{(-1+x) \log (-1+x) \log \left (4+4 x-4 x^2\right )} \, dx+\int \frac {x}{(-1+x) \log (-1+x) \log \left (4+4 x-4 x^2\right )} \, dx\\ &=-\left (2 \int \frac {1}{\log (-1+x) \log ^2\left (4+4 x-4 x^2\right )} \, dx\right )+\int \left (-\frac {2}{\left (-1-x+x^2\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )}-\frac {x}{\left (-1-x+x^2\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )}\right ) \, dx-\int \left (\frac {1}{\log ^2(-1+x) \log \left (4+4 x-4 x^2\right )}+\frac {1}{(-1+x) \log ^2(-1+x) \log \left (4+4 x-4 x^2\right )}\right ) \, dx+\int \left (\frac {1}{\log (-1+x) \log \left (4+4 x-4 x^2\right )}+\frac {1}{(-1+x) \log (-1+x) \log \left (4+4 x-4 x^2\right )}\right ) \, dx-\int \frac {1}{(-1+x) \log (-1+x) \log \left (4+4 x-4 x^2\right )} \, dx\\ &=-\left (2 \int \frac {1}{\log (-1+x) \log ^2\left (4+4 x-4 x^2\right )} \, dx\right )-2 \int \frac {1}{\left (-1-x+x^2\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )} \, dx-\int \frac {x}{\left (-1-x+x^2\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )} \, dx-\int \frac {1}{\log ^2(-1+x) \log \left (4+4 x-4 x^2\right )} \, dx-\int \frac {1}{(-1+x) \log ^2(-1+x) \log \left (4+4 x-4 x^2\right )} \, dx+\int \frac {1}{\log (-1+x) \log \left (4+4 x-4 x^2\right )} \, dx\\ &=-\left (2 \int \left (-\frac {2}{\sqrt {5} \left (1+\sqrt {5}-2 x\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )}-\frac {2}{\sqrt {5} \left (-1+\sqrt {5}+2 x\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )}\right ) \, dx\right )-2 \int \frac {1}{\log (-1+x) \log ^2\left (4+4 x-4 x^2\right )} \, dx-\int \left (\frac {1+\frac {1}{\sqrt {5}}}{\left (-1-\sqrt {5}+2 x\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )}+\frac {1-\frac {1}{\sqrt {5}}}{\left (-1+\sqrt {5}+2 x\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )}\right ) \, dx-\int \frac {1}{\log ^2(-1+x) \log \left (4+4 x-4 x^2\right )} \, dx-\int \frac {1}{(-1+x) \log ^2(-1+x) \log \left (4+4 x-4 x^2\right )} \, dx+\int \frac {1}{\log (-1+x) \log \left (4+4 x-4 x^2\right )} \, dx\\ &=-\left (2 \int \frac {1}{\log (-1+x) \log ^2\left (4+4 x-4 x^2\right )} \, dx\right )+\frac {4 \int \frac {1}{\left (1+\sqrt {5}-2 x\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )} \, dx}{\sqrt {5}}+\frac {4 \int \frac {1}{\left (-1+\sqrt {5}+2 x\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )} \, dx}{\sqrt {5}}-\frac {1}{5} \left (5-\sqrt {5}\right ) \int \frac {1}{\left (-1+\sqrt {5}+2 x\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )} \, dx-\frac {1}{5} \left (5+\sqrt {5}\right ) \int \frac {1}{\left (-1-\sqrt {5}+2 x\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )} \, dx-\int \frac {1}{\log ^2(-1+x) \log \left (4+4 x-4 x^2\right )} \, dx-\int \frac {1}{(-1+x) \log ^2(-1+x) \log \left (4+4 x-4 x^2\right )} \, dx+\int \frac {1}{\log (-1+x) \log \left (4+4 x-4 x^2\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.05, size = 21, normalized size = 0.95 \begin {gather*} \frac {x}{\log (-1+x) \log \left (4+4 x-4 x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 21, normalized size = 0.95 \begin {gather*} \frac {x}{\log \left (-4 \, x^{2} + 4 \, x + 4\right ) \log \left (x - 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 21, normalized size = 0.95 \begin {gather*} \frac {x}{\log \left (-4 \, x^{2} + 4 \, x + 4\right ) \log \left (x - 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 22, normalized size = 1.00
method | result | size |
risch | \(\frac {x}{\ln \left (x -1\right ) \ln \left (-4 x^{2}+4 x +4\right )}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.52, size = 32, normalized size = 1.45 \begin {gather*} \frac {x}{{\left (i \, \pi + 2 \, \log \relax (2)\right )} \log \left (x - 1\right ) + \log \left (x^{2} - x - 1\right ) \log \left (x - 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.96, size = 349, normalized size = 15.86 \begin {gather*} \frac {x}{4}-\frac {5\,\ln \left (x-1\right )}{8}+\frac {\frac {x}{\ln \left (x-1\right )}-\frac {\ln \left (-4\,x^2+4\,x+4\right )\,\left (-x^2+x+1\right )\,\left (x+\ln \left (x-1\right )-x\,\ln \left (x-1\right )\right )}{{\ln \left (x-1\right )}^2\,\left (2\,x-1\right )\,\left (x-1\right )}}{\ln \left (-4\,x^2+4\,x+4\right )}+\frac {\frac {\ln \left (x-1\right )\,\left (4\,x^4-11\,x^3+10\,x^2-2\right )}{2\,{\left (2\,x-1\right )}^2\,\left (x-1\right )}+\frac {x\,\left (-x^2+x+1\right )}{\left (2\,x-1\right )\,\left (x-1\right )}-\frac {{\ln \left (x-1\right )}^2\,\left (x-1\right )\,\left (2\,x^2-2\,x+3\right )}{2\,{\left (2\,x-1\right )}^2}}{{\ln \left (x-1\right )}^2}+\frac {\frac {13\,x^3}{16}-\frac {39\,x^2}{32}+\frac {3\,x}{8}+\frac {3}{32}}{x^4-\frac {5\,x^3}{2}+\frac {9\,x^2}{4}-\frac {7\,x}{8}+\frac {1}{8}}+\frac {\frac {x^3-6\,x^2+4\,x}{2\,{\left (2\,x-1\right )}^2\,\left (x-1\right )}+\frac {{\ln \left (x-1\right )}^2\,\left (x-1\right )\,\left (-4\,x^3+6\,x^2+2\,x-7\right )}{2\,{\left (2\,x-1\right )}^3}+\frac {\ln \left (x-1\right )\,\left (-8\,x^3+7\,x^2+4\,x-4\right )}{2\,{\left (2\,x-1\right )}^3\,\left (x-1\right )}}{\ln \left (x-1\right )}+\frac {\ln \left (x-1\right )\,\left (\frac {x^4}{4}-\frac {11\,x^2}{16}+\frac {33\,x}{32}-\frac {33}{64}\right )}{x^3-\frac {3\,x^2}{2}+\frac {3\,x}{4}-\frac {1}{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 17, normalized size = 0.77 \begin {gather*} \frac {x}{\log {\left (x - 1 \right )} \log {\left (- 4 x^{2} + 4 x + 4 \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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