Integrand size = 109, antiderivative size = 21 \[ \int \frac {27-54 x+(8-8 x) \log \left (x^2\right )+(-27+54 x) \log ^2\left (x^2\right )+(9-18 x) \log ^4\left (x^2\right )+(-1+2 x) \log ^6\left (x^2\right )}{27 x-27 x^2+\left (-27 x+27 x^2\right ) \log ^2\left (x^2\right )+\left (9 x-9 x^2\right ) \log ^4\left (x^2\right )+\left (-x+x^2\right ) \log ^6\left (x^2\right )} \, dx=\log ((2-2 x) x)+\frac {1}{\left (3-\log ^2\left (x^2\right )\right )^2} \]
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Time = 0.37 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.055, Rules used = {6820, 6874, 78, 205, 213, 267} \[ \int \frac {27-54 x+(8-8 x) \log \left (x^2\right )+(-27+54 x) \log ^2\left (x^2\right )+(9-18 x) \log ^4\left (x^2\right )+(-1+2 x) \log ^6\left (x^2\right )}{27 x-27 x^2+\left (-27 x+27 x^2\right ) \log ^2\left (x^2\right )+\left (9 x-9 x^2\right ) \log ^4\left (x^2\right )+\left (-x+x^2\right ) \log ^6\left (x^2\right )} \, dx=\frac {1}{\left (3-\log ^2\left (x^2\right )\right )^2}+\log (1-x)+\log (x) \]
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Rule 78
Rule 205
Rule 213
Rule 267
Rule 6820
Rule 6874
Rubi steps \begin{align*} \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{align*}
Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {27-54 x+(8-8 x) \log \left (x^2\right )+(-27+54 x) \log ^2\left (x^2\right )+(9-18 x) \log ^4\left (x^2\right )+(-1+2 x) \log ^6\left (x^2\right )}{27 x-27 x^2+\left (-27 x+27 x^2\right ) \log ^2\left (x^2\right )+\left (9 x-9 x^2\right ) \log ^4\left (x^2\right )+\left (-x+x^2\right ) \log ^6\left (x^2\right )} \, dx=\log (1-x)+\log (x)+\frac {1}{\left (-3+\log ^2\left (x^2\right )\right )^2} \]
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Time = 0.63 (sec) , antiderivative size = 20, normalized size of antiderivative = 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 | \(\frac {-3 \ln \left (x^{2}\right )^{3}+\frac {\ln \left (x^{2}\right )^{5}}{2}+\frac {9 \ln \left (x^{2}\right )}{2}+1}{{\left (\ln \left (x^{2}\right )^{2}-3\right )}^{2}}+\ln \left (-1+x \right )\) | \(41\) |
parallelrisch | \(\frac {2+2 \ln \left (-1+x \right ) \ln \left (x^{2}\right )^{4}+\ln \left (x^{2}\right )^{5}-12 \ln \left (-1+x \right ) \ln \left (x^{2}\right )^{2}-6 \ln \left (x^{2}\right )^{3}+18 \ln \left (-1+x \right )+9 \ln \left (x^{2}\right )}{2 \ln \left (x^{2}\right )^{4}-12 \ln \left (x^{2}\right )^{2}+18}\) | \(73\) |
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Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (18) = 36\).
Time = 0.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.95 \[ \int \frac {27-54 x+(8-8 x) \log \left (x^2\right )+(-27+54 x) \log ^2\left (x^2\right )+(9-18 x) \log ^4\left (x^2\right )+(-1+2 x) \log ^6\left (x^2\right )}{27 x-27 x^2+\left (-27 x+27 x^2\right ) \log ^2\left (x^2\right )+\left (9 x-9 x^2\right ) \log ^4\left (x^2\right )+\left (-x+x^2\right ) \log ^6\left (x^2\right )} \, dx=\frac {\log \left (x^{2} - x\right ) \log \left (x^{2}\right )^{4} - 6 \, \log \left (x^{2} - x\right ) \log \left (x^{2}\right )^{2} + 9 \, \log \left (x^{2} - x\right ) + 1}{\log \left (x^{2}\right )^{4} - 6 \, \log \left (x^{2}\right )^{2} + 9} \]
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Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {27-54 x+(8-8 x) \log \left (x^2\right )+(-27+54 x) \log ^2\left (x^2\right )+(9-18 x) \log ^4\left (x^2\right )+(-1+2 x) \log ^6\left (x^2\right )}{27 x-27 x^2+\left (-27 x+27 x^2\right ) \log ^2\left (x^2\right )+\left (9 x-9 x^2\right ) \log ^4\left (x^2\right )+\left (-x+x^2\right ) \log ^6\left (x^2\right )} \, dx=\log {\left (x^{2} - x \right )} + \frac {1}{\log {\left (x^{2} \right )}^{4} - 6 \log {\left (x^{2} \right )}^{2} + 9} \]
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Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {27-54 x+(8-8 x) \log \left (x^2\right )+(-27+54 x) \log ^2\left (x^2\right )+(9-18 x) \log ^4\left (x^2\right )+(-1+2 x) \log ^6\left (x^2\right )}{27 x-27 x^2+\left (-27 x+27 x^2\right ) \log ^2\left (x^2\right )+\left (9 x-9 x^2\right ) \log ^4\left (x^2\right )+\left (-x+x^2\right ) \log ^6\left (x^2\right )} \, dx=\frac {1}{16 \, \log \left (x\right )^{4} - 24 \, \log \left (x\right )^{2} + 9} + \log \left (x - 1\right ) + \log \left (x\right ) \]
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Time = 0.41 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {27-54 x+(8-8 x) \log \left (x^2\right )+(-27+54 x) \log ^2\left (x^2\right )+(9-18 x) \log ^4\left (x^2\right )+(-1+2 x) \log ^6\left (x^2\right )}{27 x-27 x^2+\left (-27 x+27 x^2\right ) \log ^2\left (x^2\right )+\left (9 x-9 x^2\right ) \log ^4\left (x^2\right )+\left (-x+x^2\right ) \log ^6\left (x^2\right )} \, dx=\frac {1}{\log \left (x^{2}\right )^{4} - 6 \, \log \left (x^{2}\right )^{2} + 9} + \log \left (x - 1\right ) + \log \left (x\right ) \]
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Time = 9.43 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {27-54 x+(8-8 x) \log \left (x^2\right )+(-27+54 x) \log ^2\left (x^2\right )+(9-18 x) \log ^4\left (x^2\right )+(-1+2 x) \log ^6\left (x^2\right )}{27 x-27 x^2+\left (-27 x+27 x^2\right ) \log ^2\left (x^2\right )+\left (9 x-9 x^2\right ) \log ^4\left (x^2\right )+\left (-x+x^2\right ) \log ^6\left (x^2\right )} \, dx=\ln \left (x\,\left (x-1\right )\right )+\frac {1}{{\left ({\ln \left (x^2\right )}^2-3\right )}^2} \]
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