Integrand size = 71, antiderivative size = 29 \[ \int \frac {-3+x+3 x^2-x^3-x^4+x^6+\left (-4+2 x-3 x^2-4 x^4+3 x^6\right ) \log (x)+\left (-3 x^2+x^4\right ) \log ^2(x)}{1-2 x^2+x^4} \, dx=-3+x+x \log (x) \left (x^2-\frac {-4+x-\log (x)}{-1+x^2}+\log (x)\right ) \]
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Leaf count is larger than twice the leaf count of optimal. \(143\) vs. \(2(29)=58\).
Time = 0.28 (sec) , antiderivative size = 143, normalized size of antiderivative = 4.93, number of steps used = 38, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.282, Rules used = {28, 6874, 205, 213, 267, 294, 272, 45, 327, 308, 2404, 2332, 2354, 2438, 2351, 31, 2352, 2341, 2333, 2355} \[ \int \frac {-3+x+3 x^2-x^3-x^4+x^6+\left (-4+2 x-3 x^2-4 x^4+3 x^6\right ) \log (x)+\left (-3 x^2+x^4\right ) \log ^2(x)}{1-2 x^2+x^4} \, dx=-4 \text {arctanh}(x)+\frac {x^3}{2}+x^3 \log (x)-\frac {1}{2} \log \left (1-x^2\right )+\frac {x^5}{2 \left (1-x^2\right )}-\frac {x^3}{2 \left (1-x^2\right )}+x-\frac {x \log ^2(x)}{2 (1-x)}-\frac {x \log ^2(x)}{2 (x+1)}+x \log ^2(x)-\frac {3 x \log (x)}{2 (1-x)}-\frac {5 x \log (x)}{2 (x+1)}-\frac {3}{2} \log (1-x)+\frac {5}{2} \log (x+1) \]
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Rule 28
Rule 31
Rule 45
Rule 205
Rule 213
Rule 267
Rule 272
Rule 294
Rule 308
Rule 327
Rule 2332
Rule 2333
Rule 2341
Rule 2351
Rule 2352
Rule 2354
Rule 2355
Rule 2404
Rule 2438
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {-3+x+3 x^2-x^3-x^4+x^6+\left (-4+2 x-3 x^2-4 x^4+3 x^6\right ) \log (x)+\left (-3 x^2+x^4\right ) \log ^2(x)}{\left (-1+x^2\right )^2} \, dx \\ & = \int \left (-\frac {3}{\left (-1+x^2\right )^2}+\frac {x}{\left (-1+x^2\right )^2}+\frac {3 x^2}{\left (-1+x^2\right )^2}-\frac {x^3}{\left (-1+x^2\right )^2}-\frac {x^4}{\left (-1+x^2\right )^2}+\frac {x^6}{\left (-1+x^2\right )^2}+\frac {\left (-4+2 x-3 x^2-4 x^4+3 x^6\right ) \log (x)}{\left (-1+x^2\right )^2}+\frac {x^2 \left (-3+x^2\right ) \log ^2(x)}{\left (-1+x^2\right )^2}\right ) \, dx \\ & = -\left (3 \int \frac {1}{\left (-1+x^2\right )^2} \, dx\right )+3 \int \frac {x^2}{\left (-1+x^2\right )^2} \, dx+\int \frac {x}{\left (-1+x^2\right )^2} \, dx-\int \frac {x^3}{\left (-1+x^2\right )^2} \, dx-\int \frac {x^4}{\left (-1+x^2\right )^2} \, dx+\int \frac {x^6}{\left (-1+x^2\right )^2} \, dx+\int \frac {\left (-4+2 x-3 x^2-4 x^4+3 x^6\right ) \log (x)}{\left (-1+x^2\right )^2} \, dx+\int \frac {x^2 \left (-3+x^2\right ) \log ^2(x)}{\left (-1+x^2\right )^2} \, dx \\ & = \frac {1}{2 \left (1-x^2\right )}-\frac {x^3}{2 \left (1-x^2\right )}+\frac {x^5}{2 \left (1-x^2\right )}-\frac {1}{2} \text {Subst}\left (\int \frac {x}{(-1+x)^2} \, dx,x,x^2\right )+2 \left (\frac {3}{2} \int \frac {1}{-1+x^2} \, dx\right )-\frac {3}{2} \int \frac {x^2}{-1+x^2} \, dx+\frac {5}{2} \int \frac {x^4}{-1+x^2} \, dx+\int \left (2 \log (x)+\frac {\log (x)}{-1-x}-\frac {3 \log (x)}{2 (-1+x)^2}+\frac {\log (x)}{-1+x}+3 x^2 \log (x)-\frac {5 \log (x)}{2 (1+x)^2}\right ) \, dx+\int \left (\log ^2(x)-\frac {\log ^2(x)}{2 (-1+x)^2}-\frac {\log ^2(x)}{2 (1+x)^2}\right ) \, dx \\ & = -\frac {3 x}{2}+\frac {1}{2 \left (1-x^2\right )}-\frac {x^3}{2 \left (1-x^2\right )}+\frac {x^5}{2 \left (1-x^2\right )}-3 \tanh ^{-1}(x)-\frac {1}{2} \int \frac {\log ^2(x)}{(-1+x)^2} \, dx-\frac {1}{2} \int \frac {\log ^2(x)}{(1+x)^2} \, dx-\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{(-1+x)^2}+\frac {1}{-1+x}\right ) \, dx,x,x^2\right )-\frac {3}{2} \int \frac {1}{-1+x^2} \, dx-\frac {3}{2} \int \frac {\log (x)}{(-1+x)^2} \, dx+2 \int \log (x) \, dx+\frac {5}{2} \int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx-\frac {5}{2} \int \frac {\log (x)}{(1+x)^2} \, dx+3 \int x^2 \log (x) \, dx+\int \frac {\log (x)}{-1-x} \, dx+\int \frac {\log (x)}{-1+x} \, dx+\int \log ^2(x) \, dx \\ & = -x+\frac {x^3}{2}-\frac {x^3}{2 \left (1-x^2\right )}+\frac {x^5}{2 \left (1-x^2\right )}-\frac {3}{2} \tanh ^{-1}(x)+2 x \log (x)-\frac {3 x \log (x)}{2 (1-x)}+x^3 \log (x)-\frac {5 x \log (x)}{2 (1+x)}+x \log ^2(x)-\frac {x \log ^2(x)}{2 (1-x)}-\frac {x \log ^2(x)}{2 (1+x)}-\log (x) \log (1+x)-\frac {1}{2} \log \left (1-x^2\right )-\text {Li}_2(1-x)-\frac {3}{2} \int \frac {1}{-1+x} \, dx-2 \int \log (x) \, dx+\frac {5}{2} \int \frac {1}{1+x} \, dx+\frac {5}{2} \int \frac {1}{-1+x^2} \, dx-\int \frac {\log (x)}{-1+x} \, dx+\int \frac {\log (x)}{1+x} \, dx+\int \frac {\log (1+x)}{x} \, dx \\ & = x+\frac {x^3}{2}-\frac {x^3}{2 \left (1-x^2\right )}+\frac {x^5}{2 \left (1-x^2\right )}-4 \tanh ^{-1}(x)-\frac {3}{2} \log (1-x)-\frac {3 x \log (x)}{2 (1-x)}+x^3 \log (x)-\frac {5 x \log (x)}{2 (1+x)}+x \log ^2(x)-\frac {x \log ^2(x)}{2 (1-x)}-\frac {x \log ^2(x)}{2 (1+x)}+\frac {5}{2} \log (1+x)-\frac {1}{2} \log \left (1-x^2\right )-\text {Li}_2(-x)-\int \frac {\log (1+x)}{x} \, dx \\ & = x+\frac {x^3}{2}-\frac {x^3}{2 \left (1-x^2\right )}+\frac {x^5}{2 \left (1-x^2\right )}-4 \tanh ^{-1}(x)-\frac {3}{2} \log (1-x)-\frac {3 x \log (x)}{2 (1-x)}+x^3 \log (x)-\frac {5 x \log (x)}{2 (1+x)}+x \log ^2(x)-\frac {x \log ^2(x)}{2 (1-x)}-\frac {x \log ^2(x)}{2 (1+x)}+\frac {5}{2} \log (1+x)-\frac {1}{2} \log \left (1-x^2\right ) \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.18 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.28 \[ \int \frac {-3+x+3 x^2-x^3-x^4+x^6+\left (-4+2 x-3 x^2-4 x^4+3 x^6\right ) \log (x)+\left (-3 x^2+x^4\right ) \log ^2(x)}{1-2 x^2+x^4} \, dx=\frac {x \left (-1+x^2\right )+\left (x \left (4-x-x^2+x^4\right )+\left (-1+x^2\right ) \log (1-x)\right ) \log (x)+x^3 \log ^2(x)}{-1+x^2}+\operatorname {PolyLog}(2,1-x)+\operatorname {PolyLog}(2,x) \]
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Time = 0.15 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.55
method | result | size |
risch | \(\frac {x^{3} \ln \left (x \right )^{2}}{x^{2}-1}+\frac {\left (x^{5}-x^{3}+4 x -1\right ) \ln \left (x \right )}{x^{2}-1}+x -\ln \left (x \right )\) | \(45\) |
norman | \(\frac {x^{3}+x^{3} \ln \left (x \right )^{2}+x^{5} \ln \left (x \right )-x +4 x \ln \left (x \right )-x^{2} \ln \left (x \right )-x^{3} \ln \left (x \right )}{x^{2}-1}\) | \(49\) |
parallelrisch | \(\frac {x^{3}+x^{3} \ln \left (x \right )^{2}+x^{5} \ln \left (x \right )-x +4 x \ln \left (x \right )-x^{2} \ln \left (x \right )-x^{3} \ln \left (x \right )}{x^{2}-1}\) | \(49\) |
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Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.48 \[ \int \frac {-3+x+3 x^2-x^3-x^4+x^6+\left (-4+2 x-3 x^2-4 x^4+3 x^6\right ) \log (x)+\left (-3 x^2+x^4\right ) \log ^2(x)}{1-2 x^2+x^4} \, dx=\frac {x^{3} \log \left (x\right )^{2} + x^{3} + {\left (x^{5} - x^{3} - x^{2} + 4 \, x\right )} \log \left (x\right ) - x}{x^{2} - 1} \]
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Time = 0.12 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \frac {-3+x+3 x^2-x^3-x^4+x^6+\left (-4+2 x-3 x^2-4 x^4+3 x^6\right ) \log (x)+\left (-3 x^2+x^4\right ) \log ^2(x)}{1-2 x^2+x^4} \, dx=\frac {x^{3} \log {\left (x \right )}^{2}}{x^{2} - 1} + x - \log {\left (x \right )} + \frac {\left (x^{5} - x^{3} + 4 x - 1\right ) \log {\left (x \right )}}{x^{2} - 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (29) = 58\).
Time = 0.23 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.59 \[ \int \frac {-3+x+3 x^2-x^3-x^4+x^6+\left (-4+2 x-3 x^2-4 x^4+3 x^6\right ) \log (x)+\left (-3 x^2+x^4\right ) \log ^2(x)}{1-2 x^2+x^4} \, dx=\frac {1}{3} \, x^{3} + x - \frac {x^{5} - 3 \, x^{3} \log \left (x\right )^{2} - x^{3} - 3 \, {\left (x^{5} - x^{3} + 4 \, x - 1\right )} \log \left (x\right )}{3 \, {\left (x^{2} - 1\right )}} - \frac {1}{2} \, \log \left (x^{2} - 1\right ) + \frac {1}{2} \, \log \left (x + 1\right ) + \frac {1}{2} \, \log \left (x - 1\right ) - \log \left (x\right ) \]
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Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45 \[ \int \frac {-3+x+3 x^2-x^3-x^4+x^6+\left (-4+2 x-3 x^2-4 x^4+3 x^6\right ) \log (x)+\left (-3 x^2+x^4\right ) \log ^2(x)}{1-2 x^2+x^4} \, dx={\left (x + \frac {x}{x^{2} - 1}\right )} \log \left (x\right )^{2} + {\left (x^{3} + \frac {4 \, x - 1}{x^{2} - 1}\right )} \log \left (x\right ) + x - \log \left (x\right ) \]
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Time = 12.49 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.48 \[ \int \frac {-3+x+3 x^2-x^3-x^4+x^6+\left (-4+2 x-3 x^2-4 x^4+3 x^6\right ) \log (x)+\left (-3 x^2+x^4\right ) \log ^2(x)}{1-2 x^2+x^4} \, dx=x^3\,\ln \left (x\right )-\ln \left (x\right )+x\,\left ({\ln \left (x\right )}^2+1\right )-\frac {\ln \left (x\right )-x\,\left ({\ln \left (x\right )}^2+4\,\ln \left (x\right )\right )}{x^2-1} \]
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