Integrand size = 86, antiderivative size = 28 \[ \int \frac {32 x^2+24 x^3-8 x^4+\left (-8 x+7 x^2+3 x^3-2 x^4\right ) \log \left (1-2 x+x^2\right ) \log \left (\log ^4\left (1-2 x+x^2\right )\right )}{\left (-5-5 x+5 x^2+5 x^3\right ) \log \left (1-2 x+x^2\right )} \, dx=\frac {(4-x) x^2 \log \left (\log ^4\left ((1-x)^2\right )\right )}{5 (1+x)} \]
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\[ \int \frac {32 x^2+24 x^3-8 x^4+\left (-8 x+7 x^2+3 x^3-2 x^4\right ) \log \left (1-2 x+x^2\right ) \log \left (\log ^4\left (1-2 x+x^2\right )\right )}{\left (-5-5 x+5 x^2+5 x^3\right ) \log \left (1-2 x+x^2\right )} \, dx=\int \frac {32 x^2+24 x^3-8 x^4+\left (-8 x+7 x^2+3 x^3-2 x^4\right ) \log \left (1-2 x+x^2\right ) \log \left (\log ^4\left (1-2 x+x^2\right )\right )}{\left (-5-5 x+5 x^2+5 x^3\right ) \log \left (1-2 x+x^2\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {32 x^2}{5 (-1+x) (1+x)^2 \log \left ((-1+x)^2\right )}+\frac {24 x^3}{5 (-1+x) (1+x)^2 \log \left ((-1+x)^2\right )}-\frac {8 x^4}{5 (-1+x) (1+x)^2 \log \left ((-1+x)^2\right )}-\frac {x \left (-8-x+2 x^2\right ) \log \left (\log ^4\left ((-1+x)^2\right )\right )}{5 (1+x)^2}\right ) \, dx \\ & = -\left (\frac {1}{5} \int \frac {x \left (-8-x+2 x^2\right ) \log \left (\log ^4\left ((-1+x)^2\right )\right )}{(1+x)^2} \, dx\right )-\frac {8}{5} \int \frac {x^4}{(-1+x) (1+x)^2 \log \left ((-1+x)^2\right )} \, dx+\frac {24}{5} \int \frac {x^3}{(-1+x) (1+x)^2 \log \left ((-1+x)^2\right )} \, dx+\frac {32}{5} \int \frac {x^2}{(-1+x) (1+x)^2 \log \left ((-1+x)^2\right )} \, dx \\ & = -\left (\frac {1}{5} \int \left (-5 \log \left (\log ^4\left ((-1+x)^2\right )\right )+2 x \log \left (\log ^4\left ((-1+x)^2\right )\right )+\frac {5 \log \left (\log ^4\left ((-1+x)^2\right )\right )}{(1+x)^2}\right ) \, dx\right )-\frac {8}{5} \int \left (-\frac {1}{\log \left ((-1+x)^2\right )}+\frac {1}{4 (-1+x) \log \left ((-1+x)^2\right )}+\frac {x}{\log \left ((-1+x)^2\right )}-\frac {1}{2 (1+x)^2 \log \left ((-1+x)^2\right )}+\frac {7}{4 (1+x) \log \left ((-1+x)^2\right )}\right ) \, dx+\frac {24}{5} \int \left (\frac {1}{\log \left ((-1+x)^2\right )}+\frac {1}{4 (-1+x) \log \left ((-1+x)^2\right )}+\frac {1}{2 (1+x)^2 \log \left ((-1+x)^2\right )}-\frac {5}{4 (1+x) \log \left ((-1+x)^2\right )}\right ) \, dx+\frac {32}{5} \int \left (\frac {1}{4 (-1+x) \log \left ((-1+x)^2\right )}-\frac {1}{2 (1+x)^2 \log \left ((-1+x)^2\right )}+\frac {3}{4 (1+x) \log \left ((-1+x)^2\right )}\right ) \, dx \\ & = -\left (\frac {2}{5} \int \frac {1}{(-1+x) \log \left ((-1+x)^2\right )} \, dx\right )-\frac {2}{5} \int x \log \left (\log ^4\left ((-1+x)^2\right )\right ) \, dx+\frac {4}{5} \int \frac {1}{(1+x)^2 \log \left ((-1+x)^2\right )} \, dx+\frac {6}{5} \int \frac {1}{(-1+x) \log \left ((-1+x)^2\right )} \, dx+\frac {8}{5} \int \frac {1}{\log \left ((-1+x)^2\right )} \, dx+\frac {8}{5} \int \frac {1}{(-1+x) \log \left ((-1+x)^2\right )} \, dx-\frac {8}{5} \int \frac {x}{\log \left ((-1+x)^2\right )} \, dx+\frac {12}{5} \int \frac {1}{(1+x)^2 \log \left ((-1+x)^2\right )} \, dx-\frac {14}{5} \int \frac {1}{(1+x) \log \left ((-1+x)^2\right )} \, dx-\frac {16}{5} \int \frac {1}{(1+x)^2 \log \left ((-1+x)^2\right )} \, dx+\frac {24}{5} \int \frac {1}{\log \left ((-1+x)^2\right )} \, dx+\frac {24}{5} \int \frac {1}{(1+x) \log \left ((-1+x)^2\right )} \, dx-6 \int \frac {1}{(1+x) \log \left ((-1+x)^2\right )} \, dx+\int \log \left (\log ^4\left ((-1+x)^2\right )\right ) \, dx-\int \frac {\log \left (\log ^4\left ((-1+x)^2\right )\right )}{(1+x)^2} \, dx \\ & = -\left (\frac {2}{5} \int x \log \left (\log ^4\left ((-1+x)^2\right )\right ) \, dx\right )-\frac {2}{5} \text {Subst}\left (\int \frac {1}{x \log \left (x^2\right )} \, dx,x,-1+x\right )+\frac {4}{5} \int \frac {1}{(1+x)^2 \log \left ((-1+x)^2\right )} \, dx+\frac {6}{5} \text {Subst}\left (\int \frac {1}{x \log \left (x^2\right )} \, dx,x,-1+x\right )-\frac {8}{5} \int \left (\frac {1}{\log \left ((-1+x)^2\right )}+\frac {-1+x}{\log \left ((-1+x)^2\right )}\right ) \, dx+\frac {8}{5} \text {Subst}\left (\int \frac {1}{\log \left (x^2\right )} \, dx,x,-1+x\right )+\frac {8}{5} \text {Subst}\left (\int \frac {1}{x \log \left (x^2\right )} \, dx,x,-1+x\right )+\frac {12}{5} \int \frac {1}{(1+x)^2 \log \left ((-1+x)^2\right )} \, dx-\frac {14}{5} \int \frac {1}{(1+x) \log \left ((-1+x)^2\right )} \, dx-\frac {16}{5} \int \frac {1}{(1+x)^2 \log \left ((-1+x)^2\right )} \, dx+\frac {24}{5} \int \frac {1}{(1+x) \log \left ((-1+x)^2\right )} \, dx+\frac {24}{5} \text {Subst}\left (\int \frac {1}{\log \left (x^2\right )} \, dx,x,-1+x\right )-6 \int \frac {1}{(1+x) \log \left ((-1+x)^2\right )} \, dx-\int \frac {\log \left (\log ^4\left ((-1+x)^2\right )\right )}{(1+x)^2} \, dx+\text {Subst}\left (\int \log \left (\log ^4\left (x^2\right )\right ) \, dx,x,-1+x\right ) \\ & = (-1+x) \log \left (\log ^4\left ((-1+x)^2\right )\right )-\frac {1}{5} \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left ((-1+x)^2\right )\right )-\frac {2}{5} \int x \log \left (\log ^4\left ((-1+x)^2\right )\right ) \, dx+\frac {3}{5} \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left ((-1+x)^2\right )\right )+\frac {4}{5} \int \frac {1}{(1+x)^2 \log \left ((-1+x)^2\right )} \, dx+\frac {4}{5} \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left ((-1+x)^2\right )\right )-\frac {8}{5} \int \frac {1}{\log \left ((-1+x)^2\right )} \, dx-\frac {8}{5} \int \frac {-1+x}{\log \left ((-1+x)^2\right )} \, dx+\frac {12}{5} \int \frac {1}{(1+x)^2 \log \left ((-1+x)^2\right )} \, dx-\frac {14}{5} \int \frac {1}{(1+x) \log \left ((-1+x)^2\right )} \, dx-\frac {16}{5} \int \frac {1}{(1+x)^2 \log \left ((-1+x)^2\right )} \, dx+\frac {24}{5} \int \frac {1}{(1+x) \log \left ((-1+x)^2\right )} \, dx-6 \int \frac {1}{(1+x) \log \left ((-1+x)^2\right )} \, dx-8 \text {Subst}\left (\int \frac {1}{\log \left (x^2\right )} \, dx,x,-1+x\right )+\frac {(4 (-1+x)) \text {Subst}\left (\int \frac {e^{x/2}}{x} \, dx,x,\log \left ((-1+x)^2\right )\right )}{5 \sqrt {(-1+x)^2}}+\frac {(12 (-1+x)) \text {Subst}\left (\int \frac {e^{x/2}}{x} \, dx,x,\log \left ((-1+x)^2\right )\right )}{5 \sqrt {(-1+x)^2}}-\int \frac {\log \left (\log ^4\left ((-1+x)^2\right )\right )}{(1+x)^2} \, dx \\ & = -\frac {16 (1-x) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \log \left ((-1+x)^2\right )\right )}{5 \sqrt {(-1+x)^2}}+\frac {6}{5} \log \left (\log \left ((-1+x)^2\right )\right )+(-1+x) \log \left (\log ^4\left ((-1+x)^2\right )\right )-\frac {2}{5} \int x \log \left (\log ^4\left ((-1+x)^2\right )\right ) \, dx+\frac {4}{5} \int \frac {1}{(1+x)^2 \log \left ((-1+x)^2\right )} \, dx-\frac {8}{5} \text {Subst}\left (\int \frac {1}{\log \left (x^2\right )} \, dx,x,-1+x\right )-\frac {8}{5} \text {Subst}\left (\int \frac {x}{\log \left (x^2\right )} \, dx,x,-1+x\right )+\frac {12}{5} \int \frac {1}{(1+x)^2 \log \left ((-1+x)^2\right )} \, dx-\frac {14}{5} \int \frac {1}{(1+x) \log \left ((-1+x)^2\right )} \, dx-\frac {16}{5} \int \frac {1}{(1+x)^2 \log \left ((-1+x)^2\right )} \, dx+\frac {24}{5} \int \frac {1}{(1+x) \log \left ((-1+x)^2\right )} \, dx-6 \int \frac {1}{(1+x) \log \left ((-1+x)^2\right )} \, dx-\frac {(4 (-1+x)) \text {Subst}\left (\int \frac {e^{x/2}}{x} \, dx,x,\log \left ((-1+x)^2\right )\right )}{\sqrt {(-1+x)^2}}-\int \frac {\log \left (\log ^4\left ((-1+x)^2\right )\right )}{(1+x)^2} \, dx \\ & = \frac {4 (1-x) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \log \left ((-1+x)^2\right )\right )}{5 \sqrt {(-1+x)^2}}+\frac {6}{5} \log \left (\log \left ((-1+x)^2\right )\right )+(-1+x) \log \left (\log ^4\left ((-1+x)^2\right )\right )-\frac {2}{5} \int x \log \left (\log ^4\left ((-1+x)^2\right )\right ) \, dx+\frac {4}{5} \int \frac {1}{(1+x)^2 \log \left ((-1+x)^2\right )} \, dx-\frac {4}{5} \text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,(-1+x)^2\right )+\frac {12}{5} \int \frac {1}{(1+x)^2 \log \left ((-1+x)^2\right )} \, dx-\frac {14}{5} \int \frac {1}{(1+x) \log \left ((-1+x)^2\right )} \, dx-\frac {16}{5} \int \frac {1}{(1+x)^2 \log \left ((-1+x)^2\right )} \, dx+\frac {24}{5} \int \frac {1}{(1+x) \log \left ((-1+x)^2\right )} \, dx-6 \int \frac {1}{(1+x) \log \left ((-1+x)^2\right )} \, dx-\frac {(4 (-1+x)) \text {Subst}\left (\int \frac {e^{x/2}}{x} \, dx,x,\log \left ((-1+x)^2\right )\right )}{5 \sqrt {(-1+x)^2}}-\int \frac {\log \left (\log ^4\left ((-1+x)^2\right )\right )}{(1+x)^2} \, dx \\ & = \frac {8 (1-x) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \log \left ((-1+x)^2\right )\right )}{5 \sqrt {(-1+x)^2}}+\frac {6}{5} \log \left (\log \left ((-1+x)^2\right )\right )+(-1+x) \log \left (\log ^4\left ((-1+x)^2\right )\right )-\frac {4}{5} \operatorname {LogIntegral}\left ((-1+x)^2\right )-\frac {2}{5} \int x \log \left (\log ^4\left ((-1+x)^2\right )\right ) \, dx+\frac {4}{5} \int \frac {1}{(1+x)^2 \log \left ((-1+x)^2\right )} \, dx+\frac {12}{5} \int \frac {1}{(1+x)^2 \log \left ((-1+x)^2\right )} \, dx-\frac {14}{5} \int \frac {1}{(1+x) \log \left ((-1+x)^2\right )} \, dx-\frac {16}{5} \int \frac {1}{(1+x)^2 \log \left ((-1+x)^2\right )} \, dx+\frac {24}{5} \int \frac {1}{(1+x) \log \left ((-1+x)^2\right )} \, dx-6 \int \frac {1}{(1+x) \log \left ((-1+x)^2\right )} \, dx-\int \frac {\log \left (\log ^4\left ((-1+x)^2\right )\right )}{(1+x)^2} \, dx \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54 \[ \int \frac {32 x^2+24 x^3-8 x^4+\left (-8 x+7 x^2+3 x^3-2 x^4\right ) \log \left (1-2 x+x^2\right ) \log \left (\log ^4\left (1-2 x+x^2\right )\right )}{\left (-5-5 x+5 x^2+5 x^3\right ) \log \left (1-2 x+x^2\right )} \, dx=\frac {1}{5} \left (-20 \log \left (\log \left ((-1+x)^2\right )\right )-\frac {\left (-5-5 x-4 x^2+x^3\right ) \log \left (\log ^4\left ((-1+x)^2\right )\right )}{1+x}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(96\) vs. \(2(26)=52\).
Time = 2.65 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.46
method | result | size |
parallelrisch | \(\frac {-x^{3} \ln \left (\ln \left (x^{2}-2 x +1\right )^{4}\right )+4 \ln \left (\ln \left (x^{2}-2 x +1\right )^{4}\right ) x^{2}-16 \ln \left (\ln \left (x^{2}-2 x +1\right )\right ) x +4 \ln \left (\ln \left (x^{2}-2 x +1\right )^{4}\right ) x -16 \ln \left (\ln \left (x^{2}-2 x +1\right )\right )+4 \ln \left (\ln \left (x^{2}-2 x +1\right )^{4}\right )}{5 x +5}\) | \(97\) |
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Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {32 x^2+24 x^3-8 x^4+\left (-8 x+7 x^2+3 x^3-2 x^4\right ) \log \left (1-2 x+x^2\right ) \log \left (\log ^4\left (1-2 x+x^2\right )\right )}{\left (-5-5 x+5 x^2+5 x^3\right ) \log \left (1-2 x+x^2\right )} \, dx=-\frac {{\left (x^{3} - 4 \, x^{2}\right )} \log \left (\log \left (x^{2} - 2 \, x + 1\right )^{4}\right )}{5 \, {\left (x + 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (20) = 40\).
Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {32 x^2+24 x^3-8 x^4+\left (-8 x+7 x^2+3 x^3-2 x^4\right ) \log \left (1-2 x+x^2\right ) \log \left (\log ^4\left (1-2 x+x^2\right )\right )}{\left (-5-5 x+5 x^2+5 x^3\right ) \log \left (1-2 x+x^2\right )} \, dx=- 4 \log {\left (\log {\left (x^{2} - 2 x + 1 \right )} \right )} + \frac {\left (- x^{3} + 4 x^{2} + 5 x + 5\right ) \log {\left (\log {\left (x^{2} - 2 x + 1 \right )}^{4} \right )}}{5 x + 5} \]
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Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (22) = 44\).
Time = 0.34 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61 \[ \int \frac {32 x^2+24 x^3-8 x^4+\left (-8 x+7 x^2+3 x^3-2 x^4\right ) \log \left (1-2 x+x^2\right ) \log \left (\log ^4\left (1-2 x+x^2\right )\right )}{\left (-5-5 x+5 x^2+5 x^3\right ) \log \left (1-2 x+x^2\right )} \, dx=-\frac {4 \, {\left (x^{3} \log \left (2\right ) - 4 \, x^{2} \log \left (2\right ) - 5 \, x \log \left (2\right ) + {\left (x^{3} - 4 \, x^{2}\right )} \log \left (\log \left (x - 1\right )\right ) - 5 \, \log \left (2\right )\right )}}{5 \, {\left (x + 1\right )}} \]
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Time = 0.65 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {32 x^2+24 x^3-8 x^4+\left (-8 x+7 x^2+3 x^3-2 x^4\right ) \log \left (1-2 x+x^2\right ) \log \left (\log ^4\left (1-2 x+x^2\right )\right )}{\left (-5-5 x+5 x^2+5 x^3\right ) \log \left (1-2 x+x^2\right )} \, dx=-\frac {1}{5} \, {\left (x^{2} - 5 \, x - \frac {5}{x + 1}\right )} \log \left (\log \left (x^{2} - 2 \, x + 1\right )^{4}\right ) - 4 \, \log \left (\log \left (x^{2} - 2 \, x + 1\right )\right ) \]
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Time = 15.78 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {32 x^2+24 x^3-8 x^4+\left (-8 x+7 x^2+3 x^3-2 x^4\right ) \log \left (1-2 x+x^2\right ) \log \left (\log ^4\left (1-2 x+x^2\right )\right )}{\left (-5-5 x+5 x^2+5 x^3\right ) \log \left (1-2 x+x^2\right )} \, dx=-\frac {x^2\,\ln \left ({\ln \left (x^2-2\,x+1\right )}^4\right )\,\left (x-4\right )}{5\,\left (x+1\right )} \]
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