Integrand size = 54, antiderivative size = 24 \[ \int \frac {48-16 x-8 x^5+\left (-64+24 x+16 x^5\right ) \log (x)-8 x^5 \log ^2(x)}{x^5-2 x^5 \log (x)+x^5 \log ^2(x)} \, dx=2 (2-x) \left (4+\frac {4}{x^3 (-x+x \log (x))}\right ) \]
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Time = 0.37 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25, number of steps used = 18, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6873, 12, 6874, 2395, 2343, 2346, 2209} \[ \int \frac {48-16 x-8 x^5+\left (-64+24 x+16 x^5\right ) \log (x)-8 x^5 \log ^2(x)}{x^5-2 x^5 \log (x)+x^5 \log ^2(x)} \, dx=-\frac {16}{x^4 (1-\log (x))}+\frac {8}{x^3 (1-\log (x))}-8 x \]
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Rule 12
Rule 2209
Rule 2343
Rule 2346
Rule 2395
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {8 \left (6-2 x-x^5-8 \log (x)+3 x \log (x)+2 x^5 \log (x)-x^5 \log ^2(x)\right )}{x^5 (1-\log (x))^2} \, dx \\ & = 8 \int \frac {6-2 x-x^5-8 \log (x)+3 x \log (x)+2 x^5 \log (x)-x^5 \log ^2(x)}{x^5 (1-\log (x))^2} \, dx \\ & = 8 \int \left (-1+\frac {-2+x}{x^5 (-1+\log (x))^2}+\frac {-8+3 x}{x^5 (-1+\log (x))}\right ) \, dx \\ & = -8 x+8 \int \frac {-2+x}{x^5 (-1+\log (x))^2} \, dx+8 \int \frac {-8+3 x}{x^5 (-1+\log (x))} \, dx \\ & = -8 x+8 \int \left (-\frac {2}{x^5 (-1+\log (x))^2}+\frac {1}{x^4 (-1+\log (x))^2}\right ) \, dx+8 \int \left (-\frac {8}{x^5 (-1+\log (x))}+\frac {3}{x^4 (-1+\log (x))}\right ) \, dx \\ & = -8 x+8 \int \frac {1}{x^4 (-1+\log (x))^2} \, dx-16 \int \frac {1}{x^5 (-1+\log (x))^2} \, dx+24 \int \frac {1}{x^4 (-1+\log (x))} \, dx-64 \int \frac {1}{x^5 (-1+\log (x))} \, dx \\ & = -8 x-\frac {16}{x^4 (1-\log (x))}+\frac {8}{x^3 (1-\log (x))}-24 \int \frac {1}{x^4 (-1+\log (x))} \, dx+24 \text {Subst}\left (\int \frac {e^{-3 x}}{-1+x} \, dx,x,\log (x)\right )+64 \int \frac {1}{x^5 (-1+\log (x))} \, dx-64 \text {Subst}\left (\int \frac {e^{-4 x}}{-1+x} \, dx,x,\log (x)\right ) \\ & = -8 x+\frac {24 \operatorname {ExpIntegralEi}(3 (1-\log (x)))}{e^3}-\frac {64 \operatorname {ExpIntegralEi}(4 (1-\log (x)))}{e^4}-\frac {16}{x^4 (1-\log (x))}+\frac {8}{x^3 (1-\log (x))}-24 \text {Subst}\left (\int \frac {e^{-3 x}}{-1+x} \, dx,x,\log (x)\right )+64 \text {Subst}\left (\int \frac {e^{-4 x}}{-1+x} \, dx,x,\log (x)\right ) \\ & = -8 x-\frac {16}{x^4 (1-\log (x))}+\frac {8}{x^3 (1-\log (x))} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71 \[ \int \frac {48-16 x-8 x^5+\left (-64+24 x+16 x^5\right ) \log (x)-8 x^5 \log ^2(x)}{x^5-2 x^5 \log (x)+x^5 \log ^2(x)} \, dx=-8 \left (x+\frac {-2+x}{x^4 (-1+\log (x))}\right ) \]
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Time = 0.18 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79
method | result | size |
risch | \(-8 x -\frac {8 \left (-2+x \right )}{x^{4} \left (\ln \left (x \right )-1\right )}\) | \(19\) |
default | \(-8 x +\frac {16}{x^{4} \left (\ln \left (x \right )-1\right )}-\frac {8}{x^{3} \left (\ln \left (x \right )-1\right )}\) | \(27\) |
norman | \(\frac {16-8 x +8 x^{5}-8 x^{5} \ln \left (x \right )}{x^{4} \left (\ln \left (x \right )-1\right )}\) | \(28\) |
parallelrisch | \(\frac {16-8 x +8 x^{5}-8 x^{5} \ln \left (x \right )}{x^{4} \left (\ln \left (x \right )-1\right )}\) | \(28\) |
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Time = 0.31 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25 \[ \int \frac {48-16 x-8 x^5+\left (-64+24 x+16 x^5\right ) \log (x)-8 x^5 \log ^2(x)}{x^5-2 x^5 \log (x)+x^5 \log ^2(x)} \, dx=-\frac {8 \, {\left (x^{5} \log \left (x\right ) - x^{5} + x - 2\right )}}{x^{4} \log \left (x\right ) - x^{4}} \]
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Time = 0.05 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71 \[ \int \frac {48-16 x-8 x^5+\left (-64+24 x+16 x^5\right ) \log (x)-8 x^5 \log ^2(x)}{x^5-2 x^5 \log (x)+x^5 \log ^2(x)} \, dx=- 8 x + \frac {16 - 8 x}{x^{4} \log {\left (x \right )} - x^{4}} \]
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Time = 0.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25 \[ \int \frac {48-16 x-8 x^5+\left (-64+24 x+16 x^5\right ) \log (x)-8 x^5 \log ^2(x)}{x^5-2 x^5 \log (x)+x^5 \log ^2(x)} \, dx=-\frac {8 \, {\left (x^{5} \log \left (x\right ) - x^{5} + x - 2\right )}}{x^{4} \log \left (x\right ) - x^{4}} \]
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Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {48-16 x-8 x^5+\left (-64+24 x+16 x^5\right ) \log (x)-8 x^5 \log ^2(x)}{x^5-2 x^5 \log (x)+x^5 \log ^2(x)} \, dx=-8 \, x - \frac {8 \, {\left (x - 2\right )}}{x^{4} \log \left (x\right ) - x^{4}} \]
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Time = 13.76 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {48-16 x-8 x^5+\left (-64+24 x+16 x^5\right ) \log (x)-8 x^5 \log ^2(x)}{x^5-2 x^5 \log (x)+x^5 \log ^2(x)} \, dx=-8\,x-\frac {8\,x-16}{x^4\,\left (\ln \left (x\right )-1\right )} \]
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