Integrand size = 102, antiderivative size = 28 \[ \int \frac {-750+750 x-247 x^2+150 x^3+3 x^4+\left (750-30 x+780 x^2-530 x^3+30 x^4\right ) \log (x)+\left (75-150 x+150 x^2-150 x^3+75 x^4\right ) \log ^2(x)}{x^2+\left (-10 x+10 x^2\right ) \log (x)+\left (25-50 x+25 x^2\right ) \log ^2(x)} \, dx=\left (3+x^2\right ) \left (x-\frac {2 x}{\frac {x}{25}+\frac {1}{5} (-1+x) \log (x)}\right ) \]
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\[ \int \frac {-750+750 x-247 x^2+150 x^3+3 x^4+\left (750-30 x+780 x^2-530 x^3+30 x^4\right ) \log (x)+\left (75-150 x+150 x^2-150 x^3+75 x^4\right ) \log ^2(x)}{x^2+\left (-10 x+10 x^2\right ) \log (x)+\left (25-50 x+25 x^2\right ) \log ^2(x)} \, dx=\int \frac {-750+750 x-247 x^2+150 x^3+3 x^4+\left (750-30 x+780 x^2-530 x^3+30 x^4\right ) \log (x)+\left (75-150 x+150 x^2-150 x^3+75 x^4\right ) \log ^2(x)}{x^2+\left (-10 x+10 x^2\right ) \log (x)+\left (25-50 x+25 x^2\right ) \log ^2(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-750+750 x-247 x^2+150 x^3+3 x^4+10 \left (75-3 x+78 x^2-53 x^3+3 x^4\right ) \log (x)+75 (-1+x)^2 \left (1+x^2\right ) \log ^2(x)}{(x+5 (-1+x) \log (x))^2} \, dx \\ & = \int \left (3 \left (1+x^2\right )+\frac {50 \left (15-33 x+20 x^2-11 x^3+5 x^4\right )}{(-1+x) (x-5 \log (x)+5 x \log (x))^2}-\frac {50 \left (-3-3 x^2+2 x^3\right )}{(-1+x) (x-5 \log (x)+5 x \log (x))}\right ) \, dx \\ & = 3 \int \left (1+x^2\right ) \, dx+50 \int \frac {15-33 x+20 x^2-11 x^3+5 x^4}{(-1+x) (x-5 \log (x)+5 x \log (x))^2} \, dx-50 \int \frac {-3-3 x^2+2 x^3}{(-1+x) (x-5 \log (x)+5 x \log (x))} \, dx \\ & = 3 x+x^3+50 \int \left (-\frac {19}{(x-5 \log (x)+5 x \log (x))^2}-\frac {4}{(-1+x) (x-5 \log (x)+5 x \log (x))^2}+\frac {14 x}{(x-5 \log (x)+5 x \log (x))^2}-\frac {6 x^2}{(x-5 \log (x)+5 x \log (x))^2}+\frac {5 x^3}{(x-5 \log (x)+5 x \log (x))^2}\right ) \, dx-50 \int \left (-\frac {1}{x-5 \log (x)+5 x \log (x)}-\frac {4}{(-1+x) (x-5 \log (x)+5 x \log (x))}-\frac {x}{x-5 \log (x)+5 x \log (x)}+\frac {2 x^2}{x-5 \log (x)+5 x \log (x)}\right ) \, dx \\ & = 3 x+x^3+50 \int \frac {1}{x-5 \log (x)+5 x \log (x)} \, dx+50 \int \frac {x}{x-5 \log (x)+5 x \log (x)} \, dx-100 \int \frac {x^2}{x-5 \log (x)+5 x \log (x)} \, dx-200 \int \frac {1}{(-1+x) (x-5 \log (x)+5 x \log (x))^2} \, dx+200 \int \frac {1}{(-1+x) (x-5 \log (x)+5 x \log (x))} \, dx+250 \int \frac {x^3}{(x-5 \log (x)+5 x \log (x))^2} \, dx-300 \int \frac {x^2}{(x-5 \log (x)+5 x \log (x))^2} \, dx+700 \int \frac {x}{(x-5 \log (x)+5 x \log (x))^2} \, dx-950 \int \frac {1}{(x-5 \log (x)+5 x \log (x))^2} \, dx \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {-750+750 x-247 x^2+150 x^3+3 x^4+\left (750-30 x+780 x^2-530 x^3+30 x^4\right ) \log (x)+\left (75-150 x+150 x^2-150 x^3+75 x^4\right ) \log ^2(x)}{x^2+\left (-10 x+10 x^2\right ) \log (x)+\left (25-50 x+25 x^2\right ) \log ^2(x)} \, dx=x \left (3+x^2-\frac {50 \left (3+x^2\right )}{x+5 (-1+x) \log (x)}\right ) \]
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Time = 1.57 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04
method | result | size |
risch | \(x^{3}+3 x -\frac {50 \left (x^{2}+3\right ) x}{5 x \ln \left (x \right )-5 \ln \left (x \right )+x}\) | \(29\) |
default | \(\frac {x^{4}-147 x -15 \ln \left (x \right )+3 x^{2}-50 x^{3}+15 x^{2} \ln \left (x \right )-5 x^{3} \ln \left (x \right )+5 x^{4} \ln \left (x \right )}{5 x \ln \left (x \right )-5 \ln \left (x \right )+x}\) | \(57\) |
norman | \(\frac {x^{4}-147 x -15 \ln \left (x \right )+3 x^{2}-50 x^{3}+15 x^{2} \ln \left (x \right )-5 x^{3} \ln \left (x \right )+5 x^{4} \ln \left (x \right )}{5 x \ln \left (x \right )-5 \ln \left (x \right )+x}\) | \(57\) |
parallelrisch | \(\frac {25 x^{4} \ln \left (x \right )+5 x^{4}-25 x^{3} \ln \left (x \right )-250 x^{3}+75 x^{2} \ln \left (x \right )+15 x^{2}-735 x -75 \ln \left (x \right )}{25 x \ln \left (x \right )-25 \ln \left (x \right )+5 x}\) | \(60\) |
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Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (22) = 44\).
Time = 0.24 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.79 \[ \int \frac {-750+750 x-247 x^2+150 x^3+3 x^4+\left (750-30 x+780 x^2-530 x^3+30 x^4\right ) \log (x)+\left (75-150 x+150 x^2-150 x^3+75 x^4\right ) \log ^2(x)}{x^2+\left (-10 x+10 x^2\right ) \log (x)+\left (25-50 x+25 x^2\right ) \log ^2(x)} \, dx=\frac {x^{4} - 50 \, x^{3} + 3 \, x^{2} + 5 \, {\left (x^{4} - x^{3} + 3 \, x^{2} - 3 \, x\right )} \log \left (x\right ) - 150 \, x}{5 \, {\left (x - 1\right )} \log \left (x\right ) + x} \]
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Time = 0.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {-750+750 x-247 x^2+150 x^3+3 x^4+\left (750-30 x+780 x^2-530 x^3+30 x^4\right ) \log (x)+\left (75-150 x+150 x^2-150 x^3+75 x^4\right ) \log ^2(x)}{x^2+\left (-10 x+10 x^2\right ) \log (x)+\left (25-50 x+25 x^2\right ) \log ^2(x)} \, dx=x^{3} + 3 x + \frac {- 50 x^{3} - 150 x}{x + \left (5 x - 5\right ) \log {\left (x \right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (22) = 44\).
Time = 0.22 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.79 \[ \int \frac {-750+750 x-247 x^2+150 x^3+3 x^4+\left (750-30 x+780 x^2-530 x^3+30 x^4\right ) \log (x)+\left (75-150 x+150 x^2-150 x^3+75 x^4\right ) \log ^2(x)}{x^2+\left (-10 x+10 x^2\right ) \log (x)+\left (25-50 x+25 x^2\right ) \log ^2(x)} \, dx=\frac {x^{4} - 50 \, x^{3} + 3 \, x^{2} + 5 \, {\left (x^{4} - x^{3} + 3 \, x^{2} - 3 \, x\right )} \log \left (x\right ) - 150 \, x}{5 \, {\left (x - 1\right )} \log \left (x\right ) + x} \]
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Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {-750+750 x-247 x^2+150 x^3+3 x^4+\left (750-30 x+780 x^2-530 x^3+30 x^4\right ) \log (x)+\left (75-150 x+150 x^2-150 x^3+75 x^4\right ) \log ^2(x)}{x^2+\left (-10 x+10 x^2\right ) \log (x)+\left (25-50 x+25 x^2\right ) \log ^2(x)} \, dx=x^{3} + 3 \, x - \frac {50 \, {\left (x^{3} + 3 \, x\right )}}{5 \, x \log \left (x\right ) + x - 5 \, \log \left (x\right )} \]
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Time = 15.74 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {-750+750 x-247 x^2+150 x^3+3 x^4+\left (750-30 x+780 x^2-530 x^3+30 x^4\right ) \log (x)+\left (75-150 x+150 x^2-150 x^3+75 x^4\right ) \log ^2(x)}{x^2+\left (-10 x+10 x^2\right ) \log (x)+\left (25-50 x+25 x^2\right ) \log ^2(x)} \, dx=x\,\left (x^2+3\right )-\frac {x^2\,\left (x^2+3\right )-x\,\left (x^2+3\right )\,\left (x-50\right )}{x-5\,\ln \left (x\right )+5\,x\,\ln \left (x\right )} \]
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