Integrand size = 59, antiderivative size = 27 \[ \int \frac {-36 x+13 x^2-x^3+\left (640-340 x+40 x^2\right ) \log (x)+\left (320-160 x+20 x^2\right ) \log \left (4 x^2\right )}{16 x-8 x^2+x^3} \, dx=5-x-5 \log (x) \left (\frac {x}{4-x}-4 \log \left (4 x^2\right )\right ) \]
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Time = 0.22 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1608, 27, 6874, 45, 2404, 2351, 31, 2338} \[ \int \frac {-36 x+13 x^2-x^3+\left (640-340 x+40 x^2\right ) \log (x)+\left (320-160 x+20 x^2\right ) \log \left (4 x^2\right )}{16 x-8 x^2+x^3} \, dx=5 \log ^2\left (4 x^2\right )-x+20 \log ^2(x)-\frac {5 x \log (x)}{4-x} \]
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Rule 27
Rule 31
Rule 45
Rule 1608
Rule 2338
Rule 2351
Rule 2404
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {-36 x+13 x^2-x^3+\left (640-340 x+40 x^2\right ) \log (x)+\left (320-160 x+20 x^2\right ) \log \left (4 x^2\right )}{x \left (16-8 x+x^2\right )} \, dx \\ & = \int \frac {-36 x+13 x^2-x^3+\left (640-340 x+40 x^2\right ) \log (x)+\left (320-160 x+20 x^2\right ) \log \left (4 x^2\right )}{(-4+x)^2 x} \, dx \\ & = \int \left (-\frac {36}{(-4+x)^2}+\frac {13 x}{(-4+x)^2}-\frac {x^2}{(-4+x)^2}+\frac {20 \left (32-17 x+2 x^2\right ) \log (x)}{(-4+x)^2 x}+\frac {20 \log \left (4 x^2\right )}{x}\right ) \, dx \\ & = -\frac {36}{4-x}+13 \int \frac {x}{(-4+x)^2} \, dx+20 \int \frac {\left (32-17 x+2 x^2\right ) \log (x)}{(-4+x)^2 x} \, dx+20 \int \frac {\log \left (4 x^2\right )}{x} \, dx-\int \frac {x^2}{(-4+x)^2} \, dx \\ & = -\frac {36}{4-x}+5 \log ^2\left (4 x^2\right )+13 \int \left (\frac {4}{(-4+x)^2}+\frac {1}{-4+x}\right ) \, dx+20 \int \left (-\frac {\log (x)}{(-4+x)^2}+\frac {2 \log (x)}{x}\right ) \, dx-\int \left (1+\frac {16}{(-4+x)^2}+\frac {8}{-4+x}\right ) \, dx \\ & = -x+5 \log (4-x)+5 \log ^2\left (4 x^2\right )-20 \int \frac {\log (x)}{(-4+x)^2} \, dx+40 \int \frac {\log (x)}{x} \, dx \\ & = -x+5 \log (4-x)-\frac {5 x \log (x)}{4-x}+20 \log ^2(x)+5 \log ^2\left (4 x^2\right )-5 \int \frac {1}{-4+x} \, dx \\ & = -x-\frac {5 x \log (x)}{4-x}+20 \log ^2(x)+5 \log ^2\left (4 x^2\right ) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {-36 x+13 x^2-x^3+\left (640-340 x+40 x^2\right ) \log (x)+\left (320-160 x+20 x^2\right ) \log \left (4 x^2\right )}{16 x-8 x^2+x^3} \, dx=-x+\frac {5 x \log (x)}{-4+x}+20 \log ^2(x)+5 \log ^2\left (4 x^2\right ) \]
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Time = 0.16 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15
method | result | size |
parts | \(5 \ln \left (4 x^{2}\right )^{2}-x +20 \ln \left (x \right )^{2}+\frac {5 \ln \left (x \right ) x}{x -4}\) | \(31\) |
default | \(-x +40 \ln \left (2\right ) \ln \left (x \right )+5 \ln \left (x^{2}\right )^{2}+20 \ln \left (x \right )^{2}+\frac {5 \ln \left (x \right ) x}{x -4}\) | \(35\) |
parallelrisch | \(\frac {16+20 x \ln \left (x \right ) \ln \left (4 x^{2}\right )-x^{2}+5 x \ln \left (x \right )-80 \ln \left (x \right ) \ln \left (4 x^{2}\right )}{x -4}\) | \(40\) |
norman | \(\frac {20 \ln \left (x \right )-10 \ln \left (4 x^{2}\right )-80 \ln \left (x \right ) \ln \left (4 x^{2}\right )+\frac {5 x \ln \left (4 x^{2}\right )}{2}+20 x \ln \left (x \right ) \ln \left (4 x^{2}\right )-x^{2}+16}{x -4}\) | \(56\) |
risch | \(40 \ln \left (x \right )^{2}+\frac {20 \ln \left (x \right )}{x -4}-x -10 i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+20 i \pi \ln \left (x \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-10 i \pi \ln \left (x \right ) \operatorname {csgn}\left (i x^{2}\right )^{3}+40 \ln \left (2\right ) \ln \left (x \right )+5 \ln \left (x \right )\) | \(85\) |
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Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {-36 x+13 x^2-x^3+\left (640-340 x+40 x^2\right ) \log (x)+\left (320-160 x+20 x^2\right ) \log \left (4 x^2\right )}{16 x-8 x^2+x^3} \, dx=\frac {40 \, {\left (x - 4\right )} \log \left (x\right )^{2} - x^{2} + 5 \, {\left (8 \, {\left (x - 4\right )} \log \left (2\right ) + x\right )} \log \left (x\right ) + 4 \, x}{x - 4} \]
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Time = 0.13 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {-36 x+13 x^2-x^3+\left (640-340 x+40 x^2\right ) \log (x)+\left (320-160 x+20 x^2\right ) \log \left (4 x^2\right )}{16 x-8 x^2+x^3} \, dx=- x + 40 \log {\left (x \right )}^{2} + 5 \cdot \left (1 + 8 \log {\left (2 \right )}\right ) \log {\left (x \right )} + \frac {20 \log {\left (x \right )}}{x - 4} \]
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Time = 0.31 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {-36 x+13 x^2-x^3+\left (640-340 x+40 x^2\right ) \log (x)+\left (320-160 x+20 x^2\right ) \log \left (4 x^2\right )}{16 x-8 x^2+x^3} \, dx=5 \, {\left (8 \, \log \left (2\right ) + 1\right )} \log \left (x\right ) - x + \frac {20 \, {\left (2 \, {\left (x - 4\right )} \log \left (x\right )^{2} + \log \left (x\right )\right )}}{x - 4} \]
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\[ \int \frac {-36 x+13 x^2-x^3+\left (640-340 x+40 x^2\right ) \log (x)+\left (320-160 x+20 x^2\right ) \log \left (4 x^2\right )}{16 x-8 x^2+x^3} \, dx=\int { -\frac {x^{3} - 13 \, x^{2} - 20 \, {\left (x^{2} - 8 \, x + 16\right )} \log \left (4 \, x^{2}\right ) - 20 \, {\left (2 \, x^{2} - 17 \, x + 32\right )} \log \left (x\right ) + 36 \, x}{x^{3} - 8 \, x^{2} + 16 \, x} \,d x } \]
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Time = 12.38 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {-36 x+13 x^2-x^3+\left (640-340 x+40 x^2\right ) \log (x)+\left (320-160 x+20 x^2\right ) \log \left (4 x^2\right )}{16 x-8 x^2+x^3} \, dx=5\,\ln \left (x\right )-x+20\,\ln \left (x^2\right )\,\ln \left (x\right )+40\,\ln \left (2\right )\,\ln \left (x\right )-\frac {20\,x^2\,\ln \left (x\right )}{4\,x^2-x^3} \]
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