Integrand size = 75, antiderivative size = 24 \[ \int \frac {20-3 x^2+8 x^3+\left (20-32 x+3 x^2-4 x^3\right ) \log \left (\frac {-20+32 x-3 x^2+4 x^3}{4 x}\right )}{-20 x^2+32 x^3-3 x^4+4 x^5} \, dx=5+\frac {\log \left (8-\frac {5}{x}+\frac {5 x}{4}+(-2+x) x\right )}{x} \]
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Time = 3.52 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83, number of steps used = 26, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {6873, 6874, 2125, 2106, 2104, 814, 648, 632, 210, 642, 2605} \[ \int \frac {20-3 x^2+8 x^3+\left (20-32 x+3 x^2-4 x^3\right ) \log \left (\frac {-20+32 x-3 x^2+4 x^3}{4 x}\right )}{-20 x^2+32 x^3-3 x^4+4 x^5} \, dx=\frac {\log \left (x^2-\frac {3 x}{4}-\frac {5}{x}+8\right )}{x} \]
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 814
Rule 2104
Rule 2106
Rule 2125
Rule 2605
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {-20+3 x^2-8 x^3-\left (20-32 x+3 x^2-4 x^3\right ) \log \left (\frac {-20+32 x-3 x^2+4 x^3}{4 x}\right )}{x^2 \left (20-32 x+3 x^2-4 x^3\right )} \, dx \\ & = \int \left (\frac {20-3 x^2+8 x^3}{x^2 \left (-20+32 x-3 x^2+4 x^3\right )}-\frac {\log \left (8-\frac {5}{x}-\frac {3 x}{4}+x^2\right )}{x^2}\right ) \, dx \\ & = \int \frac {20-3 x^2+8 x^3}{x^2 \left (-20+32 x-3 x^2+4 x^3\right )} \, dx-\int \frac {\log \left (8-\frac {5}{x}-\frac {3 x}{4}+x^2\right )}{x^2} \, dx \\ & = \frac {\log \left (8-\frac {5}{x}-\frac {3 x}{4}+x^2\right )}{x}-\int \frac {\frac {3}{4}-\frac {5}{x^2}-2 x}{5-8 x+\frac {3 x^2}{4}-x^3} \, dx+\int \left (-\frac {1}{x^2}-\frac {8}{5 x}+\frac {2 \left (113+18 x+16 x^2\right )}{5 \left (-20+32 x-3 x^2+4 x^3\right )}\right ) \, dx \\ & = \frac {1}{x}-\frac {8 \log (x)}{5}+\frac {\log \left (8-\frac {5}{x}-\frac {3 x}{4}+x^2\right )}{x}+\frac {2}{5} \int \frac {113+18 x+16 x^2}{-20+32 x-3 x^2+4 x^3} \, dx-\int \left (-\frac {1}{x^2}-\frac {8}{5 x}+\frac {2 \left (113+18 x+16 x^2\right )}{5 \left (-20+32 x-3 x^2+4 x^3\right )}\right ) \, dx \\ & = \frac {\log \left (8-\frac {5}{x}-\frac {3 x}{4}+x^2\right )}{x}+\frac {8}{15} \log \left (-20+32 x-3 x^2+4 x^3\right )+\frac {1}{30} \int \frac {844+312 x}{-20+32 x-3 x^2+4 x^3} \, dx-\frac {2}{5} \int \frac {113+18 x+16 x^2}{-20+32 x-3 x^2+4 x^3} \, dx \\ & = \frac {\log \left (8-\frac {5}{x}-\frac {3 x}{4}+x^2\right )}{x}-\frac {1}{30} \int \frac {844+312 x}{-20+32 x-3 x^2+4 x^3} \, dx+\frac {1}{30} \text {Subst}\left (\int \frac {922+312 x}{-\frac {97}{8}+\frac {125 x}{4}+4 x^3} \, dx,x,-\frac {1}{4}+x\right ) \\ & = \frac {\log \left (8-\frac {5}{x}-\frac {3 x}{4}+x^2\right )}{x}-\frac {1}{30} \text {Subst}\left (\int \frac {922+312 x}{-\frac {97}{8}+\frac {125 x}{4}+4 x^3} \, dx,x,-\frac {1}{4}+x\right )+\frac {8}{15} \text {Subst}\left (\int \frac {922+312 x}{\left (\frac {1}{3} \left (\frac {125\ 3^{2/3}}{\sqrt [3]{873+8 \sqrt {103461}}}-\sqrt [3]{3 \left (873+8 \sqrt {103461}\right )}\right )+4 x\right ) \left (\frac {1}{9} \left (375+\frac {46875 \sqrt [3]{3}}{\left (873+8 \sqrt {103461}\right )^{2/3}}+\left (3 \left (873+8 \sqrt {103461}\right )\right )^{2/3}\right )-\frac {4 \left (125 \sqrt [3]{\frac {3}{873+8 \sqrt {103461}}}-\sqrt [3]{873+8 \sqrt {103461}}\right ) x}{3^{2/3}}+16 x^2\right )} \, dx,x,-\frac {1}{4}+x\right ) \\ & = \frac {\log \left (8-\frac {5}{x}-\frac {3 x}{4}+x^2\right )}{x}-\frac {8}{15} \text {Subst}\left (\int \frac {922+312 x}{\left (\frac {1}{3} \left (\frac {125\ 3^{2/3}}{\sqrt [3]{873+8 \sqrt {103461}}}-\sqrt [3]{3 \left (873+8 \sqrt {103461}\right )}\right )+4 x\right ) \left (\frac {1}{9} \left (375+\frac {46875 \sqrt [3]{3}}{\left (873+8 \sqrt {103461}\right )^{2/3}}+\left (3 \left (873+8 \sqrt {103461}\right )\right )^{2/3}\right )-\frac {4 \left (125 \sqrt [3]{\frac {3}{873+8 \sqrt {103461}}}-\sqrt [3]{873+8 \sqrt {103461}}\right ) x}{3^{2/3}}+16 x^2\right )} \, dx,x,-\frac {1}{4}+x\right )+\frac {8}{15} \text {Subst}\left (\int \left (\frac {6 \left (873+8 \sqrt {103461}\right )^{2/3} \left (-1625 3^{2/3}+461 \sqrt [3]{873+8 \sqrt {103461}}+13 \sqrt [3]{3} \left (873+8 \sqrt {103461}\right )^{2/3}\right )}{\left (15625 \sqrt [3]{3}+\left (291\ 3^{2/3}+8 \sqrt [6]{3} \sqrt {34487}\right ) \sqrt [3]{873+8 \sqrt {103461}}-125 \left (873+8 \sqrt {103461}\right )^{2/3}\right ) \left (125\ 3^{2/3}-\sqrt [3]{3} \left (873+8 \sqrt {103461}\right )^{2/3}+12 \sqrt [3]{873+8 \sqrt {103461}} x\right )}+\frac {2 \left (873+8 \sqrt {103461}\right )^{2/3} \left (\left (126599\ 3^{2/3}+312 \sqrt [6]{3} \sqrt {34487}\right ) \sqrt [3]{873+8 \sqrt {103461}}+4875 \left (873+8 \sqrt {103461}\right )^{2/3}-\sqrt [3]{3} \left (195531+7376 \sqrt {103461}\right )+12 \sqrt [3]{873+8 \sqrt {103461}} \left (1625\ 3^{2/3}-461 \sqrt [3]{873+8 \sqrt {103461}}-13 \sqrt [3]{3} \left (873+8 \sqrt {103461}\right )^{2/3}\right ) x\right )}{\left (15625 \sqrt [3]{3}+\left (291\ 3^{2/3}+8 \sqrt [6]{3} \sqrt {34487}\right ) \sqrt [3]{873+8 \sqrt {103461}}-125 \left (873+8 \sqrt {103461}\right )^{2/3}\right ) \left (15625 \sqrt [3]{3}+\left (291\ 3^{2/3}+8 \sqrt [6]{3} \sqrt {34487}\right ) \sqrt [3]{873+8 \sqrt {103461}}+125 \left (873+8 \sqrt {103461}\right )^{2/3}+4\ 3^{2/3} \left (291\ 3^{2/3}+8 \sqrt [6]{3} \sqrt {34487}-125 \sqrt [3]{873+8 \sqrt {103461}}\right ) x+48 \left (873+8 \sqrt {103461}\right )^{2/3} x^2\right )}\right ) \, dx,x,-\frac {1}{4}+x\right ) \\ & = -\frac {4 \sqrt [3]{873+8 \sqrt {103461}} \left (1625\ 3^{2/3}-461 \sqrt [3]{873+8 \sqrt {103461}}-13 \sqrt [3]{3} \left (873+8 \sqrt {103461}\right )^{2/3}\right ) \log \left (125 \sqrt [3]{3}-\left (873+8 \sqrt {103461}\right )^{2/3}-3^{2/3} \sqrt [3]{873+8 \sqrt {103461}} (1-4 x)\right )}{15 \left (15625 \sqrt [3]{3}+\left (291\ 3^{2/3}+8 \sqrt [6]{3} \sqrt {34487}\right ) \sqrt [3]{873+8 \sqrt {103461}}-125 \left (873+8 \sqrt {103461}\right )^{2/3}\right )}+\frac {\log \left (8-\frac {5}{x}-\frac {3 x}{4}+x^2\right )}{x}-\frac {8}{15} \text {Subst}\left (\int \left (\frac {6 \left (873+8 \sqrt {103461}\right )^{2/3} \left (-1625 3^{2/3}+461 \sqrt [3]{873+8 \sqrt {103461}}+13 \sqrt [3]{3} \left (873+8 \sqrt {103461}\right )^{2/3}\right )}{\left (15625 \sqrt [3]{3}+\left (291\ 3^{2/3}+8 \sqrt [6]{3} \sqrt {34487}\right ) \sqrt [3]{873+8 \sqrt {103461}}-125 \left (873+8 \sqrt {103461}\right )^{2/3}\right ) \left (125\ 3^{2/3}-\sqrt [3]{3} \left (873+8 \sqrt {103461}\right )^{2/3}+12 \sqrt [3]{873+8 \sqrt {103461}} x\right )}+\frac {2 \left (873+8 \sqrt {103461}\right )^{2/3} \left (\left (126599\ 3^{2/3}+312 \sqrt [6]{3} \sqrt {34487}\right ) \sqrt [3]{873+8 \sqrt {103461}}+4875 \left (873+8 \sqrt {103461}\right )^{2/3}-\sqrt [3]{3} \left (195531+7376 \sqrt {103461}\right )+12 \sqrt [3]{873+8 \sqrt {103461}} \left (1625\ 3^{2/3}-461 \sqrt [3]{873+8 \sqrt {103461}}-13 \sqrt [3]{3} \left (873+8 \sqrt {103461}\right )^{2/3}\right ) x\right )}{\left (15625 \sqrt [3]{3}+\left (291\ 3^{2/3}+8 \sqrt [6]{3} \sqrt {34487}\right ) \sqrt [3]{873+8 \sqrt {103461}}-125 \left (873+8 \sqrt {103461}\right )^{2/3}\right ) \left (15625 \sqrt [3]{3}+\left (291\ 3^{2/3}+8 \sqrt [6]{3} \sqrt {34487}\right ) \sqrt [3]{873+8 \sqrt {103461}}+125 \left (873+8 \sqrt {103461}\right )^{2/3}+4\ 3^{2/3} \left (291\ 3^{2/3}+8 \sqrt [6]{3} \sqrt {34487}-125 \sqrt [3]{873+8 \sqrt {103461}}\right ) x+48 \left (873+8 \sqrt {103461}\right )^{2/3} x^2\right )}\right ) \, dx,x,-\frac {1}{4}+x\right )+\frac {\left (16 \left (873+8 \sqrt {103461}\right )^{2/3}\right ) \text {Subst}\left (\int \frac {\left (126599\ 3^{2/3}+312 \sqrt [6]{3} \sqrt {34487}\right ) \sqrt [3]{873+8 \sqrt {103461}}+4875 \left (873+8 \sqrt {103461}\right )^{2/3}-\sqrt [3]{3} \left (195531+7376 \sqrt {103461}\right )+12 \sqrt [3]{873+8 \sqrt {103461}} \left (1625\ 3^{2/3}-461 \sqrt [3]{873+8 \sqrt {103461}}-13 \sqrt [3]{3} \left (873+8 \sqrt {103461}\right )^{2/3}\right ) x}{15625 \sqrt [3]{3}+\left (291\ 3^{2/3}+8 \sqrt [6]{3} \sqrt {34487}\right ) \sqrt [3]{873+8 \sqrt {103461}}+125 \left (873+8 \sqrt {103461}\right )^{2/3}+4\ 3^{2/3} \left (291\ 3^{2/3}+8 \sqrt [6]{3} \sqrt {34487}-125 \sqrt [3]{873+8 \sqrt {103461}}\right ) x+48 \left (873+8 \sqrt {103461}\right )^{2/3} x^2} \, dx,x,-\frac {1}{4}+x\right )}{15 \left (15625 \sqrt [3]{3}+\left (291\ 3^{2/3}+8 \sqrt [6]{3} \sqrt {34487}\right ) \sqrt [3]{873+8 \sqrt {103461}}-125 \left (873+8 \sqrt {103461}\right )^{2/3}\right )} \\ & = \frac {\log \left (8-\frac {5}{x}-\frac {3 x}{4}+x^2\right )}{x}-\frac {\left (16 \left (873+8 \sqrt {103461}\right )^{2/3}\right ) \text {Subst}\left (\int \frac {\left (126599\ 3^{2/3}+312 \sqrt [6]{3} \sqrt {34487}\right ) \sqrt [3]{873+8 \sqrt {103461}}+4875 \left (873+8 \sqrt {103461}\right )^{2/3}-\sqrt [3]{3} \left (195531+7376 \sqrt {103461}\right )+12 \sqrt [3]{873+8 \sqrt {103461}} \left (1625\ 3^{2/3}-461 \sqrt [3]{873+8 \sqrt {103461}}-13 \sqrt [3]{3} \left (873+8 \sqrt {103461}\right )^{2/3}\right ) x}{15625 \sqrt [3]{3}+\left (291\ 3^{2/3}+8 \sqrt [6]{3} \sqrt {34487}\right ) \sqrt [3]{873+8 \sqrt {103461}}+125 \left (873+8 \sqrt {103461}\right )^{2/3}+4\ 3^{2/3} \left (291\ 3^{2/3}+8 \sqrt [6]{3} \sqrt {34487}-125 \sqrt [3]{873+8 \sqrt {103461}}\right ) x+48 \left (873+8 \sqrt {103461}\right )^{2/3} x^2} \, dx,x,-\frac {1}{4}+x\right )}{15 \left (15625 \sqrt [3]{3}+\left (291\ 3^{2/3}+8 \sqrt [6]{3} \sqrt {34487}\right ) \sqrt [3]{873+8 \sqrt {103461}}-125 \left (873+8 \sqrt {103461}\right )^{2/3}\right )}+\frac {\left (16 \left (873+8 \sqrt {103461}\right )^{2/3} \left (\sqrt [3]{3} \left (103461-1844 \sqrt {103461}\right )+\left (34487\ 3^{2/3}+156 \sqrt [6]{3} \sqrt {34487}\right ) \sqrt [3]{873+8 \sqrt {103461}}\right )\right ) \text {Subst}\left (\int \frac {1}{15625 \sqrt [3]{3}+\left (291\ 3^{2/3}+8 \sqrt [6]{3} \sqrt {34487}\right ) \sqrt [3]{873+8 \sqrt {103461}}+125 \left (873+8 \sqrt {103461}\right )^{2/3}+4\ 3^{2/3} \left (291\ 3^{2/3}+8 \sqrt [6]{3} \sqrt {34487}-125 \sqrt [3]{873+8 \sqrt {103461}}\right ) x+48 \left (873+8 \sqrt {103461}\right )^{2/3} x^2} \, dx,x,-\frac {1}{4}+x\right )}{5 \left (15625 \sqrt [3]{3}+\left (291\ 3^{2/3}+8 \sqrt [6]{3} \sqrt {34487}\right ) \sqrt [3]{873+8 \sqrt {103461}}-125 \left (873+8 \sqrt {103461}\right )^{2/3}\right )}-\frac {\left (2 \left (873+8 \sqrt {103461}\right )^{2/3} \left (461-\frac {1625\ 3^{2/3}}{\sqrt [3]{873+8 \sqrt {103461}}}+13 \sqrt [3]{3 \left (873+8 \sqrt {103461}\right )}\right )\right ) \text {Subst}\left (\int \frac {4\ 3^{2/3} \left (291\ 3^{2/3}+8 \sqrt [6]{3} \sqrt {34487}-125 \sqrt [3]{873+8 \sqrt {103461}}\right )+96 \left (873+8 \sqrt {103461}\right )^{2/3} x}{15625 \sqrt [3]{3}+\left (291\ 3^{2/3}+8 \sqrt [6]{3} \sqrt {34487}\right ) \sqrt [3]{873+8 \sqrt {103461}}+125 \left (873+8 \sqrt {103461}\right )^{2/3}+4\ 3^{2/3} \left (291\ 3^{2/3}+8 \sqrt [6]{3} \sqrt {34487}-125 \sqrt [3]{873+8 \sqrt {103461}}\right ) x+48 \left (873+8 \sqrt {103461}\right )^{2/3} x^2} \, dx,x,-\frac {1}{4}+x\right )}{15 \left (15625 \sqrt [3]{3}+\left (291\ 3^{2/3}+8 \sqrt [6]{3} \sqrt {34487}\right ) \sqrt [3]{873+8 \sqrt {103461}}-125 \left (873+8 \sqrt {103461}\right )^{2/3}\right )} \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {20-3 x^2+8 x^3+\left (20-32 x+3 x^2-4 x^3\right ) \log \left (\frac {-20+32 x-3 x^2+4 x^3}{4 x}\right )}{-20 x^2+32 x^3-3 x^4+4 x^5} \, dx=\frac {\log \left (8-\frac {5}{x}-\frac {3 x}{4}+x^2\right )}{x} \]
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Time = 0.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08
method | result | size |
norman | \(\frac {\ln \left (\frac {4 x^{3}-3 x^{2}+32 x -20}{4 x}\right )}{x}\) | \(26\) |
risch | \(\frac {\ln \left (\frac {4 x^{3}-3 x^{2}+32 x -20}{4 x}\right )}{x}\) | \(26\) |
parallelrisch | \(\frac {\ln \left (\frac {4 x^{3}-3 x^{2}+32 x -20}{4 x}\right )}{x}\) | \(26\) |
default | \(\frac {\ln \left (\frac {4 x^{3}-3 x^{2}+32 x -20}{x}\right )}{x}-\frac {2 \ln \left (2\right )}{x}\) | \(33\) |
parts | \(\frac {\ln \left (\frac {4 x^{3}-3 x^{2}+32 x -20}{x}\right )}{x}-\frac {2 \ln \left (2\right )}{x}\) | \(33\) |
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Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {20-3 x^2+8 x^3+\left (20-32 x+3 x^2-4 x^3\right ) \log \left (\frac {-20+32 x-3 x^2+4 x^3}{4 x}\right )}{-20 x^2+32 x^3-3 x^4+4 x^5} \, dx=\frac {\log \left (\frac {4 \, x^{3} - 3 \, x^{2} + 32 \, x - 20}{4 \, x}\right )}{x} \]
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Time = 0.10 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {20-3 x^2+8 x^3+\left (20-32 x+3 x^2-4 x^3\right ) \log \left (\frac {-20+32 x-3 x^2+4 x^3}{4 x}\right )}{-20 x^2+32 x^3-3 x^4+4 x^5} \, dx=\frac {\log {\left (\frac {x^{3} - \frac {3 x^{2}}{4} + 8 x - 5}{x} \right )}}{x} \]
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Time = 0.32 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25 \[ \int \frac {20-3 x^2+8 x^3+\left (20-32 x+3 x^2-4 x^3\right ) \log \left (\frac {-20+32 x-3 x^2+4 x^3}{4 x}\right )}{-20 x^2+32 x^3-3 x^4+4 x^5} \, dx=-\frac {2 \, \log \left (2\right ) - \log \left (4 \, x^{3} - 3 \, x^{2} + 32 \, x - 20\right ) + \log \left (x\right )}{x} \]
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Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {20-3 x^2+8 x^3+\left (20-32 x+3 x^2-4 x^3\right ) \log \left (\frac {-20+32 x-3 x^2+4 x^3}{4 x}\right )}{-20 x^2+32 x^3-3 x^4+4 x^5} \, dx=\frac {\log \left (\frac {4 \, x^{3} - 3 \, x^{2} + 32 \, x - 20}{4 \, x}\right )}{x} \]
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Time = 11.45 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {20-3 x^2+8 x^3+\left (20-32 x+3 x^2-4 x^3\right ) \log \left (\frac {-20+32 x-3 x^2+4 x^3}{4 x}\right )}{-20 x^2+32 x^3-3 x^4+4 x^5} \, dx=\frac {\ln \left (\frac {x^3-\frac {3\,x^2}{4}+8\,x-5}{x}\right )}{x} \]
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