Integrand size = 97, antiderivative size = 29 \[ \int \frac {24 x+6 x^2-21 x^3+5 x^4+5 x^5+\left (4+6 x-5 x^2-5 x^3\right ) \log \left (\frac {-100+100 x}{\left (16+80 x+140 x^2+100 x^3+25 x^4\right ) \log (2)}\right )}{-4 x^2-6 x^3+5 x^4+5 x^5} \, dx=5+x+\frac {\log \left (\frac {4 (-1+x)}{\left (\frac {4}{5}+2 x+x^2\right )^2 \log (2)}\right )}{x} \]
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Leaf count is larger than twice the leaf count of optimal. \(382\) vs. \(2(29)=58\).
Time = 0.76 (sec) , antiderivative size = 382, normalized size of antiderivative = 13.17, number of steps used = 34, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.082, Rules used = {6873, 6874, 2083, 646, 31, 2099, 2605, 6860} \[ \int \frac {24 x+6 x^2-21 x^3+5 x^4+5 x^5+\left (4+6 x-5 x^2-5 x^3\right ) \log \left (\frac {-100+100 x}{\left (16+80 x+140 x^2+100 x^3+25 x^4\right ) \log (2)}\right )}{-4 x^2-6 x^3+5 x^4+5 x^5} \, dx=\frac {\log \left (-\frac {100 (1-x)}{\left (5 x^2+10 x+4\right )^2 \log (2)}\right )}{x}+x+\frac {3}{19} \left (15+11 \sqrt {5}\right ) \log \left (-\sqrt {5} x-\sqrt {5}+1\right )-\frac {3}{19} \left (1+2 \sqrt {5}\right ) \log \left (-\sqrt {5} x-\sqrt {5}+1\right )-\frac {1}{2} \left (5+\sqrt {5}\right ) \log \left (-\sqrt {5} x-\sqrt {5}+1\right )+\frac {1}{19} \left (7-5 \sqrt {5}\right ) \log \left (-\sqrt {5} x-\sqrt {5}+1\right )-\frac {4}{95} \left (15-8 \sqrt {5}\right ) \log \left (-\sqrt {5} x-\sqrt {5}+1\right )+\frac {21}{190} \left (5-9 \sqrt {5}\right ) \log \left (-\sqrt {5} x-\sqrt {5}+1\right )+\frac {21}{190} \left (5+9 \sqrt {5}\right ) \log \left (\sqrt {5} x+\sqrt {5}+1\right )-\frac {4}{95} \left (15+8 \sqrt {5}\right ) \log \left (\sqrt {5} x+\sqrt {5}+1\right )+\frac {1}{19} \left (7+5 \sqrt {5}\right ) \log \left (\sqrt {5} x+\sqrt {5}+1\right )-\frac {1}{2} \left (5-\sqrt {5}\right ) \log \left (\sqrt {5} x+\sqrt {5}+1\right )-\frac {3}{19} \left (1-2 \sqrt {5}\right ) \log \left (\sqrt {5} x+\sqrt {5}+1\right )+\frac {3}{19} \left (15-11 \sqrt {5}\right ) \log \left (\sqrt {5} x+\sqrt {5}+1\right ) \]
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Rule 31
Rule 646
Rule 2083
Rule 2099
Rule 2605
Rule 6860
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {-24 x-6 x^2+21 x^3-5 x^4-5 x^5-\left (4+6 x-5 x^2-5 x^3\right ) \log \left (\frac {-100+100 x}{\left (16+80 x+140 x^2+100 x^3+25 x^4\right ) \log (2)}\right )}{x^2 \left (4+6 x-5 x^2-5 x^3\right )} \, dx \\ & = \int \left (\frac {6}{-4-6 x+5 x^2+5 x^3}+\frac {24}{x \left (-4-6 x+5 x^2+5 x^3\right )}-\frac {21 x}{-4-6 x+5 x^2+5 x^3}+\frac {5 x^2}{-4-6 x+5 x^2+5 x^3}+\frac {5 x^3}{-4-6 x+5 x^2+5 x^3}-\frac {\log \left (\frac {100 (-1+x)}{\left (4+10 x+5 x^2\right )^2 \log (2)}\right )}{x^2}\right ) \, dx \\ & = 5 \int \frac {x^2}{-4-6 x+5 x^2+5 x^3} \, dx+5 \int \frac {x^3}{-4-6 x+5 x^2+5 x^3} \, dx+6 \int \frac {1}{-4-6 x+5 x^2+5 x^3} \, dx-21 \int \frac {x}{-4-6 x+5 x^2+5 x^3} \, dx+24 \int \frac {1}{x \left (-4-6 x+5 x^2+5 x^3\right )} \, dx-\int \frac {\log \left (\frac {100 (-1+x)}{\left (4+10 x+5 x^2\right )^2 \log (2)}\right )}{x^2} \, dx \\ & = \frac {\log \left (-\frac {100 (1-x)}{\left (4+10 x+5 x^2\right )^2 \log (2)}\right )}{x}+5 \int \left (\frac {1}{19 (-1+x)}+\frac {2 (2+7 x)}{19 \left (4+10 x+5 x^2\right )}\right ) \, dx+5 \int \left (\frac {1}{5}+\frac {1}{19 (-1+x)}-\frac {8 (7+15 x)}{95 \left (4+10 x+5 x^2\right )}\right ) \, dx+6 \int \left (\frac {1}{19 (-1+x)}-\frac {5 (3+x)}{19 \left (4+10 x+5 x^2\right )}\right ) \, dx-21 \int \left (\frac {1}{19 (-1+x)}+\frac {4-5 x}{19 \left (4+10 x+5 x^2\right )}\right ) \, dx+24 \int \left (\frac {1}{19 (-1+x)}-\frac {1}{4 x}+\frac {5 (26+15 x)}{76 \left (4+10 x+5 x^2\right )}\right ) \, dx-\int \frac {-24-10 x+15 x^2}{(1-x) x \left (4+10 x+5 x^2\right )} \, dx \\ & = x+\log (1-x)-6 \log (x)+\frac {\log \left (-\frac {100 (1-x)}{\left (4+10 x+5 x^2\right )^2 \log (2)}\right )}{x}-\frac {8}{19} \int \frac {7+15 x}{4+10 x+5 x^2} \, dx+\frac {10}{19} \int \frac {2+7 x}{4+10 x+5 x^2} \, dx-\frac {21}{19} \int \frac {4-5 x}{4+10 x+5 x^2} \, dx-\frac {30}{19} \int \frac {3+x}{4+10 x+5 x^2} \, dx+\frac {30}{19} \int \frac {26+15 x}{4+10 x+5 x^2} \, dx-\int \left (\frac {1}{-1+x}-\frac {6}{x}+\frac {5 (6+5 x)}{4+10 x+5 x^2}\right ) \, dx \\ & = x+\frac {\log \left (-\frac {100 (1-x)}{\left (4+10 x+5 x^2\right )^2 \log (2)}\right )}{x}-5 \int \frac {6+5 x}{4+10 x+5 x^2} \, dx+\frac {1}{19} \left (15 \left (15-11 \sqrt {5}\right )\right ) \int \frac {1}{5+\sqrt {5}+5 x} \, dx+\frac {1}{38} \left (21 \left (5-9 \sqrt {5}\right )\right ) \int \frac {1}{5-\sqrt {5}+5 x} \, dx-\frac {1}{19} \left (4 \left (15-8 \sqrt {5}\right )\right ) \int \frac {1}{5-\sqrt {5}+5 x} \, dx+\frac {1}{19} \left (5 \left (7-5 \sqrt {5}\right )\right ) \int \frac {1}{5-\sqrt {5}+5 x} \, dx-\frac {1}{19} \left (15 \left (1-2 \sqrt {5}\right )\right ) \int \frac {1}{5+\sqrt {5}+5 x} \, dx-\frac {1}{19} \left (15 \left (1+2 \sqrt {5}\right )\right ) \int \frac {1}{5-\sqrt {5}+5 x} \, dx+\frac {1}{19} \left (5 \left (7+5 \sqrt {5}\right )\right ) \int \frac {1}{5+\sqrt {5}+5 x} \, dx-\frac {1}{19} \left (4 \left (15+8 \sqrt {5}\right )\right ) \int \frac {1}{5+\sqrt {5}+5 x} \, dx+\frac {1}{38} \left (21 \left (5+9 \sqrt {5}\right )\right ) \int \frac {1}{5+\sqrt {5}+5 x} \, dx+\frac {1}{19} \left (15 \left (15+11 \sqrt {5}\right )\right ) \int \frac {1}{5-\sqrt {5}+5 x} \, dx \\ & = x+\frac {21}{190} \left (5-9 \sqrt {5}\right ) \log \left (1-\sqrt {5}-\sqrt {5} x\right )-\frac {4}{95} \left (15-8 \sqrt {5}\right ) \log \left (1-\sqrt {5}-\sqrt {5} x\right )+\frac {1}{19} \left (7-5 \sqrt {5}\right ) \log \left (1-\sqrt {5}-\sqrt {5} x\right )-\frac {3}{19} \left (1+2 \sqrt {5}\right ) \log \left (1-\sqrt {5}-\sqrt {5} x\right )+\frac {3}{19} \left (15+11 \sqrt {5}\right ) \log \left (1-\sqrt {5}-\sqrt {5} x\right )+\frac {3}{19} \left (15-11 \sqrt {5}\right ) \log \left (1+\sqrt {5}+\sqrt {5} x\right )-\frac {3}{19} \left (1-2 \sqrt {5}\right ) \log \left (1+\sqrt {5}+\sqrt {5} x\right )+\frac {1}{19} \left (7+5 \sqrt {5}\right ) \log \left (1+\sqrt {5}+\sqrt {5} x\right )-\frac {4}{95} \left (15+8 \sqrt {5}\right ) \log \left (1+\sqrt {5}+\sqrt {5} x\right )+\frac {21}{190} \left (5+9 \sqrt {5}\right ) \log \left (1+\sqrt {5}+\sqrt {5} x\right )+\frac {\log \left (-\frac {100 (1-x)}{\left (4+10 x+5 x^2\right )^2 \log (2)}\right )}{x}-\frac {1}{2} \left (5 \left (5-\sqrt {5}\right )\right ) \int \frac {1}{5+\sqrt {5}+5 x} \, dx-\frac {1}{2} \left (5 \left (5+\sqrt {5}\right )\right ) \int \frac {1}{5-\sqrt {5}+5 x} \, dx \\ & = x+\frac {21}{190} \left (5-9 \sqrt {5}\right ) \log \left (1-\sqrt {5}-\sqrt {5} x\right )-\frac {4}{95} \left (15-8 \sqrt {5}\right ) \log \left (1-\sqrt {5}-\sqrt {5} x\right )+\frac {1}{19} \left (7-5 \sqrt {5}\right ) \log \left (1-\sqrt {5}-\sqrt {5} x\right )-\frac {1}{2} \left (5+\sqrt {5}\right ) \log \left (1-\sqrt {5}-\sqrt {5} x\right )-\frac {3}{19} \left (1+2 \sqrt {5}\right ) \log \left (1-\sqrt {5}-\sqrt {5} x\right )+\frac {3}{19} \left (15+11 \sqrt {5}\right ) \log \left (1-\sqrt {5}-\sqrt {5} x\right )+\frac {3}{19} \left (15-11 \sqrt {5}\right ) \log \left (1+\sqrt {5}+\sqrt {5} x\right )-\frac {3}{19} \left (1-2 \sqrt {5}\right ) \log \left (1+\sqrt {5}+\sqrt {5} x\right )-\frac {1}{2} \left (5-\sqrt {5}\right ) \log \left (1+\sqrt {5}+\sqrt {5} x\right )+\frac {1}{19} \left (7+5 \sqrt {5}\right ) \log \left (1+\sqrt {5}+\sqrt {5} x\right )-\frac {4}{95} \left (15+8 \sqrt {5}\right ) \log \left (1+\sqrt {5}+\sqrt {5} x\right )+\frac {21}{190} \left (5+9 \sqrt {5}\right ) \log \left (1+\sqrt {5}+\sqrt {5} x\right )+\frac {\log \left (-\frac {100 (1-x)}{\left (4+10 x+5 x^2\right )^2 \log (2)}\right )}{x} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \frac {24 x+6 x^2-21 x^3+5 x^4+5 x^5+\left (4+6 x-5 x^2-5 x^3\right ) \log \left (\frac {-100+100 x}{\left (16+80 x+140 x^2+100 x^3+25 x^4\right ) \log (2)}\right )}{-4 x^2-6 x^3+5 x^4+5 x^5} \, dx=x+\frac {\log \left (\frac {100 (-1+x)}{\left (4+10 x+5 x^2\right )^2 \log (2)}\right )}{x} \]
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Time = 0.34 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.38
method | result | size |
risch | \(\frac {\ln \left (\frac {100 x -100}{\left (25 x^{4}+100 x^{3}+140 x^{2}+80 x +16\right ) \ln \left (2\right )}\right )}{x}+x\) | \(40\) |
norman | \(\frac {x^{2}+\ln \left (\frac {100 x -100}{\left (25 x^{4}+100 x^{3}+140 x^{2}+80 x +16\right ) \ln \left (2\right )}\right )}{x}\) | \(42\) |
default | \(-\frac {\ln \left (\ln \left (2\right )\right )}{x}+x +\frac {2 \ln \left (10\right )}{x}+\frac {\ln \left (\frac {-1+x}{25 x^{4}+100 x^{3}+140 x^{2}+80 x +16}\right )}{x}\) | \(49\) |
parallelrisch | \(-\frac {-50 x^{2}+100 x -50 \ln \left (\frac {100 x -100}{\left (25 x^{4}+100 x^{3}+140 x^{2}+80 x +16\right ) \ln \left (2\right )}\right )}{50 x}\) | \(49\) |
parts | \(-\frac {\ln \left (\ln \left (2\right )\right )}{x}+x +\frac {2 \ln \left (10\right )}{x}+\frac {\ln \left (\frac {-1+x}{25 x^{4}+100 x^{3}+140 x^{2}+80 x +16}\right )}{x}\) | \(49\) |
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Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.38 \[ \int \frac {24 x+6 x^2-21 x^3+5 x^4+5 x^5+\left (4+6 x-5 x^2-5 x^3\right ) \log \left (\frac {-100+100 x}{\left (16+80 x+140 x^2+100 x^3+25 x^4\right ) \log (2)}\right )}{-4 x^2-6 x^3+5 x^4+5 x^5} \, dx=\frac {x^{2} + \log \left (\frac {100 \, {\left (x - 1\right )}}{{\left (25 \, x^{4} + 100 \, x^{3} + 140 \, x^{2} + 80 \, x + 16\right )} \log \left (2\right )}\right )}{x} \]
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Time = 0.15 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \frac {24 x+6 x^2-21 x^3+5 x^4+5 x^5+\left (4+6 x-5 x^2-5 x^3\right ) \log \left (\frac {-100+100 x}{\left (16+80 x+140 x^2+100 x^3+25 x^4\right ) \log (2)}\right )}{-4 x^2-6 x^3+5 x^4+5 x^5} \, dx=x + \frac {\log {\left (\frac {100 x - 100}{\left (25 x^{4} + 100 x^{3} + 140 x^{2} + 80 x + 16\right ) \log {\left (2 \right )}} \right )}}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (29) = 58\).
Time = 0.36 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.21 \[ \int \frac {24 x+6 x^2-21 x^3+5 x^4+5 x^5+\left (4+6 x-5 x^2-5 x^3\right ) \log \left (\frac {-100+100 x}{\left (16+80 x+140 x^2+100 x^3+25 x^4\right ) \log (2)}\right )}{-4 x^2-6 x^3+5 x^4+5 x^5} \, dx=x - \frac {{\left (5 \, x + 4\right )} \log \left (5 \, x^{2} + 10 \, x + 4\right ) + 2 \, {\left (x - 1\right )} \log \left (x - 1\right ) - 4 \, \log \left (5\right ) - 4 \, \log \left (2\right ) + 2 \, \log \left (\log \left (2\right )\right )}{2 \, x} + \frac {5}{2} \, \log \left (5 \, x^{2} + 10 \, x + 4\right ) + \log \left (x - 1\right ) \]
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Time = 0.35 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.93 \[ \int \frac {24 x+6 x^2-21 x^3+5 x^4+5 x^5+\left (4+6 x-5 x^2-5 x^3\right ) \log \left (\frac {-100+100 x}{\left (16+80 x+140 x^2+100 x^3+25 x^4\right ) \log (2)}\right )}{-4 x^2-6 x^3+5 x^4+5 x^5} \, dx=x + \frac {2 \, \log \left (2\right )}{x} - \frac {\log \left (25 \, x^{4} \log \left (2\right ) + 100 \, x^{3} \log \left (2\right ) + 140 \, x^{2} \log \left (2\right ) + 80 \, x \log \left (2\right ) + 16 \, \log \left (2\right )\right )}{x} + \frac {\log \left (25 \, x - 25\right )}{x} \]
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Time = 14.19 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31 \[ \int \frac {24 x+6 x^2-21 x^3+5 x^4+5 x^5+\left (4+6 x-5 x^2-5 x^3\right ) \log \left (\frac {-100+100 x}{\left (16+80 x+140 x^2+100 x^3+25 x^4\right ) \log (2)}\right )}{-4 x^2-6 x^3+5 x^4+5 x^5} \, dx=x+\frac {\ln \left (\frac {100\,\left (x-1\right )}{\ln \left (2\right )\,\left (25\,x^4+100\,x^3+140\,x^2+80\,x+16\right )}\right )}{x} \]
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