Integrand size = 44, antiderivative size = 38 \[ \int \frac {5-3 x+6 x^2+5 x^3-x^4}{-1+x+2 x^2-2 x^3-x^4+x^5} \, dx=-\frac {3}{2 (1-x)^2}+\frac {2}{1-x}+\frac {1}{1+x}+\log (1-x)-2 \log (1+x) \]
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Time = 0.05 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {2099} \[ \int \frac {5-3 x+6 x^2+5 x^3-x^4}{-1+x+2 x^2-2 x^3-x^4+x^5} \, dx=\frac {2}{1-x}+\frac {1}{x+1}-\frac {3}{2 (1-x)^2}+\log (1-x)-2 \log (x+1) \]
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Rule 2099
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3}{(-1+x)^3}+\frac {2}{(-1+x)^2}+\frac {1}{-1+x}-\frac {1}{(1+x)^2}-\frac {2}{1+x}\right ) \, dx \\ & = -\frac {3}{2 (1-x)^2}+\frac {2}{1-x}+\frac {1}{1+x}+\log (1-x)-2 \log (1+x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.84 \[ \int \frac {5-3 x+6 x^2+5 x^3-x^4}{-1+x+2 x^2-2 x^3-x^4+x^5} \, dx=-\frac {3}{2 (-1+x)^2}-\frac {2}{-1+x}+\frac {1}{1+x}+\log (-1+x)-2 \log (1+x) \]
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Time = 0.05 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.82
method | result | size |
default | \(\ln \left (-1+x \right )-\frac {3}{2 \left (-1+x \right )^{2}}-\frac {2}{-1+x}+\frac {1}{1+x}-2 \ln \left (1+x \right )\) | \(31\) |
norman | \(\frac {-x^{2}-\frac {7}{2} x +\frac {3}{2}}{\left (-1+x \right )^{2} \left (1+x \right )}-2 \ln \left (1+x \right )+\ln \left (-1+x \right )\) | \(33\) |
risch | \(\frac {-x^{2}-\frac {7}{2} x +\frac {3}{2}}{x^{3}-x^{2}-x +1}-2 \ln \left (1+x \right )+\ln \left (-1+x \right )\) | \(38\) |
parallelrisch | \(\frac {2 \ln \left (-1+x \right ) x^{3}-4 \ln \left (1+x \right ) x^{3}+3-2 \ln \left (-1+x \right ) x^{2}+4 \ln \left (1+x \right ) x^{2}-2 \ln \left (-1+x \right ) x +4 \ln \left (1+x \right ) x -2 x^{2}+2 \ln \left (-1+x \right )-4 \ln \left (1+x \right )-7 x}{2 x^{3}-2 x^{2}-2 x +2}\) | \(90\) |
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (32) = 64\).
Time = 0.23 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.71 \[ \int \frac {5-3 x+6 x^2+5 x^3-x^4}{-1+x+2 x^2-2 x^3-x^4+x^5} \, dx=-\frac {2 \, x^{2} + 4 \, {\left (x^{3} - x^{2} - x + 1\right )} \log \left (x + 1\right ) - 2 \, {\left (x^{3} - x^{2} - x + 1\right )} \log \left (x - 1\right ) + 7 \, x - 3}{2 \, {\left (x^{3} - x^{2} - x + 1\right )}} \]
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Time = 0.06 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.95 \[ \int \frac {5-3 x+6 x^2+5 x^3-x^4}{-1+x+2 x^2-2 x^3-x^4+x^5} \, dx=- \frac {2 x^{2} + 7 x - 3}{2 x^{3} - 2 x^{2} - 2 x + 2} + \log {\left (x - 1 \right )} - 2 \log {\left (x + 1 \right )} \]
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none
Time = 0.22 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {5-3 x+6 x^2+5 x^3-x^4}{-1+x+2 x^2-2 x^3-x^4+x^5} \, dx=-\frac {2 \, x^{2} + 7 \, x - 3}{2 \, {\left (x^{3} - x^{2} - x + 1\right )}} - 2 \, \log \left (x + 1\right ) + \log \left (x - 1\right ) \]
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none
Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92 \[ \int \frac {5-3 x+6 x^2+5 x^3-x^4}{-1+x+2 x^2-2 x^3-x^4+x^5} \, dx=-\frac {2 \, x^{2} + 7 \, x - 3}{2 \, {\left (x + 1\right )} {\left (x - 1\right )}^{2}} - 2 \, \log \left ({\left | x + 1 \right |}\right ) + \log \left ({\left | x - 1 \right |}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.87 \[ \int \frac {5-3 x+6 x^2+5 x^3-x^4}{-1+x+2 x^2-2 x^3-x^4+x^5} \, dx=\ln \left (x-1\right )-2\,\ln \left (x+1\right )+\frac {x^2+\frac {7\,x}{2}-\frac {3}{2}}{-x^3+x^2+x-1} \]
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