Integrand size = 44, antiderivative size = 33 \[ \int \frac {-3+25 x+23 x^2+32 x^3+15 x^4+7 x^5+x^6}{\left (1+x^2\right )^2 \left (2+x+x^2\right )^2} \, dx=-\frac {3}{1+x^2}+\frac {1}{2+x+x^2}+\log \left (1+x^2\right )-\log \left (2+x+x^2\right ) \]
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Time = 0.11 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {6874, 267, 266, 643, 642} \[ \int \frac {-3+25 x+23 x^2+32 x^3+15 x^4+7 x^5+x^6}{\left (1+x^2\right )^2 \left (2+x+x^2\right )^2} \, dx=-\frac {3}{x^2+1}+\frac {1}{x^2+x+2}+\log \left (x^2+1\right )-\log \left (x^2+x+2\right ) \]
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Rule 266
Rule 267
Rule 642
Rule 643
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {6 x}{\left (1+x^2\right )^2}+\frac {2 x}{1+x^2}+\frac {-1-2 x}{\left (2+x+x^2\right )^2}+\frac {-1-2 x}{2+x+x^2}\right ) \, dx \\ & = 2 \int \frac {x}{1+x^2} \, dx+6 \int \frac {x}{\left (1+x^2\right )^2} \, dx+\int \frac {-1-2 x}{\left (2+x+x^2\right )^2} \, dx+\int \frac {-1-2 x}{2+x+x^2} \, dx \\ & = -\frac {3}{1+x^2}+\frac {1}{2+x+x^2}+\log \left (1+x^2\right )-\log \left (2+x+x^2\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {-3+25 x+23 x^2+32 x^3+15 x^4+7 x^5+x^6}{\left (1+x^2\right )^2 \left (2+x+x^2\right )^2} \, dx=-\frac {3}{1+x^2}+\frac {1}{2+x+x^2}+\log \left (1+x^2\right )-\log \left (2+x+x^2\right ) \]
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Time = 1.33 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03
method | result | size |
default | \(-\frac {3}{x^{2}+1}+\frac {1}{x^{2}+x +2}+\ln \left (x^{2}+1\right )-\ln \left (x^{2}+x +2\right )\) | \(34\) |
norman | \(\frac {-2 x^{2}-3 x -5}{\left (x^{2}+x +2\right ) \left (x^{2}+1\right )}-\ln \left (x^{2}+x +2\right )+\ln \left (x^{2}+1\right )\) | \(43\) |
risch | \(\frac {-2 x^{2}-3 x -5}{\left (x^{2}+x +2\right ) \left (x^{2}+1\right )}-\ln \left (x^{2}+x +2\right )+\ln \left (x^{2}+1\right )\) | \(43\) |
parallelrisch | \(\frac {\ln \left (x^{2}+1\right ) x^{4}-\ln \left (x^{2}+x +2\right ) x^{4}-5+\ln \left (x^{2}+1\right ) x^{3}-\ln \left (x^{2}+x +2\right ) x^{3}+3 \ln \left (x^{2}+1\right ) x^{2}-3 \ln \left (x^{2}+x +2\right ) x^{2}+\ln \left (x^{2}+1\right ) x -\ln \left (x^{2}+x +2\right ) x -2 x^{2}+2 \ln \left (x^{2}+1\right )-2 \ln \left (x^{2}+x +2\right )-3 x}{\left (x^{2}+1\right ) \left (x^{2}+x +2\right )}\) | \(129\) |
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Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (33) = 66\).
Time = 0.28 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.18 \[ \int \frac {-3+25 x+23 x^2+32 x^3+15 x^4+7 x^5+x^6}{\left (1+x^2\right )^2 \left (2+x+x^2\right )^2} \, dx=-\frac {2 \, x^{2} + {\left (x^{4} + x^{3} + 3 \, x^{2} + x + 2\right )} \log \left (x^{2} + x + 2\right ) - {\left (x^{4} + x^{3} + 3 \, x^{2} + x + 2\right )} \log \left (x^{2} + 1\right ) + 3 \, x + 5}{x^{4} + x^{3} + 3 \, x^{2} + x + 2} \]
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Time = 0.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.24 \[ \int \frac {-3+25 x+23 x^2+32 x^3+15 x^4+7 x^5+x^6}{\left (1+x^2\right )^2 \left (2+x+x^2\right )^2} \, dx=\frac {- 2 x^{2} - 3 x - 5}{x^{4} + x^{3} + 3 x^{2} + x + 2} + \log {\left (x^{2} + 1 \right )} - \log {\left (x^{2} + x + 2 \right )} \]
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none
Time = 0.19 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33 \[ \int \frac {-3+25 x+23 x^2+32 x^3+15 x^4+7 x^5+x^6}{\left (1+x^2\right )^2 \left (2+x+x^2\right )^2} \, dx=-\frac {2 \, x^{2} + 3 \, x + 5}{x^{4} + x^{3} + 3 \, x^{2} + x + 2} - \log \left (x^{2} + x + 2\right ) + \log \left (x^{2} + 1\right ) \]
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Time = 0.31 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33 \[ \int \frac {-3+25 x+23 x^2+32 x^3+15 x^4+7 x^5+x^6}{\left (1+x^2\right )^2 \left (2+x+x^2\right )^2} \, dx=-\frac {2 \, x^{2} + 3 \, x + 5}{x^{4} + x^{3} + 3 \, x^{2} + x + 2} - \log \left (x^{2} + x + 2\right ) + \log \left (x^{2} + 1\right ) \]
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Time = 9.39 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.70 \[ \int \frac {-3+25 x+23 x^2+32 x^3+15 x^4+7 x^5+x^6}{\left (1+x^2\right )^2 \left (2+x+x^2\right )^2} \, dx=-\frac {2\,x^2+3\,x+5}{x^4+x^3+3\,x^2+x+2}+\mathrm {atan}\left (\frac {\frac {x\,224{}\mathrm {i}}{11}+\frac {224}{11}{}\mathrm {i}}{44\,x^2+16\,x+60}-\frac {3}{11}{}\mathrm {i}\right )\,2{}\mathrm {i} \]
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