Integrand size = 31, antiderivative size = 44 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x \left (2+3 x^2+x^4\right )^2} \, dx=-\frac {11+12 x^2}{2 \left (2+3 x^2+x^4\right )}+\log (x)-\frac {9}{2} \log \left (1+x^2\right )+4 \log \left (2+x^2\right ) \]
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Time = 0.05 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {1677, 1660, 814} \[ \int \frac {4+x^2+3 x^4+5 x^6}{x \left (2+3 x^2+x^4\right )^2} \, dx=-\frac {9}{2} \log \left (x^2+1\right )+4 \log \left (x^2+2\right )-\frac {12 x^2+11}{2 \left (x^4+3 x^2+2\right )}+\log (x) \]
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Rule 814
Rule 1660
Rule 1677
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {4+x+3 x^2+5 x^3}{x \left (2+3 x+x^2\right )^2} \, dx,x,x^2\right ) \\ & = -\frac {11+12 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {1}{2} \text {Subst}\left (\int \frac {-2+7 x}{x \left (2+3 x+x^2\right )} \, dx,x,x^2\right ) \\ & = -\frac {11+12 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {1}{2} \text {Subst}\left (\int \left (-\frac {1}{x}+\frac {9}{1+x}-\frac {8}{2+x}\right ) \, dx,x,x^2\right ) \\ & = -\frac {11+12 x^2}{2 \left (2+3 x^2+x^4\right )}+\log (x)-\frac {9}{2} \log \left (1+x^2\right )+4 \log \left (2+x^2\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x \left (2+3 x^2+x^4\right )^2} \, dx=\frac {-11-12 x^2}{2 \left (2+3 x^2+x^4\right )}+\log (x)-\frac {9}{2} \log \left (1+x^2\right )+4 \log \left (2+x^2\right ) \]
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Time = 0.07 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.86
method | result | size |
default | \(4 \ln \left (x^{2}+2\right )-\frac {13}{2 \left (x^{2}+2\right )}+\ln \left (x \right )-\frac {9 \ln \left (x^{2}+1\right )}{2}+\frac {1}{2 x^{2}+2}\) | \(38\) |
norman | \(\frac {-6 x^{2}-\frac {11}{2}}{x^{4}+3 x^{2}+2}-\frac {9 \ln \left (x^{2}+1\right )}{2}+4 \ln \left (x^{2}+2\right )+\ln \left (x \right )\) | \(40\) |
risch | \(\frac {-6 x^{2}-\frac {11}{2}}{x^{4}+3 x^{2}+2}-\frac {9 \ln \left (x^{2}+1\right )}{2}+4 \ln \left (x^{2}+2\right )+\ln \left (x \right )\) | \(40\) |
parallelrisch | \(\frac {2 \ln \left (x \right ) x^{4}-9 \ln \left (x^{2}+1\right ) x^{4}+8 \ln \left (x^{2}+2\right ) x^{4}-11+6 \ln \left (x \right ) x^{2}-27 \ln \left (x^{2}+1\right ) x^{2}+24 \ln \left (x^{2}+2\right ) x^{2}-12 x^{2}+4 \ln \left (x \right )-18 \ln \left (x^{2}+1\right )+16 \ln \left (x^{2}+2\right )}{2 x^{4}+6 x^{2}+4}\) | \(100\) |
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Time = 0.25 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.61 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x \left (2+3 x^2+x^4\right )^2} \, dx=-\frac {12 \, x^{2} - 8 \, {\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 2\right ) + 9 \, {\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 1\right ) - 2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x\right ) + 11}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} \]
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Time = 0.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.93 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x \left (2+3 x^2+x^4\right )^2} \, dx=\frac {- 12 x^{2} - 11}{2 x^{4} + 6 x^{2} + 4} + \log {\left (x \right )} - \frac {9 \log {\left (x^{2} + 1 \right )}}{2} + 4 \log {\left (x^{2} + 2 \right )} \]
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Time = 0.19 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x \left (2+3 x^2+x^4\right )^2} \, dx=-\frac {12 \, x^{2} + 11}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} + 4 \, \log \left (x^{2} + 2\right ) - \frac {9}{2} \, \log \left (x^{2} + 1\right ) + \frac {1}{2} \, \log \left (x^{2}\right ) \]
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Time = 0.29 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.07 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x \left (2+3 x^2+x^4\right )^2} \, dx=\frac {x^{4} - 21 \, x^{2} - 20}{4 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} + 4 \, \log \left (x^{2} + 2\right ) - \frac {9}{2} \, \log \left (x^{2} + 1\right ) + \frac {1}{2} \, \log \left (x^{2}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.91 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x \left (2+3 x^2+x^4\right )^2} \, dx=4\,\ln \left (x^2+2\right )-\frac {9\,\ln \left (x^2+1\right )}{2}+\ln \left (x\right )-\frac {6\,x^2+\frac {11}{2}}{x^4+3\,x^2+2} \]
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