Integrand size = 16, antiderivative size = 26 \[ \int \frac {x^{11}}{2+3 x^4+x^8} \, dx=\frac {x^4}{4}+\frac {1}{4} \log \left (1+x^4\right )-\log \left (2+x^4\right ) \]
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Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1371, 717, 646, 31} \[ \int \frac {x^{11}}{2+3 x^4+x^8} \, dx=\frac {x^4}{4}+\frac {1}{4} \log \left (x^4+1\right )-\log \left (x^4+2\right ) \]
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Rule 31
Rule 646
Rule 717
Rule 1371
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {x^2}{2+3 x+x^2} \, dx,x,x^4\right ) \\ & = \frac {x^4}{4}+\frac {1}{4} \text {Subst}\left (\int \frac {-2-3 x}{2+3 x+x^2} \, dx,x,x^4\right ) \\ & = \frac {x^4}{4}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,x^4\right )-\text {Subst}\left (\int \frac {1}{2+x} \, dx,x,x^4\right ) \\ & = \frac {x^4}{4}+\frac {1}{4} \log \left (1+x^4\right )-\log \left (2+x^4\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {x^{11}}{2+3 x^4+x^8} \, dx=\frac {x^4}{4}+\frac {1}{4} \log \left (1+x^4\right )-\log \left (2+x^4\right ) \]
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Time = 0.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88
method | result | size |
default | \(\frac {x^{4}}{4}+\frac {\ln \left (x^{4}+1\right )}{4}-\ln \left (x^{4}+2\right )\) | \(23\) |
norman | \(\frac {x^{4}}{4}+\frac {\ln \left (x^{4}+1\right )}{4}-\ln \left (x^{4}+2\right )\) | \(23\) |
risch | \(\frac {x^{4}}{4}+\frac {\ln \left (x^{4}+1\right )}{4}-\ln \left (x^{4}+2\right )\) | \(23\) |
parallelrisch | \(\frac {x^{4}}{4}+\frac {\ln \left (x^{4}+1\right )}{4}-\ln \left (x^{4}+2\right )\) | \(23\) |
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none
Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {x^{11}}{2+3 x^4+x^8} \, dx=\frac {1}{4} \, x^{4} - \log \left (x^{4} + 2\right ) + \frac {1}{4} \, \log \left (x^{4} + 1\right ) \]
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Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73 \[ \int \frac {x^{11}}{2+3 x^4+x^8} \, dx=\frac {x^{4}}{4} + \frac {\log {\left (x^{4} + 1 \right )}}{4} - \log {\left (x^{4} + 2 \right )} \]
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none
Time = 0.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {x^{11}}{2+3 x^4+x^8} \, dx=\frac {1}{4} \, x^{4} - \log \left (x^{4} + 2\right ) + \frac {1}{4} \, \log \left (x^{4} + 1\right ) \]
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none
Time = 0.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {x^{11}}{2+3 x^4+x^8} \, dx=\frac {1}{4} \, x^{4} - \log \left (x^{4} + 2\right ) + \frac {1}{4} \, \log \left (x^{4} + 1\right ) \]
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Time = 8.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {x^{11}}{2+3 x^4+x^8} \, dx=\frac {\ln \left (x^4+1\right )}{4}-\ln \left (x^4+2\right )+\frac {x^4}{4} \]
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