Integrand size = 16, antiderivative size = 30 \[ \int \frac {x^9}{1+2 x^4+x^8} \, dx=\frac {3 x^2}{4}-\frac {x^6}{4 \left (1+x^4\right )}-\frac {3 \arctan \left (x^2\right )}{4} \]
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Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {28, 281, 294, 327, 209} \[ \int \frac {x^9}{1+2 x^4+x^8} \, dx=-\frac {3 \arctan \left (x^2\right )}{4}+\frac {3 x^2}{4}-\frac {x^6}{4 \left (x^4+1\right )} \]
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Rule 28
Rule 209
Rule 281
Rule 294
Rule 327
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^9}{\left (1+x^4\right )^2} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {x^4}{\left (1+x^2\right )^2} \, dx,x,x^2\right ) \\ & = -\frac {x^6}{4 \left (1+x^4\right )}+\frac {3}{4} \text {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,x^2\right ) \\ & = \frac {3 x^2}{4}-\frac {x^6}{4 \left (1+x^4\right )}-\frac {3}{4} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,x^2\right ) \\ & = \frac {3 x^2}{4}-\frac {x^6}{4 \left (1+x^4\right )}-\frac {3}{4} \tan ^{-1}\left (x^2\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {x^9}{1+2 x^4+x^8} \, dx=\frac {1}{4} \left (x^2 \left (2+\frac {1}{1+x^4}\right )-3 \arctan \left (x^2\right )\right ) \]
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Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83
method | result | size |
default | \(\frac {x^{2}}{2}+\frac {x^{2}}{4 x^{4}+4}-\frac {3 \arctan \left (x^{2}\right )}{4}\) | \(25\) |
risch | \(\frac {x^{2}}{2}+\frac {x^{2}}{4 x^{4}+4}-\frac {3 \arctan \left (x^{2}\right )}{4}\) | \(25\) |
parallelrisch | \(\frac {3 i \ln \left (x^{2}-i\right ) x^{4}-3 i \ln \left (x^{2}+i\right ) x^{4}+4 x^{6}+3 i \ln \left (x^{2}-i\right )-3 i \ln \left (x^{2}+i\right )+6 x^{2}}{8 x^{4}+8}\) | \(67\) |
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Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {x^9}{1+2 x^4+x^8} \, dx=\frac {2 \, x^{6} + 3 \, x^{2} - 3 \, {\left (x^{4} + 1\right )} \arctan \left (x^{2}\right )}{4 \, {\left (x^{4} + 1\right )}} \]
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Time = 0.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73 \[ \int \frac {x^9}{1+2 x^4+x^8} \, dx=\frac {x^{2}}{2} + \frac {x^{2}}{4 x^{4} + 4} - \frac {3 \operatorname {atan}{\left (x^{2} \right )}}{4} \]
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Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {x^9}{1+2 x^4+x^8} \, dx=\frac {1}{2} \, x^{2} + \frac {x^{2}}{4 \, {\left (x^{4} + 1\right )}} - \frac {3}{4} \, \arctan \left (x^{2}\right ) \]
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Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {x^9}{1+2 x^4+x^8} \, dx=\frac {1}{2} \, x^{2} + \frac {x^{2}}{4 \, {\left (x^{4} + 1\right )}} - \frac {3}{4} \, \arctan \left (x^{2}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \frac {x^9}{1+2 x^4+x^8} \, dx=\frac {x^2}{4\,\left (x^4+1\right )}-\frac {3\,\mathrm {atan}\left (x^2\right )}{4}+\frac {x^2}{2} \]
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