Integrand size = 11, antiderivative size = 12 \[ \int \frac {x^4}{16+x^{10}} \, dx=\frac {1}{20} \arctan \left (\frac {x^5}{4}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {281, 209} \[ \int \frac {x^4}{16+x^{10}} \, dx=\frac {1}{20} \arctan \left (\frac {x^5}{4}\right ) \]
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Rule 209
Rule 281
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \text {Subst}\left (\int \frac {1}{16+x^2} \, dx,x,x^5\right ) \\ & = \frac {1}{20} \arctan \left (\frac {x^5}{4}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {x^4}{16+x^{10}} \, dx=\frac {1}{20} \arctan \left (\frac {x^5}{4}\right ) \]
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Time = 0.21 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.75
method | result | size |
default | \(\frac {\arctan \left (\frac {x^{5}}{4}\right )}{20}\) | \(9\) |
meijerg | \(\frac {\arctan \left (\frac {x^{5}}{4}\right )}{20}\) | \(9\) |
risch | \(\frac {\arctan \left (\frac {x^{5}}{4}\right )}{20}\) | \(9\) |
parallelrisch | \(\frac {i \ln \left (x^{5}+4 i\right )}{40}-\frac {i \ln \left (x^{5}-4 i\right )}{40}\) | \(22\) |
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none
Time = 0.25 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {x^4}{16+x^{10}} \, dx=\frac {1}{20} \, \arctan \left (\frac {1}{4} \, x^{5}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.58 \[ \int \frac {x^4}{16+x^{10}} \, dx=\frac {\operatorname {atan}{\left (\frac {x^{5}}{4} \right )}}{20} \]
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none
Time = 0.28 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {x^4}{16+x^{10}} \, dx=\frac {1}{20} \, \arctan \left (\frac {1}{4} \, x^{5}\right ) \]
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none
Time = 0.29 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {x^4}{16+x^{10}} \, dx=\frac {1}{20} \, \arctan \left (\frac {1}{4} \, x^{5}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {x^4}{16+x^{10}} \, dx=\frac {\mathrm {atan}\left (\frac {x^5}{4}\right )}{20} \]
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