Integrand size = 11, antiderivative size = 18 \[ \int \frac {x^9}{1+x^5} \, dx=\frac {x^5}{5}-\frac {1}{5} \log \left (1+x^5\right ) \]
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Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {272, 45} \[ \int \frac {x^9}{1+x^5} \, dx=\frac {x^5}{5}-\frac {1}{5} \log \left (x^5+1\right ) \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \text {Subst}\left (\int \frac {x}{1+x} \, dx,x,x^5\right ) \\ & = \frac {1}{5} \text {Subst}\left (\int \left (1+\frac {1}{-1-x}\right ) \, dx,x,x^5\right ) \\ & = \frac {x^5}{5}-\frac {1}{5} \log \left (1+x^5\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {x^9}{1+x^5} \, dx=\frac {x^5}{5}-\frac {1}{5} \log \left (1+x^5\right ) \]
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Time = 4.35 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83
method | result | size |
default | \(\frac {x^{5}}{5}-\frac {\ln \left (x^{5}+1\right )}{5}\) | \(15\) |
meijerg | \(\frac {x^{5}}{5}-\frac {\ln \left (x^{5}+1\right )}{5}\) | \(15\) |
risch | \(\frac {x^{5}}{5}-\frac {\ln \left (x^{5}+1\right )}{5}\) | \(15\) |
norman | \(\frac {x^{5}}{5}-\frac {\ln \left (1+x \right )}{5}-\frac {\ln \left (x^{4}-x^{3}+x^{2}-x +1\right )}{5}\) | \(32\) |
parallelrisch | \(\frac {x^{5}}{5}-\frac {\ln \left (1+x \right )}{5}-\frac {\ln \left (x^{4}-x^{3}+x^{2}-x +1\right )}{5}\) | \(32\) |
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none
Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {x^9}{1+x^5} \, dx=\frac {1}{5} \, x^{5} - \frac {1}{5} \, \log \left (x^{5} + 1\right ) \]
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Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67 \[ \int \frac {x^9}{1+x^5} \, dx=\frac {x^{5}}{5} - \frac {\log {\left (x^{5} + 1 \right )}}{5} \]
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none
Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {x^9}{1+x^5} \, dx=\frac {1}{5} \, x^{5} - \frac {1}{5} \, \log \left (x^{5} + 1\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {x^9}{1+x^5} \, dx=\frac {1}{5} \, x^{5} - \frac {1}{5} \, \log \left ({\left | x^{5} + 1 \right |}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {x^9}{1+x^5} \, dx=\frac {x^5}{5}-\frac {\ln \left (x^5+1\right )}{5} \]
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