Integrand size = 13, antiderivative size = 12 \[ \int \frac {x^4}{a^5+x^5} \, dx=\frac {1}{5} \log \left (a^5+x^5\right ) \]
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Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {266} \[ \int \frac {x^4}{a^5+x^5} \, dx=\frac {1}{5} \log \left (a^5+x^5\right ) \]
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Rule 266
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \log \left (a^5+x^5\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {x^4}{a^5+x^5} \, dx=\frac {1}{5} \log \left (a^5+x^5\right ) \]
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Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92
method | result | size |
derivativedivides | \(\frac {\ln \left (a^{5}+x^{5}\right )}{5}\) | \(11\) |
default | \(\frac {\ln \left (a^{5}+x^{5}\right )}{5}\) | \(11\) |
risch | \(\frac {\ln \left (a^{5}+x^{5}\right )}{5}\) | \(11\) |
norman | \(\frac {\ln \left (a +x \right )}{5}+\frac {\ln \left (a^{4}-a^{3} x +a^{2} x^{2}-a \,x^{3}+x^{4}\right )}{5}\) | \(37\) |
parallelrisch | \(\frac {\ln \left (a +x \right )}{5}+\frac {\ln \left (a^{4}-a^{3} x +a^{2} x^{2}-a \,x^{3}+x^{4}\right )}{5}\) | \(37\) |
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none
Time = 0.23 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {x^4}{a^5+x^5} \, dx=\frac {1}{5} \, \log \left (a^{5} + x^{5}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {x^4}{a^5+x^5} \, dx=\frac {\log {\left (a^{5} + x^{5} \right )}}{5} \]
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none
Time = 0.20 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {x^4}{a^5+x^5} \, dx=\frac {1}{5} \, \log \left (a^{5} + x^{5}\right ) \]
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none
Time = 0.30 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92 \[ \int \frac {x^4}{a^5+x^5} \, dx=\frac {1}{5} \, \log \left ({\left | a^{5} + x^{5} \right |}\right ) \]
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Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {x^4}{a^5+x^5} \, dx=\frac {\ln \left (a^5+x^5\right )}{5} \]
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