Integrand size = 13, antiderivative size = 13 \[ \int \frac {x^3}{\left (a^4+x^4\right )^3} \, dx=-\frac {1}{8 \left (a^4+x^4\right )^2} \]
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Time = 0.00 (sec) , antiderivative size = 13, 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 = {267} \[ \int \frac {x^3}{\left (a^4+x^4\right )^3} \, dx=-\frac {1}{8 \left (a^4+x^4\right )^2} \]
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Rule 267
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{8 \left (a^4+x^4\right )^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {x^3}{\left (a^4+x^4\right )^3} \, dx=-\frac {1}{8 \left (a^4+x^4\right )^2} \]
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Time = 0.22 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92
method | result | size |
gosper | \(-\frac {1}{8 \left (a^{4}+x^{4}\right )^{2}}\) | \(12\) |
derivativedivides | \(-\frac {1}{8 \left (a^{4}+x^{4}\right )^{2}}\) | \(12\) |
default | \(-\frac {1}{8 \left (a^{4}+x^{4}\right )^{2}}\) | \(12\) |
norman | \(-\frac {1}{8 \left (a^{4}+x^{4}\right )^{2}}\) | \(12\) |
risch | \(-\frac {1}{8 \left (a^{4}+x^{4}\right )^{2}}\) | \(12\) |
parallelrisch | \(-\frac {1}{8 \left (a^{4}+x^{4}\right )^{2}}\) | \(12\) |
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none
Time = 0.23 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.46 \[ \int \frac {x^3}{\left (a^4+x^4\right )^3} \, dx=-\frac {1}{8 \, {\left (a^{8} + 2 \, a^{4} x^{4} + x^{8}\right )}} \]
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Time = 0.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.54 \[ \int \frac {x^3}{\left (a^4+x^4\right )^3} \, dx=- \frac {1}{8 a^{8} + 16 a^{4} x^{4} + 8 x^{8}} \]
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none
Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {x^3}{\left (a^4+x^4\right )^3} \, dx=-\frac {1}{8 \, {\left (a^{4} + x^{4}\right )}^{2}} \]
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none
Time = 0.27 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {x^3}{\left (a^4+x^4\right )^3} \, dx=-\frac {1}{8 \, {\left (a^{4} + x^{4}\right )}^{2}} \]
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Time = 0.23 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {x^3}{\left (a^4+x^4\right )^3} \, dx=-\frac {1}{8\,{\left (a^4+x^4\right )}^2} \]
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