Integrand size = 15, antiderivative size = 15 \[ \int \frac {x^3}{\left (1-x^4\right )^{3/2}} \, dx=\frac {1}{2 \sqrt {1-x^4}} \]
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Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {267} \[ \int \frac {x^3}{\left (1-x^4\right )^{3/2}} \, dx=\frac {1}{2 \sqrt {1-x^4}} \]
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Rule 267
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2 \sqrt {1-x^4}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {x^3}{\left (1-x^4\right )^{3/2}} \, dx=\frac {1}{2 \sqrt {1-x^4}} \]
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Time = 4.32 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(\frac {1}{2 \sqrt {-x^{4}+1}}\) | \(12\) |
default | \(\frac {1}{2 \sqrt {-x^{4}+1}}\) | \(12\) |
risch | \(\frac {1}{2 \sqrt {-x^{4}+1}}\) | \(12\) |
elliptic | \(\frac {1}{2 \sqrt {-x^{4}+1}}\) | \(12\) |
pseudoelliptic | \(\frac {1}{2 \sqrt {-x^{4}+1}}\) | \(12\) |
trager | \(-\frac {\sqrt {-x^{4}+1}}{2 \left (x^{4}-1\right )}\) | \(19\) |
gosper | \(-\frac {\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}{2 \left (-x^{4}+1\right )^{\frac {3}{2}}}\) | \(23\) |
meijerg | \(-\frac {\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {-x^{4}+1}}}{2 \sqrt {\pi }}\) | \(24\) |
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none
Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.20 \[ \int \frac {x^3}{\left (1-x^4\right )^{3/2}} \, dx=-\frac {\sqrt {-x^{4} + 1}}{2 \, {\left (x^{4} - 1\right )}} \]
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Time = 0.16 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \frac {x^3}{\left (1-x^4\right )^{3/2}} \, dx=\frac {1}{2 \sqrt {1 - x^{4}}} \]
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none
Time = 0.18 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \frac {x^3}{\left (1-x^4\right )^{3/2}} \, dx=\frac {1}{2 \, \sqrt {-x^{4} + 1}} \]
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none
Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \frac {x^3}{\left (1-x^4\right )^{3/2}} \, dx=\frac {1}{2 \, \sqrt {-x^{4} + 1}} \]
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Time = 5.82 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \frac {x^3}{\left (1-x^4\right )^{3/2}} \, dx=\frac {1}{2\,\sqrt {1-x^4}} \]
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