Integrand size = 13, antiderivative size = 16 \[ \int \frac {1}{x^4 \sqrt {1+x^6}} \, dx=-\frac {\sqrt {1+x^6}}{3 x^3} \]
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Time = 0.00 (sec) , antiderivative size = 16, 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 = {270} \[ \int \frac {1}{x^4 \sqrt {1+x^6}} \, dx=-\frac {\sqrt {x^6+1}}{3 x^3} \]
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Rule 270
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1+x^6}}{3 x^3} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^4 \sqrt {1+x^6}} \, dx=-\frac {\sqrt {1+x^6}}{3 x^3} \]
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Time = 0.78 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
trager | \(-\frac {\sqrt {x^{6}+1}}{3 x^{3}}\) | \(13\) |
meijerg | \(-\frac {\sqrt {x^{6}+1}}{3 x^{3}}\) | \(13\) |
risch | \(-\frac {\sqrt {x^{6}+1}}{3 x^{3}}\) | \(13\) |
pseudoelliptic | \(-\frac {\sqrt {x^{6}+1}}{3 x^{3}}\) | \(13\) |
gosper | \(-\frac {\left (x^{2}+1\right ) \left (x^{4}-x^{2}+1\right )}{3 x^{3} \sqrt {x^{6}+1}}\) | \(28\) |
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none
Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^4 \sqrt {1+x^6}} \, dx=-\frac {x^{3} + \sqrt {x^{6} + 1}}{3 \, x^{3}} \]
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Time = 0.34 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^4 \sqrt {1+x^6}} \, dx=- \frac {\sqrt {1 + \frac {1}{x^{6}}}}{3} \]
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none
Time = 0.19 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^4 \sqrt {1+x^6}} \, dx=-\frac {\sqrt {x^{6} + 1}}{3 \, x^{3}} \]
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none
Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {1}{x^4 \sqrt {1+x^6}} \, dx=-\frac {\sqrt {\frac {1}{x^{6}} + 1}}{3 \, \mathrm {sgn}\left (x\right )} + \frac {1}{3} \, \mathrm {sgn}\left (x\right ) \]
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Time = 4.86 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^4 \sqrt {1+x^6}} \, dx=-\frac {\sqrt {x^6+1}}{3\,x^3} \]
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