Integrand size = 13, antiderivative size = 18 \[ \int \frac {x^2}{\sqrt {1+x^6}} \, dx=\frac {1}{3} \log \left (x^3+\sqrt {1+x^6}\right ) \]
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Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.44, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {281, 221} \[ \int \frac {x^2}{\sqrt {1+x^6}} \, dx=\frac {\text {arcsinh}\left (x^3\right )}{3} \]
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Rule 221
Rule 281
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^3\right ) \\ & = \frac {\text {arcsinh}\left (x^3\right )}{3} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\sqrt {1+x^6}} \, dx=\frac {1}{3} \log \left (x^3+\sqrt {1+x^6}\right ) \]
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Time = 0.80 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.39
method | result | size |
meijerg | \(\frac {\operatorname {arcsinh}\left (x^{3}\right )}{3}\) | \(7\) |
pseudoelliptic | \(\frac {\operatorname {arcsinh}\left (x^{3}\right )}{3}\) | \(7\) |
trager | \(-\frac {\ln \left (x^{3}-\sqrt {x^{6}+1}\right )}{3}\) | \(17\) |
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none
Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {x^2}{\sqrt {1+x^6}} \, dx=-\frac {1}{3} \, \log \left (-x^{3} + \sqrt {x^{6} + 1}\right ) \]
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Time = 0.45 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.28 \[ \int \frac {x^2}{\sqrt {1+x^6}} \, dx=\frac {\operatorname {asinh}{\left (x^{3} \right )}}{3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (14) = 28\).
Time = 0.18 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.83 \[ \int \frac {x^2}{\sqrt {1+x^6}} \, dx=\frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} + 1}}{x^{3}} + 1\right ) - \frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} + 1}}{x^{3}} - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (14) = 28\).
Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.61 \[ \int \frac {x^2}{\sqrt {1+x^6}} \, dx=\frac {1}{6} \, \sqrt {x^{6} + 1} x^{3} - \frac {1}{6} \, \log \left (-x^{3} + \sqrt {x^{6} + 1}\right ) \]
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Timed out. \[ \int \frac {x^2}{\sqrt {1+x^6}} \, dx=\int \frac {x^2}{\sqrt {x^6+1}} \,d x \]
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