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 = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {281, 223, 212} \[ \int \frac {x^2}{\sqrt {-1+x^6}} \, dx=\frac {1}{3} \text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right ) \]
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Rule 212
Rule 223
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 {1}{3} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right ) \\ & = \frac {1}{3} \text {arctanh}\left (\frac {x^3}{\sqrt {-1+x^6}}\right ) \\ \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 = 0.81 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83
method | result | size |
pseudoelliptic | \(\frac {\ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{3}\) | \(15\) |
trager | \(-\frac {\ln \left (x^{3}-\sqrt {x^{6}-1}\right )}{3}\) | \(17\) |
meijerg | \(\frac {\sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \arcsin \left (x^{3}\right )}{3 \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}\) | \(25\) |
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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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Result contains complex when optimal does not.
Time = 0.48 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {x^2}{\sqrt {-1+x^6}} \, dx=\begin {cases} \frac {\operatorname {acosh}{\left (x^{3} \right )}}{3} & \text {for}\: \left |{x^{6}}\right | > 1 \\- \frac {i \operatorname {asin}{\left (x^{3} \right )}}{3} & \text {otherwise} \end {cases} \]
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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. 30 vs. \(2 (14) = 28\).
Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.67 \[ \int \frac {x^2}{\sqrt {-1+x^6}} \, dx=\frac {1}{6} \, \sqrt {x^{6} - 1} x^{3} + \frac {1}{6} \, \log \left ({\left | -x^{3} + \sqrt {x^{6} - 1} \right |}\right ) \]
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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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