Integrand size = 13, antiderivative size = 35 \[ \int x^2 \sqrt {-1+x^6} \, dx=\frac {1}{6} x^3 \sqrt {-1+x^6}-\frac {1}{6} \log \left (x^3+\sqrt {-1+x^6}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {281, 201, 223, 212} \[ \int x^2 \sqrt {-1+x^6} \, dx=\frac {1}{6} x^3 \sqrt {x^6-1}-\frac {1}{6} \text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right ) \]
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Rule 201
Rule 212
Rule 223
Rule 281
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \sqrt {-1+x^2} \, dx,x,x^3\right ) \\ & = \frac {1}{6} x^3 \sqrt {-1+x^6}-\frac {1}{6} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^3\right ) \\ & = \frac {1}{6} x^3 \sqrt {-1+x^6}-\frac {1}{6} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right ) \\ & = \frac {1}{6} x^3 \sqrt {-1+x^6}-\frac {1}{6} \text {arctanh}\left (\frac {x^3}{\sqrt {-1+x^6}}\right ) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int x^2 \sqrt {-1+x^6} \, dx=\frac {1}{6} x^3 \sqrt {-1+x^6}-\frac {1}{6} \log \left (x^3+\sqrt {-1+x^6}\right ) \]
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Time = 1.20 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.80
method | result | size |
trager | \(\frac {x^{3} \sqrt {x^{6}-1}}{6}-\frac {\ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{6}\) | \(28\) |
pseudoelliptic | \(\frac {x^{3} \sqrt {x^{6}-1}}{6}-\frac {\ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{6}\) | \(28\) |
risch | \(\frac {x^{3} \sqrt {x^{6}-1}}{6}-\frac {\sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \arcsin \left (x^{3}\right )}{6 \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}\) | \(38\) |
meijerg | \(\frac {i \sqrt {\operatorname {signum}\left (x^{6}-1\right )}\, \left (-2 i \sqrt {\pi }\, x^{3} \sqrt {-x^{6}+1}-2 i \sqrt {\pi }\, \arcsin \left (x^{3}\right )\right )}{12 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}}\) | \(54\) |
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none
Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int x^2 \sqrt {-1+x^6} \, dx=\frac {1}{6} \, \sqrt {x^{6} - 1} x^{3} + \frac {1}{6} \, \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) \]
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Result contains complex when optimal does not.
Time = 0.81 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.74 \[ \int x^2 \sqrt {-1+x^6} \, dx=\begin {cases} \frac {x^{3} \sqrt {x^{6} - 1}}{6} - \frac {\operatorname {acosh}{\left (x^{3} \right )}}{6} & \text {for}\: \left |{x^{6}}\right | > 1 \\- \frac {i x^{9}}{6 \sqrt {1 - x^{6}}} + \frac {i x^{3}}{6 \sqrt {1 - x^{6}}} + \frac {i \operatorname {asin}{\left (x^{3} \right )}}{6} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (27) = 54\).
Time = 0.20 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.66 \[ \int x^2 \sqrt {-1+x^6} \, dx=-\frac {\sqrt {x^{6} - 1}}{6 \, x^{3} {\left (\frac {x^{6} - 1}{x^{6}} - 1\right )}} - \frac {1}{12} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} + 1\right ) + \frac {1}{12} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} - 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.86 \[ \int x^2 \sqrt {-1+x^6} \, dx=\frac {1}{6} \, \sqrt {x^{6} - 1} x^{3} + \frac {1}{6} \, \log \left ({\left | -x^{3} + \sqrt {x^{6} - 1} \right |}\right ) \]
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Timed out. \[ \int x^2 \sqrt {-1+x^6} \, dx=\int x^2\,\sqrt {x^6-1} \,d x \]
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