Integrand size = 15, antiderivative size = 27 \[ \int \frac {x^5}{\sqrt {1-x^4}} \, dx=-\frac {1}{4} x^2 \sqrt {1-x^4}+\frac {\arcsin \left (x^2\right )}{4} \]
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Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {281, 327, 222} \[ \int \frac {x^5}{\sqrt {1-x^4}} \, dx=\frac {\arcsin \left (x^2\right )}{4}-\frac {1}{4} x^2 \sqrt {1-x^4} \]
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Rule 222
Rule 281
Rule 327
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^2}} \, dx,x,x^2\right ) \\ & = -\frac {1}{4} x^2 \sqrt {1-x^4}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,x^2\right ) \\ & = -\frac {1}{4} x^2 \sqrt {1-x^4}+\frac {1}{4} \sin ^{-1}\left (x^2\right ) \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{\sqrt {1-x^4}} \, dx=-\frac {1}{4} x^2 \sqrt {1-x^4}+\frac {\arcsin \left (x^2\right )}{4} \]
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Time = 4.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81
method | result | size |
default | \(\frac {\arcsin \left (x^{2}\right )}{4}-\frac {x^{2} \sqrt {-x^{4}+1}}{4}\) | \(22\) |
elliptic | \(\frac {\arcsin \left (x^{2}\right )}{4}-\frac {x^{2} \sqrt {-x^{4}+1}}{4}\) | \(22\) |
pseudoelliptic | \(\frac {\arcsin \left (x^{2}\right )}{4}-\frac {x^{2} \sqrt {-x^{4}+1}}{4}\) | \(22\) |
risch | \(\frac {x^{2} \left (x^{4}-1\right )}{4 \sqrt {-x^{4}+1}}+\frac {\arcsin \left (x^{2}\right )}{4}\) | \(27\) |
meijerg | \(\frac {i \left (i \sqrt {\pi }\, x^{2} \sqrt {-x^{4}+1}-i \sqrt {\pi }\, \arcsin \left (x^{2}\right )\right )}{4 \sqrt {\pi }}\) | \(36\) |
trager | \(-\frac {x^{2} \sqrt {-x^{4}+1}}{4}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{4}+1}+x^{2}\right )}{4}\) | \(45\) |
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none
Time = 0.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {x^5}{\sqrt {1-x^4}} \, dx=-\frac {1}{4} \, \sqrt {-x^{4} + 1} x^{2} - \frac {1}{2} \, \arctan \left (\frac {\sqrt {-x^{4} + 1} - 1}{x^{2}}\right ) \]
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Result contains complex when optimal does not.
Time = 1.00 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.26 \[ \int \frac {x^5}{\sqrt {1-x^4}} \, dx=\begin {cases} - \frac {i x^{2} \sqrt {x^{4} - 1}}{4} - \frac {i \operatorname {acosh}{\left (x^{2} \right )}}{4} & \text {for}\: \left |{x^{4}}\right | > 1 \\\frac {x^{6}}{4 \sqrt {1 - x^{4}}} - \frac {x^{2}}{4 \sqrt {1 - x^{4}}} + \frac {\operatorname {asin}{\left (x^{2} \right )}}{4} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (21) = 42\).
Time = 0.29 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.63 \[ \int \frac {x^5}{\sqrt {1-x^4}} \, dx=\frac {\sqrt {-x^{4} + 1}}{4 \, x^{2} {\left (\frac {x^{4} - 1}{x^{4}} - 1\right )}} - \frac {1}{4} \, \arctan \left (\frac {\sqrt {-x^{4} + 1}}{x^{2}}\right ) \]
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none
Time = 0.32 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {x^5}{\sqrt {1-x^4}} \, dx=-\frac {1}{4} \, \sqrt {-x^{4} + 1} x^{2} + \frac {1}{4} \, \arcsin \left (x^{2}\right ) \]
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Timed out. \[ \int \frac {x^5}{\sqrt {1-x^4}} \, dx=\int \frac {x^5}{\sqrt {1-x^4}} \,d x \]
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