Integrand size = 13, antiderivative size = 20 \[ \int \frac {1}{\sqrt {-x^2+x^8}} \, dx=-\frac {1}{3} \arctan \left (\frac {x}{\sqrt {-x^2+x^8}}\right ) \]
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Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2033, 209} \[ \int \frac {1}{\sqrt {-x^2+x^8}} \, dx=-\frac {1}{3} \arctan \left (\frac {x}{\sqrt {x^8-x^2}}\right ) \]
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Rule 209
Rule 2033
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{3} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt {-x^2+x^8}}\right )\right ) \\ & = -\frac {1}{3} \arctan \left (\frac {x}{\sqrt {-x^2+x^8}}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.85 \[ \int \frac {1}{\sqrt {-x^2+x^8}} \, dx=\frac {x \sqrt {-1+x^6} \arctan \left (\sqrt {-1+x^6}\right )}{3 \sqrt {x^2 \left (-1+x^6\right )}} \]
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Time = 1.58 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.35
method | result | size |
pseudoelliptic | \(-\frac {\arcsin \left (\frac {1}{x^{3}}\right )}{3}\) | \(7\) |
trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +\sqrt {x^{8}-x^{2}}}{x^{4}}\right )}{3}\) | \(35\) |
meijerg | \(\frac {\sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )+\left (-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }\right )}{6 \sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}\) | \(61\) |
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none
Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\sqrt {-x^2+x^8}} \, dx=\frac {1}{3} \, \arctan \left (\frac {\sqrt {x^{8} - x^{2}}}{x}\right ) \]
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\[ \int \frac {1}{\sqrt {-x^2+x^8}} \, dx=\int \frac {1}{\sqrt {x^{8} - x^{2}}}\, dx \]
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\[ \int \frac {1}{\sqrt {-x^2+x^8}} \, dx=\int { \frac {1}{\sqrt {x^{8} - x^{2}}} \,d x } \]
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Exception generated. \[ \int \frac {1}{\sqrt {-x^2+x^8}} \, dx=\text {Exception raised: NotImplementedError} \]
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Timed out. \[ \int \frac {1}{\sqrt {-x^2+x^8}} \, dx=\int \frac {1}{\sqrt {x^8-x^2}} \,d x \]
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