Integrand size = 13, antiderivative size = 28 \[ \int \frac {\sqrt {-1+x^6}}{x} \, dx=\frac {1}{3} \sqrt {-1+x^6}-\frac {1}{3} \arctan \left (\sqrt {-1+x^6}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 28, 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 = {272, 52, 65, 209} \[ \int \frac {\sqrt {-1+x^6}}{x} \, dx=\frac {\sqrt {x^6-1}}{3}-\frac {1}{3} \arctan \left (\sqrt {x^6-1}\right ) \]
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Rule 52
Rule 65
Rule 209
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} \text {Subst}\left (\int \frac {\sqrt {-1+x}}{x} \, dx,x,x^6\right ) \\ & = \frac {1}{3} \sqrt {-1+x^6}-\frac {1}{6} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right ) \\ & = \frac {1}{3} \sqrt {-1+x^6}-\frac {1}{3} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right ) \\ & = \frac {1}{3} \sqrt {-1+x^6}-\frac {1}{3} \arctan \left (\sqrt {-1+x^6}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-1+x^6}}{x} \, dx=\frac {1}{3} \sqrt {-1+x^6}-\frac {1}{3} \arctan \left (\sqrt {-1+x^6}\right ) \]
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Time = 0.90 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75
method | result | size |
pseudoelliptic | \(\frac {\sqrt {x^{6}-1}}{3}+\frac {\arctan \left (\frac {1}{\sqrt {x^{6}-1}}\right )}{3}\) | \(21\) |
trager | \(\frac {\sqrt {x^{6}-1}}{3}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{3}\) | \(40\) |
meijerg | \(-\frac {\sqrt {\operatorname {signum}\left (x^{6}-1\right )}\, \left (4 \sqrt {\pi }-4 \sqrt {\pi }\, \sqrt {-x^{6}+1}+4 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )-2 \left (2-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }\right )}{12 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}}\) | \(82\) |
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Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {-1+x^6}}{x} \, dx=\frac {1}{3} \, \sqrt {x^{6} - 1} - \frac {1}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \]
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Result contains complex when optimal does not.
Time = 0.71 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.14 \[ \int \frac {\sqrt {-1+x^6}}{x} \, dx=\begin {cases} - \frac {i x^{3}}{3 \sqrt {-1 + \frac {1}{x^{6}}}} - \frac {i \operatorname {acosh}{\left (\frac {1}{x^{3}} \right )}}{3} + \frac {i}{3 x^{3} \sqrt {-1 + \frac {1}{x^{6}}}} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\\frac {x^{3}}{3 \sqrt {1 - \frac {1}{x^{6}}}} + \frac {\operatorname {asin}{\left (\frac {1}{x^{3}} \right )}}{3} - \frac {1}{3 x^{3} \sqrt {1 - \frac {1}{x^{6}}}} & \text {otherwise} \end {cases} \]
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Time = 0.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {-1+x^6}}{x} \, dx=\frac {1}{3} \, \sqrt {x^{6} - 1} - \frac {1}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {-1+x^6}}{x} \, dx=\frac {1}{3} \, \sqrt {x^{6} - 1} - \frac {1}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \]
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Time = 5.14 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {-1+x^6}}{x} \, dx=\frac {\sqrt {x^6-1}}{3}-\frac {\mathrm {atan}\left (\sqrt {x^6-1}\right )}{3} \]
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