Integrand size = 13, antiderivative size = 14 \[ \int \frac {1}{x \sqrt {-1+x^6}} \, dx=\frac {1}{3} \arctan \left (\sqrt {-1+x^6}\right ) \]
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Time = 0.00 (sec) , antiderivative size = 14, 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 = {272, 65, 209} \[ \int \frac {1}{x \sqrt {-1+x^6}} \, dx=\frac {1}{3} \arctan \left (\sqrt {x^6-1}\right ) \]
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Rule 65
Rule 209
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right ) \\ & = \frac {1}{3} \arctan \left (\sqrt {-1+x^6}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {-1+x^6}} \, dx=\frac {1}{3} \arctan \left (\sqrt {-1+x^6}\right ) \]
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Time = 1.37 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.79
method | result | size |
pseudoelliptic | \(-\frac {\arctan \left (\frac {1}{\sqrt {x^{6}-1}}\right )}{3}\) | \(11\) |
trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{3}\) | \(28\) |
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 = 0.23 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \frac {1}{x \sqrt {-1+x^6}} \, dx=\frac {1}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \]
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Result contains complex when optimal does not.
Time = 0.46 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.71 \[ \int \frac {1}{x \sqrt {-1+x^6}} \, dx=\begin {cases} \frac {i \operatorname {acosh}{\left (\frac {1}{x^{3}} \right )}}{3} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\- \frac {\operatorname {asin}{\left (\frac {1}{x^{3}} \right )}}{3} & \text {otherwise} \end {cases} \]
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Time = 0.25 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \frac {1}{x \sqrt {-1+x^6}} \, dx=\frac {1}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \frac {1}{x \sqrt {-1+x^6}} \, dx=\frac {1}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \]
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Time = 4.95 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \frac {1}{x \sqrt {-1+x^6}} \, dx=\frac {\mathrm {atan}\left (\sqrt {x^6-1}\right )}{3} \]
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