Integrand size = 13, antiderivative size = 77 \[ \int \frac {1}{x \sqrt [4]{-1+x^6}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{-1+x^6}}{-1+\sqrt {-1+x^6}}\right )}{3 \sqrt {2}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-1+x^6}}{1+\sqrt {-1+x^6}}\right )}{3 \sqrt {2}} \]
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Time = 0.10 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.68, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {272, 65, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {1}{x \sqrt [4]{-1+x^6}} \, dx=-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{x^6-1}\right )}{3 \sqrt {2}}+\frac {\arctan \left (\sqrt {2} \sqrt [4]{x^6-1}+1\right )}{3 \sqrt {2}}+\frac {\log \left (\sqrt {x^6-1}-\sqrt {2} \sqrt [4]{x^6-1}+1\right )}{6 \sqrt {2}}-\frac {\log \left (\sqrt {x^6-1}+\sqrt {2} \sqrt [4]{x^6-1}+1\right )}{6 \sqrt {2}} \]
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Rule 65
Rule 210
Rule 272
Rule 303
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x} x} \, dx,x,x^6\right ) \\ & = \frac {2}{3} \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt [4]{-1+x^6}\right ) \\ & = -\left (\frac {1}{3} \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt [4]{-1+x^6}\right )\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt [4]{-1+x^6}\right ) \\ & = \frac {1}{6} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+x^6}\right )+\frac {1}{6} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+x^6}\right )+\frac {\text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt [4]{-1+x^6}\right )}{6 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt [4]{-1+x^6}\right )}{6 \sqrt {2}} \\ & = \frac {\log \left (1-\sqrt {2} \sqrt [4]{-1+x^6}+\sqrt {-1+x^6}\right )}{6 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} \sqrt [4]{-1+x^6}+\sqrt {-1+x^6}\right )}{6 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt [4]{-1+x^6}\right )}{3 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt [4]{-1+x^6}\right )}{3 \sqrt {2}} \\ & = -\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{-1+x^6}\right )}{3 \sqrt {2}}+\frac {\arctan \left (1+\sqrt {2} \sqrt [4]{-1+x^6}\right )}{3 \sqrt {2}}+\frac {\log \left (1-\sqrt {2} \sqrt [4]{-1+x^6}+\sqrt {-1+x^6}\right )}{6 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} \sqrt [4]{-1+x^6}+\sqrt {-1+x^6}\right )}{6 \sqrt {2}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x \sqrt [4]{-1+x^6}} \, dx=\frac {\arctan \left (\frac {-1+\sqrt {-1+x^6}}{\sqrt {2} \sqrt [4]{-1+x^6}}\right )-\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-1+x^6}}{1+\sqrt {-1+x^6}}\right )}{3 \sqrt {2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 8.92 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.03
method | result | size |
meijerg | \(\frac {\sqrt {2}\, \Gamma \left (\frac {3}{4}\right ) {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{4}} \left (\frac {\pi \sqrt {2}\, x^{6} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{4}\right ], \left [2, 2\right ], x^{6}\right )}{4 \Gamma \left (\frac {3}{4}\right )}+\frac {\left (-3 \ln \left (2\right )-\frac {\pi }{2}+6 \ln \left (x \right )+i \pi \right ) \pi \sqrt {2}}{\Gamma \left (\frac {3}{4}\right )}\right )}{12 \pi \operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{4}}}\) | \(79\) |
pseudoelliptic | \(\frac {\sqrt {2}\, \left (\ln \left (\frac {\sqrt {x^{6}-1}-\left (x^{6}-1\right )^{\frac {1}{4}} \sqrt {2}+1}{\sqrt {x^{6}-1}+\left (x^{6}-1\right )^{\frac {1}{4}} \sqrt {2}+1}\right )+2 \arctan \left (\left (x^{6}-1\right )^{\frac {1}{4}} \sqrt {2}+1\right )+2 \arctan \left (\left (x^{6}-1\right )^{\frac {1}{4}} \sqrt {2}-1\right )\right )}{12}\) | \(84\) |
trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{6}+2 \sqrt {x^{6}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}+2 \left (x^{6}-1\right )^{\frac {3}{4}}-2 \left (x^{6}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{6}}\right )}{6}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{6}+2 \left (x^{6}-1\right )^{\frac {3}{4}}-2 \sqrt {x^{6}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )+2 \left (x^{6}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{x^{6}}\right )}{6}\) | \(159\) |
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Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x \sqrt [4]{-1+x^6}} \, dx=\left (\frac {1}{12} i - \frac {1}{12}\right ) \, \sqrt {2} \log \left (\left (i + 1\right ) \, \sqrt {2} + 2 \, {\left (x^{6} - 1\right )}^{\frac {1}{4}}\right ) - \left (\frac {1}{12} i + \frac {1}{12}\right ) \, \sqrt {2} \log \left (-\left (i - 1\right ) \, \sqrt {2} + 2 \, {\left (x^{6} - 1\right )}^{\frac {1}{4}}\right ) + \left (\frac {1}{12} i + \frac {1}{12}\right ) \, \sqrt {2} \log \left (\left (i - 1\right ) \, \sqrt {2} + 2 \, {\left (x^{6} - 1\right )}^{\frac {1}{4}}\right ) - \left (\frac {1}{12} i - \frac {1}{12}\right ) \, \sqrt {2} \log \left (-\left (i + 1\right ) \, \sqrt {2} + 2 \, {\left (x^{6} - 1\right )}^{\frac {1}{4}}\right ) \]
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Result contains complex when optimal does not.
Time = 0.48 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.44 \[ \int \frac {1}{x \sqrt [4]{-1+x^6}} \, dx=- \frac {\Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{6}}} \right )}}{6 x^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.32 \[ \int \frac {1}{x \sqrt [4]{-1+x^6}} \, dx=\frac {1}{6} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (x^{6} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {1}{6} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (x^{6} - 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{12} \, \sqrt {2} \log \left (\sqrt {2} {\left (x^{6} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{6} - 1} + 1\right ) + \frac {1}{12} \, \sqrt {2} \log \left (-\sqrt {2} {\left (x^{6} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{6} - 1} + 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.32 \[ \int \frac {1}{x \sqrt [4]{-1+x^6}} \, dx=\frac {1}{6} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (x^{6} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {1}{6} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (x^{6} - 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{12} \, \sqrt {2} \log \left (\sqrt {2} {\left (x^{6} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{6} - 1} + 1\right ) + \frac {1}{12} \, \sqrt {2} \log \left (-\sqrt {2} {\left (x^{6} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{6} - 1} + 1\right ) \]
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Time = 6.13 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.58 \[ \int \frac {1}{x \sqrt [4]{-1+x^6}} \, dx=\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (x^6-1\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{6}-\frac {1}{6}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (x^6-1\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{6}+\frac {1}{6}{}\mathrm {i}\right ) \]
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