Integrand size = 13, antiderivative size = 38 \[ \int \frac {1}{x^{13} \sqrt {-1+x^6}} \, dx=\frac {\sqrt {-1+x^6} \left (2+3 x^6\right )}{24 x^{12}}+\frac {1}{8} \arctan \left (\sqrt {-1+x^6}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.24, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {272, 44, 65, 209} \[ \int \frac {1}{x^{13} \sqrt {-1+x^6}} \, dx=\frac {1}{8} \arctan \left (\sqrt {x^6-1}\right )+\frac {\sqrt {x^6-1}}{8 x^6}+\frac {\sqrt {x^6-1}}{12 x^{12}} \]
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Rule 44
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^3} \, dx,x,x^6\right ) \\ & = \frac {\sqrt {-1+x^6}}{12 x^{12}}+\frac {1}{8} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^2} \, dx,x,x^6\right ) \\ & = \frac {\sqrt {-1+x^6}}{12 x^{12}}+\frac {\sqrt {-1+x^6}}{8 x^6}+\frac {1}{16} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right ) \\ & = \frac {\sqrt {-1+x^6}}{12 x^{12}}+\frac {\sqrt {-1+x^6}}{8 x^6}+\frac {1}{8} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right ) \\ & = \frac {\sqrt {-1+x^6}}{12 x^{12}}+\frac {\sqrt {-1+x^6}}{8 x^6}+\frac {1}{8} \arctan \left (\sqrt {-1+x^6}\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^{13} \sqrt {-1+x^6}} \, dx=\frac {\sqrt {-1+x^6} \left (2+3 x^6\right )}{24 x^{12}}+\frac {1}{8} \arctan \left (\sqrt {-1+x^6}\right ) \]
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Time = 1.32 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.08
method | result | size |
pseudoelliptic | \(\frac {-3 \arctan \left (\frac {1}{\sqrt {x^{6}-1}}\right ) x^{12}+3 \sqrt {x^{6}-1}\, x^{6}+2 \sqrt {x^{6}-1}}{24 x^{12}}\) | \(41\) |
trager | \(\frac {\sqrt {x^{6}-1}\, \left (3 x^{6}+2\right )}{24 x^{12}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{8}\) | \(50\) |
risch | \(\frac {3 x^{12}-x^{6}-2}{24 x^{12} \sqrt {x^{6}-1}}+\frac {\sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )+\left (-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }\right )}{16 \sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}\) | \(86\) |
meijerg | \(\frac {\sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \left (\frac {\sqrt {\pi }\, \left (-7 x^{12}+8 x^{6}+8\right )}{16 x^{12}}-\frac {\sqrt {\pi }\, \left (12 x^{6}+8\right ) \sqrt {-x^{6}+1}}{16 x^{12}}-\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )}{4}+\frac {3 \left (\frac {7}{6}-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{8}-\frac {\sqrt {\pi }}{2 x^{12}}-\frac {\sqrt {\pi }}{2 x^{6}}\right )}{6 \sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}\) | \(123\) |
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Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^{13} \sqrt {-1+x^6}} \, dx=\frac {3 \, x^{12} \arctan \left (\sqrt {x^{6} - 1}\right ) + {\left (3 \, x^{6} + 2\right )} \sqrt {x^{6} - 1}}{24 \, x^{12}} \]
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Result contains complex when optimal does not.
Time = 2.20 (sec) , antiderivative size = 124, normalized size of antiderivative = 3.26 \[ \int \frac {1}{x^{13} \sqrt {-1+x^6}} \, dx=\begin {cases} \frac {i \operatorname {acosh}{\left (\frac {1}{x^{3}} \right )}}{8} - \frac {i}{8 x^{3} \sqrt {-1 + \frac {1}{x^{6}}}} + \frac {i}{24 x^{9} \sqrt {-1 + \frac {1}{x^{6}}}} + \frac {i}{12 x^{15} \sqrt {-1 + \frac {1}{x^{6}}}} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\- \frac {\operatorname {asin}{\left (\frac {1}{x^{3}} \right )}}{8} + \frac {1}{8 x^{3} \sqrt {1 - \frac {1}{x^{6}}}} - \frac {1}{24 x^{9} \sqrt {1 - \frac {1}{x^{6}}}} - \frac {1}{12 x^{15} \sqrt {1 - \frac {1}{x^{6}}}} & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.26 \[ \int \frac {1}{x^{13} \sqrt {-1+x^6}} \, dx=\frac {3 \, {\left (x^{6} - 1\right )}^{\frac {3}{2}} + 5 \, \sqrt {x^{6} - 1}}{24 \, {\left (2 \, x^{6} + {\left (x^{6} - 1\right )}^{2} - 1\right )}} + \frac {1}{8} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^{13} \sqrt {-1+x^6}} \, dx=\frac {3 \, {\left (x^{6} - 1\right )}^{\frac {3}{2}} + 5 \, \sqrt {x^{6} - 1}}{24 \, x^{12}} + \frac {1}{8} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \]
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Time = 5.15 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^{13} \sqrt {-1+x^6}} \, dx=\frac {\mathrm {atan}\left (\sqrt {x^6-1}\right )}{8}+\frac {\sqrt {x^6-1}}{8\,x^6}+\frac {\sqrt {x^6-1}}{12\,x^{12}} \]
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