Integrand size = 13, antiderivative size = 23 \[ \int \frac {1}{x^{10} \sqrt {-1+x^6}} \, dx=\frac {\sqrt {-1+x^6} \left (1+2 x^6\right )}{9 x^9} \]
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Time = 0.00 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {277, 270} \[ \int \frac {1}{x^{10} \sqrt {-1+x^6}} \, dx=\frac {\sqrt {x^6-1}}{9 x^9}+\frac {2 \sqrt {x^6-1}}{9 x^3} \]
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Rule 270
Rule 277
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-1+x^6}}{9 x^9}+\frac {2}{3} \int \frac {1}{x^4 \sqrt {-1+x^6}} \, dx \\ & = \frac {\sqrt {-1+x^6}}{9 x^9}+\frac {2 \sqrt {-1+x^6}}{9 x^3} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^{10} \sqrt {-1+x^6}} \, dx=\frac {\sqrt {-1+x^6} \left (1+2 x^6\right )}{9 x^9} \]
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Time = 0.90 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87
method | result | size |
trager | \(\frac {\sqrt {x^{6}-1}\, \left (2 x^{6}+1\right )}{9 x^{9}}\) | \(20\) |
pseudoelliptic | \(\frac {\sqrt {x^{6}-1}\, \left (2 x^{6}+1\right )}{9 x^{9}}\) | \(20\) |
risch | \(\frac {2 x^{12}-x^{6}-1}{9 x^{9} \sqrt {x^{6}-1}}\) | \(25\) |
gosper | \(\frac {\left (x -1\right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right ) \left (2 x^{6}+1\right )}{9 x^{9} \sqrt {x^{6}-1}}\) | \(40\) |
meijerg | \(-\frac {\sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \left (2 x^{6}+1\right ) \sqrt {-x^{6}+1}}{9 \sqrt {\operatorname {signum}\left (x^{6}-1\right )}\, x^{9}}\) | \(40\) |
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none
Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {1}{x^{10} \sqrt {-1+x^6}} \, dx=\frac {2 \, x^{9} + {\left (2 \, x^{6} + 1\right )} \sqrt {x^{6} - 1}}{9 \, x^{9}} \]
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Result contains complex when optimal does not.
Time = 0.55 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.74 \[ \int \frac {1}{x^{10} \sqrt {-1+x^6}} \, dx=\begin {cases} \frac {2 \sqrt {x^{6} - 1}}{9 x^{3}} + \frac {\sqrt {x^{6} - 1}}{9 x^{9}} & \text {for}\: \left |{x^{6}}\right | > 1 \\\frac {2 i \sqrt {1 - x^{6}}}{9 x^{3}} + \frac {i \sqrt {1 - x^{6}}}{9 x^{9}} & \text {otherwise} \end {cases} \]
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none
Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^{10} \sqrt {-1+x^6}} \, dx=\frac {\sqrt {x^{6} - 1}}{3 \, x^{3}} - \frac {{\left (x^{6} - 1\right )}^{\frac {3}{2}}}{9 \, x^{9}} \]
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Exception generated. \[ \int \frac {1}{x^{10} \sqrt {-1+x^6}} \, dx=\text {Exception raised: TypeError} \]
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Time = 5.72 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^{10} \sqrt {-1+x^6}} \, dx=\frac {\sqrt {x^6-1}+2\,x^6\,\sqrt {x^6-1}}{9\,x^9} \]
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