Integrand size = 18, antiderivative size = 13 \[ \int \frac {-2+x^6}{x^3 \sqrt {1+x^6}} \, dx=\frac {\sqrt {1+x^6}}{x^2} \]
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Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {460} \[ \int \frac {-2+x^6}{x^3 \sqrt {1+x^6}} \, dx=\frac {\sqrt {x^6+1}}{x^2} \]
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Rule 460
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+x^6}}{x^2} \\ \end{align*}
Time = 1.56 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {-2+x^6}{x^3 \sqrt {1+x^6}} \, dx=\frac {\sqrt {1+x^6}}{x^2} \]
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Time = 1.66 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92
method | result | size |
trager | \(\frac {\sqrt {x^{6}+1}}{x^{2}}\) | \(12\) |
risch | \(\frac {\sqrt {x^{6}+1}}{x^{2}}\) | \(12\) |
pseudoelliptic | \(\frac {\sqrt {x^{6}+1}}{x^{2}}\) | \(12\) |
gosper | \(\frac {\left (x^{4}-x^{2}+1\right ) \left (x^{2}+1\right )}{x^{2} \sqrt {x^{6}+1}}\) | \(27\) |
meijerg | \(\frac {x^{4} \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -x^{6}\right )}{4}+\frac {\operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {1}{2}\right ], \left [\frac {2}{3}\right ], -x^{6}\right )}{x^{2}}\) | \(33\) |
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none
Time = 0.24 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {-2+x^6}{x^3 \sqrt {1+x^6}} \, dx=\frac {\sqrt {x^{6} + 1}}{x^{2}} \]
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Result contains complex when optimal does not.
Time = 0.99 (sec) , antiderivative size = 63, normalized size of antiderivative = 4.85 \[ \int \frac {-2+x^6}{x^3 \sqrt {1+x^6}} \, dx=\frac {x^{4} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{6 \Gamma \left (\frac {5}{3}\right )} - \frac {\Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {1}{2} \\ \frac {2}{3} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{3 x^{2} \Gamma \left (\frac {2}{3}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 23 vs. \(2 (11) = 22\).
Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.77 \[ \int \frac {-2+x^6}{x^3 \sqrt {1+x^6}} \, dx=\frac {\sqrt {x^{4} - x^{2} + 1} \sqrt {x^{2} + 1}}{x^{2}} \]
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\[ \int \frac {-2+x^6}{x^3 \sqrt {1+x^6}} \, dx=\int { \frac {x^{6} - 2}{\sqrt {x^{6} + 1} x^{3}} \,d x } \]
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Time = 0.07 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {-2+x^6}{x^3 \sqrt {1+x^6}} \, dx=\frac {\sqrt {x^6+1}}{x^2} \]
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