Integrand size = 18, antiderivative size = 13 \[ \int \frac {-1+x^4}{x^2 \sqrt {1+x^4}} \, dx=\frac {\sqrt {1+x^4}}{x} \]
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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} \[ \int \frac {-1+x^4}{x^2 \sqrt {1+x^4}} \, dx=\frac {\sqrt {x^4+1}}{x} \]
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Rule 460
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+x^4}}{x} \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^4}{x^2 \sqrt {1+x^4}} \, dx=\frac {\sqrt {1+x^4}}{x} \]
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Time = 0.84 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92
method | result | size |
gosper | \(\frac {\sqrt {x^{4}+1}}{x}\) | \(12\) |
default | \(\frac {\sqrt {x^{4}+1}}{x}\) | \(12\) |
trager | \(\frac {\sqrt {x^{4}+1}}{x}\) | \(12\) |
risch | \(\frac {\sqrt {x^{4}+1}}{x}\) | \(12\) |
elliptic | \(\frac {\sqrt {x^{4}+1}}{x}\) | \(12\) |
pseudoelliptic | \(\frac {\sqrt {x^{4}+1}}{x}\) | \(12\) |
meijerg | \(\frac {x^{3} \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], -x^{4}\right )}{3}+\frac {\operatorname {hypergeom}\left (\left [-\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{4}\right ], -x^{4}\right )}{x}\) | \(33\) |
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none
Time = 0.24 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {-1+x^4}{x^2 \sqrt {1+x^4}} \, dx=\frac {\sqrt {x^{4} + 1}}{x} \]
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Result contains complex when optimal does not.
Time = 0.77 (sec) , antiderivative size = 61, normalized size of antiderivative = 4.69 \[ \int \frac {-1+x^4}{x^2 \sqrt {1+x^4}} \, dx=\frac {x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} - \frac {\Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} \]
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none
Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {-1+x^4}{x^2 \sqrt {1+x^4}} \, dx=\frac {\sqrt {x^{4} + 1}}{x} \]
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\[ \int \frac {-1+x^4}{x^2 \sqrt {1+x^4}} \, dx=\int { \frac {x^{4} - 1}{\sqrt {x^{4} + 1} x^{2}} \,d x } \]
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Time = 0.06 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {-1+x^4}{x^2 \sqrt {1+x^4}} \, dx=\frac {\sqrt {x^4+1}}{x} \]
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