Integrand size = 18, antiderivative size = 14 \[ \int \frac {3+x^4}{x^4 \sqrt {1+x^4}} \, dx=-\frac {\sqrt {1+x^4}}{x^3} \]
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Time = 0.00 (sec) , antiderivative size = 14, 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} \[ \int \frac {3+x^4}{x^4 \sqrt {1+x^4}} \, dx=-\frac {\sqrt {x^4+1}}{x^3} \]
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Rule 460
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1+x^4}}{x^3} \\ \end{align*}
Time = 0.55 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {3+x^4}{x^4 \sqrt {1+x^4}} \, dx=-\frac {\sqrt {1+x^4}}{x^3} \]
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Time = 0.92 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
gosper | \(-\frac {\sqrt {x^{4}+1}}{x^{3}}\) | \(13\) |
default | \(-\frac {\sqrt {x^{4}+1}}{x^{3}}\) | \(13\) |
trager | \(-\frac {\sqrt {x^{4}+1}}{x^{3}}\) | \(13\) |
risch | \(-\frac {\sqrt {x^{4}+1}}{x^{3}}\) | \(13\) |
elliptic | \(-\frac {\sqrt {x^{4}+1}}{x^{3}}\) | \(13\) |
meijerg | \(x \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {5}{4}\right ], -x^{4}\right )-\frac {\operatorname {hypergeom}\left (\left [-\frac {3}{4}, \frac {1}{2}\right ], \left [\frac {1}{4}\right ], -x^{4}\right )}{x^{3}}\) | \(31\) |
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none
Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {3+x^4}{x^4 \sqrt {1+x^4}} \, dx=-\frac {\sqrt {x^{4} + 1}}{x^{3}} \]
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Result contains complex when optimal does not.
Time = 0.76 (sec) , antiderivative size = 63, normalized size of antiderivative = 4.50 \[ \int \frac {3+x^4}{x^4 \sqrt {1+x^4}} \, dx=\frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {3 \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 x^{3} \Gamma \left (\frac {1}{4}\right )} \]
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none
Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {3+x^4}{x^4 \sqrt {1+x^4}} \, dx=-\frac {\sqrt {x^{4} + 1}}{x^{3}} \]
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none
Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {3+x^4}{x^4 \sqrt {1+x^4}} \, dx=-\frac {\sqrt {\frac {1}{x^{4}} + 1}}{x} \]
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Time = 4.75 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {3+x^4}{x^4 \sqrt {1+x^4}} \, dx=-\frac {\sqrt {x^4+1}}{x^3} \]
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