Integrand size = 17, antiderivative size = 70 \[ \int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx=\frac {x}{2 \sqrt {x^2+\sqrt {1+x^4}}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}} \]
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\[ \int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx=\int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx=\frac {x}{2 \sqrt {x^2+\sqrt {1+x^4}}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}} \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 0.16 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.73
method | result | size |
meijerg | \(-\frac {-\frac {\sqrt {\pi }\, \sqrt {2}\, \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{4}, \frac {7}{4}\right ], \left [2, 2, \frac {5}{2}\right ], -\frac {1}{x^{4}}\right )}{4 x^{4}}+2 \left (-4 \ln \left (2\right )-2-4 \ln \left (x \right )\right ) \sqrt {\pi }\, \sqrt {2}}{16 \sqrt {\pi }}\) | \(51\) |
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none
Time = 0.42 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.29 \[ \int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx=-\frac {1}{2} \, {\left (x^{3} - \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + \frac {1}{8} \, \sqrt {2} \log \left (4 \, x^{4} + 4 \, \sqrt {x^{4} + 1} x^{2} + 2 \, {\left (\sqrt {2} x^{3} + \sqrt {2} \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + 1\right ) \]
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Time = 1.75 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.21 \[ \int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx=\frac {{G_{3, 3}^{2, 2}\left (\begin {matrix} 1, 1 & \frac {3}{2} \\\frac {1}{4}, \frac {3}{4} & 0 \end {matrix} \middle | {x^{4}} \right )}}{16 \sqrt {\pi }} \]
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\[ \int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx=\int { \frac {1}{\sqrt {x^{2} + \sqrt {x^{4} + 1}}} \,d x } \]
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\[ \int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx=\int { \frac {1}{\sqrt {x^{2} + \sqrt {x^{4} + 1}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx=\int \frac {1}{\sqrt {\sqrt {x^4+1}+x^2}} \,d x \]
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