Integrand size = 21, antiderivative size = 70 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^2} \, dx=-\frac {\sqrt {x^2+\sqrt {1+x^4}}}{x}+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \]
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\[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^2} \, dx=\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^2} \, dx \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^2} \, dx=-\frac {\sqrt {x^2+\sqrt {1+x^4}}}{x}+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \]
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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 | \(\frac {-\frac {\sqrt {\pi }\, \sqrt {2}\, \operatorname {hypergeom}\left (\left [\frac {3}{4}, 1, 1, \frac {5}{4}\right ], \left [\frac {3}{2}, 2, 2\right ], -\frac {1}{x^{4}}\right )}{2 x^{4}}-4 \left (-4 \ln \left (2\right )+4-4 \ln \left (x \right )\right ) \sqrt {\pi }\, \sqrt {2}}{16 \sqrt {\pi }}\) | \(51\) |
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none
Time = 0.52 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.16 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^2} \, dx=\frac {\sqrt {2} x \log \left (4 \, x^{4} + 4 \, \sqrt {x^{4} + 1} x^{2} + 2 \, {\left (\sqrt {2} x^{3} + \sqrt {2} \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + 1\right ) - 4 \, \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{4 \, x} \]
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Result contains complex when optimal does not.
Time = 1.83 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^2} \, dx=- \frac {\log {\left (\frac {1}{x^{4}} \right )} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right )}{4 \pi } - \frac {\Gamma \left (\frac {3}{4}\right ) \Gamma \left (\frac {5}{4}\right ) {{}_{4}F_{3}\left (\begin {matrix} \frac {3}{4}, 1, 1, \frac {5}{4} \\ \frac {3}{2}, 2, 2 \end {matrix}\middle | {\frac {e^{i \pi }}{x^{4}}} \right )}}{8 \pi x^{4}} \]
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\[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^2} \, dx=\int { \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{x^{2}} \,d x } \]
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\[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^2} \, dx=\int { \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^2} \, dx=\int \frac {\sqrt {\sqrt {x^4+1}+x^2}}{x^2} \,d x \]
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