Integrand size = 13, antiderivative size = 28 \[ \int \frac {\sqrt {1+x^4}}{x} \, dx=\frac {\sqrt {1+x^4}}{2}-\frac {1}{2} \text {arctanh}\left (\sqrt {1+x^4}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {272, 52, 65, 213} \[ \int \frac {\sqrt {1+x^4}}{x} \, dx=\frac {\sqrt {x^4+1}}{2}-\frac {1}{2} \text {arctanh}\left (\sqrt {x^4+1}\right ) \]
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Rule 52
Rule 65
Rule 213
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {\sqrt {1+x}}{x} \, dx,x,x^4\right ) \\ & = \frac {\sqrt {1+x^4}}{2}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^4\right ) \\ & = \frac {\sqrt {1+x^4}}{2}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^4}\right ) \\ & = \frac {\sqrt {1+x^4}}{2}-\frac {1}{2} \text {arctanh}\left (\sqrt {1+x^4}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+x^4}}{x} \, dx=\frac {\sqrt {1+x^4}}{2}-\frac {1}{2} \text {arctanh}\left (\sqrt {1+x^4}\right ) \]
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Time = 0.82 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75
method | result | size |
default | \(\frac {\sqrt {x^{4}+1}}{2}-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{4}+1}}\right )}{2}\) | \(21\) |
elliptic | \(\frac {\sqrt {x^{4}+1}}{2}-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{4}+1}}\right )}{2}\) | \(21\) |
pseudoelliptic | \(\frac {\sqrt {x^{4}+1}}{2}-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{4}+1}}\right )}{2}\) | \(21\) |
trager | \(\frac {\sqrt {x^{4}+1}}{2}-\frac {\ln \left (\frac {1+\sqrt {x^{4}+1}}{x^{2}}\right )}{2}\) | \(27\) |
meijerg | \(-\frac {4 \sqrt {\pi }-4 \sqrt {\pi }\, \sqrt {x^{4}+1}+4 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{4}+1}}{2}\right )-2 \left (2-2 \ln \left (2\right )+4 \ln \left (x \right )\right ) \sqrt {\pi }}{8 \sqrt {\pi }}\) | \(56\) |
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Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt {1+x^4}}{x} \, dx=\frac {1}{2} \, \sqrt {x^{4} + 1} - \frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} + 1\right ) + \frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} - 1\right ) \]
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Time = 0.66 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {\sqrt {1+x^4}}{x} \, dx=\frac {x^{2}}{2 \sqrt {1 + \frac {1}{x^{4}}}} - \frac {\operatorname {asinh}{\left (\frac {1}{x^{2}} \right )}}{2} + \frac {1}{2 x^{2} \sqrt {1 + \frac {1}{x^{4}}}} \]
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Time = 0.22 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt {1+x^4}}{x} \, dx=\frac {1}{2} \, \sqrt {x^{4} + 1} - \frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} + 1\right ) + \frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} - 1\right ) \]
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Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt {1+x^4}}{x} \, dx=\frac {1}{2} \, \sqrt {x^{4} + 1} - \frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} + 1\right ) + \frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} - 1\right ) \]
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Time = 5.39 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {1+x^4}}{x} \, dx=\frac {\sqrt {x^4+1}}{2}-\frac {\mathrm {atanh}\left (\sqrt {x^4+1}\right )}{2} \]
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