Integrand size = 13, antiderivative size = 14 \[ \int \frac {1}{x \sqrt {1+x^4}} \, dx=-\frac {1}{2} \text {arctanh}\left (\sqrt {1+x^4}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {272, 65, 213} \[ \int \frac {1}{x \sqrt {1+x^4}} \, dx=-\frac {1}{2} \text {arctanh}\left (\sqrt {x^4+1}\right ) \]
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Rule 65
Rule 213
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^4\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^4}\right ) \\ & = -\frac {1}{2} \text {arctanh}\left (\sqrt {1+x^4}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {1+x^4}} \, dx=-\frac {1}{2} \text {arctanh}\left (\sqrt {1+x^4}\right ) \]
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Time = 0.23 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.79
method | result | size |
default | \(-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{4}+1}}\right )}{2}\) | \(11\) |
elliptic | \(-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{4}+1}}\right )}{2}\) | \(11\) |
pseudoelliptic | \(-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{4}+1}}\right )}{2}\) | \(11\) |
trager | \(\frac {\ln \left (\frac {-1+\sqrt {x^{4}+1}}{x^{2}}\right )}{2}\) | \(17\) |
meijerg | \(\frac {\left (-2 \ln \left (2\right )+4 \ln \left (x \right )\right ) \sqrt {\pi }-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{4}+1}}{2}\right )}{4 \sqrt {\pi }}\) | \(37\) |
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Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (10) = 20\).
Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.79 \[ \int \frac {1}{x \sqrt {1+x^4}} \, dx=-\frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} + 1\right ) + \frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} - 1\right ) \]
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Time = 0.48 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.57 \[ \int \frac {1}{x \sqrt {1+x^4}} \, dx=- \frac {\operatorname {asinh}{\left (\frac {1}{x^{2}} \right )}}{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (10) = 20\).
Time = 0.21 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.79 \[ \int \frac {1}{x \sqrt {1+x^4}} \, dx=-\frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} + 1\right ) + \frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (10) = 20\).
Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.79 \[ \int \frac {1}{x \sqrt {1+x^4}} \, dx=-\frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} + 1\right ) + \frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} - 1\right ) \]
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Time = 0.16 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \frac {1}{x \sqrt {1+x^4}} \, dx=-\frac {\mathrm {atanh}\left (\sqrt {x^4+1}\right )}{2} \]
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