Integrand size = 15, antiderivative size = 27 \[ \int \frac {1}{x \sqrt {b+a x^4}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {b+a x^4}}{\sqrt {b}}\right )}{2 \sqrt {b}} \]
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Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {272, 65, 214} \[ \int \frac {1}{x \sqrt {b+a x^4}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a x^4+b}}{\sqrt {b}}\right )}{2 \sqrt {b}} \]
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Rule 65
Rule 214
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {1}{x \sqrt {b+a x}} \, dx,x,x^4\right ) \\ & = \frac {\text {Subst}\left (\int \frac {1}{-\frac {b}{a}+\frac {x^2}{a}} \, dx,x,\sqrt {b+a x^4}\right )}{2 a} \\ & = -\frac {\text {arctanh}\left (\frac {\sqrt {b+a x^4}}{\sqrt {b}}\right )}{2 \sqrt {b}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {b+a x^4}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {b+a x^4}}{\sqrt {b}}\right )}{2 \sqrt {b}} \]
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Time = 1.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74
method | result | size |
pseudoelliptic | \(-\frac {\operatorname {arctanh}\left (\frac {\sqrt {a \,x^{4}+b}}{\sqrt {b}}\right )}{2 \sqrt {b}}\) | \(20\) |
default | \(-\frac {\ln \left (\frac {2 b +2 \sqrt {a \,x^{4}+b}\, \sqrt {b}}{x^{2}}\right )}{2 \sqrt {b}}\) | \(29\) |
elliptic | \(-\frac {\ln \left (\frac {2 b +2 \sqrt {a \,x^{4}+b}\, \sqrt {b}}{x^{2}}\right )}{2 \sqrt {b}}\) | \(29\) |
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none
Time = 0.25 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.33 \[ \int \frac {1}{x \sqrt {b+a x^4}} \, dx=\left [\frac {\log \left (\frac {a x^{4} - 2 \, \sqrt {a x^{4} + b} \sqrt {b} + 2 \, b}{x^{4}}\right )}{4 \, \sqrt {b}}, \frac {\sqrt {-b} \arctan \left (\frac {\sqrt {a x^{4} + b} \sqrt {-b}}{b}\right )}{2 \, b}\right ] \]
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Time = 0.52 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {1}{x \sqrt {b+a x^4}} \, dx=- \frac {\operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} x^{2}} \right )}}{2 \sqrt {b}} \]
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none
Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {1}{x \sqrt {b+a x^4}} \, dx=\frac {\log \left (\frac {\sqrt {a x^{4} + b} - \sqrt {b}}{\sqrt {a x^{4} + b} + \sqrt {b}}\right )}{4 \, \sqrt {b}} \]
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none
Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x \sqrt {b+a x^4}} \, dx=\frac {\arctan \left (\frac {\sqrt {a x^{4} + b}}{\sqrt {-b}}\right )}{2 \, \sqrt {-b}} \]
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Time = 5.42 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {1}{x \sqrt {b+a x^4}} \, dx=-\frac {\mathrm {atanh}\left (\frac {\sqrt {a\,x^4+b}}{\sqrt {b}}\right )}{2\,\sqrt {b}} \]
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