Integrand size = 17, antiderivative size = 29 \[ \int \frac {1}{x \sqrt {-b+a x^4}} \, dx=\frac {\arctan \left (\frac {\sqrt {-b+a x^4}}{\sqrt {b}}\right )}{2 \sqrt {b}} \]
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Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {272, 65, 211} \[ \int \frac {1}{x \sqrt {-b+a x^4}} \, dx=\frac {\arctan \left (\frac {\sqrt {a x^4-b}}{\sqrt {b}}\right )}{2 \sqrt {b}} \]
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Rule 65
Rule 211
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 {\arctan \left (\frac {\sqrt {-b+a x^4}}{\sqrt {b}}\right )}{2 \sqrt {b}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {-b+a x^4}} \, dx=\frac {\arctan \left (\frac {\sqrt {-b+a x^4}}{\sqrt {b}}\right )}{2 \sqrt {b}} \]
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Time = 2.65 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76
method | result | size |
pseudoelliptic | \(\frac {\arctan \left (\frac {\sqrt {a \,x^{4}-b}}{\sqrt {b}}\right )}{2 \sqrt {b}}\) | \(22\) |
default | \(-\frac {\ln \left (\frac {-2 b +2 \sqrt {-b}\, \sqrt {a \,x^{4}-b}}{x^{2}}\right )}{2 \sqrt {-b}}\) | \(35\) |
elliptic | \(-\frac {\ln \left (\frac {-2 b +2 \sqrt {-b}\, \sqrt {a \,x^{4}-b}}{x^{2}}\right )}{2 \sqrt {-b}}\) | \(35\) |
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none
Time = 0.25 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.21 \[ \int \frac {1}{x \sqrt {-b+a x^4}} \, dx=\left [-\frac {\sqrt {-b} \log \left (\frac {a x^{4} - 2 \, \sqrt {a x^{4} - b} \sqrt {-b} - 2 \, b}{x^{4}}\right )}{4 \, b}, \frac {\arctan \left (\frac {\sqrt {a x^{4} - b}}{\sqrt {b}}\right )}{2 \, \sqrt {b}}\right ] \]
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Result contains complex when optimal does not.
Time = 0.52 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83 \[ \int \frac {1}{x \sqrt {-b+a x^4}} \, dx=\begin {cases} \frac {i \operatorname {acosh}{\left (\frac {\sqrt {b}}{\sqrt {a} x^{2}} \right )}}{2 \sqrt {b}} & \text {for}\: \left |{\frac {b}{a x^{4}}}\right | > 1 \\- \frac {\operatorname {asin}{\left (\frac {\sqrt {b}}{\sqrt {a} x^{2}} \right )}}{2 \sqrt {b}} & \text {otherwise} \end {cases} \]
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none
Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {1}{x \sqrt {-b+a x^4}} \, dx=\frac {\arctan \left (\frac {\sqrt {a x^{4} - b}}{\sqrt {b}}\right )}{2 \, \sqrt {b}} \]
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none
Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {1}{x \sqrt {-b+a x^4}} \, dx=\frac {\arctan \left (\frac {\sqrt {a x^{4} - b}}{\sqrt {b}}\right )}{2 \, \sqrt {b}} \]
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Time = 5.45 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {1}{x \sqrt {-b+a x^4}} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {a\,x^4-b}}{\sqrt {b}}\right )}{2\,\sqrt {b}} \]
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