Integrand size = 17, antiderivative size = 29 \[ \int \frac {1}{x \sqrt {-b+a x^3}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {-b+a x^3}}{\sqrt {b}}\right )}{3 \sqrt {b}} \]
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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} \[ \int \frac {1}{x \sqrt {-b+a x^3}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {a x^3-b}}{\sqrt {b}}\right )}{3 \sqrt {b}} \]
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Rule 65
Rule 211
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {1}{x \sqrt {-b+a x}} \, dx,x,x^3\right ) \\ & = \frac {2 \text {Subst}\left (\int \frac {1}{\frac {b}{a}+\frac {x^2}{a}} \, dx,x,\sqrt {-b+a x^3}\right )}{3 a} \\ & = \frac {2 \arctan \left (\frac {\sqrt {-b+a x^3}}{\sqrt {b}}\right )}{3 \sqrt {b}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {-b+a x^3}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {-b+a x^3}}{\sqrt {b}}\right )}{3 \sqrt {b}} \]
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Time = 2.17 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76
method | result | size |
pseudoelliptic | \(\frac {2 \arctan \left (\frac {\sqrt {a \,x^{3}-b}}{\sqrt {b}}\right )}{3 \sqrt {b}}\) | \(22\) |
default | \(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {a \,x^{3}-b}}{\sqrt {-b}}\right )}{3 \sqrt {-b}}\) | \(26\) |
elliptic | \(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {a \,x^{3}-b}}{\sqrt {-b}}\right )}{3 \sqrt {-b}}\) | \(26\) |
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none
Time = 0.25 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.21 \[ \int \frac {1}{x \sqrt {-b+a x^3}} \, dx=\left [-\frac {\sqrt {-b} \log \left (\frac {a x^{3} - 2 \, \sqrt {a x^{3} - b} \sqrt {-b} - 2 \, b}{x^{3}}\right )}{3 \, b}, \frac {2 \, \arctan \left (\frac {\sqrt {a x^{3} - b}}{\sqrt {b}}\right )}{3 \, \sqrt {b}}\right ] \]
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Result contains complex when optimal does not.
Time = 0.55 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.07 \[ \int \frac {1}{x \sqrt {-b+a x^3}} \, dx=\begin {cases} \frac {2 i \operatorname {acosh}{\left (\frac {\sqrt {b}}{\sqrt {a} x^{\frac {3}{2}}} \right )}}{3 \sqrt {b}} & \text {for}\: \left |{\frac {b}{a x^{3}}}\right | > 1 \\- \frac {2 \operatorname {asin}{\left (\frac {\sqrt {b}}{\sqrt {a} x^{\frac {3}{2}}} \right )}}{3 \sqrt {b}} & \text {otherwise} \end {cases} \]
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none
Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {1}{x \sqrt {-b+a x^3}} \, dx=\frac {2 \, \arctan \left (\frac {\sqrt {a x^{3} - b}}{\sqrt {b}}\right )}{3 \, \sqrt {b}} \]
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none
Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {1}{x \sqrt {-b+a x^3}} \, dx=\frac {2 \, \arctan \left (\frac {\sqrt {a x^{3} - b}}{\sqrt {b}}\right )}{3 \, \sqrt {b}} \]
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Time = 5.75 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \frac {1}{x \sqrt {-b+a x^3}} \, dx=\frac {\ln \left (\frac {a\,x^3-2\,b+\sqrt {b}\,\sqrt {a\,x^3-b}\,2{}\mathrm {i}}{x^3}\right )\,1{}\mathrm {i}}{3\,\sqrt {b}} \]
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