Integrand size = 13, antiderivative size = 28 \[ \int \frac {\sqrt {-1+x^3}}{x} \, dx=\frac {2}{3} \sqrt {-1+x^3}-\frac {2}{3} \arctan \left (\sqrt {-1+x^3}\right ) \]
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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, 209} \[ \int \frac {\sqrt {-1+x^3}}{x} \, dx=\frac {2 \sqrt {x^3-1}}{3}-\frac {2}{3} \arctan \left (\sqrt {x^3-1}\right ) \]
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Rule 52
Rule 65
Rule 209
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {\sqrt {-1+x}}{x} \, dx,x,x^3\right ) \\ & = \frac {2}{3} \sqrt {-1+x^3}-\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^3\right ) \\ & = \frac {2}{3} \sqrt {-1+x^3}-\frac {2}{3} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^3}\right ) \\ & = \frac {2}{3} \sqrt {-1+x^3}-\frac {2}{3} \arctan \left (\sqrt {-1+x^3}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-1+x^3}}{x} \, dx=\frac {2}{3} \sqrt {-1+x^3}-\frac {2}{3} \arctan \left (\sqrt {-1+x^3}\right ) \]
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Time = 2.53 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75
method | result | size |
default | \(\frac {2 \sqrt {x^{3}-1}}{3}-\frac {2 \arctan \left (\sqrt {x^{3}-1}\right )}{3}\) | \(21\) |
elliptic | \(\frac {2 \sqrt {x^{3}-1}}{3}-\frac {2 \arctan \left (\sqrt {x^{3}-1}\right )}{3}\) | \(21\) |
pseudoelliptic | \(\frac {2 \sqrt {x^{3}-1}}{3}-\frac {2 \arctan \left (\sqrt {x^{3}-1}\right )}{3}\) | \(21\) |
trager | \(\frac {2 \sqrt {x^{3}-1}}{3}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \sqrt {x^{3}-1}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{3}}\right )}{3}\) | \(53\) |
meijerg | \(-\frac {\sqrt {\operatorname {signum}\left (x^{3}-1\right )}\, \left (4 \sqrt {\pi }-4 \sqrt {\pi }\, \sqrt {-x^{3}+1}+4 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{3}+1}}{2}\right )-2 \left (2-2 \ln \left (2\right )+3 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }\right )}{6 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{3}-1\right )}}\) | \(82\) |
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none
Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {-1+x^3}}{x} \, dx=\frac {2}{3} \, \sqrt {x^{3} - 1} - \frac {2}{3} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \]
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Result contains complex when optimal does not.
Time = 0.71 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.04 \[ \int \frac {\sqrt {-1+x^3}}{x} \, dx=\begin {cases} \frac {2 \sqrt {x^{3} - 1}}{3} - \frac {2 i \log {\left (x^{\frac {3}{2}} \right )}}{3} + \frac {i \log {\left (x^{3} \right )}}{3} + \frac {2 \operatorname {asin}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{3} & \text {for}\: \left |{x^{3}}\right | > 1 \\\frac {2 i \sqrt {1 - x^{3}}}{3} + \frac {i \log {\left (x^{3} \right )}}{3} - \frac {2 i \log {\left (\sqrt {1 - x^{3}} + 1 \right )}}{3} & \text {otherwise} \end {cases} \]
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none
Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {-1+x^3}}{x} \, dx=\frac {2}{3} \, \sqrt {x^{3} - 1} - \frac {2}{3} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {-1+x^3}}{x} \, dx=\frac {2}{3} \, \sqrt {x^{3} - 1} - \frac {2}{3} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \]
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Time = 5.36 (sec) , antiderivative size = 174, normalized size of antiderivative = 6.21 \[ \int \frac {\sqrt {-1+x^3}}{x} \, dx=\frac {2\,\sqrt {x^3-1}}{3}+\frac {2\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
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