Optimal. Leaf size=28 \[ \frac {2 \sqrt {x^3-1}}{3}-\frac {2}{3} \tan ^{-1}\left (\sqrt {x^3-1}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {266, 50, 63, 203} \begin {gather*} \frac {2 \sqrt {x^3-1}}{3}-\frac {2}{3} \tan ^{-1}\left (\sqrt {x^3-1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 203
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt {-1+x^3}}{x} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{x} \, dx,x,x^3\right )\\ &=\frac {2}{3} \sqrt {-1+x^3}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^3\right )\\ &=\frac {2}{3} \sqrt {-1+x^3}-\frac {2}{3} \operatorname {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} \tan ^{-1}\left (\sqrt {-1+x^3}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 28, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {x^3-1}}{3}-\frac {2}{3} \tan ^{-1}\left (\sqrt {x^3-1}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 28, normalized size = 1.00 \begin {gather*} \frac {2}{3} \sqrt {-1+x^3}-\frac {2}{3} \tan ^{-1}\left (\sqrt {-1+x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 20, normalized size = 0.71 \begin {gather*} \frac {2}{3} \, \sqrt {x^{3} - 1} - \frac {2}{3} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.65, size = 20, normalized size = 0.71 \begin {gather*} \frac {2}{3} \, \sqrt {x^{3} - 1} - \frac {2}{3} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 21, normalized size = 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 | \(\frac {2 \sqrt {x^{3}-1}}{3}-\frac {2 \arctan \left (\sqrt {x^{3}-1}\right )}{3}\) | \(21\) |
trager | \(\frac {2 \sqrt {x^{3}-1}}{3}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \sqrt {x^{3}-1}-2 \RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{3}}\right )}{3}\) | \(53\) |
meijerg | \(-\frac {\sqrt {\mathrm {signum}\left (x^{3}-1\right )}\, \left (-2 \left (2-2 \ln \relax (2)+3 \ln \relax (x )+i \pi \right ) \sqrt {\pi }+4 \sqrt {\pi }-4 \sqrt {\pi }\, \sqrt {-x^{3}+1}+4 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{3}+1}}{2}\right )\right )}{6 \sqrt {-\mathrm {signum}\left (x^{3}-1\right )}\, \sqrt {\pi }}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 20, normalized size = 0.71 \begin {gather*} \frac {2}{3} \, \sqrt {x^{3} - 1} - \frac {2}{3} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 174, normalized size = 6.21 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.04, size = 85, normalized size = 3.04 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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