Optimal. Leaf size=28 \[ \frac {\sqrt {x^6-1}}{3}-\frac {1}{3} \tan ^{-1}\left (\sqrt {x^6-1}\right ) \]
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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 {\sqrt {x^6-1}}{3}-\frac {1}{3} \tan ^{-1}\left (\sqrt {x^6-1}\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^6}}{x} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{x} \, dx,x,x^6\right )\\ &=\frac {1}{3} \sqrt {-1+x^6}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )\\ &=\frac {1}{3} \sqrt {-1+x^6}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right )\\ &=\frac {1}{3} \sqrt {-1+x^6}-\frac {1}{3} \tan ^{-1}\left (\sqrt {-1+x^6}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 28, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x^6-1}}{3}-\frac {1}{3} \tan ^{-1}\left (\sqrt {x^6-1}\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 {1}{3} \sqrt {-1+x^6}-\frac {1}{3} \tan ^{-1}\left (\sqrt {-1+x^6}\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 {1}{3} \, \sqrt {x^{6} - 1} - \frac {1}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 20, normalized size = 0.71 \begin {gather*} \frac {1}{3} \, \sqrt {x^{6} - 1} - \frac {1}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.35, size = 40, normalized size = 1.43
method | result | size |
trager | \(\frac {\sqrt {x^{6}-1}}{3}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{3}\) | \(40\) |
meijerg | \(-\frac {\sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, \left (-2 \left (2-2 \ln \relax (2)+6 \ln \relax (x )+i \pi \right ) \sqrt {\pi }+4 \sqrt {\pi }-4 \sqrt {\pi }\, \sqrt {-x^{6}+1}+4 \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right ) \sqrt {\pi }\right )}{12 \sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \sqrt {\pi }}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 20, normalized size = 0.71 \begin {gather*} \frac {1}{3} \, \sqrt {x^{6} - 1} - \frac {1}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 20, normalized size = 0.71 \begin {gather*} \frac {\sqrt {x^6-1}}{3}-\frac {\mathrm {atan}\left (\sqrt {x^6-1}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.03, size = 88, normalized size = 3.14 \begin {gather*} \begin {cases} - \frac {i x^{3}}{3 \sqrt {-1 + \frac {1}{x^{6}}}} - \frac {i \operatorname {acosh}{\left (\frac {1}{x^{3}} \right )}}{3} + \frac {i}{3 x^{3} \sqrt {-1 + \frac {1}{x^{6}}}} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\\frac {x^{3}}{3 \sqrt {1 - \frac {1}{x^{6}}}} + \frac {\operatorname {asin}{\left (\frac {1}{x^{3}} \right )}}{3} - \frac {1}{3 x^{3} \sqrt {1 - \frac {1}{x^{6}}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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