Optimal. Leaf size=14 \[ \frac {1}{3} \tan ^{-1}\left (\sqrt {x^6-1}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {266, 63, 203} \begin {gather*} \frac {1}{3} \tan ^{-1}\left (\sqrt {x^6-1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {-1+x^6}} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right )\\ &=\frac {1}{3} \tan ^{-1}\left (\sqrt {-1+x^6}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {1}{3} \tan ^{-1}\left (\sqrt {x^6-1}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} \frac {1}{3} \tan ^{-1}\left (\sqrt {-1+x^6}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 10, normalized size = 0.71 \begin {gather*} \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 = 0.46, size = 10, normalized size = 0.71 \begin {gather*} \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 [A] time = 0.38, size = 7, normalized size = 0.50
method | result | size |
default | \(-\frac {\arcsin \left (\frac {1}{x^{3}}\right )}{3}\) | \(7\) |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{3}\) | \(28\) |
meijerg | \(\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \left (\left (-2 \ln \relax (2)+6 \ln \relax (x )+i \pi \right ) \sqrt {\pi }-2 \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right ) \sqrt {\pi }\right )}{6 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, \sqrt {\pi }}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 10, normalized size = 0.71 \begin {gather*} \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 = 0.18, size = 10, normalized size = 0.71 \begin {gather*} \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 = 0.82, size = 24, normalized size = 1.71 \begin {gather*} \begin {cases} \frac {i \operatorname {acosh}{\left (\frac {1}{x^{3}} \right )}}{3} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\- \frac {\operatorname {asin}{\left (\frac {1}{x^{3}} \right )}}{3} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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