Optimal. Leaf size=40 \[ -\frac {1}{3} \text {ArcTan}\left (\sqrt {-1+x^6}\right )+\frac {2 \text {ArcTan}\left (\frac {\sqrt {-1+x^6}}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
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Rubi [A]
time = 0.03, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {587, 162, 65,
209} \begin {gather*} \frac {2 \text {ArcTan}\left (\frac {\sqrt {x^6-1}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{3} \text {ArcTan}\left (\sqrt {x^6-1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 162
Rule 209
Rule 587
Rubi steps
\begin {align*} \int \frac {-2+x^6}{x \sqrt {-1+x^6} \left (2+x^6\right )} \, dx &=\frac {1}{6} \text {Subst}\left (\int \frac {-2+x}{\sqrt {-1+x} x (2+x)} \, dx,x,x^6\right )\\ &=-\left (\frac {1}{6} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} (2+x)} \, dx,x,x^6\right )\\ &=-\left (\frac {1}{3} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right )\right )+\frac {2}{3} \text {Subst}\left (\int \frac {1}{3+x^2} \, dx,x,\sqrt {-1+x^6}\right )\\ &=-\frac {1}{3} \tan ^{-1}\left (\sqrt {-1+x^6}\right )+\frac {2 \tan ^{-1}\left (\frac {\sqrt {-1+x^6}}{\sqrt {3}}\right )}{3 \sqrt {3}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 40, normalized size = 1.00 \begin {gather*} -\frac {1}{3} \text {ArcTan}\left (\sqrt {-1+x^6}\right )+\frac {2 \text {ArcTan}\left (\frac {\sqrt {-1+x^6}}{\sqrt {3}}\right )}{3 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.25, size = 76, normalized size = 1.90
method | result | size |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{3}-\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) x^{6}+6 \sqrt {x^{6}-1}-4 \RootOf \left (\textit {\_Z}^{2}+3\right )}{x^{6}+2}\right )}{9}\) | \(76\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 29, normalized size = 0.72 \begin {gather*} \frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} \sqrt {x^{6} - 1}\right ) - \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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Sympy [A]
time = 9.99, size = 36, normalized size = 0.90 \begin {gather*} \frac {2 \sqrt {3} \operatorname {atan}{\left (\frac {\sqrt {3} \sqrt {x^{6} - 1}}{3} \right )}}{9} - \frac {\operatorname {atan}{\left (\sqrt {x^{6} - 1} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 29, normalized size = 0.72 \begin {gather*} \frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} \sqrt {x^{6} - 1}\right ) - \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 = 0.39, size = 29, normalized size = 0.72 \begin {gather*} \frac {2\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,\sqrt {x^6-1}}{3}\right )}{9}-\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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