Optimal. Leaf size=37 \[ -\frac {2}{3} \text {ArcTan}\left (x^3+\sqrt {-1+x^6}\right )+\frac {2}{3} \log \left (x^3+\sqrt {-1+x^6}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 33, normalized size of antiderivative = 0.89, number of steps
used = 7, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1489, 858, 223,
212, 272, 65, 209} \begin {gather*} \frac {2}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-1}}\right )-\frac {1}{3} \text {ArcTan}\left (\sqrt {x^6-1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 209
Rule 212
Rule 223
Rule 272
Rule 858
Rule 1489
Rubi steps
\begin {align*} \int \frac {-1+2 x^3}{x \sqrt {-1+x^6}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {-1+2 x}{x \sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=-\left (\frac {1}{3} \text {Subst}\left (\int \frac {1}{x \sqrt {-1+x^2}} \, dx,x,x^3\right )\right )+\frac {2}{3} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=-\left (\frac {1}{6} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )\right )+\frac {2}{3} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right )\\ &=\frac {2}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )-\frac {1}{3} \text {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 )+\frac {2}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 39, normalized size = 1.05 \begin {gather*} \frac {2}{3} \text {ArcTan}\left (x^3-\sqrt {-1+x^6}\right )+\frac {2}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-1+x^6}}\right ) \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 = 0.65, size = 45, normalized size = 1.22
method | result | size |
trager | \(\frac {2 \ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{3}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{3}\) | \(45\) |
meijerg | \(\frac {2 \sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \arcsin \left (x^{3}\right )}{3 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}}-\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )+\left (-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }\right )}{6 \sqrt {\pi }\, \sqrt {\mathrm {signum}\left (x^{6}-1\right )}}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 43, normalized size = 1.16 \begin {gather*} -\frac {1}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) + \frac {1}{3} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} + 1\right ) - \frac {1}{3} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 33, normalized size = 0.89 \begin {gather*} -\frac {2}{3} \, \arctan \left (-x^{3} + \sqrt {x^{6} - 1}\right ) - \frac {2}{3} \, \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.34, size = 24, normalized size = 0.65 \begin {gather*} - \frac {\begin {cases} \operatorname {acos}{\left (\frac {1}{x^{3}} \right )} & \text {for}\: x^{3} > -1 \wedge x^{3} < 1 \end {cases}}{3} + \frac {2 \operatorname {acosh}{\left (x^{3} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.49, size = 25, normalized size = 0.68 \begin {gather*} \frac {2\,\ln \left (\sqrt {x^6-1}+x^3\right )}{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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