Optimal. Leaf size=35 \[ \frac {1}{6} x^3 \sqrt {-2+x^6}-\frac {1}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-2+x^6}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 35, 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 = {281, 201, 223,
212} \begin {gather*} \frac {1}{6} x^3 \sqrt {x^6-2}-\frac {1}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 212
Rule 223
Rule 281
Rubi steps
\begin {align*} \int x^2 \sqrt {-2+x^6} \, dx &=\frac {1}{3} \text {Subst}\left (\int \sqrt {-2+x^2} \, dx,x,x^3\right )\\ &=\frac {1}{6} x^3 \sqrt {-2+x^6}-\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt {-2+x^2}} \, dx,x,x^3\right )\\ &=\frac {1}{6} x^3 \sqrt {-2+x^6}-\frac {1}{3} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-2+x^6}}\right )\\ &=\frac {1}{6} x^3 \sqrt {-2+x^6}-\frac {1}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-2+x^6}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 35, normalized size = 1.00 \begin {gather*} \frac {1}{6} x^3 \sqrt {-2+x^6}-\frac {1}{3} \tanh ^{-1}\left (\frac {\sqrt {-2+x^6}}{x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 30, normalized size = 0.86
method | result | size |
trager | \(\frac {x^{3} \sqrt {x^{6}-2}}{6}+\frac {\ln \left (x^{3}-\sqrt {x^{6}-2}\right )}{3}\) | \(30\) |
risch | \(\frac {x^{3} \sqrt {x^{6}-2}}{6}-\frac {\sqrt {-\mathrm {signum}\left (-1+\frac {x^{6}}{2}\right )}\, \arcsin \left (\frac {x^{3} \sqrt {2}}{2}\right )}{3 \sqrt {\mathrm {signum}\left (-1+\frac {x^{6}}{2}\right )}}\) | \(47\) |
meijerg | \(\frac {i \sqrt {\mathrm {signum}\left (-1+\frac {x^{6}}{2}\right )}\, \left (-i \sqrt {\pi }\, x^{3} \sqrt {2}\, \sqrt {-\frac {x^{6}}{2}+1}-2 i \sqrt {\pi }\, \arcsin \left (\frac {x^{3} \sqrt {2}}{2}\right )\right )}{6 \sqrt {\pi }\, \sqrt {-\mathrm {signum}\left (-1+\frac {x^{6}}{2}\right )}}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs.
\(2 (27) = 54\).
time = 0.29, size = 58, normalized size = 1.66 \begin {gather*} -\frac {\sqrt {x^{6} - 2}}{3 \, x^{3} {\left (\frac {x^{6} - 2}{x^{6}} - 1\right )}} - \frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} - 2}}{x^{3}} + 1\right ) + \frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} - 2}}{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 = 29, normalized size = 0.83 \begin {gather*} \frac {1}{6} \, \sqrt {x^{6} - 2} x^{3} + \frac {1}{3} \, \log \left (-x^{3} + \sqrt {x^{6} - 2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.76, size = 88, normalized size = 2.51 \begin {gather*} \begin {cases} \frac {x^{9}}{6 \sqrt {x^{6} - 2}} - \frac {x^{3}}{3 \sqrt {x^{6} - 2}} - \frac {\operatorname {acosh}{\left (\frac {\sqrt {2} x^{3}}{2} \right )}}{3} & \text {for}\: \left |{x^{6}}\right | > 2 \\- \frac {i x^{9}}{6 \sqrt {2 - x^{6}}} + \frac {i x^{3}}{3 \sqrt {2 - x^{6}}} + \frac {i \operatorname {asin}{\left (\frac {\sqrt {2} x^{3}}{2} \right )}}{3} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.45, size = 30, normalized size = 0.86 \begin {gather*} \frac {1}{6} \, \sqrt {x^{6} - 2} x^{3} + \frac {1}{3} \, \log \left ({\left | -x^{3} + \sqrt {x^{6} - 2} \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int x^2\,\sqrt {x^6-2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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