Optimal. Leaf size=31 \[ \frac {1}{6} x^3 \sqrt {2+x^6}-\frac {1}{3} \sinh ^{-1}\left (\frac {x^3}{\sqrt {2}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 31, 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 = {281, 327, 221}
\begin {gather*} \frac {1}{6} x^3 \sqrt {x^6+2}-\frac {1}{3} \sinh ^{-1}\left (\frac {x^3}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 281
Rule 327
Rubi steps
\begin {align*} \int \frac {x^8}{\sqrt {2+x^6}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^2}{\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} \sinh ^{-1}\left (\frac {x^3}{\sqrt {2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 35, normalized size = 1.13 \begin {gather*} \frac {1}{6} x^3 \sqrt {2+x^6}-\frac {1}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {2+x^6}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 25, normalized size = 0.81
method | result | size |
risch | \(-\frac {\arcsinh \left (\frac {x^{3} \sqrt {2}}{2}\right )}{3}+\frac {x^{3} \sqrt {x^{6}+2}}{6}\) | \(25\) |
trager | \(\frac {x^{3} \sqrt {x^{6}+2}}{6}+\frac {\ln \left (x^{3}-\sqrt {x^{6}+2}\right )}{3}\) | \(30\) |
meijerg | \(\frac {\frac {\sqrt {\pi }\, x^{3} \sqrt {2}\, \sqrt {1+\frac {x^{6}}{2}}}{2}-\sqrt {\pi }\, \arcsinh \left (\frac {x^{3} \sqrt {2}}{2}\right )}{3 \sqrt {\pi }}\) | \(41\) |
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.
\(2 (24) = 48\).
time = 0.30, size = 58, normalized size = 1.87 \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 = 0.36, size = 29, normalized size = 0.94 \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 [A]
time = 0.99, size = 39, normalized size = 1.26 \begin {gather*} \frac {x^{9}}{6 \sqrt {x^{6} + 2}} + \frac {x^{3}}{3 \sqrt {x^{6} + 2}} - \frac {\operatorname {asinh}{\left (\frac {\sqrt {2} x^{3}}{2} \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 [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x^8}{\sqrt {x^6+2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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