Optimal. Leaf size=14 \[ \frac {1}{3} \sinh ^{-1}\left (\frac {x^3}{\sqrt {2}}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {281, 221}
\begin {gather*} \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
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {2+x^6}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt {2+x^2}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \sinh ^{-1}\left (\frac {x^3}{\sqrt {2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 18, normalized size = 1.29 \begin {gather*} \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 = 0.16, size = 12, normalized size = 0.86
method | result | size |
meijerg | \(\frac {\arcsinh \left (\frac {x^{3} \sqrt {2}}{2}\right )}{3}\) | \(12\) |
trager | \(-\frac {\ln \left (x^{3}-\sqrt {x^{6}+2}\right )}{3}\) | \(17\) |
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. 33 vs.
\(2 (11) = 22\).
time = 0.32, size = 33, normalized size = 2.36 \begin {gather*} \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 = 16, normalized size = 1.14 \begin {gather*} -\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 = 0.40, size = 12, normalized size = 0.86 \begin {gather*} \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 29 vs.
\(2 (11) = 22\).
time = 1.43, size = 29, normalized size = 2.07 \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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.07 \begin {gather*} \int \frac {x^2}{\sqrt {x^6+2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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