Optimal. Leaf size=31 \[ -\frac {x^3}{3 \sqrt {2+x^6}}+\frac {1}{3} \sinh ^{-1}\left (\frac {x^3}{\sqrt {2}}\right ) \]
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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, 294, 221}
\begin {gather*} \frac {1}{3} \sinh ^{-1}\left (\frac {x^3}{\sqrt {2}}\right )-\frac {x^3}{3 \sqrt {x^6+2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 281
Rule 294
Rubi steps
\begin {align*} \int \frac {x^8}{\left (2+x^6\right )^{3/2}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^2}{\left (2+x^2\right )^{3/2}} \, dx,x,x^3\right )\\ &=-\frac {x^3}{3 \sqrt {2+x^6}}+\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt {2+x^2}} \, dx,x,x^3\right )\\ &=-\frac {x^3}{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 = 0.13, size = 35, normalized size = 1.13 \begin {gather*} -\frac {x^3}{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 | \(\frac {\arcsinh \left (\frac {x^{3} \sqrt {2}}{2}\right )}{3}-\frac {x^{3}}{3 \sqrt {x^{6}+2}}\) | \(25\) |
trager | \(-\frac {x^{3}}{3 \sqrt {x^{6}+2}}-\frac {\ln \left (x^{3}-\sqrt {x^{6}+2}\right )}{3}\) | \(30\) |
meijerg | \(\frac {-\frac {\sqrt {\pi }\, x^{3} \sqrt {2}}{2 \sqrt {1+\frac {x^{6}}{2}}}+\sqrt {\pi }\, \arcsinh \left (\frac {x^{3} \sqrt {2}}{2}\right )}{3 \sqrt {\pi }}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 45, normalized size = 1.45 \begin {gather*} -\frac {x^{3}}{3 \, \sqrt {x^{6} + 2}} + \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 = 0.37, size = 45, normalized size = 1.45 \begin {gather*} -\frac {x^{6} + \sqrt {x^{6} + 2} x^{3} + {\left (x^{6} + 2\right )} \log \left (-x^{3} + \sqrt {x^{6} + 2}\right ) + 2}{3 \, {\left (x^{6} + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.72, size = 26, normalized size = 0.84 \begin {gather*} - \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}{{\left (x^6+2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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