Optimal. Leaf size=33 \[ -\frac {1}{6 x^3 \sqrt {2+x^6}}-\frac {x^3}{6 \sqrt {2+x^6}} \]
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Rubi [A]
time = 0.00, antiderivative size = 33, 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 = {277, 270}
\begin {gather*} -\frac {x^3}{6 \sqrt {x^6+2}}-\frac {1}{6 \sqrt {x^6+2} x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 277
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (2+x^6\right )^{3/2}} \, dx &=-\frac {1}{6 x^3 \sqrt {2+x^6}}-\int \frac {x^2}{\left (2+x^6\right )^{3/2}} \, dx\\ &=-\frac {1}{6 x^3 \sqrt {2+x^6}}-\frac {x^3}{6 \sqrt {2+x^6}}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 23, normalized size = 0.70 \begin {gather*} \frac {-1-x^6}{6 x^3 \sqrt {2+x^6}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 18, normalized size = 0.55
method | result | size |
gosper | \(-\frac {x^{6}+1}{6 x^{3} \sqrt {x^{6}+2}}\) | \(18\) |
trager | \(-\frac {x^{6}+1}{6 x^{3} \sqrt {x^{6}+2}}\) | \(18\) |
risch | \(-\frac {x^{6}+1}{6 x^{3} \sqrt {x^{6}+2}}\) | \(18\) |
meijerg | \(-\frac {\sqrt {2}\, \left (x^{6}+1\right )}{12 x^{3} \sqrt {1+\frac {x^{6}}{2}}}\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 25, normalized size = 0.76 \begin {gather*} -\frac {x^{3}}{12 \, \sqrt {x^{6} + 2}} - \frac {\sqrt {x^{6} + 2}}{12 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 35, normalized size = 1.06 \begin {gather*} -\frac {x^{9} + 2 \, x^{3} + \sqrt {x^{6} + 2} {\left (x^{6} + 1\right )}}{6 \, {\left (x^{9} + 2 \, x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.41, size = 31, normalized size = 0.94 \begin {gather*} - \frac {1}{6 \sqrt {1 + \frac {2}{x^{6}}}} - \frac {1}{6 x^{6} \sqrt {1 + \frac {2}{x^{6}}}} \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 = 1.15, size = 17, normalized size = 0.52 \begin {gather*} -\frac {x^6+1}{6\,x^3\,\sqrt {x^6+2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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