Optimal. Leaf size=49 \[ -\frac {\sqrt {2+x^2}}{10 x^5}+\frac {\sqrt {2+x^2}}{15 x^3}-\frac {\sqrt {2+x^2}}{15 x} \]
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Rubi [A]
time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 3, 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 {\sqrt {x^2+2}}{15 x}-\frac {\sqrt {x^2+2}}{10 x^5}+\frac {\sqrt {x^2+2}}{15 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 277
Rubi steps
\begin {align*} \int \frac {1}{x^6 \sqrt {2+x^2}} \, dx &=-\frac {\sqrt {2+x^2}}{10 x^5}-\frac {2}{5} \int \frac {1}{x^4 \sqrt {2+x^2}} \, dx\\ &=-\frac {\sqrt {2+x^2}}{10 x^5}+\frac {\sqrt {2+x^2}}{15 x^3}+\frac {2}{15} \int \frac {1}{x^2 \sqrt {2+x^2}} \, dx\\ &=-\frac {\sqrt {2+x^2}}{10 x^5}+\frac {\sqrt {2+x^2}}{15 x^3}-\frac {\sqrt {2+x^2}}{15 x}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 28, normalized size = 0.57 \begin {gather*} \frac {\sqrt {2+x^2} \left (-3+2 x^2-2 x^4\right )}{30 x^5} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 3.16, size = 28, normalized size = 0.57 \begin {gather*} \frac {\left (-3+2 x^2-2 x^4\right ) \sqrt {\frac {2+x^2}{x^2}}}{30 x^4} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.06, size = 38, normalized size = 0.78
method | result | size |
gosper | \(-\frac {\sqrt {x^{2}+2}\, \left (2 x^{4}-2 x^{2}+3\right )}{30 x^{5}}\) | \(25\) |
trager | \(-\frac {\sqrt {x^{2}+2}\, \left (2 x^{4}-2 x^{2}+3\right )}{30 x^{5}}\) | \(25\) |
meijerg | \(-\frac {\sqrt {2}\, \left (\frac {2}{3} x^{4}-\frac {2}{3} x^{2}+1\right ) \sqrt {1+\frac {x^{2}}{2}}}{10 x^{5}}\) | \(30\) |
risch | \(-\frac {2 x^{6}+2 x^{4}-x^{2}+6}{30 x^{5} \sqrt {x^{2}+2}}\) | \(30\) |
default | \(-\frac {\sqrt {x^{2}+2}}{10 x^{5}}+\frac {\sqrt {x^{2}+2}}{15 x^{3}}-\frac {\sqrt {x^{2}+2}}{15 x}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 37, normalized size = 0.76 \begin {gather*} -\frac {\sqrt {x^{2} + 2}}{15 \, x} + \frac {\sqrt {x^{2} + 2}}{15 \, x^{3}} - \frac {\sqrt {x^{2} + 2}}{10 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 31, normalized size = 0.63 \begin {gather*} -\frac {2 \, x^{5} + {\left (2 \, x^{4} - 2 \, x^{2} + 3\right )} \sqrt {x^{2} + 2}}{30 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.46, size = 41, normalized size = 0.84 \begin {gather*} - \frac {\sqrt {1 + \frac {2}{x^{2}}}}{15} + \frac {\sqrt {1 + \frac {2}{x^{2}}}}{15 x^{2}} - \frac {\sqrt {1 + \frac {2}{x^{2}}}}{10 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 59, normalized size = 1.20 \begin {gather*} -\frac {64 \left (-5 \left (\sqrt {x^{2}+2}-x\right )^{4}+5 \left (\sqrt {x^{2}+2}-x\right )^{2}-2\right )}{30 \left (\left (\sqrt {x^{2}+2}-x\right )^{2}-2\right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 25, normalized size = 0.51 \begin {gather*} -\sqrt {x^2+2}\,\left (\frac {1}{15\,x}-\frac {1}{15\,x^3}+\frac {1}{10\,x^5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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