Optimal. Leaf size=55 \[ -\frac {1}{8 \sqrt {2+x^6}}-\frac {1}{12 x^6 \sqrt {2+x^6}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2+x^6}}{\sqrt {2}}\right )}{8 \sqrt {2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {272, 44, 53, 65,
213} \begin {gather*} -\frac {1}{12 x^6 \sqrt {x^6+2}}-\frac {1}{8 \sqrt {x^6+2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {x^6+2}}{\sqrt {2}}\right )}{8 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 53
Rule 65
Rule 213
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x^7 \left (2+x^6\right )^{3/2}} \, dx &=\frac {1}{6} \text {Subst}\left (\int \frac {1}{x^2 (2+x)^{3/2}} \, dx,x,x^6\right )\\ &=\frac {1}{6 x^6 \sqrt {2+x^6}}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{x^2 \sqrt {2+x}} \, dx,x,x^6\right )\\ &=\frac {1}{6 x^6 \sqrt {2+x^6}}-\frac {\sqrt {2+x^6}}{8 x^6}-\frac {1}{16} \text {Subst}\left (\int \frac {1}{x \sqrt {2+x}} \, dx,x,x^6\right )\\ &=\frac {1}{6 x^6 \sqrt {2+x^6}}-\frac {\sqrt {2+x^6}}{8 x^6}-\frac {1}{8} \text {Subst}\left (\int \frac {1}{-2+x^2} \, dx,x,\sqrt {2+x^6}\right )\\ &=\frac {1}{6 x^6 \sqrt {2+x^6}}-\frac {\sqrt {2+x^6}}{8 x^6}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2+x^6}}{\sqrt {2}}\right )}{8 \sqrt {2}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 49, normalized size = 0.89 \begin {gather*} \frac {-2-3 x^6}{24 x^6 \sqrt {2+x^6}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2+x^6}}{\sqrt {2}}\right )}{8 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.35, size = 46, normalized size = 0.84
method | result | size |
risch | \(-\frac {3 x^{6}+2}{24 x^{6} \sqrt {x^{6}+2}}-\frac {\sqrt {2}\, \ln \left (\frac {\sqrt {x^{6}+2}-\sqrt {2}}{\sqrt {x^{6}}}\right )}{16}\) | \(46\) |
trager | \(-\frac {3 x^{6}+2}{24 x^{6} \sqrt {x^{6}+2}}+\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {\sqrt {x^{6}+2}+\RootOf \left (\textit {\_Z}^{2}-2\right )}{x^{3}}\right )}{16}\) | \(48\) |
meijerg | \(\frac {\sqrt {2}\, \left (\frac {\sqrt {\pi }\, \left (10 x^{6}+8\right )}{8 x^{6}}-\frac {\sqrt {\pi }\, \left (12 x^{6}+8\right )}{8 x^{6} \sqrt {1+\frac {x^{6}}{2}}}+\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1+\frac {x^{6}}{2}}}{2}\right )}{2}-\frac {3 \left (\frac {5}{3}-3 \ln \left (2\right )+6 \ln \left (x \right )\right ) \sqrt {\pi }}{4}-\frac {\sqrt {\pi }}{x^{6}}\right )}{24 \sqrt {\pi }}\) | \(91\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 63, normalized size = 1.15 \begin {gather*} -\frac {1}{32} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {x^{6} + 2}}{\sqrt {2} + \sqrt {x^{6} + 2}}\right ) - \frac {3 \, x^{6} + 2}{24 \, {\left ({\left (x^{6} + 2\right )}^{\frac {3}{2}} - 2 \, \sqrt {x^{6} + 2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 66, normalized size = 1.20 \begin {gather*} \frac {3 \, \sqrt {2} {\left (x^{12} + 2 \, x^{6}\right )} \log \left (\frac {x^{6} + 2 \, \sqrt {2} \sqrt {x^{6} + 2} + 4}{x^{6}}\right ) - 4 \, {\left (3 \, x^{6} + 2\right )} \sqrt {x^{6} + 2}}{96 \, {\left (x^{12} + 2 \, x^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.48, size = 49, normalized size = 0.89 \begin {gather*} \frac {\sqrt {2} \operatorname {asinh}{\left (\frac {\sqrt {2}}{x^{3}} \right )}}{16} - \frac {1}{8 x^{3} \sqrt {1 + \frac {2}{x^{6}}}} - \frac {1}{12 x^{9} \sqrt {1 + \frac {2}{x^{6}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.54, size = 63, normalized size = 1.15 \begin {gather*} -\frac {1}{32} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {x^{6} + 2}}{\sqrt {2} + \sqrt {x^{6} + 2}}\right ) - \frac {3 \, x^{6} + 2}{24 \, {\left ({\left (x^{6} + 2\right )}^{\frac {3}{2}} - 2 \, \sqrt {x^{6} + 2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.30, size = 40, normalized size = 0.73 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {x^6+2}}{2}\right )}{16}-\frac {1}{8\,\sqrt {x^6+2}}-\frac {1}{12\,x^6\,\sqrt {x^6+2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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