Optimal. Leaf size=39 \[ \frac {1}{6 \sqrt {2+x^6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2+x^6}}{\sqrt {2}}\right )}{6 \sqrt {2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {272, 53, 65,
213} \begin {gather*} \frac {1}{6 \sqrt {x^6+2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {x^6+2}}{\sqrt {2}}\right )}{6 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 65
Rule 213
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x \left (2+x^6\right )^{3/2}} \, dx &=\frac {1}{6} \text {Subst}\left (\int \frac {1}{x (2+x)^{3/2}} \, dx,x,x^6\right )\\ &=\frac {1}{6 \sqrt {2+x^6}}+\frac {1}{12} \text {Subst}\left (\int \frac {1}{x \sqrt {2+x}} \, dx,x,x^6\right )\\ &=\frac {1}{6 \sqrt {2+x^6}}+\frac {1}{6} \text {Subst}\left (\int \frac {1}{-2+x^2} \, dx,x,\sqrt {2+x^6}\right )\\ &=\frac {1}{6 \sqrt {2+x^6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2+x^6}}{\sqrt {2}}\right )}{6 \sqrt {2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 39, normalized size = 1.00 \begin {gather*} \frac {1}{6 \sqrt {2+x^6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2+x^6}}{\sqrt {2}}\right )}{6 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.34, size = 36, normalized size = 0.92
method | result | size |
risch | \(\frac {1}{6 \sqrt {x^{6}+2}}+\frac {\sqrt {2}\, \ln \left (\frac {\sqrt {x^{6}+2}-\sqrt {2}}{\sqrt {x^{6}}}\right )}{12}\) | \(36\) |
trager | \(\frac {1}{6 \sqrt {x^{6}+2}}-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {\sqrt {x^{6}+2}+\RootOf \left (\textit {\_Z}^{2}-2\right )}{x^{3}}\right )}{12}\) | \(38\) |
meijerg | \(\frac {\sqrt {2}\, \left (-\sqrt {\pi }+\frac {\sqrt {\pi }}{\sqrt {1+\frac {x^{6}}{2}}}-\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1+\frac {x^{6}}{2}}}{2}\right )+\frac {\left (2-3 \ln \left (2\right )+6 \ln \left (x \right )\right ) \sqrt {\pi }}{2}\right )}{12 \sqrt {\pi }}\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 44, normalized size = 1.13 \begin {gather*} \frac {1}{24} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {x^{6} + 2}}{\sqrt {2} + \sqrt {x^{6} + 2}}\right ) + \frac {1}{6 \, \sqrt {x^{6} + 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 50, normalized size = 1.28 \begin {gather*} \frac {\sqrt {2} {\left (x^{6} + 2\right )} \log \left (\frac {x^{6} - 2 \, \sqrt {2} \sqrt {x^{6} + 2} + 4}{x^{6}}\right ) + 4 \, \sqrt {x^{6} + 2}}{24 \, {\left (x^{6} + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 194 vs.
\(2 (34) = 68\).
time = 0.66, size = 194, normalized size = 4.97 \begin {gather*} \frac {x^{6} \log {\left (x^{6} \right )}}{12 \sqrt {2} x^{6} + 24 \sqrt {2}} - \frac {2 x^{6} \log {\left (\sqrt {\frac {x^{6}}{2} + 1} + 1 \right )}}{12 \sqrt {2} x^{6} + 24 \sqrt {2}} - \frac {x^{6} \log {\left (2 \right )}}{12 \sqrt {2} x^{6} + 24 \sqrt {2}} + \frac {2 \sqrt {2} \sqrt {x^{6} + 2}}{12 \sqrt {2} x^{6} + 24 \sqrt {2}} + \frac {2 \log {\left (x^{6} \right )}}{12 \sqrt {2} x^{6} + 24 \sqrt {2}} - \frac {4 \log {\left (\sqrt {\frac {x^{6}}{2} + 1} + 1 \right )}}{12 \sqrt {2} x^{6} + 24 \sqrt {2}} - \frac {2 \log {\left (2 \right )}}{12 \sqrt {2} x^{6} + 24 \sqrt {2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.38, size = 44, normalized size = 1.13 \begin {gather*} \frac {1}{24} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {x^{6} + 2}}{\sqrt {2} + \sqrt {x^{6} + 2}}\right ) + \frac {1}{6 \, \sqrt {x^{6} + 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.22, size = 28, normalized size = 0.72 \begin {gather*} \frac {1}{6\,\sqrt {x^6+2}}-\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {x^6+2}}{2}\right )}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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