Optimal. Leaf size=58 \[ -\frac {\sqrt {2+x^6}}{24 x^{12}}+\frac {\sqrt {2+x^6}}{32 x^6}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2+x^6}}{\sqrt {2}}\right )}{32 \sqrt {2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {272, 44, 65,
213} \begin {gather*} \frac {\sqrt {x^6+2}}{32 x^6}-\frac {\tanh ^{-1}\left (\frac {\sqrt {x^6+2}}{\sqrt {2}}\right )}{32 \sqrt {2}}-\frac {\sqrt {x^6+2}}{24 x^{12}} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 213
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x^{13} \sqrt {2+x^6}} \, dx &=\frac {1}{6} \text {Subst}\left (\int \frac {1}{x^3 \sqrt {2+x}} \, dx,x,x^6\right )\\ &=-\frac {\sqrt {2+x^6}}{24 x^{12}}-\frac {1}{16} \text {Subst}\left (\int \frac {1}{x^2 \sqrt {2+x}} \, dx,x,x^6\right )\\ &=-\frac {\sqrt {2+x^6}}{24 x^{12}}+\frac {\sqrt {2+x^6}}{32 x^6}+\frac {1}{64} \text {Subst}\left (\int \frac {1}{x \sqrt {2+x}} \, dx,x,x^6\right )\\ &=-\frac {\sqrt {2+x^6}}{24 x^{12}}+\frac {\sqrt {2+x^6}}{32 x^6}+\frac {1}{32} \text {Subst}\left (\int \frac {1}{-2+x^2} \, dx,x,\sqrt {2+x^6}\right )\\ &=-\frac {\sqrt {2+x^6}}{24 x^{12}}+\frac {\sqrt {2+x^6}}{32 x^6}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2+x^6}}{\sqrt {2}}\right )}{32 \sqrt {2}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 49, normalized size = 0.84 \begin {gather*} \frac {\sqrt {2+x^6} \left (-4+3 x^6\right )}{96 x^{12}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2+x^6}}{\sqrt {2}}\right )}{32 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.35, size = 48, normalized size = 0.83
method | result | size |
trager | \(\frac {\left (3 x^{6}-4\right ) \sqrt {x^{6}+2}}{96 x^{12}}-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {\sqrt {x^{6}+2}+\RootOf \left (\textit {\_Z}^{2}-2\right )}{x^{3}}\right )}{64}\) | \(48\) |
risch | \(\frac {3 x^{12}+2 x^{6}-8}{96 x^{12} \sqrt {x^{6}+2}}+\frac {\sqrt {2}\, \ln \left (\frac {\sqrt {x^{6}+2}-\sqrt {2}}{\sqrt {x^{6}}}\right )}{64}\) | \(51\) |
meijerg | \(\frac {\sqrt {2}\, \left (\frac {\sqrt {\pi }\, \left (-\frac {7}{4} x^{12}-4 x^{6}+8\right )}{4 x^{12}}-\frac {\sqrt {\pi }\, \left (-6 x^{6}+8\right ) \sqrt {1+\frac {x^{6}}{2}}}{4 x^{12}}-\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1+\frac {x^{6}}{2}}}{2}\right )}{4}+\frac {3 \left (\frac {7}{6}-3 \ln \left (2\right )+6 \ln \left (x \right )\right ) \sqrt {\pi }}{8}-\frac {2 \sqrt {\pi }}{x^{12}}+\frac {\sqrt {\pi }}{x^{6}}\right )}{48 \sqrt {\pi }}\) | \(103\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 74, normalized size = 1.28 \begin {gather*} \frac {1}{128} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {x^{6} + 2}}{\sqrt {2} + \sqrt {x^{6} + 2}}\right ) - \frac {3 \, {\left (x^{6} + 2\right )}^{\frac {3}{2}} - 10 \, \sqrt {x^{6} + 2}}{96 \, {\left (4 \, x^{6} - {\left (x^{6} + 2\right )}^{2} + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 52, normalized size = 0.90 \begin {gather*} \frac {3 \, \sqrt {2} x^{12} \log \left (\frac {x^{6} - 2 \, \sqrt {2} \sqrt {x^{6} + 2} + 4}{x^{6}}\right ) + 4 \, {\left (3 \, x^{6} - 4\right )} \sqrt {x^{6} + 2}}{384 \, x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.41, size = 66, normalized size = 1.14 \begin {gather*} - \frac {\sqrt {2} \operatorname {asinh}{\left (\frac {\sqrt {2}}{x^{3}} \right )}}{64} + \frac {1}{32 x^{3} \sqrt {1 + \frac {2}{x^{6}}}} + \frac {1}{48 x^{9} \sqrt {1 + \frac {2}{x^{6}}}} - \frac {1}{12 x^{15} \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 = 1.20, size = 59, normalized size = 1.02 \begin {gather*} \frac {1}{128} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {x^{6} + 2}}{\sqrt {2} + \sqrt {x^{6} + 2}}\right ) + \frac {3 \, {\left (x^{6} + 2\right )}^{\frac {3}{2}} - 10 \, \sqrt {x^{6} + 2}}{96 \, x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.28, size = 57, normalized size = 0.98 \begin {gather*} \frac {\frac {5\,\sqrt {x^6+2}}{48}-\frac {{\left (x^6+2\right )}^{3/2}}{32}}{4\,x^6-{\left (x^6+2\right )}^2+4}-\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {x^6+2}}{2}\right )}{64} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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