Optimal. Leaf size=58 \[ \frac {\sqrt {x^6+2}}{32 x^6}-\frac {\tanh ^{-1}\left (\frac {\sqrt {x^6+2}}{\sqrt {2}}\right )}{32 \sqrt {2}}-\frac {\sqrt {x^6+2}}{24 x^{12}} \]
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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 = {266, 51, 63, 207} \[ \frac {\sqrt {x^6+2}}{32 x^6}-\frac {\sqrt {x^6+2}}{24 x^{12}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {x^6+2}}{\sqrt {2}}\right )}{32 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 207
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^{13} \sqrt {2+x^6}} \, dx &=\frac {1}{6} \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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 [C] time = 0.01, size = 30, normalized size = 0.52 \[ -\frac {1}{24} \sqrt {x^6+2} \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};\frac {x^6}{2}+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 52, normalized size = 0.90 \[ \frac {3 \, \sqrt {2} x^{12} \log \left (\frac {x^{6} - 2 \, \sqrt {2} \sqrt {x^{6} + 2} + 4}{x^{6}}\right ) + 4 \, {\left (3 \, x^{6} - 4\right )} \sqrt {x^{6} + 2}}{384 \, x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 59, normalized size = 1.02 \[ \frac {1}{128} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {x^{6} + 2}}{\sqrt {2} + \sqrt {x^{6} + 2}}\right ) + \frac {3 \, {\left (x^{6} + 2\right )}^{\frac {3}{2}} - 10 \, \sqrt {x^{6} + 2}}{96 \, x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 51, normalized size = 0.88 \[ \frac {\sqrt {2}\, \ln \left (\frac {\sqrt {x^{6}+2}-\sqrt {2}}{\sqrt {x^{6}}}\right )}{64}+\frac {3 x^{12}+2 x^{6}-8}{96 \sqrt {x^{6}+2}\, x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.30, size = 74, normalized size = 1.28 \[ \frac {1}{128} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {x^{6} + 2}}{\sqrt {2} + \sqrt {x^{6} + 2}}\right ) - \frac {3 \, {\left (x^{6} + 2\right )}^{\frac {3}{2}} - 10 \, \sqrt {x^{6} + 2}}{96 \, {\left (4 \, x^{6} - {\left (x^{6} + 2\right )}^{2} + 4\right )}} \]
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 \[ \frac {\frac {5\,\sqrt {x^6+2}}{48}-\frac {{\left (x^6+2\right )}^{3/2}}{32}}{4\,x^6-{\left (x^6+2\right )}^2+4}-\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {x^6+2}}{2}\right )}{64} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.15, size = 66, normalized size = 1.14 \[ - \frac {\sqrt {2} \operatorname {asinh}{\left (\frac {\sqrt {2}}{x^{3}} \right )}}{64} + \frac {1}{32 x^{3} \sqrt {1 + \frac {2}{x^{6}}}} + \frac {1}{48 x^{9} \sqrt {1 + \frac {2}{x^{6}}}} - \frac {1}{12 x^{15} \sqrt {1 + \frac {2}{x^{6}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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