Optimal. Leaf size=55 \[ -\frac {1}{12 x^6 \sqrt {x^6+2}}-\frac {1}{8 \sqrt {x^6+2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {x^6+2}}{\sqrt {2}}\right )}{8 \sqrt {2}} \]
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Rubi [A] time = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.05, 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} \[ -\frac {\sqrt {x^6+2}}{8 x^6}+\frac {1}{6 x^6 \sqrt {x^6+2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {x^6+2}}{\sqrt {2}}\right )}{8 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 207
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^7 \left (2+x^6\right )^{3/2}} \, dx &=\frac {1}{6} \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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 [C] time = 0.01, size = 30, normalized size = 0.55 \[ -\frac {\, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};\frac {x^6}{2}+1\right )}{12 \sqrt {x^6+2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 66, normalized size = 1.20 \[ \frac {3 \, \sqrt {2} {\left (x^{12} + 2 \, x^{6}\right )} \log \left (\frac {x^{6} + 2 \, \sqrt {2} \sqrt {x^{6} + 2} + 4}{x^{6}}\right ) - 4 \, {\left (3 \, x^{6} + 2\right )} \sqrt {x^{6} + 2}}{96 \, {\left (x^{12} + 2 \, x^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 63, normalized size = 1.15 \[ -\frac {1}{32} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {x^{6} + 2}}{\sqrt {2} + \sqrt {x^{6} + 2}}\right ) - \frac {3 \, x^{6} + 2}{24 \, {\left ({\left (x^{6} + 2\right )}^{\frac {3}{2}} - 2 \, \sqrt {x^{6} + 2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 46, normalized size = 0.84 \[ -\frac {\sqrt {2}\, \ln \left (\frac {\sqrt {x^{6}+2}-\sqrt {2}}{\sqrt {x^{6}}}\right )}{16}-\frac {3 x^{6}+2}{24 \sqrt {x^{6}+2}\, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.39, size = 63, normalized size = 1.15 \[ -\frac {1}{32} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {x^{6} + 2}}{\sqrt {2} + \sqrt {x^{6} + 2}}\right ) - \frac {3 \, x^{6} + 2}{24 \, {\left ({\left (x^{6} + 2\right )}^{\frac {3}{2}} - 2 \, \sqrt {x^{6} + 2}\right )}} \]
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 \[ \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {x^6+2}}{2}\right )}{16}-\frac {1}{8\,\sqrt {x^6+2}}-\frac {1}{12\,x^6\,\sqrt {x^6+2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.84, size = 49, normalized size = 0.89 \[ \frac {\sqrt {2} \operatorname {asinh}{\left (\frac {\sqrt {2}}{x^{3}} \right )}}{16} - \frac {1}{8 x^{3} \sqrt {1 + \frac {2}{x^{6}}}} - \frac {1}{12 x^{9} \sqrt {1 + \frac {2}{x^{6}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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