Optimal. Leaf size=31 \[ -\frac {\sqrt {x^6+1}}{6 x^6}-\frac {1}{6} \tanh ^{-1}\left (\sqrt {x^6+1}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 31, 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 = {266, 47, 63, 207} \begin {gather*} -\frac {\sqrt {x^6+1}}{6 x^6}-\frac {1}{6} \tanh ^{-1}\left (\sqrt {x^6+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 207
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt {1+x^6}}{x^7} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {\sqrt {1+x}}{x^2} \, dx,x,x^6\right )\\ &=-\frac {\sqrt {1+x^6}}{6 x^6}+\frac {1}{12} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^6\right )\\ &=-\frac {\sqrt {1+x^6}}{6 x^6}+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^6}\right )\\ &=-\frac {\sqrt {1+x^6}}{6 x^6}-\frac {1}{6} \tanh ^{-1}\left (\sqrt {1+x^6}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 44, normalized size = 1.42 \begin {gather*} -\frac {1}{6 x^6 \sqrt {x^6+1}}-\frac {1}{6 \sqrt {x^6+1}}-\frac {1}{6} \tanh ^{-1}\left (\sqrt {x^6+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 31, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {x^6+1}}{6 x^6}-\frac {1}{6} \tanh ^{-1}\left (\sqrt {x^6+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 44, normalized size = 1.42 \begin {gather*} -\frac {x^{6} \log \left (\sqrt {x^{6} + 1} + 1\right ) - x^{6} \log \left (\sqrt {x^{6} + 1} - 1\right ) + 2 \, \sqrt {x^{6} + 1}}{12 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 37, normalized size = 1.19 \begin {gather*} -\frac {\sqrt {x^{6} + 1}}{6 \, x^{6}} - \frac {1}{12} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) + \frac {1}{12} \, \log \left (\sqrt {x^{6} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 32, normalized size = 1.03 \begin {gather*} -\frac {\sqrt {x^{6}+1}}{6 x^{6}}+\frac {\ln \left (\frac {\sqrt {x^{6}+1}-1}{\sqrt {x^{6}}}\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 37, normalized size = 1.19 \begin {gather*} -\frac {\sqrt {x^{6} + 1}}{6 \, x^{6}} - \frac {1}{12} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) + \frac {1}{12} \, \log \left (\sqrt {x^{6} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.33, size = 23, normalized size = 0.74 \begin {gather*} -\frac {\mathrm {atanh}\left (\sqrt {x^6+1}\right )}{6}-\frac {\sqrt {x^6+1}}{6\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.35, size = 24, normalized size = 0.77 \begin {gather*} - \frac {\operatorname {asinh}{\left (\frac {1}{x^{3}} \right )}}{6} - \frac {\sqrt {1 + \frac {1}{x^{6}}}}{6 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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