Optimal. Leaf size=31 \[ \frac {\sqrt {x^6+1}}{6 x^6}-\frac {1}{2} \tanh ^{-1}\left (\sqrt {x^6+1}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {446, 78, 63, 207} \begin {gather*} \frac {\sqrt {x^6+1}}{6 x^6}-\frac {1}{2} \tanh ^{-1}\left (\sqrt {x^6+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 78
Rule 207
Rule 446
Rubi steps
\begin {align*} \int \frac {-1+x^6}{x^7 \sqrt {1+x^6}} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {-1+x}{x^2 \sqrt {1+x}} \, dx,x,x^6\right )\\ &=\frac {\sqrt {1+x^6}}{6 x^6}+\frac {1}{4} \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}{2} \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}{2} \tanh ^{-1}\left (\sqrt {1+x^6}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 0.97 \begin {gather*} \frac {1}{6} \left (\frac {\sqrt {x^6+1}}{x^6}-3 \tanh ^{-1}\left (\sqrt {x^6+1}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 31, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x^6+1}}{6 x^6}-\frac {1}{2} \tanh ^{-1}\left (\sqrt {x^6+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 45, normalized size = 1.45 \begin {gather*} -\frac {3 \, x^{6} \log \left (\sqrt {x^{6} + 1} + 1\right ) - 3 \, 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 = 0.32, size = 37, normalized size = 1.19 \begin {gather*} \frac {\sqrt {x^{6} + 1}}{6 \, x^{6}} - \frac {1}{4} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) + \frac {1}{4} \, \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 = 0.04, 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 )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 37, normalized size = 1.19 \begin {gather*} \frac {\sqrt {x^{6} + 1}}{6 \, x^{6}} - \frac {1}{4} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) + \frac {1}{4} \, \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 = 0.52, size = 23, normalized size = 0.74 \begin {gather*} \frac {\sqrt {x^6+1}}{6\,x^6}-\frac {\mathrm {atanh}\left (\sqrt {x^6+1}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 98.97, size = 56, normalized size = 1.81 \begin {gather*} \frac {\log {\left (-1 + \frac {1}{\sqrt {x^{6} + 1}} \right )}}{4} - \frac {\log {\left (1 + \frac {1}{\sqrt {x^{6} + 1}} \right )}}{4} - \frac {1}{12 \left (1 + \frac {1}{\sqrt {x^{6} + 1}}\right )} - \frac {1}{12 \left (-1 + \frac {1}{\sqrt {x^{6} + 1}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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