Optimal. Leaf size=14 \[ -\frac {1}{4} \tanh ^{-1}\left (\sqrt {x^8+1}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {266, 63, 207} \[ -\frac {1}{4} \tanh ^{-1}\left (\sqrt {x^8+1}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 207
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {1+x^8}} \, dx &=\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^8\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^8}\right )\\ &=-\frac {1}{4} \tanh ^{-1}\left (\sqrt {1+x^8}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 14, normalized size = 1.00 \[ -\frac {1}{4} \tanh ^{-1}\left (\sqrt {x^8+1}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 25, normalized size = 1.79 \[ -\frac {1}{8} \, \log \left (\sqrt {x^{8} + 1} + 1\right ) + \frac {1}{8} \, \log \left (\sqrt {x^{8} + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.78, size = 25, normalized size = 1.79 \[ -\frac {1}{8} \, \log \left (\sqrt {x^{8} + 1} + 1\right ) + \frac {1}{8} \, \log \left (\sqrt {x^{8} + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 19, normalized size = 1.36 \[ \frac {\ln \left (\frac {\sqrt {x^{8}+1}-1}{\sqrt {x^{8}}}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 25, normalized size = 1.79 \[ -\frac {1}{8} \, \log \left (\sqrt {x^{8} + 1} + 1\right ) + \frac {1}{8} \, \log \left (\sqrt {x^{8} + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 10, normalized size = 0.71 \[ -\frac {\mathrm {atanh}\left (\sqrt {x^8+1}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.92, size = 8, normalized size = 0.57 \[ - \frac {\operatorname {asinh}{\left (\frac {1}{x^{4}} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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