Optimal. Leaf size=16 \[ -\frac {1}{2} \tanh ^{-1}\left (\sqrt {1-x^4}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {266, 63, 206} \[ -\frac {1}{2} \tanh ^{-1}\left (\sqrt {1-x^4}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {1-x^4}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,x^4\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x^4}\right )\right )\\ &=-\frac {1}{2} \tanh ^{-1}\left (\sqrt {1-x^4}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 16, normalized size = 1.00 \[ -\frac {1}{2} \tanh ^{-1}\left (\sqrt {1-x^4}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.68, size = 29, normalized size = 1.81 \[ -\frac {1}{4} \, \log \left (\sqrt {-x^{4} + 1} + 1\right ) + \frac {1}{4} \, \log \left (\sqrt {-x^{4} + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 31, normalized size = 1.94 \[ -\frac {1}{4} \, \log \left (\sqrt {-x^{4} + 1} + 1\right ) + \frac {1}{4} \, \log \left (-\sqrt {-x^{4} + 1} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 13, normalized size = 0.81 \[ -\frac {\arctanh \left (\frac {1}{\sqrt {-x^{4}+1}}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.36, size = 29, normalized size = 1.81 \[ -\frac {1}{4} \, \log \left (\sqrt {-x^{4} + 1} + 1\right ) + \frac {1}{4} \, \log \left (\sqrt {-x^{4} + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.31, size = 12, normalized size = 0.75 \[ -\frac {\mathrm {atanh}\left (\sqrt {1-x^4}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.92, size = 24, normalized size = 1.50 \[ \begin {cases} - \frac {\operatorname {acosh}{\left (\frac {1}{x^{2}} \right )}}{2} & \text {for}\: \frac {1}{\left |{x^{4}}\right |} > 1 \\\frac {i \operatorname {asin}{\left (\frac {1}{x^{2}} \right )}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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