Optimal. Leaf size=14 \[ \frac {1}{2} \tan ^{-1}\left (\sqrt {x^4-1}\right ) \]
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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, 203} \[ \frac {1}{2} \tan ^{-1}\left (\sqrt {x^4-1}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 203
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 )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^4}\right )\\ &=\frac {1}{2} \tan ^{-1}\left (\sqrt {-1+x^4}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 14, normalized size = 1.00 \[ \frac {1}{2} \tan ^{-1}\left (\sqrt {x^4-1}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 10, normalized size = 0.71 \[ \frac {1}{2} \, \arctan \left (\sqrt {x^{4} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 10, normalized size = 0.71 \[ \frac {1}{2} \, \arctan \left (\sqrt {x^{4} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 11, normalized size = 0.79 \[ -\frac {\arctan \left (\frac {1}{\sqrt {x^{4}-1}}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.00, size = 10, normalized size = 0.71 \[ \frac {1}{2} \, \arctan \left (\sqrt {x^{4} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 10, normalized size = 0.71 \[ \frac {\mathrm {atan}\left (\sqrt {x^4-1}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.56, size = 24, normalized size = 1.71 \[ \begin {cases} \frac {i \operatorname {acosh}{\left (\frac {1}{x^{2}} \right )}}{2} & \text {for}\: \frac {1}{\left |{x^{4}}\right |} > 1 \\- \frac {\operatorname {asin}{\left (\frac {1}{x^{2}} \right )}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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