Optimal. Leaf size=30 \[ -\frac {1}{2} \tan ^{-1}\left (\frac {2-x^2}{2 \sqrt {-x^4+x^2-1}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1114, 724, 204} \[ -\frac {1}{2} \tan ^{-1}\left (\frac {2-x^2}{2 \sqrt {-x^4+x^2-1}}\right ) \]
Antiderivative was successfully verified.
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Rule 204
Rule 724
Rule 1114
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {-1+x^2-x^4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {-1+x-x^2}} \, dx,x,x^2\right )\\ &=-\operatorname {Subst}\left (\int \frac {1}{-4-x^2} \, dx,x,\frac {-2+x^2}{\sqrt {-1+x^2-x^4}}\right )\\ &=\frac {1}{2} \tan ^{-1}\left (\frac {-2+x^2}{2 \sqrt {-1+x^2-x^4}}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 28, normalized size = 0.93 \[ \frac {1}{2} \tan ^{-1}\left (\frac {x^2-2}{2 \sqrt {-x^4+x^2-1}}\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 0.41, size = 55, normalized size = 1.83 \[ \frac {1}{4} i \, \log \left (\frac {x^{2} + 2 i \, \sqrt {-x^{4} + x^{2} - 1} - 2}{2 \, x^{2}}\right ) - \frac {1}{4} i \, \log \left (\frac {x^{2} - 2 i \, \sqrt {-x^{4} + x^{2} - 1} - 2}{2 \, x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-x^{4} + x^{2} - 1} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 23, normalized size = 0.77 \[ \frac {\arctan \left (\frac {x^{2}-2}{2 \sqrt {-x^{4}+x^{2}-1}}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.01, size = 17, normalized size = 0.57 \[ -\frac {1}{2} i \, \operatorname {arsinh}\left (-\frac {1}{3} \, \sqrt {3} + \frac {2 \, \sqrt {3}}{3 \, x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.54, size = 32, normalized size = 1.07 \[ \frac {\ln \left (\frac {1}{x^2}\right )\,1{}\mathrm {i}}{2}+\frac {\ln \left (x^2-2+\sqrt {-x^4+x^2-1}\,2{}\mathrm {i}\right )\,1{}\mathrm {i}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {- x^{4} + x^{2} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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