Optimal. Leaf size=14 \[ \frac {\tanh ^{-1}\left (\sqrt {2} x\right )}{\sqrt {2}} \]
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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 = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {28, 21, 206} \begin {gather*} \frac {\tanh ^{-1}\left (\sqrt {2} x\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 28
Rule 206
Rubi steps
\begin {align*} \int \frac {1-2 x^2}{1-4 x^2+4 x^4} \, dx &=4 \int \frac {1-2 x^2}{\left (-2+4 x^2\right )^2} \, dx\\ &=\int \frac {1}{1-2 x^2} \, dx\\ &=\frac {\tanh ^{-1}\left (\sqrt {2} x\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [B] time = 0.01, size = 32, normalized size = 2.29 \begin {gather*} \frac {\log \left (2 x+\sqrt {2}\right )-\log \left (\sqrt {2}-2 x\right )}{2 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-2 x^2}{1-4 x^2+4 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.61, size = 29, normalized size = 2.07 \begin {gather*} \frac {1}{4} \, \sqrt {2} \log \left (\frac {2 \, x^{2} + 2 \, \sqrt {2} x + 1}{2 \, x^{2} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 29, normalized size = 2.07 \begin {gather*} \frac {1}{4} \, \sqrt {2} \log \left ({\left | x + \frac {1}{2} \, \sqrt {2} \right |}\right ) - \frac {1}{4} \, \sqrt {2} \log \left ({\left | x - \frac {1}{2} \, \sqrt {2} \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 12, normalized size = 0.86 \begin {gather*} \frac {\sqrt {2}\, \arctanh \left (\sqrt {2}\, x \right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.35, size = 25, normalized size = 1.79 \begin {gather*} -\frac {1}{4} \, \sqrt {2} \log \left (\frac {2 \, x - \sqrt {2}}{2 \, x + \sqrt {2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.33, size = 11, normalized size = 0.79 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,x\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.11, size = 32, normalized size = 2.29 \begin {gather*} - \frac {\sqrt {2} \log {\left (x - \frac {\sqrt {2}}{2} \right )}}{4} + \frac {\sqrt {2} \log {\left (x + \frac {\sqrt {2}}{2} \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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