Optimal. Leaf size=29 \[ \frac {1}{2} \log \left (2 x^2+x+1\right )-\frac {1}{2} \log \left (2 x^2-x+1\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1164, 628} \begin {gather*} \frac {1}{2} \log \left (2 x^2+x+1\right )-\frac {1}{2} \log \left (2 x^2-x+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 628
Rule 1164
Rubi steps
\begin {align*} \int \frac {1-2 x^2}{1+3 x^2+4 x^4} \, dx &=-\left (\frac {1}{2} \int \frac {\frac {1}{2}+2 x}{-\frac {1}{2}-\frac {x}{2}-x^2} \, dx\right )-\frac {1}{2} \int \frac {\frac {1}{2}-2 x}{-\frac {1}{2}+\frac {x}{2}-x^2} \, dx\\ &=-\frac {1}{2} \log \left (1-x+2 x^2\right )+\frac {1}{2} \log \left (1+x+2 x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 1.00 \begin {gather*} \frac {1}{2} \log \left (2 x^2+x+1\right )-\frac {1}{2} \log \left (2 x^2-x+1\right ) \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+3 x^2+4 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.87, size = 25, normalized size = 0.86 \begin {gather*} \frac {1}{2} \, \log \left (2 \, x^{2} + x + 1\right ) - \frac {1}{2} \, \log \left (2 \, x^{2} - x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 25, normalized size = 0.86 \begin {gather*} \frac {1}{2} \, \log \left (2 \, x^{2} + x + 1\right ) - \frac {1}{2} \, \log \left (2 \, x^{2} - x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 26, normalized size = 0.90 \begin {gather*} -\frac {\ln \left (2 x^{2}-x +1\right )}{2}+\frac {\ln \left (2 x^{2}+x +1\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 25, normalized size = 0.86 \begin {gather*} \frac {1}{2} \, \log \left (2 \, x^{2} + x + 1\right ) - \frac {1}{2} \, \log \left (2 \, x^{2} - x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 12, normalized size = 0.41 \begin {gather*} \mathrm {atanh}\left (\frac {x}{2\,x^2+1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 26, normalized size = 0.90 \begin {gather*} - \frac {\log {\left (x^{2} - \frac {x}{2} + \frac {1}{2} \right )}}{2} + \frac {\log {\left (x^{2} + \frac {x}{2} + \frac {1}{2} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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