Optimal. Leaf size=31 \[ \frac {1}{4} \log \left (2 x^2+2 x+1\right )-\frac {1}{4} \log \left (2 x^2-2 x+1\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1165, 628} \begin {gather*} \frac {1}{4} \log \left (2 x^2+2 x+1\right )-\frac {1}{4} \log \left (2 x^2-2 x+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 628
Rule 1165
Rubi steps
\begin {align*} \int \frac {1-2 x^2}{1+4 x^4} \, dx &=-\left (\frac {1}{4} \int \frac {1+2 x}{-\frac {1}{2}-x-x^2} \, dx\right )-\frac {1}{4} \int \frac {1-2 x}{-\frac {1}{2}+x-x^2} \, dx\\ &=-\frac {1}{4} \log \left (1-2 x+2 x^2\right )+\frac {1}{4} \log \left (1+2 x+2 x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 31, normalized size = 1.00 \begin {gather*} \frac {1}{4} \log \left (2 x^2+2 x+1\right )-\frac {1}{4} \log \left (2 x^2-2 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+4 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.66, size = 27, normalized size = 0.87 \begin {gather*} \frac {1}{4} \, \log \left (2 \, x^{2} + 2 \, x + 1\right ) - \frac {1}{4} \, \log \left (2 \, x^{2} - 2 \, x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 34, normalized size = 1.10 \begin {gather*} \frac {1}{4} \, \log \left (x^{2} + \sqrt {2} \left (\frac {1}{4}\right )^{\frac {1}{4}} x + \frac {1}{2}\right ) - \frac {1}{4} \, \log \left (x^{2} - \sqrt {2} \left (\frac {1}{4}\right )^{\frac {1}{4}} x + \frac {1}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 28, normalized size = 0.90 \begin {gather*} -\frac {\ln \left (2 x^{2}-2 x +1\right )}{4}+\frac {\ln \left (2 x^{2}+2 x +1\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 27, normalized size = 0.87 \begin {gather*} \frac {1}{4} \, \log \left (2 \, x^{2} + 2 \, x + 1\right ) - \frac {1}{4} \, \log \left (2 \, x^{2} - 2 \, x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 15, normalized size = 0.48 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {2\,x}{2\,x^2+1}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 22, normalized size = 0.71 \begin {gather*} - \frac {\log {\left (x^{2} - x + \frac {1}{2} \right )}}{4} + \frac {\log {\left (x^{2} + x + \frac {1}{2} \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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