Optimal. Leaf size=50 \[ \frac {\log \left (2 x^2+\sqrt {3} x+1\right )}{2 \sqrt {3}}-\frac {\log \left (2 x^2-\sqrt {3} x+1\right )}{2 \sqrt {3}} \]
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Rubi [A] time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1164, 628} \begin {gather*} \frac {\log \left (2 x^2+\sqrt {3} x+1\right )}{2 \sqrt {3}}-\frac {\log \left (2 x^2-\sqrt {3} x+1\right )}{2 \sqrt {3}} \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+x^2+4 x^4} \, dx &=-\frac {\int \frac {\frac {\sqrt {3}}{2}+2 x}{-\frac {1}{2}-\frac {\sqrt {3} x}{2}-x^2} \, dx}{2 \sqrt {3}}-\frac {\int \frac {\frac {\sqrt {3}}{2}-2 x}{-\frac {1}{2}+\frac {\sqrt {3} x}{2}-x^2} \, dx}{2 \sqrt {3}}\\ &=-\frac {\log \left (1-\sqrt {3} x+2 x^2\right )}{2 \sqrt {3}}+\frac {\log \left (1+\sqrt {3} x+2 x^2\right )}{2 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 42, normalized size = 0.84 \begin {gather*} \frac {\log \left (2 x^2+\sqrt {3} x+1\right )-\log \left (-2 x^2+\sqrt {3} x-1\right )}{2 \sqrt {3}} \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+x^2+4 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.81, size = 43, normalized size = 0.86 \begin {gather*} \frac {1}{6} \, \sqrt {3} \log \left (\frac {4 \, x^{4} + 7 \, x^{2} + 2 \, \sqrt {3} {\left (2 \, x^{3} + x\right )} + 1}{4 \, x^{4} + x^{2} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 41, normalized size = 0.82 \begin {gather*} \frac {1}{6} \, \sqrt {3} \log \left (x^{2} + \frac {1}{2} \, \sqrt {6} \left (\frac {1}{4}\right )^{\frac {1}{4}} x + \frac {1}{2}\right ) - \frac {1}{6} \, \sqrt {3} \log \left (x^{2} - \frac {1}{2} \, \sqrt {6} \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 = 0.01, size = 39, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {3}\, \ln \left (2 x^{2}-\sqrt {3}\, x +1\right )}{6}+\frac {\sqrt {3}\, \ln \left (2 x^{2}+\sqrt {3}\, x +1\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {2 \, x^{2} - 1}{4 \, x^{4} + x^{2} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 20, normalized size = 0.40 \begin {gather*} \frac {\sqrt {3}\,\mathrm {atanh}\left (\frac {\sqrt {3}\,x}{2\,x^2+1}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 46, normalized size = 0.92 \begin {gather*} - \frac {\sqrt {3} \log {\left (x^{2} - \frac {\sqrt {3} x}{2} + \frac {1}{2} \right )}}{6} + \frac {\sqrt {3} \log {\left (x^{2} + \frac {\sqrt {3} x}{2} + \frac {1}{2} \right )}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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