Optimal. Leaf size=50 \[ \frac {\log \left (2 x^2+\sqrt {2} x+1\right )}{2 \sqrt {2}}-\frac {\log \left (2 x^2-\sqrt {2} x+1\right )}{2 \sqrt {2}} \]
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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 = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1164, 628} \[ \frac {\log \left (2 x^2+\sqrt {2} x+1\right )}{2 \sqrt {2}}-\frac {\log \left (2 x^2-\sqrt {2} x+1\right )}{2 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 628
Rule 1164
Rubi steps
\begin {align*} \int \frac {1-2 x^2}{1+2 x^2+4 x^4} \, dx &=-\frac {\int \frac {\frac {1}{\sqrt {2}}+2 x}{-\frac {1}{2}-\frac {x}{\sqrt {2}}-x^2} \, dx}{2 \sqrt {2}}-\frac {\int \frac {\frac {1}{\sqrt {2}}-2 x}{-\frac {1}{2}+\frac {x}{\sqrt {2}}-x^2} \, dx}{2 \sqrt {2}}\\ &=-\frac {\log \left (1-\sqrt {2} x+2 x^2\right )}{2 \sqrt {2}}+\frac {\log \left (1+\sqrt {2} x+2 x^2\right )}{2 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 42, normalized size = 0.84 \[ \frac {\log \left (2 x^2+\sqrt {2} x+1\right )-\log \left (-2 x^2+\sqrt {2} x-1\right )}{2 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 45, normalized size = 0.90 \[ \frac {1}{4} \, \sqrt {2} \log \left (\frac {4 \, x^{4} + 6 \, x^{2} + 2 \, \sqrt {2} {\left (2 \, x^{3} + x\right )} + 1}{4 \, x^{4} + 2 \, x^{2} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 34, normalized size = 0.68 \[ \frac {1}{4} \, \sqrt {2} \log \left (x^{2} + \left (\frac {1}{4}\right )^{\frac {1}{4}} x + \frac {1}{2}\right ) - \frac {1}{4} \, \sqrt {2} \log \left (x^{2} - \left (\frac {1}{4}\right )^{\frac {1}{4}} x + \frac {1}{2}\right ) \]
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 \[ -\frac {\sqrt {2}\, \ln \left (2 x^{2}-\sqrt {2}\, x +1\right )}{4}+\frac {\sqrt {2}\, \ln \left (2 x^{2}+\sqrt {2}\, x +1\right )}{4} \]
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 \[ -\int \frac {2 \, x^{2} - 1}{4 \, x^{4} + 2 \, x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.37, size = 20, normalized size = 0.40 \[ \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,x}{2\,x^2+1}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 46, normalized size = 0.92 \[ - \frac {\sqrt {2} \log {\left (x^{2} - \frac {\sqrt {2} x}{2} + \frac {1}{2} \right )}}{4} + \frac {\sqrt {2} \log {\left (x^{2} + \frac {\sqrt {2} x}{2} + \frac {1}{2} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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