Optimal. Leaf size=46 \[ \frac {\log \left (x^2+\sqrt {2} x+1\right )}{2 \sqrt {2}}-\frac {\log \left (x^2-\sqrt {2} x+1\right )}{2 \sqrt {2}} \]
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Rubi [A] time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1165, 628} \begin {gather*} \frac {\log \left (x^2+\sqrt {2} x+1\right )}{2 \sqrt {2}}-\frac {\log \left (x^2-\sqrt {2} x+1\right )}{2 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 628
Rule 1165
Rubi steps
\begin {align*} \int \frac {1-x^2}{1+x^4} \, dx &=-\frac {\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx}{2 \sqrt {2}}\\ &=-\frac {\log \left (1-\sqrt {2} x+x^2\right )}{2 \sqrt {2}}+\frac {\log \left (1+\sqrt {2} x+x^2\right )}{2 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 40, normalized size = 0.87 \begin {gather*} \frac {\log \left (x^2+\sqrt {2} x+1\right )-\log \left (-x^2+\sqrt {2} x-1\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-x^2}{1+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.00, size = 34, normalized size = 0.74 \begin {gather*} \frac {1}{4} \, \sqrt {2} \log \left (\frac {x^{4} + 4 \, x^{2} + 2 \, \sqrt {2} {\left (x^{3} + x\right )} + 1}{x^{4} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 34, normalized size = 0.74 \begin {gather*} \frac {1}{4} \, \sqrt {2} \log \left (x^{2} + \sqrt {2} x + 1\right ) - \frac {1}{4} \, \sqrt {2} \log \left (x^{2} - \sqrt {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 = 62, normalized size = 1.35 \begin {gather*} -\frac {\sqrt {2}\, \ln \left (\frac {x^{2}-\sqrt {2}\, x +1}{x^{2}+\sqrt {2}\, x +1}\right )}{8}+\frac {\sqrt {2}\, \ln \left (\frac {x^{2}+\sqrt {2}\, x +1}{x^{2}-\sqrt {2}\, x +1}\right )}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.26, size = 34, normalized size = 0.74 \begin {gather*} \frac {1}{4} \, \sqrt {2} \log \left (x^{2} + \sqrt {2} x + 1\right ) - \frac {1}{4} \, \sqrt {2} \log \left (x^{2} - \sqrt {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 = 18, normalized size = 0.39 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,x}{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 = 39, normalized size = 0.85 \begin {gather*} - \frac {\sqrt {2} \log {\left (x^{2} - \sqrt {2} x + 1 \right )}}{4} + \frac {\sqrt {2} \log {\left (x^{2} + \sqrt {2} x + 1 \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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