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 = 0.0194266, antiderivative size = 46, normalized size of antiderivative = 1., 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} \[ \frac{\log \left (x^2+\sqrt{2} x+1\right )}{2 \sqrt{2}}-\frac{\log \left (x^2-\sqrt{2} x+1\right )}{2 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1165
Rule 628
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*}
Mathematica [A] time = 0.0109317, size = 40, normalized size = 0.87 \[ \frac{\log \left (x^2+\sqrt{2} x+1\right )-\log \left (-x^2+\sqrt{2} x-1\right )}{2 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 62, normalized size = 1.4 \begin{align*}{\frac{\sqrt{2}}{8}\ln \left ({\frac{1+{x}^{2}+x\sqrt{2}}{1+{x}^{2}-x\sqrt{2}}} \right ) }-{\frac{\sqrt{2}}{8}\ln \left ({\frac{1+{x}^{2}-x\sqrt{2}}{1+{x}^{2}+x\sqrt{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45449, size = 46, normalized size = 1. \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36964, size = 92, normalized size = 2. \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.097355, size = 39, normalized size = 0.85 \begin{align*} - \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10006, size = 46, normalized size = 1. \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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