Optimal. Leaf size=65 \[ \frac{1}{2} \log \left (-2 x-\sqrt{5}+1\right )+\frac{1}{2} \log \left (-2 x+\sqrt{5}+1\right )-\frac{1}{2} \log \left (2 x-\sqrt{5}+1\right )-\frac{1}{2} \log \left (2 x+\sqrt{5}+1\right ) \]
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Rubi [A] time = 0.0319407, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1161, 616, 31} \[ \frac{1}{2} \log \left (-2 x-\sqrt{5}+1\right )+\frac{1}{2} \log \left (-2 x+\sqrt{5}+1\right )-\frac{1}{2} \log \left (2 x-\sqrt{5}+1\right )-\frac{1}{2} \log \left (2 x+\sqrt{5}+1\right ) \]
Antiderivative was successfully verified.
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Rule 1161
Rule 616
Rule 31
Rubi steps
\begin{align*} \int \frac{1+x^2}{1-3 x^2+x^4} \, dx &=\frac{1}{2} \int \frac{1}{1-\sqrt{5} x+x^2} \, dx+\frac{1}{2} \int \frac{1}{1+\sqrt{5} x+x^2} \, dx\\ &=\frac{1}{2} \int \frac{1}{\frac{1}{2} \left (-1-\sqrt{5}\right )+x} \, dx-\frac{1}{2} \int \frac{1}{\frac{1}{2} \left (1-\sqrt{5}\right )+x} \, dx+\frac{1}{2} \int \frac{1}{\frac{1}{2} \left (-1+\sqrt{5}\right )+x} \, dx-\frac{1}{2} \int \frac{1}{\frac{1}{2} \left (1+\sqrt{5}\right )+x} \, dx\\ &=\frac{1}{2} \log \left (1-\sqrt{5}-2 x\right )+\frac{1}{2} \log \left (1+\sqrt{5}-2 x\right )-\frac{1}{2} \log \left (1-\sqrt{5}+2 x\right )-\frac{1}{2} \log \left (1+\sqrt{5}+2 x\right )\\ \end{align*}
Mathematica [A] time = 0.0057495, size = 29, normalized size = 0.45 \[ \frac{1}{2} \log \left (-x^2+x+1\right )-\frac{1}{2} \log \left (-x^2-x+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 22, normalized size = 0.3 \begin{align*} -{\frac{\ln \left ({x}^{2}+x-1 \right ) }{2}}+{\frac{\ln \left ({x}^{2}-x-1 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.953934, size = 28, normalized size = 0.43 \begin{align*} -\frac{1}{2} \, \log \left (x^{2} + x - 1\right ) + \frac{1}{2} \, \log \left (x^{2} - x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27954, size = 62, normalized size = 0.95 \begin{align*} -\frac{1}{2} \, \log \left (x^{2} + x - 1\right ) + \frac{1}{2} \, \log \left (x^{2} - x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.097323, size = 19, normalized size = 0.29 \begin{align*} \frac{\log{\left (x^{2} - x - 1 \right )}}{2} - \frac{\log{\left (x^{2} + x - 1 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13753, size = 58, normalized size = 0.89 \begin{align*} -\frac{1}{4} \, \log \left ({\left | x + \frac{1}{x - \frac{1}{x}} - \frac{1}{x} + 2 \right |}\right ) + \frac{1}{4} \, \log \left ({\left | x + \frac{1}{x - \frac{1}{x}} - \frac{1}{x} - 2 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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