Optimal. Leaf size=45 \[ \frac{1}{4} \left (2+\sqrt{2}\right ) \log \left (x-\sqrt{2}+1\right )+\frac{1}{4} \left (2-\sqrt{2}\right ) \log \left (x+\sqrt{2}+1\right ) \]
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Rubi [A] time = 0.0209227, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2055, 632, 31} \[ \frac{1}{4} \left (2+\sqrt{2}\right ) \log \left (x-\sqrt{2}+1\right )+\frac{1}{4} \left (2-\sqrt{2}\right ) \log \left (x+\sqrt{2}+1\right ) \]
Antiderivative was successfully verified.
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Rule 2055
Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{-4+x^2}{2-5 x+x^3} \, dx &=\int \frac{2+x}{-1+2 x+x^2} \, dx\\ &=-\left (\frac{1}{4} \left (-2+\sqrt{2}\right ) \int \frac{1}{1+\sqrt{2}+x} \, dx\right )+\frac{1}{4} \left (2+\sqrt{2}\right ) \int \frac{1}{1-\sqrt{2}+x} \, dx\\ &=\frac{1}{4} \left (2+\sqrt{2}\right ) \log \left (1-\sqrt{2}+x\right )+\frac{1}{4} \left (2-\sqrt{2}\right ) \log \left (1+\sqrt{2}+x\right )\\ \end{align*}
Mathematica [A] time = 0.0051534, size = 42, normalized size = 0.93 \[ \frac{1}{4} \left (\left (2+\sqrt{2}\right ) \log \left (-x+\sqrt{2}-1\right )-\left (\sqrt{2}-2\right ) \log \left (x+\sqrt{2}+1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 29, normalized size = 0.6 \begin{align*}{\frac{\ln \left ({x}^{2}+2\,x-1 \right ) }{2}}-{\frac{\sqrt{2}}{2}{\it Artanh} \left ({\frac{ \left ( 2+2\,x \right ) \sqrt{2}}{4}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45851, size = 47, normalized size = 1.04 \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (\frac{x - \sqrt{2} + 1}{x + \sqrt{2} + 1}\right ) + \frac{1}{2} \, \log \left (x^{2} + 2 \, x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.959254, size = 128, normalized size = 2.84 \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (\frac{x^{2} - 2 \, \sqrt{2}{\left (x + 1\right )} + 2 \, x + 3}{x^{2} + 2 \, x - 1}\right ) + \frac{1}{2} \, \log \left (x^{2} + 2 \, x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.108295, size = 39, normalized size = 0.87 \begin{align*} \left (\frac{1}{2} - \frac{\sqrt{2}}{4}\right ) \log{\left (x + 1 + \sqrt{2} \right )} + \left (\frac{\sqrt{2}}{4} + \frac{1}{2}\right ) \log{\left (x - \sqrt{2} + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11646, size = 59, normalized size = 1.31 \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (\frac{{\left | 2 \, x - 2 \, \sqrt{2} + 2 \right |}}{{\left | 2 \, x + 2 \, \sqrt{2} + 2 \right |}}\right ) + \frac{1}{2} \, \log \left ({\left | x^{2} + 2 \, x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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