Optimal. Leaf size=48 \[ x-\frac{1}{6} \left (9-5 \sqrt{6}\right ) \log \left (x-\sqrt{6}+1\right )-\frac{1}{6} \left (9+5 \sqrt{6}\right ) \log \left (x+\sqrt{6}+1\right ) \]
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Rubi [A] time = 0.0446019, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {1657, 632, 31} \[ x-\frac{1}{6} \left (9-5 \sqrt{6}\right ) \log \left (x-\sqrt{6}+1\right )-\frac{1}{6} \left (9+5 \sqrt{6}\right ) \log \left (x+\sqrt{6}+1\right ) \]
Antiderivative was successfully verified.
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Rule 1657
Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{2-x+x^2}{-5+2 x+x^2} \, dx &=\int \left (1+\frac{7-3 x}{-5+2 x+x^2}\right ) \, dx\\ &=x+\int \frac{7-3 x}{-5+2 x+x^2} \, dx\\ &=x+\frac{1}{6} \left (-9+5 \sqrt{6}\right ) \int \frac{1}{1-\sqrt{6}+x} \, dx-\frac{1}{6} \left (9+5 \sqrt{6}\right ) \int \frac{1}{1+\sqrt{6}+x} \, dx\\ &=x-\frac{1}{6} \left (9-5 \sqrt{6}\right ) \log \left (1-\sqrt{6}+x\right )-\frac{1}{6} \left (9+5 \sqrt{6}\right ) \log \left (1+\sqrt{6}+x\right )\\ \end{align*}
Mathematica [A] time = 0.038973, size = 48, normalized size = 1. \[ x+\frac{1}{6} \left (5 \sqrt{6}-9\right ) \log \left (-x+\sqrt{6}-1\right )+\frac{1}{6} \left (-9-5 \sqrt{6}\right ) \log \left (x+\sqrt{6}+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 30, normalized size = 0.6 \begin{align*} x-{\frac{3\,\ln \left ({x}^{2}+2\,x-5 \right ) }{2}}-{\frac{5\,\sqrt{6}}{3}{\it Artanh} \left ({\frac{ \left ( 2\,x+2 \right ) \sqrt{6}}{12}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.57666, size = 49, normalized size = 1.02 \begin{align*} \frac{5}{6} \, \sqrt{6} \log \left (\frac{x - \sqrt{6} + 1}{x + \sqrt{6} + 1}\right ) + x - \frac{3}{2} \, \log \left (x^{2} + 2 \, x - 5\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76941, size = 157, normalized size = 3.27 \begin{align*} \frac{5}{6} \, \sqrt{3} \sqrt{2} \log \left (-\frac{2 \, \sqrt{3} \sqrt{2}{\left (x + 1\right )} - x^{2} - 2 \, x - 7}{x^{2} + 2 \, x - 5}\right ) + x - \frac{3}{2} \, \log \left (x^{2} + 2 \, x - 5\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.112266, size = 46, normalized size = 0.96 \begin{align*} x + \left (- \frac{5 \sqrt{6}}{6} - \frac{3}{2}\right ) \log{\left (x + 1 + \sqrt{6} \right )} + \left (- \frac{3}{2} + \frac{5 \sqrt{6}}{6}\right ) \log{\left (x - \sqrt{6} + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16208, size = 61, normalized size = 1.27 \begin{align*} \frac{5}{6} \, \sqrt{6} \log \left (\frac{{\left | 2 \, x - 2 \, \sqrt{6} + 2 \right |}}{{\left | 2 \, x + 2 \, \sqrt{6} + 2 \right |}}\right ) + x - \frac{3}{2} \, \log \left ({\left | x^{2} + 2 \, x - 5 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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