Optimal. Leaf size=72 \[ \frac{1}{2} \log \left (x^2+\left (1+\sqrt{3}\right ) x-\sqrt{3}+2\right )+\sqrt{\frac{1}{23} \left (13+8 \sqrt{3}\right )} \tanh ^{-1}\left (\frac{2 x+\sqrt{3}+1}{\sqrt{2 \left (3 \sqrt{3}-2\right )}}\right ) \]
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Rubi [A] time = 0.201693, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{1}{2} \log \left (x^2+\left (1+\sqrt{3}\right ) x-\sqrt{3}+2\right )+\sqrt{\frac{1}{23} \left (13+8 \sqrt{3}\right )} \tanh ^{-1}\left (\frac{2 x+\sqrt{3}+1}{\sqrt{2 \left (3 \sqrt{3}-2\right )}}\right ) \]
Antiderivative was successfully verified.
[In] Int[x/(2 - Sqrt[3] + (1 + Sqrt[3])*x + x^2),x]
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Rubi in Sympy [A] time = 4.72877, size = 78, normalized size = 1.08 \[ \frac{\log{\left (x^{2} + x \left (1 + \sqrt{3}\right ) - \sqrt{3} + 2 \right )}}{2} + \frac{\sqrt{2} \left (\frac{1}{2} + \frac{\sqrt{3}}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{2} \left (x + \frac{1}{2} + \frac{\sqrt{3}}{2}\right )}{\sqrt{-2 + 3 \sqrt{3}}} \right )}}{\sqrt{-2 + 3 \sqrt{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(2+x**2-3**(1/2)+x*(1+3**(1/2))),x)
[Out]
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Mathematica [A] time = 0.160082, size = 72, normalized size = 1. \[ \frac{1}{2} \log \left (x^2+\sqrt{3} x+x-\sqrt{3}+2\right )+\frac{\left (1+\sqrt{3}\right ) \tanh ^{-1}\left (\frac{2 x+\sqrt{3}+1}{\sqrt{6 \sqrt{3}-4}}\right )}{\sqrt{6 \sqrt{3}-4}} \]
Antiderivative was successfully verified.
[In] Integrate[x/(2 - Sqrt[3] + (1 + Sqrt[3])*x + x^2),x]
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Maple [A] time = 0.023, size = 82, normalized size = 1.1 \[{\frac{\ln \left ( x\sqrt{3}+{x}^{2}-\sqrt{3}+x+2 \right ) }{2}}+{\frac{1}{\sqrt{-4+6\,\sqrt{3}}}{\it Artanh} \left ({\frac{1+2\,x+\sqrt{3}}{\sqrt{-4+6\,\sqrt{3}}}} \right ) }+{\frac{\sqrt{3}}{\sqrt{-4+6\,\sqrt{3}}}{\it Artanh} \left ({\frac{1+2\,x+\sqrt{3}}{\sqrt{-4+6\,\sqrt{3}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(2+x^2-3^(1/2)+x*(1+3^(1/2))),x)
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Maxima [A] time = 0.828038, size = 104, normalized size = 1.44 \[ -\frac{{\left (\sqrt{3} + 1\right )} \log \left (\frac{2 \, x + \sqrt{3} - \sqrt{6 \, \sqrt{3} - 4} + 1}{2 \, x + \sqrt{3} + \sqrt{6 \, \sqrt{3} - 4} + 1}\right )}{2 \, \sqrt{6 \, \sqrt{3} - 4}} + \frac{1}{2} \, \log \left (x^{2} + x{\left (\sqrt{3} + 1\right )} - \sqrt{3} + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(x^2 + x*(sqrt(3) + 1) - sqrt(3) + 2),x, algorithm="maxima")
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(x^2 + x*(sqrt(3) + 1) - sqrt(3) + 2),x, algorithm="fricas")
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Sympy [A] time = 4.22278, size = 168, normalized size = 2.33 \[ \left (\frac{1}{2} - \frac{\sqrt{11 + 64 \sqrt{3}}}{2 \left (-31 + 12 \sqrt{3}\right )}\right ) \log{\left (x - \frac{-521 + 287 \sqrt{3}}{11 + 64 \sqrt{3}} + \frac{\left (\frac{1}{2} - \frac{\sqrt{11 + 64 \sqrt{3}}}{2 \left (-31 + 12 \sqrt{3}\right )}\right ) \left (269 + 459 \sqrt{3}\right )}{214 + 139 \sqrt{3}} \right )} + \left (\frac{\sqrt{11 + 64 \sqrt{3}}}{2 \left (-31 + 12 \sqrt{3}\right )} + \frac{1}{2}\right ) \log{\left (x + \frac{\left (269 + 459 \sqrt{3}\right ) \left (\frac{\sqrt{11 + 64 \sqrt{3}}}{2 \left (-31 + 12 \sqrt{3}\right )} + \frac{1}{2}\right )}{214 + 139 \sqrt{3}} - \frac{-521 + 287 \sqrt{3}}{11 + 64 \sqrt{3}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(2+x**2-3**(1/2)+x*(1+3**(1/2))),x)
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GIAC/XCAS [A] time = 0.285437, size = 108, normalized size = 1.5 \[ -\frac{{\left (\sqrt{3} + 1\right )}{\rm ln}\left (\frac{{\left | 2 \, x + \sqrt{3} - \sqrt{6 \, \sqrt{3} - 4} + 1 \right |}}{{\left | 2 \, x + \sqrt{3} + \sqrt{6 \, \sqrt{3} - 4} + 1 \right |}}\right )}{2 \, \sqrt{6 \, \sqrt{3} - 4}} + \frac{1}{2} \,{\rm ln}\left ({\left | x^{2} + x{\left (\sqrt{3} + 1\right )} - \sqrt{3} + 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(x^2 + x*(sqrt(3) + 1) - sqrt(3) + 2),x, algorithm="giac")
[Out]