3.817 \(\int \frac{x}{2-\sqrt{3}+\left (1+\sqrt{3}\right ) x+x^2} \, dx\)

Optimal. Leaf size=72 \[ \frac{1}{2} \log \left (x^2+\left (1+\sqrt{3}\right ) x-\sqrt{3}+2\right )+\sqrt{\frac{1}{23} \left (13+8 \sqrt{3}\right )} \tanh ^{-1}\left (\frac{2 x+\sqrt{3}+1}{\sqrt{2 \left (3 \sqrt{3}-2\right )}}\right ) \]

[Out]

Sqrt[(13 + 8*Sqrt[3])/23]*ArcTanh[(1 + Sqrt[3] + 2*x)/Sqrt[2*(-2 + 3*Sqrt[3])]]
+ Log[2 - Sqrt[3] + (1 + Sqrt[3])*x + x^2]/2

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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]

[Out]

Sqrt[(13 + 8*Sqrt[3])/23]*ArcTanh[(1 + Sqrt[3] + 2*x)/Sqrt[2*(-2 + 3*Sqrt[3])]]
+ Log[2 - Sqrt[3] + (1 + Sqrt[3])*x + x^2]/2

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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]

log(x**2 + x*(1 + sqrt(3)) - sqrt(3) + 2)/2 + sqrt(2)*(1/2 + sqrt(3)/2)*atanh(sq
rt(2)*(x + 1/2 + sqrt(3)/2)/sqrt(-2 + 3*sqrt(3)))/sqrt(-2 + 3*sqrt(3))

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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]

[Out]

((1 + Sqrt[3])*ArcTanh[(1 + Sqrt[3] + 2*x)/Sqrt[-4 + 6*Sqrt[3]]])/Sqrt[-4 + 6*Sq
rt[3]] + Log[2 - Sqrt[3] + x + Sqrt[3]*x + x^2]/2

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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)

[Out]

1/2*ln(x*3^(1/2)+x^2-3^(1/2)+x+2)+1/(-4+6*3^(1/2))^(1/2)*arctanh((1+2*x+3^(1/2))
/(-4+6*3^(1/2))^(1/2))+1/(-4+6*3^(1/2))^(1/2)*arctanh((1+2*x+3^(1/2))/(-4+6*3^(1
/2))^(1/2))*3^(1/2)

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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")

[Out]

-1/2*(sqrt(3) + 1)*log((2*x + sqrt(3) - sqrt(6*sqrt(3) - 4) + 1)/(2*x + sqrt(3)
+ sqrt(6*sqrt(3) - 4) + 1))/sqrt(6*sqrt(3) - 4) + 1/2*log(x^2 + x*(sqrt(3) + 1)
- sqrt(3) + 2)

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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")

[Out]

Exception raised: NotImplementedError

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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)

[Out]

(1/2 - sqrt(11 + 64*sqrt(3))/(2*(-31 + 12*sqrt(3))))*log(x - (-521 + 287*sqrt(3)
)/(11 + 64*sqrt(3)) + (1/2 - sqrt(11 + 64*sqrt(3))/(2*(-31 + 12*sqrt(3))))*(269
+ 459*sqrt(3))/(214 + 139*sqrt(3))) + (sqrt(11 + 64*sqrt(3))/(2*(-31 + 12*sqrt(3
))) + 1/2)*log(x + (269 + 459*sqrt(3))*(sqrt(11 + 64*sqrt(3))/(2*(-31 + 12*sqrt(
3))) + 1/2)/(214 + 139*sqrt(3)) - (-521 + 287*sqrt(3))/(11 + 64*sqrt(3)))

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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]

-1/2*(sqrt(3) + 1)*ln(abs(2*x + sqrt(3) - sqrt(6*sqrt(3) - 4) + 1)/abs(2*x + sqr
t(3) + sqrt(6*sqrt(3) - 4) + 1))/sqrt(6*sqrt(3) - 4) + 1/2*ln(abs(x^2 + x*(sqrt(
3) + 1) - sqrt(3) + 2))