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

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 = 0.103707, 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, Rules used = {634, 618, 206, 628} \[ \frac{1}{2} \log \left (x^2+\left (1+\sqrt{3}\right ) x-\sqrt{3}+2\right )+\sqrt{\frac{1}{23} \left (13+8 \sqrt{3}\right )} \tanh ^{-1}\left (\frac{2 x+\sqrt{3}+1}{\sqrt{2 \left (3 \sqrt{3}-2\right )}}\right ) \]

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

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{x}{2-\sqrt{3}+\left (1+\sqrt{3}\right ) x+x^2} \, dx &=\frac{1}{2} \int \frac{1+\sqrt{3}+2 x}{2-\sqrt{3}+\left (1+\sqrt{3}\right ) x+x^2} \, dx+\frac{1}{2} \left (-1-\sqrt{3}\right ) \int \frac{1}{2-\sqrt{3}+\left (1+\sqrt{3}\right ) x+x^2} \, dx\\ &=\frac{1}{2} \log \left (2-\sqrt{3}+\left (1+\sqrt{3}\right ) x+x^2\right )+\left (1+\sqrt{3}\right ) \operatorname{Subst}\left (\int \frac{1}{-2 \left (2-3 \sqrt{3}\right )-x^2} \, dx,x,1+\sqrt{3}+2 x\right )\\ &=\sqrt{\frac{1}{23} \left (13+8 \sqrt{3}\right )} \tanh ^{-1}\left (\frac{1+\sqrt{3}+2 x}{\sqrt{2 \left (-2+3 \sqrt{3}\right )}}\right )+\frac{1}{2} \log \left (2-\sqrt{3}+\left (1+\sqrt{3}\right ) x+x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0925707, 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*Sqrt[3]] + Log[2 - Sqrt[3] + x +
Sqrt[3]*x + x^2]/2

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Maple [A]  time = 0.013, size = 82, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( x\sqrt{3}+{x}^{2}-\sqrt{3}+x+2 \right ) }{2}}+{\frac{\sqrt{3}}{\sqrt{-4+6\,\sqrt{3}}}{\it Artanh} \left ({\frac{1+2\,x+\sqrt{3}}{\sqrt{-4+6\,\sqrt{3}}}} \right ) }+{\frac{1}{\sqrt{-4+6\,\sqrt{3}}}{\it Artanh} \left ({\frac{1+2\,x+\sqrt{3}}{\sqrt{-4+6\,\sqrt{3}}}} \right ) } \end{align*}

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

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Maxima [A]  time = 1.83463, size = 104, normalized size = 1.44 \begin{align*} -\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 ) \end{align*}

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

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Fricas [A]  time = 1.58244, size = 320, normalized size = 4.44 \begin{align*} \frac{1}{46} \, \sqrt{23} \sqrt{8 \, \sqrt{3} + 13} \log \left (-\frac{\sqrt{23} \sqrt{8 \, \sqrt{3} + 13}{\left (5 \, \sqrt{3} - 11\right )} - 46 \, x - 23 \, \sqrt{3} - 23}{\sqrt{23} \sqrt{8 \, \sqrt{3} + 13}{\left (5 \, \sqrt{3} - 11\right )} + 46 \, x + 23 \, \sqrt{3} + 23}\right ) + \frac{1}{2} \, \log \left (x^{2} + x{\left (\sqrt{3} + 1\right )} - \sqrt{3} + 2\right ) \end{align*}

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, algorithm="fricas")

[Out]

1/46*sqrt(23)*sqrt(8*sqrt(3) + 13)*log(-(sqrt(23)*sqrt(8*sqrt(3) + 13)*(5*sqrt(3) - 11) - 46*x - 23*sqrt(3) -
23)/(sqrt(23)*sqrt(8*sqrt(3) + 13)*(5*sqrt(3) - 11) + 46*x + 23*sqrt(3) + 23)) + 1/2*log(x^2 + x*(sqrt(3) + 1)
- sqrt(3) + 2)

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Sympy [B]  time = 1.0805, size = 168, normalized size = 2.33 \begin{align*} \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 )} \end{align*}

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 - s
qrt(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 [A]  time = 1.19527, size = 108, normalized size = 1.5 \begin{align*} -\frac{{\left (\sqrt{3} + 1\right )} \log \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} \, \log \left ({\left | x^{2} + x{\left (\sqrt{3} + 1\right )} - \sqrt{3} + 2 \right |}\right ) \end{align*}

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, algorithm="giac")

[Out]

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