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]

1/2*ln(2+x^2-3^(1/2)+x*(1+3^(1/2)))+1/23*arctanh((1+2*x+3^(1/2))/(-4+6*3^(1/2))^(1/2))*(299+184*3^(1/2))^(1/2)

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Rubi [A]  time = 0.10, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, 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 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 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 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]

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]

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*}

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Mathematica [A]  time = 0.10, size = 72, normalized size = 1.00 \[ \frac {1}{2} \log \left (x^2+\sqrt {3} x+x-\sqrt {3}+2\right )+\frac {\left (1+\sqrt {3}\right ) \tanh ^{-1}\left (\frac {2 x+\sqrt {3}+1}{\sqrt {6 \sqrt {3}-4}}\right )}{\sqrt {6 \sqrt {3}-4}} \]

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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fricas [A]  time = 0.62, size = 100, normalized size = 1.39 \[ \frac {1}{46} \, \sqrt {23} \sqrt {8 \, \sqrt {3} + 13} \log \left (-\frac {\sqrt {23} \sqrt {8 \, \sqrt {3} + 13} {\left (5 \, \sqrt {3} - 11\right )} - 46 \, x - 23 \, \sqrt {3} - 23}{\sqrt {23} \sqrt {8 \, \sqrt {3} + 13} {\left (5 \, \sqrt {3} - 11\right )} + 46 \, x + 23 \, \sqrt {3} + 23}\right ) + \frac {1}{2} \, \log \left (x^{2} + x {\left (\sqrt {3} + 1\right )} - \sqrt {3} + 2\right ) \]

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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giac [A]  time = 0.43, size = 80, normalized size = 1.11 \[ -\frac {{\left (\sqrt {3} + 1\right )} \log \left (\frac {{\left | 2 \, x + \sqrt {3} - \sqrt {6 \, \sqrt {3} - 4} + 1 \right |}}{{\left | 2 \, x + \sqrt {3} + \sqrt {6 \, \sqrt {3} - 4} + 1 \right |}}\right )}{2 \, \sqrt {6 \, \sqrt {3} - 4}} + \frac {1}{2} \, \log \left ({\left | x^{2} + x {\left (\sqrt {3} + 1\right )} - \sqrt {3} + 2 \right |}\right ) \]

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))

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maple [A]  time = 0.02, size = 82, normalized size = 1.14 \[ \frac {\arctanh \left (\frac {2 x +1+\sqrt {3}}{\sqrt {-4+6 \sqrt {3}}}\right )}{\sqrt {-4+6 \sqrt {3}}}+\frac {\sqrt {3}\, \arctanh \left (\frac {2 x +1+\sqrt {3}}{\sqrt {-4+6 \sqrt {3}}}\right )}{\sqrt {-4+6 \sqrt {3}}}+\frac {\ln \left (x^{2}+\sqrt {3}\, x +x -\sqrt {3}+2\right )}{2} \]

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(3^(1/2)*x+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 = 1.00, size = 77, normalized size = 1.07 \[ -\frac {{\left (\sqrt {3} + 1\right )} \log \left (\frac {2 \, x + \sqrt {3} - \sqrt {6 \, \sqrt {3} - 4} + 1}{2 \, x + \sqrt {3} + \sqrt {6 \, \sqrt {3} - 4} + 1}\right )}{2 \, \sqrt {6 \, \sqrt {3} - 4}} + \frac {1}{2} \, \log \left (x^{2} + x {\left (\sqrt {3} + 1\right )} - \sqrt {3} + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(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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mupad [B]  time = 4.23, size = 233, normalized size = 3.24 \[ \ln \left (x-\left (\frac {\frac {\sqrt {\left (\sqrt {3}-1\right )\,\left (\sqrt {3}+7\right )}}{2}+\frac {\sqrt {3}\,\sqrt {\left (\sqrt {3}-1\right )\,\left (\sqrt {3}+7\right )}}{2}}{\left (\sqrt {3}-1\right )\,\left (\sqrt {3}+7\right )}+\frac {1}{2}\right )\,\left (2\,x+\sqrt {3}+1\right )\right )\,\left (\frac {\frac {\sqrt {\left (\sqrt {3}-1\right )\,\left (\sqrt {3}+7\right )}}{2}+\frac {\sqrt {3}\,\sqrt {\left (\sqrt {3}-1\right )\,\left (\sqrt {3}+7\right )}}{2}}{\left (\sqrt {3}-1\right )\,\left (\sqrt {3}+7\right )}+\frac {1}{2}\right )-\ln \left (x+\left (\frac {\frac {\sqrt {\left (\sqrt {3}-1\right )\,\left (\sqrt {3}+7\right )}}{2}+\frac {\sqrt {3}\,\sqrt {\left (\sqrt {3}-1\right )\,\left (\sqrt {3}+7\right )}}{2}}{\left (\sqrt {3}-1\right )\,\left (\sqrt {3}+7\right )}-\frac {1}{2}\right )\,\left (2\,x+\sqrt {3}+1\right )\right )\,\left (\frac {\frac {\sqrt {\left (\sqrt {3}-1\right )\,\left (\sqrt {3}+7\right )}}{2}+\frac {\sqrt {3}\,\sqrt {\left (\sqrt {3}-1\right )\,\left (\sqrt {3}+7\right )}}{2}}{\left (\sqrt {3}-1\right )\,\left (\sqrt {3}+7\right )}-\frac {1}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x*(3^(1/2) + 1) - 3^(1/2) + x^2 + 2),x)

[Out]

log(x - ((((3^(1/2) - 1)*(3^(1/2) + 7))^(1/2)/2 + (3^(1/2)*((3^(1/2) - 1)*(3^(1/2) + 7))^(1/2))/2)/((3^(1/2) -
1)*(3^(1/2) + 7)) + 1/2)*(2*x + 3^(1/2) + 1))*((((3^(1/2) - 1)*(3^(1/2) + 7))^(1/2)/2 + (3^(1/2)*((3^(1/2) -
1)*(3^(1/2) + 7))^(1/2))/2)/((3^(1/2) - 1)*(3^(1/2) + 7)) + 1/2) - log(x + ((((3^(1/2) - 1)*(3^(1/2) + 7))^(1/
2)/2 + (3^(1/2)*((3^(1/2) - 1)*(3^(1/2) + 7))^(1/2))/2)/((3^(1/2) - 1)*(3^(1/2) + 7)) - 1/2)*(2*x + 3^(1/2) +
1))*((((3^(1/2) - 1)*(3^(1/2) + 7))^(1/2)/2 + (3^(1/2)*((3^(1/2) - 1)*(3^(1/2) + 7))^(1/2))/2)/((3^(1/2) - 1)*
(3^(1/2) + 7)) - 1/2)

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sympy [B]  time = 1.55, size = 202, normalized size = 2.81 \[ \left (\frac {1}{2} - \frac {\sqrt {11 + 64 \sqrt {3}}}{2 \left (-31 + 12 \sqrt {3}\right )}\right ) \log {\left (x - \frac {287 \sqrt {3}}{11 + 64 \sqrt {3}} + \left (\frac {1}{2} - \frac {\sqrt {11 + 64 \sqrt {3}}}{2 \left (-31 + 12 \sqrt {3}\right )}\right ) \left (\frac {269}{214 + 139 \sqrt {3}} + \frac {459 \sqrt {3}}{214 + 139 \sqrt {3}}\right ) + \frac {521}{11 + 64 \sqrt {3}} \right )} + \left (\frac {\sqrt {11 + 64 \sqrt {3}}}{2 \left (-31 + 12 \sqrt {3}\right )} + \frac {1}{2}\right ) \log {\left (x - \frac {287 \sqrt {3}}{11 + 64 \sqrt {3}} + \left (\frac {\sqrt {11 + 64 \sqrt {3}}}{2 \left (-31 + 12 \sqrt {3}\right )} + \frac {1}{2}\right ) \left (\frac {269}{214 + 139 \sqrt {3}} + \frac {459 \sqrt {3}}{214 + 139 \sqrt {3}}\right ) + \frac {521}{11 + 64 \sqrt {3}} \right )} \]

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

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