3.411 \(\int \frac{1+\sqrt{3}+2 x}{(1-\sqrt{3}+2 x) \sqrt{-1-4 \sqrt{3} x^2+4 x^4}} \, dx\)

Optimal. Leaf size=70 \[ -\frac{1}{3} \sqrt{3+2 \sqrt{3}} \tan ^{-1}\left (\frac{\left (2 x+\sqrt{3}+1\right )^2}{2 \sqrt{3 \left (3+2 \sqrt{3}\right )} \sqrt{4 x^4-4 \sqrt{3} x^2-1}}\right ) \]

[Out]

-(Sqrt[3 + 2*Sqrt[3]]*ArcTan[(1 + Sqrt[3] + 2*x)^2/(2*Sqrt[3*(3 + 2*Sqrt[3])]*Sqrt[-1 - 4*Sqrt[3]*x^2 + 4*x^4]
)])/3

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Rubi [A]  time = 0.129127, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {1740, 203} \[ -\frac{1}{3} \sqrt{3+2 \sqrt{3}} \tan ^{-1}\left (\frac{\left (2 x+\sqrt{3}+1\right )^2}{2 \sqrt{3 \left (3+2 \sqrt{3}\right )} \sqrt{4 x^4-4 \sqrt{3} x^2-1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + Sqrt[3] + 2*x)/((1 - Sqrt[3] + 2*x)*Sqrt[-1 - 4*Sqrt[3]*x^2 + 4*x^4]),x]

[Out]

-(Sqrt[3 + 2*Sqrt[3]]*ArcTan[(1 + Sqrt[3] + 2*x)^2/(2*Sqrt[3*(3 + 2*Sqrt[3])]*Sqrt[-1 - 4*Sqrt[3]*x^2 + 4*x^4]
)])/3

Rule 1740

Int[((A_) + (B_.)*(x_))/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> -Dist[(A^
2*(B*d + A*e))/e, Subst[Int[1/(6*A^3*B*d + 3*A^4*e - a*e*x^2), x], x, (A + B*x)^2/Sqrt[a + b*x^2 + c*x^4]], x]
/; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[B*d - A*e, 0] && EqQ[c^2*d^6 + a*e^4*(13*c*d^2 + b*e^2), 0] && EqQ[
b^2*e^4 - 12*c*d^2*(c*d^2 - b*e^2), 0] && EqQ[4*A*c*d*e + B*(2*c*d^2 - b*e^2), 0]

Rule 203

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

Rubi steps

\begin{align*} \int \frac{1+\sqrt{3}+2 x}{\left (1-\sqrt{3}+2 x\right ) \sqrt{-1-4 \sqrt{3} x^2+4 x^4}} \, dx &=-\left (\left (4 \left (2+\sqrt{3}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{12 \left (1-\sqrt{3}\right ) \left (1+\sqrt{3}\right )^3+6 \left (1+\sqrt{3}\right )^4+2 x^2} \, dx,x,\frac{\left (1+\sqrt{3}+2 x\right )^2}{\sqrt{-1-4 \sqrt{3} x^2+4 x^4}}\right )\right )\\ &=-\frac{1}{3} \sqrt{3+2 \sqrt{3}} \tan ^{-1}\left (\frac{\left (1+\sqrt{3}+2 x\right )^2}{2 \sqrt{3 \left (3+2 \sqrt{3}\right )} \sqrt{-1-4 \sqrt{3} x^2+4 x^4}}\right )\\ \end{align*}

Mathematica [C]  time = 5.81788, size = 881, normalized size = 12.59 \[ -\frac{\sqrt{\frac{\sqrt{3}-1-\frac{4}{-2 x+\sqrt{3}+1}}{-3+\sqrt{3}-i \sqrt{4-2 \sqrt{3}}}} \left (-2 x+\sqrt{3}+1\right )^2 \left (\left (-\frac{2 i \left (2 \sqrt{3} \sqrt{i \left (\sqrt{3}+1-\frac{8}{-2 x+\sqrt{3}+1}\right )+\sqrt{4-2 \sqrt{3}}}-i \sqrt{6} \sqrt{-\frac{2 i \left (\left (-1+\sqrt{3}\right ) x-4 \sqrt{3}+7\right )}{-2 x+\sqrt{3}+1}+2 \sqrt{4-2 \sqrt{3}}-\sqrt{12-6 \sqrt{3}}}-i \sqrt{-\frac{4 i \left (\left (-1+\sqrt{3}\right ) x-4 \sqrt{3}+7\right )}{-2 x+\sqrt{3}+1}+4 \sqrt{4-2 \sqrt{3}}-2 \sqrt{12-6 \sqrt{3}}}\right )}{-2 x+\sqrt{3}+1}+i \sqrt{3} \sqrt{i \left (\sqrt{3}+1-\frac{8}{-2 x+\sqrt{3}+1}\right )+\sqrt{4-2 \sqrt{3}}}+i \sqrt{i \left (\sqrt{3}+1-\frac{8}{-2 x+\sqrt{3}+1}\right )+\sqrt{4-2 \sqrt{3}}}+\sqrt{-\frac{4 i \left (\left (-1+\sqrt{3}\right ) x-4 \sqrt{3}+7\right )}{-2 x+\sqrt{3}+1}+4 \sqrt{4-2 \sqrt{3}}-2 \sqrt{12-6 \sqrt{3}}}\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt{4-2 \sqrt{3}}-i \left (\sqrt{3}+1-\frac{8}{-2 x+\sqrt{3}+1}\right )}}{2^{3/4} \sqrt [4]{2-\sqrt{3}}}\right ),\frac{2 \sqrt{4-2 \sqrt{3}}}{\sqrt{4-2 \sqrt{3}}+i \left (-3+\sqrt{3}\right )}\right )+4 \sqrt{\sqrt{4-2 \sqrt{3}}-i \left (\sqrt{3}+1-\frac{8}{-2 x+\sqrt{3}+1}\right )} \sqrt{\frac{6 x^2-3 \sqrt{3}+6}{\left (-2 x+\sqrt{3}+1\right )^2}} \Pi \left (\frac{2 \sqrt{4-2 \sqrt{3}}}{\sqrt{4-2 \sqrt{3}}-i \left (-3+\sqrt{3}\right )};\sin ^{-1}\left (\frac{\sqrt{\sqrt{4-2 \sqrt{3}}-i \left (\sqrt{3}+1-\frac{8}{-2 x+\sqrt{3}+1}\right )}}{2^{3/4} \sqrt [4]{2-\sqrt{3}}}\right )|\frac{2 \sqrt{4-2 \sqrt{3}}}{\sqrt{4-2 \sqrt{3}}+i \left (-3+\sqrt{3}\right )}\right )\right )}{\sqrt{2} \left (\sqrt{4-2 \sqrt{3}}-i \left (-3+\sqrt{3}\right )\right ) \sqrt{\sqrt{4-2 \sqrt{3}}-i \left (\sqrt{3}+1-\frac{8}{-2 x+\sqrt{3}+1}\right )} \sqrt{4 x^4-4 \sqrt{3} x^2-1}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(1 + Sqrt[3] + 2*x)/((1 - Sqrt[3] + 2*x)*Sqrt[-1 - 4*Sqrt[3]*x^2 + 4*x^4]),x]

[Out]

-((Sqrt[(-1 + Sqrt[3] - 4/(1 + Sqrt[3] - 2*x))/(-3 + Sqrt[3] - I*Sqrt[4 - 2*Sqrt[3]])]*(1 + Sqrt[3] - 2*x)^2*(
(I*Sqrt[Sqrt[4 - 2*Sqrt[3]] + I*(1 + Sqrt[3] - 8/(1 + Sqrt[3] - 2*x))] + I*Sqrt[3]*Sqrt[Sqrt[4 - 2*Sqrt[3]] +
I*(1 + Sqrt[3] - 8/(1 + Sqrt[3] - 2*x))] + Sqrt[-2*Sqrt[12 - 6*Sqrt[3]] + 4*Sqrt[4 - 2*Sqrt[3]] - ((4*I)*(7 -
4*Sqrt[3] + (-1 + Sqrt[3])*x))/(1 + Sqrt[3] - 2*x)] - ((2*I)*(2*Sqrt[3]*Sqrt[Sqrt[4 - 2*Sqrt[3]] + I*(1 + Sqrt
[3] - 8/(1 + Sqrt[3] - 2*x))] - I*Sqrt[6]*Sqrt[-Sqrt[12 - 6*Sqrt[3]] + 2*Sqrt[4 - 2*Sqrt[3]] - ((2*I)*(7 - 4*S
qrt[3] + (-1 + Sqrt[3])*x))/(1 + Sqrt[3] - 2*x)] - I*Sqrt[-2*Sqrt[12 - 6*Sqrt[3]] + 4*Sqrt[4 - 2*Sqrt[3]] - ((
4*I)*(7 - 4*Sqrt[3] + (-1 + Sqrt[3])*x))/(1 + Sqrt[3] - 2*x)]))/(1 + Sqrt[3] - 2*x))*EllipticF[ArcSin[Sqrt[Sqr
t[4 - 2*Sqrt[3]] - I*(1 + Sqrt[3] - 8/(1 + Sqrt[3] - 2*x))]/(2^(3/4)*(2 - Sqrt[3])^(1/4))], (2*Sqrt[4 - 2*Sqrt
[3]])/(Sqrt[4 - 2*Sqrt[3]] + I*(-3 + Sqrt[3]))] + 4*Sqrt[Sqrt[4 - 2*Sqrt[3]] - I*(1 + Sqrt[3] - 8/(1 + Sqrt[3]
- 2*x))]*Sqrt[(6 - 3*Sqrt[3] + 6*x^2)/(1 + Sqrt[3] - 2*x)^2]*EllipticPi[(2*Sqrt[4 - 2*Sqrt[3]])/(Sqrt[4 - 2*S
qrt[3]] - I*(-3 + Sqrt[3])), ArcSin[Sqrt[Sqrt[4 - 2*Sqrt[3]] - I*(1 + Sqrt[3] - 8/(1 + Sqrt[3] - 2*x))]/(2^(3/
4)*(2 - Sqrt[3])^(1/4))], (2*Sqrt[4 - 2*Sqrt[3]])/(Sqrt[4 - 2*Sqrt[3]] + I*(-3 + Sqrt[3]))]))/(Sqrt[2]*(Sqrt[4
- 2*Sqrt[3]] - I*(-3 + Sqrt[3]))*Sqrt[Sqrt[4 - 2*Sqrt[3]] - I*(1 + Sqrt[3] - 8/(1 + Sqrt[3] - 2*x))]*Sqrt[-1
- 4*Sqrt[3]*x^2 + 4*x^4]))

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Maple [C]  time = 0.134, size = 337, normalized size = 4.8 \begin{align*}{\frac{{\it EllipticF} \left ( x \left ( i+i\sqrt{3} \right ) ,i\sqrt{1-\sqrt{3} \left ( -2\,\sqrt{3}+4 \right ) } \right ) }{i+i\sqrt{3}}\sqrt{1- \left ( -2\,\sqrt{3}-4 \right ){x}^{2}}\sqrt{1- \left ( -2\,\sqrt{3}+4 \right ){x}^{2}}{\frac{1}{\sqrt{-1+4\,{x}^{4}-4\,{x}^{2}\sqrt{3}}}}}+2\,\sqrt{3} \left ( -1/4\,{\frac{1}{\sqrt{4\, \left ( 1/2\,\sqrt{3}-1/2 \right ) ^{4}-4\,\sqrt{3} \left ( 1/2\,\sqrt{3}-1/2 \right ) ^{2}-1}}{\it Artanh} \left ( 1/2\,{\frac{-4\,\sqrt{3} \left ( 1/2\,\sqrt{3}-1/2 \right ) ^{2}-2-4\,{x}^{2}\sqrt{3}+8\,{x}^{2} \left ( 1/2\,\sqrt{3}-1/2 \right ) ^{2}}{\sqrt{4\, \left ( 1/2\,\sqrt{3}-1/2 \right ) ^{4}-4\,\sqrt{3} \left ( 1/2\,\sqrt{3}-1/2 \right ) ^{2}-1}\sqrt{-1+4\,{x}^{4}-4\,{x}^{2}\sqrt{3}}}} \right ) }-1/2\,{\frac{\sqrt{1- \left ( -2\,\sqrt{3}-4 \right ){x}^{2}}\sqrt{1- \left ( -2\,\sqrt{3}+4 \right ){x}^{2}}}{\sqrt{-2\,\sqrt{3}-4} \left ( 1/2\,\sqrt{3}-1/2 \right ) \sqrt{-1+4\,{x}^{4}-4\,{x}^{2}\sqrt{3}}}{\it EllipticPi} \left ( \sqrt{-2\,\sqrt{3}-4}x,{\frac{1}{ \left ( -2\,\sqrt{3}-4 \right ) \left ( 1/2\,\sqrt{3}-1/2 \right ) ^{2}}},{\frac{\sqrt{-2\,\sqrt{3}+4}}{\sqrt{-2\,\sqrt{3}-4}}} \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+2*x+3^(1/2))/(1+2*x-3^(1/2))/(-1+4*x^4-4*x^2*3^(1/2))^(1/2),x)

[Out]

1/(I+I*3^(1/2))*(1-(-2*3^(1/2)-4)*x^2)^(1/2)*(1-(-2*3^(1/2)+4)*x^2)^(1/2)/(-1+4*x^4-4*x^2*3^(1/2))^(1/2)*Ellip
ticF(x*(I+I*3^(1/2)),I*(1-3^(1/2)*(-2*3^(1/2)+4))^(1/2))+2*3^(1/2)*(-1/4/(4*(1/2*3^(1/2)-1/2)^4-4*3^(1/2)*(1/2
*3^(1/2)-1/2)^2-1)^(1/2)*arctanh(1/2*(-4*3^(1/2)*(1/2*3^(1/2)-1/2)^2-2-4*x^2*3^(1/2)+8*x^2*(1/2*3^(1/2)-1/2)^2
)/(4*(1/2*3^(1/2)-1/2)^4-4*3^(1/2)*(1/2*3^(1/2)-1/2)^2-1)^(1/2)/(-1+4*x^4-4*x^2*3^(1/2))^(1/2))-1/2/(-2*3^(1/2
)-4)^(1/2)/(1/2*3^(1/2)-1/2)*(1-(-2*3^(1/2)-4)*x^2)^(1/2)*(1-(-2*3^(1/2)+4)*x^2)^(1/2)/(-1+4*x^4-4*x^2*3^(1/2)
)^(1/2)*EllipticPi((-2*3^(1/2)-4)^(1/2)*x,1/(-2*3^(1/2)-4)/(1/2*3^(1/2)-1/2)^2,(-2*3^(1/2)+4)^(1/2)/(-2*3^(1/2
)-4)^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{2 \, x + \sqrt{3} + 1}{\sqrt{4 \, x^{4} - 4 \, \sqrt{3} x^{2} - 1}{\left (2 \, x - \sqrt{3} + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x+3^(1/2))/(1+2*x-3^(1/2))/(-1+4*x^4-4*3^(1/2)*x^2)^(1/2),x, algorithm="maxima")

[Out]

integrate((2*x + sqrt(3) + 1)/(sqrt(4*x^4 - 4*sqrt(3)*x^2 - 1)*(2*x - sqrt(3) + 1)), x)

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Fricas [B]  time = 2.68323, size = 302, normalized size = 4.31 \begin{align*} \frac{1}{6} \, \sqrt{2 \, \sqrt{3} + 3} \arctan \left (-\frac{{\left (36 \, x^{4} - 60 \, x^{3} + 18 \, x^{2} - \sqrt{3}{\left (16 \, x^{4} - 40 \, x^{3} + 6 \, x^{2} - 10 \, x + 1\right )} + 6\right )} \sqrt{4 \, x^{4} - 4 \, \sqrt{3} x^{2} - 1} \sqrt{2 \, \sqrt{3} + 3}}{88 \, x^{6} - 168 \, x^{5} + 132 \, x^{4} - 176 \, x^{3} - 66 \, x^{2} - 42 \, x - 11}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x+3^(1/2))/(1+2*x-3^(1/2))/(-1+4*x^4-4*3^(1/2)*x^2)^(1/2),x, algorithm="fricas")

[Out]

1/6*sqrt(2*sqrt(3) + 3)*arctan(-(36*x^4 - 60*x^3 + 18*x^2 - sqrt(3)*(16*x^4 - 40*x^3 + 6*x^2 - 10*x + 1) + 6)*
sqrt(4*x^4 - 4*sqrt(3)*x^2 - 1)*sqrt(2*sqrt(3) + 3)/(88*x^6 - 168*x^5 + 132*x^4 - 176*x^3 - 66*x^2 - 42*x - 11
))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{2 x + 1 + \sqrt{3}}{\left (2 x - \sqrt{3} + 1\right ) \sqrt{4 x^{4} - 4 \sqrt{3} x^{2} - 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x+3**(1/2))/(1+2*x-3**(1/2))/(-1+4*x**4-4*3**(1/2)*x**2)**(1/2),x)

[Out]

Integral((2*x + 1 + sqrt(3))/((2*x - sqrt(3) + 1)*sqrt(4*x**4 - 4*sqrt(3)*x**2 - 1)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{2 \, x + \sqrt{3} + 1}{\sqrt{4 \, x^{4} - 4 \, \sqrt{3} x^{2} - 1}{\left (2 \, x - \sqrt{3} + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x+3^(1/2))/(1+2*x-3^(1/2))/(-1+4*x^4-4*3^(1/2)*x^2)^(1/2),x, algorithm="giac")

[Out]

integrate((2*x + sqrt(3) + 1)/(sqrt(4*x^4 - 4*sqrt(3)*x^2 - 1)*(2*x - sqrt(3) + 1)), x)