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

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

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Rubi [A]  time = 0.198258, 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 \[ -\frac{1}{3} \sqrt{3+2 \sqrt{3}} \tan ^{-1}\left (\frac{\left (2 x+\sqrt{3}+1\right )^2}{2 \sqrt{3 \left (3+2 \sqrt{3}\right )} \sqrt{4 x^4-4 \sqrt{3} x^2-1}}\right ) \]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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]

Timed out

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Mathematica [C]  time = 6.2223, size = 1198, normalized size = 17.11 \[ -\frac{\left (2 x-\sqrt{3}-1\right )^2 \sqrt{\frac{\sqrt{3}-1+\frac{4}{2 x-\sqrt{3}-1}}{-3+\sqrt{3}-i \sqrt{4-2 \sqrt{3}}}} \sqrt{\left (2 x-\sqrt{3}+1\right )^3+\left (-2+4 \sqrt{3}\right ) \left (2 x-\sqrt{3}+1\right )^2+\left (20-8 \sqrt{3}\right ) \left (2 x-\sqrt{3}+1\right )+16 \sqrt{3}-24} \left (\left (\frac{2 \left (2 i \sqrt{3} \sqrt{i \left (1+\sqrt{3}\right )+\sqrt{4-2 \sqrt{3}}+\frac{8 i}{2 x-\sqrt{3}-1}}+\sqrt{6} \sqrt{2 \sqrt{4-2 \sqrt{3}}-\sqrt{12-6 \sqrt{3}}+i \sqrt{3}-i-\frac{8 i \left (-2+\sqrt{3}\right )}{2 x-\sqrt{3}-1}}+\sqrt{4 \sqrt{4-2 \sqrt{3}}-2 \sqrt{12-6 \sqrt{3}}+2 i \sqrt{3}-2 i-\frac{16 i \left (-2+\sqrt{3}\right )}{2 x-\sqrt{3}-1}}\right )}{2 x-\sqrt{3}-1}+i \sqrt{3} \sqrt{i \left (1+\sqrt{3}\right )+\sqrt{4-2 \sqrt{3}}+\frac{8 i}{2 x-\sqrt{3}-1}}+i \sqrt{i \left (1+\sqrt{3}\right )+\sqrt{4-2 \sqrt{3}}+\frac{8 i}{2 x-\sqrt{3}-1}}+\sqrt{4 \sqrt{4-2 \sqrt{3}}-2 \sqrt{12-6 \sqrt{3}}+2 i \sqrt{3}-2 i-\frac{16 i \left (-2+\sqrt{3}\right )}{2 x-\sqrt{3}-1}}\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-i \left (1+\sqrt{3}\right )+\sqrt{4-2 \sqrt{3}}-\frac{8 i}{2 x-\sqrt{3}-1}}}{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 )+2 \sqrt{6} \sqrt{-i \left (1+\sqrt{3}\right )+\sqrt{4-2 \sqrt{3}}-\frac{8 i}{2 x-\sqrt{3}-1}} \sqrt{1+\frac{2 \left (1+\sqrt{3}\right )}{2 x-\sqrt{3}-1}+\frac{8}{\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{-i \left (1+\sqrt{3}\right )+\sqrt{4-2 \sqrt{3}}-\frac{8 i}{2 x-\sqrt{3}-1}}}{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 )}{2 \left (\sqrt{4-2 \sqrt{3}}-i \left (-3+\sqrt{3}\right )\right ) \sqrt{-i \left (1+\sqrt{3}\right )+\sqrt{4-2 \sqrt{3}}-\frac{8 i}{2 x-\sqrt{3}-1}} \sqrt{\frac{1}{2} \left (2 x-\sqrt{3}-1\right )^3+2 \left (1+\sqrt{3}\right ) \left (2 x-\sqrt{3}-1\right )^2+4 \left (3+\sqrt{3}\right ) \left (2 x-\sqrt{3}-1\right )+8 \left (1+\sqrt{3}\right )} \sqrt{\frac{1}{4} \left (2 x-\sqrt{3}+1\right )^4+\sqrt{3} \left (2 x-\sqrt{3}+1\right )^3-\left (2 x-\sqrt{3}+1\right )^3-4 \sqrt{3} \left (2 x-\sqrt{3}+1\right )^2+6 \left (2 x-\sqrt{3}+1\right )^2+8 \sqrt{3} \left (2 x-\sqrt{3}+1\right )-16 \left (2 x-\sqrt{3}+1\right )-8 \sqrt{3}+12}} \]

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]

-((-1 - Sqrt[3] + 2*x)^2*Sqrt[(-1 + Sqrt[3] + 4/(-1 - Sqrt[3] + 2*x))/(-3 + Sqrt
[3] - I*Sqrt[4 - 2*Sqrt[3]])]*Sqrt[-24 + 16*Sqrt[3] + (20 - 8*Sqrt[3])*(1 - Sqrt
[3] + 2*x) + (-2 + 4*Sqrt[3])*(1 - Sqrt[3] + 2*x)^2 + (1 - Sqrt[3] + 2*x)^3]*((I
*Sqrt[Sqrt[4 - 2*Sqrt[3]] + I*(1 + Sqrt[3]) + (8*I)/(-1 - Sqrt[3] + 2*x)] + I*Sq
rt[3]*Sqrt[Sqrt[4 - 2*Sqrt[3]] + I*(1 + Sqrt[3]) + (8*I)/(-1 - Sqrt[3] + 2*x)] +
Sqrt[-2*I + (2*I)*Sqrt[3] - 2*Sqrt[12 - 6*Sqrt[3]] + 4*Sqrt[4 - 2*Sqrt[3]] - ((
16*I)*(-2 + Sqrt[3]))/(-1 - Sqrt[3] + 2*x)] + (2*((2*I)*Sqrt[3]*Sqrt[Sqrt[4 - 2*
Sqrt[3]] + I*(1 + Sqrt[3]) + (8*I)/(-1 - Sqrt[3] + 2*x)] + Sqrt[6]*Sqrt[-I + I*S
qrt[3] - Sqrt[12 - 6*Sqrt[3]] + 2*Sqrt[4 - 2*Sqrt[3]] - ((8*I)*(-2 + Sqrt[3]))/(
-1 - Sqrt[3] + 2*x)] + Sqrt[-2*I + (2*I)*Sqrt[3] - 2*Sqrt[12 - 6*Sqrt[3]] + 4*Sq
rt[4 - 2*Sqrt[3]] - ((16*I)*(-2 + Sqrt[3]))/(-1 - Sqrt[3] + 2*x)]))/(-1 - Sqrt[3
] + 2*x))*EllipticF[ArcSin[Sqrt[Sqrt[4 - 2*Sqrt[3]] - I*(1 + Sqrt[3]) - (8*I)/(-
1 - Sqrt[3] + 2*x)]/(2^(3/4)*(2 - Sqrt[3])^(1/4))], (2*Sqrt[4 - 2*Sqrt[3]])/(Sqr
t[4 - 2*Sqrt[3]] + I*(-3 + Sqrt[3]))] + 2*Sqrt[6]*Sqrt[Sqrt[4 - 2*Sqrt[3]] - I*(
1 + Sqrt[3]) - (8*I)/(-1 - Sqrt[3] + 2*x)]*Sqrt[1 + 8/(-1 - Sqrt[3] + 2*x)^2 + (
2*(1 + Sqrt[3]))/(-1 - Sqrt[3] + 2*x)]*EllipticPi[(2*Sqrt[4 - 2*Sqrt[3]])/(Sqrt[
4 - 2*Sqrt[3]] - I*(-3 + Sqrt[3])), ArcSin[Sqrt[Sqrt[4 - 2*Sqrt[3]] - I*(1 + Sqr
t[3]) - (8*I)/(-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]))]))/(2*(Sqrt[4 - 2*Sqrt[3]]
- I*(-3 + Sqrt[3]))*Sqrt[Sqrt[4 - 2*Sqrt[3]] - I*(1 + Sqrt[3]) - (8*I)/(-1 - Sqr
t[3] + 2*x)]*Sqrt[8*(1 + Sqrt[3]) + 4*(3 + Sqrt[3])*(-1 - Sqrt[3] + 2*x) + 2*(1
+ Sqrt[3])*(-1 - Sqrt[3] + 2*x)^2 + (-1 - Sqrt[3] + 2*x)^3/2]*Sqrt[12 - 8*Sqrt[3
] - 16*(1 - Sqrt[3] + 2*x) + 8*Sqrt[3]*(1 - Sqrt[3] + 2*x) + 6*(1 - Sqrt[3] + 2*
x)^2 - 4*Sqrt[3]*(1 - Sqrt[3] + 2*x)^2 - (1 - Sqrt[3] + 2*x)^3 + Sqrt[3]*(1 - Sq
rt[3] + 2*x)^3 + (1 - Sqrt[3] + 2*x)^4/4])

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Maple [C]  time = 0.217, size = 337, normalized size = 4.8 \[{\frac{{\it EllipticF} \left ( x \left ( i+i\sqrt{3} \right ) ,i\sqrt{1-\sqrt{3} \left ( -2\,\sqrt{3}+4 \right ) } \right ) }{i+i\sqrt{3}}\sqrt{1- \left ( -2\,\sqrt{3}-4 \right ){x}^{2}}\sqrt{1- \left ( -2\,\sqrt{3}+4 \right ){x}^{2}}{\frac{1}{\sqrt{-1+4\,{x}^{4}-4\,{x}^{2}\sqrt{3}}}}}+2\,\sqrt{3} \left ( -1/4\,{\frac{1}{\sqrt{4\, \left ( 1/2\,\sqrt{3}-1/2 \right ) ^{4}-4\,\sqrt{3} \left ( 1/2\,\sqrt{3}-1/2 \right ) ^{2}-1}}{\it Artanh} \left ( 1/2\,{\frac{-4\,\sqrt{3} \left ( 1/2\,\sqrt{3}-1/2 \right ) ^{2}-2-4\,{x}^{2}\sqrt{3}+8\,{x}^{2} \left ( 1/2\,\sqrt{3}-1/2 \right ) ^{2}}{\sqrt{4\, \left ( 1/2\,\sqrt{3}-1/2 \right ) ^{4}-4\,\sqrt{3} \left ( 1/2\,\sqrt{3}-1/2 \right ) ^{2}-1}\sqrt{-1+4\,{x}^{4}-4\,{x}^{2}\sqrt{3}}}} \right ) }-1/2\,{\frac{\sqrt{1- \left ( -2\,\sqrt{3}-4 \right ){x}^{2}}\sqrt{1- \left ( -2\,\sqrt{3}+4 \right ){x}^{2}}}{\sqrt{-2\,\sqrt{3}-4} \left ( 1/2\,\sqrt{3}-1/2 \right ) \sqrt{-1+4\,{x}^{4}-4\,{x}^{2}\sqrt{3}}}{\it EllipticPi} \left ( \sqrt{-2\,\sqrt{3}-4}x,{\frac{1}{ \left ( -2\,\sqrt{3}-4 \right ) \left ( 1/2\,\sqrt{3}-1/2 \right ) ^{2}}},{\frac{\sqrt{-2\,\sqrt{3}+4}}{\sqrt{-2\,\sqrt{3}-4}}} \right ) } \right ) \]

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)*EllipticF(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)*El
lipticPi((-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. \[ \int \frac{2 \, x + \sqrt{3} + 1}{\sqrt{4 \, x^{4} - 4 \, \sqrt{3} x^{2} - 1}{\left (2 \, x - \sqrt{3} + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x + sqrt(3) + 1)/(sqrt(4*x^4 - 4*sqrt(3)*x^2 - 1)*(2*x - sqrt(3) + 1)),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 [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Sympy [A]  time = 11.2237, size = 0, normalized size = 0. \[ \mathrm{NaN} \]

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]

nan

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{2 \, x + \sqrt{3} + 1}{\sqrt{4 \, x^{4} - 4 \, \sqrt{3} x^{2} - 1}{\left (2 \, x - \sqrt{3} + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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