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

Optimal. Leaf size=63 \[ -\frac{1}{3} \sqrt{3+2 \sqrt{3}} \tan ^{-1}\left (\frac{\left (x+\sqrt{3}+1\right )^2}{\sqrt{3 \left (3+2 \sqrt{3}\right )} \sqrt{x^4-4 \sqrt{3} x^2-4}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.228188, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{1}{3} \sqrt{3+2 \sqrt{3}} \tan ^{-1}\left (\frac{\left (x+\sqrt{3}+1\right )^2}{\sqrt{3 \left (3+2 \sqrt{3}\right )} \sqrt{x^4-4 \sqrt{3} x^2-4}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Timed out

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Mathematica [C]  time = 6.28109, size = 1137, normalized size = 18.05 \[ -\frac{\left (x-\sqrt{3}-1\right )^2 \sqrt{\frac{\sqrt{3}-1+\frac{4}{x-\sqrt{3}-1}}{-3+\sqrt{3}-i \sqrt{4-2 \sqrt{3}}}} \sqrt{\left (x-\sqrt{3}+1\right )^3+\left (-2+4 \sqrt{3}\right ) \left (x-\sqrt{3}+1\right )^2+\left (20-8 \sqrt{3}\right ) \left (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}{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 )}{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 )}{x-\sqrt{3}-1}}\right )}{x-\sqrt{3}-1}+i \sqrt{3} \sqrt{i \left (1+\sqrt{3}\right )+\sqrt{4-2 \sqrt{3}}+\frac{8 i}{x-\sqrt{3}-1}}+i \sqrt{i \left (1+\sqrt{3}\right )+\sqrt{4-2 \sqrt{3}}+\frac{8 i}{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 )}{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}{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}{x-\sqrt{3}-1}} \sqrt{1+\frac{2 \left (1+\sqrt{3}\right )}{x-\sqrt{3}-1}+\frac{8}{\left (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}{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 )}{\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}{x-\sqrt{3}-1}} \sqrt{\frac{1}{2} \left (x-\sqrt{3}-1\right )^3+2 \left (1+\sqrt{3}\right ) \left (x-\sqrt{3}-1\right )^2+4 \left (3+\sqrt{3}\right ) \left (x-\sqrt{3}-1\right )+8 \left (1+\sqrt{3}\right )} \sqrt{\left (x-\sqrt{3}+1\right )^4+4 \sqrt{3} \left (x-\sqrt{3}+1\right )^3-4 \left (x-\sqrt{3}+1\right )^3-16 \sqrt{3} \left (x-\sqrt{3}+1\right )^2+24 \left (x-\sqrt{3}+1\right )^2+32 \sqrt{3} \left (x-\sqrt{3}+1\right )-64 \left (x-\sqrt{3}+1\right )-32 \sqrt{3}+48}} \]

Warning: Unable to verify antiderivative.

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

[Out]

-(((-1 - Sqrt[3] + x)^2*Sqrt[(-1 + Sqrt[3] + 4/(-1 - Sqrt[3] + x))/(-3 + Sqrt[3]
- I*Sqrt[4 - 2*Sqrt[3]])]*Sqrt[-24 + 16*Sqrt[3] + (20 - 8*Sqrt[3])*(1 - Sqrt[3]
+ x) + (-2 + 4*Sqrt[3])*(1 - Sqrt[3] + x)^2 + (1 - Sqrt[3] + x)^3]*((I*Sqrt[Sqr
t[4 - 2*Sqrt[3]] + I*(1 + Sqrt[3]) + (8*I)/(-1 - Sqrt[3] + x)] + I*Sqrt[3]*Sqrt[
Sqrt[4 - 2*Sqrt[3]] + I*(1 + Sqrt[3]) + (8*I)/(-1 - Sqrt[3] + x)] + Sqrt[-2*I +
(2*I)*Sqrt[3] - 2*Sqrt[12 - 6*Sqrt[3]] + 4*Sqrt[4 - 2*Sqrt[3]] - ((16*I)*(-2 + S
qrt[3]))/(-1 - Sqrt[3] + x)] + (2*((2*I)*Sqrt[3]*Sqrt[Sqrt[4 - 2*Sqrt[3]] + I*(1
+ Sqrt[3]) + (8*I)/(-1 - Sqrt[3] + x)] + Sqrt[6]*Sqrt[-I + I*Sqrt[3] - Sqrt[12
- 6*Sqrt[3]] + 2*Sqrt[4 - 2*Sqrt[3]] - ((8*I)*(-2 + Sqrt[3]))/(-1 - Sqrt[3] + 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] + x)]))/(-1 - Sqrt[3] + x))*EllipticF[Arc
Sin[Sqrt[Sqrt[4 - 2*Sqrt[3]] - I*(1 + Sqrt[3]) - (8*I)/(-1 - Sqrt[3] + 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[6]*Sqrt[Sqrt[4 - 2*Sqrt[3]] - I*(1 + Sqrt[3]) - (8*I)/(-1
- Sqrt[3] + x)]*Sqrt[1 + 8/(-1 - Sqrt[3] + x)^2 + (2*(1 + Sqrt[3]))/(-1 - Sqrt[3
] + 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 + Sqrt[3]) - (8*I)/(-1 - Sqrt[3] + 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[4 - 2*Sqrt[3]] - I*(-3 + Sqrt[3]))*Sqrt[Sqrt[4 - 2
*Sqrt[3]] - I*(1 + Sqrt[3]) - (8*I)/(-1 - Sqrt[3] + x)]*Sqrt[8*(1 + Sqrt[3]) + 4
*(3 + Sqrt[3])*(-1 - Sqrt[3] + x) + 2*(1 + Sqrt[3])*(-1 - Sqrt[3] + x)^2 + (-1 -
Sqrt[3] + x)^3/2]*Sqrt[48 - 32*Sqrt[3] - 64*(1 - Sqrt[3] + x) + 32*Sqrt[3]*(1 -
Sqrt[3] + x) + 24*(1 - Sqrt[3] + x)^2 - 16*Sqrt[3]*(1 - Sqrt[3] + x)^2 - 4*(1 -
Sqrt[3] + x)^3 + 4*Sqrt[3]*(1 - Sqrt[3] + x)^3 + (1 - Sqrt[3] + x)^4]))

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Maple [C]  time = 0.232, size = 311, normalized size = 4.9 \[{\frac{{\it EllipticF} \left ( x \left ({\frac{i}{2}}+{\frac{i}{2}}\sqrt{3} \right ) ,i\sqrt{1-4\,\sqrt{3} \left ( -1/2\,\sqrt{3}+1 \right ) } \right ) }{{\frac{i}{2}}+{\frac{i}{2}}\sqrt{3}}\sqrt{1- \left ( -1-{\frac{\sqrt{3}}{2}} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{\sqrt{3}}{2}}+1 \right ){x}^{2}}{\frac{1}{\sqrt{-4+{x}^{4}-4\,{x}^{2}\sqrt{3}}}}}+2\,\sqrt{3} \left ( -1/2\,{\frac{1}{\sqrt{ \left ( \sqrt{3}-1 \right ) ^{4}-4\,\sqrt{3} \left ( \sqrt{3}-1 \right ) ^{2}-4}}{\it Artanh} \left ( 1/2\,{\frac{-4\,\sqrt{3} \left ( \sqrt{3}-1 \right ) ^{2}-8-4\,{x}^{2}\sqrt{3}+2\,{x}^{2} \left ( \sqrt{3}-1 \right ) ^{2}}{\sqrt{ \left ( \sqrt{3}-1 \right ) ^{4}-4\,\sqrt{3} \left ( \sqrt{3}-1 \right ) ^{2}-4}\sqrt{-4+{x}^{4}-4\,{x}^{2}\sqrt{3}}}} \right ) }-{\frac{\sqrt{1- \left ( -1-1/2\,\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( -1/2\,\sqrt{3}+1 \right ){x}^{2}}}{\sqrt{-1-1/2\,\sqrt{3}} \left ( \sqrt{3}-1 \right ) \sqrt{-4+{x}^{4}-4\,{x}^{2}\sqrt{3}}}{\it EllipticPi} \left ( \sqrt{-1-1/2\,\sqrt{3}}x,{\frac{1}{ \left ( -1-1/2\,\sqrt{3} \right ) \left ( \sqrt{3}-1 \right ) ^{2}}},{\frac{\sqrt{-1/2\,\sqrt{3}+1}}{\sqrt{-1-1/2\,\sqrt{3}}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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