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

Optimal. Leaf size=44 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{3}-3} (1-x)}{\sqrt{x^3-1}}\right )}{\sqrt{2 \sqrt{3}-3}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.177109, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{3}-3} (1-x)}{\sqrt{x^3-1}}\right )}{\sqrt{2 \sqrt{3}-3}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 20.1712, size = 76, normalized size = 1.73 \[ \frac{2 \tilde{\infty } \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \left (- x + 1\right ) F\left (\operatorname{asin}{\left (\frac{- x + 1 + \sqrt{3}}{- x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{\sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{x^{3} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*zoo*sqrt((x**2 + x + 1)/(-x - sqrt(3) + 1)**2)*(-x + 1)*elliptic_f(asin((-x +
1 + sqrt(3))/(-x - sqrt(3) + 1)), -7 + 4*sqrt(3))/(sqrt((x - 1)/(-x - sqrt(3) +
1)**2)*sqrt(x**3 - 1))

_______________________________________________________________________________________

Mathematica [C]  time = 0.475992, size = 267, normalized size = 6.07 \[ \frac{2 \sqrt{6} \sqrt{\frac{i (x-1)}{\sqrt{3}-3 i}} \left (4 \sqrt{-2 i x+\sqrt{3}-i} \sqrt{x^2+x+1} \Pi \left (\frac{2 \sqrt{3}}{-3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right )+\sqrt{2 i x+\sqrt{3}+i} \left (\left ((1+2 i)-i \sqrt{3}\right ) x-\sqrt{3}+(2+i)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right )\right )}{\left ((1+2 i) \sqrt{3}-3 i\right ) \sqrt{-2 i x+\sqrt{3}-i} \sqrt{x^3-1}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*Sqrt[6]*Sqrt[(I*(-1 + x))/(-3*I + Sqrt[3])]*(Sqrt[I + Sqrt[3] + (2*I)*x]*((2
+ I) - Sqrt[3] + ((1 + 2*I) - I*Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[-I + Sqrt[3] -
(2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-3*I + Sqrt[3])] + 4*Sqrt[-I + Sqrt[3
] - (2*I)*x]*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(-3*I + (1 + 2*I)*Sqrt[3])
, ArcSin[Sqrt[-I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-3*I + Sq
rt[3])]))/((-3*I + (1 + 2*I)*Sqrt[3])*Sqrt[-I + Sqrt[3] - (2*I)*x]*Sqrt[-1 + x^3
])

_______________________________________________________________________________________

Maple [C]  time = 0.053, size = 245, normalized size = 5.6 \[ 2\,{\frac{-3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}-1}}\sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) }-4\,{\frac{-3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}-1}}\sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticPi} \left ( \sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}},1/3\, \left ( 3/2+i/2\sqrt{3} \right ) \sqrt{3},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.355566, size = 140, normalized size = 3.18 \[ \frac{1}{3} \, \sqrt{3} \sqrt{2 \, \sqrt{3} + 3} \arctan \left (\frac{2340 \, x^{4} + 4680 \, x^{3} + 6516 \, x^{2} + \sqrt{3}{\left (1351 \, x^{4} + 2702 \, x^{3} + 3762 \, x^{2} + 3284 \, x + 1060\right )} + 5688 \, x + 1836}{2 \, \sqrt{x^{3} - 1}{\left (627 \, x^{2} + 2 \, \sqrt{3}{\left (181 \, x^{2} + 265 \, x + 97\right )} + 918 \, x + 336\right )} \sqrt{2 \, \sqrt{3} + 3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*sqrt(3)*sqrt(2*sqrt(3) + 3)*arctan(1/2*(2340*x^4 + 4680*x^3 + 6516*x^2 + sqr
t(3)*(1351*x^4 + 2702*x^3 + 3762*x^2 + 3284*x + 1060) + 5688*x + 1836)/(sqrt(x^3
- 1)*(627*x^2 + 2*sqrt(3)*(181*x^2 + 265*x + 97) + 918*x + 336)*sqrt(2*sqrt(3)
+ 3)))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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