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

Optimal. Leaf size=46 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{2 \sqrt{3}-3} (1-x)}{\sqrt{1-x^3}}\right )}{\sqrt{2 \sqrt{3}-3}} \]

[Out]

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

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Rubi [A]  time = 0.19175, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{2 \sqrt{3}-3} (1-x)}{\sqrt{1-x^3}}\right )}{\sqrt{2 \sqrt{3}-3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 36.4102, size = 133, normalized size = 2.89 \[ \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \left (- x + 1\right ) \operatorname{atanh}{\left (\frac{\sqrt{1 - \frac{\left (x - 1 + \sqrt{3}\right )^{2}}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \left (- \sqrt{3} + 2\right )}{\sqrt{- 4 \sqrt{3} + 7 + \frac{\left (x - 1 + \sqrt{3}\right )^{2}}{\left (- x + 1 + \sqrt{3}\right )^{2}}}} \right )}}{3 \sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \sqrt{- x^{3} + 1}} \]

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*3**(3/4)*sqrt((x**2 + x + 1)/(-x + 1 + sqrt(3))**2)*(-x + 1)*atanh(sqrt(1 - (x
- 1 + sqrt(3))**2/(-x + 1 + sqrt(3))**2)*(-sqrt(3) + 2)/sqrt(-4*sqrt(3) + 7 + (
x - 1 + sqrt(3))**2/(-x + 1 + sqrt(3))**2))/(3*sqrt((-x + 1)/(-x + 1 + sqrt(3))*
*2)*sqrt(-sqrt(3) + 2)*sqrt(-x**3 + 1))

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Mathematica [C]  time = 0.514982, size = 269, normalized size = 5.85 \[ \frac{2 \sqrt{6} \sqrt{\frac{i (x-1)}{\sqrt{3}-3 i}} \left (4 \sqrt{-2 i x+\sqrt{3}-i} \sqrt{x^2+x+1} \Pi \left (\frac{2 \sqrt{3}}{-3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right )+\sqrt{2 i x+\sqrt{3}+i} \left (\left ((1+2 i)-i \sqrt{3}\right ) x-\sqrt{3}+(2+i)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right )\right )}{\left ((1+2 i) \sqrt{3}-3 i\right ) \sqrt{-2 i x+\sqrt{3}-i} \sqrt{1-x^3}} \]

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]
)

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Maple [C]  time = 0.096, size = 243, normalized size = 5.3 \[{-{\frac{2\,i}{3}}\sqrt{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}}+{\frac{4\,i}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}+\sqrt{3}}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}+\sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}} \]

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

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

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)

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Fricas [A]  time = 0.355209, size = 366, normalized size = 7.96 \[ \frac{1}{6} \, \sqrt{3} \sqrt{2 \, \sqrt{3} + 3} \log \left (\frac{6322680 \, x^{8} + 13553256 \, x^{7} + 26133432 \, x^{6} + 63422352 \, x^{5} + 113743056 \, x^{4} + 136435776 \, x^{3} + 102727296 \, x^{2} + 4 \,{\left (1694157 \, x^{6} + 5868732 \, x^{5} + 10586298 \, x^{4} + 12840912 \, x^{3} + 9886740 \, x^{2} + 2 \, \sqrt{3}{\left (489061 \, x^{6} + 1694157 \, x^{5} + 3056001 \, x^{4} + 3706852 \, x^{3} + 2854056 \, x^{2} + 1198884 \, x + 205636\right )} + 4153056 \, x + 712344\right )} \sqrt{-x^{3} + 1} \sqrt{2 \, \sqrt{3} + 3} + \sqrt{3}{\left (3650401 \, x^{8} + 7824976 \, x^{7} + 15088144 \, x^{6} + 36616912 \, x^{5} + 65669584 \, x^{4} + 78771232 \, x^{3} + 59309632 \, x^{2} + 24558208 \, x + 4193392\right )} + 42536064 \, x + 7263168}{6322680 \, x^{8} + 37028184 \, x^{7} + 94872792 \, x^{6} + 138903408 \, x^{5} + 127105440 \, x^{4} + 74438112 \, x^{3} + 27246240 \, x^{2} + \sqrt{3}{\left (3650401 \, x^{8} + 21378232 \, x^{7} + 54774832 \, x^{6} + 80195920 \, x^{5} + 73384360 \, x^{4} + 42976864 \, x^{3} + 15730624 \, x^{2} + 3290176 \, x + 301072\right )} + 5698752 \, x + 521472}\right ) \]

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]

1/6*sqrt(3)*sqrt(2*sqrt(3) + 3)*log((6322680*x^8 + 13553256*x^7 + 26133432*x^6 +
63422352*x^5 + 113743056*x^4 + 136435776*x^3 + 102727296*x^2 + 4*(1694157*x^6 +
5868732*x^5 + 10586298*x^4 + 12840912*x^3 + 9886740*x^2 + 2*sqrt(3)*(489061*x^6
+ 1694157*x^5 + 3056001*x^4 + 3706852*x^3 + 2854056*x^2 + 1198884*x + 205636) +
4153056*x + 712344)*sqrt(-x^3 + 1)*sqrt(2*sqrt(3) + 3) + sqrt(3)*(3650401*x^8 +
7824976*x^7 + 15088144*x^6 + 36616912*x^5 + 65669584*x^4 + 78771232*x^3 + 59309
632*x^2 + 24558208*x + 4193392) + 42536064*x + 7263168)/(6322680*x^8 + 37028184*
x^7 + 94872792*x^6 + 138903408*x^5 + 127105440*x^4 + 74438112*x^3 + 27246240*x^2
+ sqrt(3)*(3650401*x^8 + 21378232*x^7 + 54774832*x^6 + 80195920*x^5 + 73384360*
x^4 + 42976864*x^3 + 15730624*x^2 + 3290176*x + 301072) + 5698752*x + 521472))

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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)

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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)