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

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

[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 = 0.175824, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} (x+1)}{\sqrt{-x^3-1}}\right )}{\sqrt{3+2 \sqrt{3}}} \]

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 = 31.1072, size = 138, normalized size = 3.14 \[ - \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (x + 1\right ) \operatorname{atanh}{\left (\frac{\sqrt{1 - \frac{\left (x + 1 + \sqrt{3}\right )^{2}}{\left (- x - 1 + \sqrt{3}\right )^{2}}} \left (\sqrt{3} + 2\right )}{\sqrt{4 \sqrt{3} + 7 + \frac{\left (x + 1 + \sqrt{3}\right )^{2}}{\left (- x - 1 + \sqrt{3}\right )^{2}}}} \right )}}{3 \sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \sqrt{- x^{3} - 1}} \]

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 - sqrt(3) + 1)**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 - sqrt(3) + 1)**2)*
sqrt(sqrt(3) + 2)*sqrt(-x**3 - 1))

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Mathematica [C]  time = 0.50524, size = 271, normalized size = 6.16 \[ \frac{2 \sqrt{6} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \left (4 \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^2-x+1} \Pi \left (\frac{2 \sqrt{3}}{3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )+\sqrt{2 i x+\sqrt{3}-i} \left (\left ((1+2 i)+i \sqrt{3}\right ) x-\sqrt{3}-(2+i)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )\right )}{\left (3 i+(1+2 i) \sqrt{3}\right ) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{-x^3-1}} \]

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]), Ar
cSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[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 = 0.022, size = 243, normalized size = 5.5 \[{-{\frac{2\,i}{3}}\sqrt{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}}+{\frac{4\,i}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}+\sqrt{3}}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}+\sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}} \]

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 = 0.319938, size = 369, normalized size = 8.39 \[ \frac{1}{6} \, \sqrt{3} \sqrt{2 \, \sqrt{3} - 3} \log \left (\frac{6322680 \, x^{8} - 13553256 \, x^{7} + 26133432 \, x^{6} - 63422352 \, x^{5} + 113743056 \, x^{4} - 136435776 \, x^{3} + 102727296 \, x^{2} + 4 \,{\left (1694157 \, x^{6} - 5868732 \, x^{5} + 10586298 \, x^{4} - 12840912 \, x^{3} + 9886740 \, x^{2} - 2 \, \sqrt{3}{\left (489061 \, x^{6} - 1694157 \, x^{5} + 3056001 \, x^{4} - 3706852 \, x^{3} + 2854056 \, x^{2} - 1198884 \, x + 205636\right )} - 4153056 \, x + 712344\right )} \sqrt{-x^{3} - 1} \sqrt{2 \, \sqrt{3} - 3} - \sqrt{3}{\left (3650401 \, x^{8} - 7824976 \, x^{7} + 15088144 \, x^{6} - 36616912 \, x^{5} + 65669584 \, x^{4} - 78771232 \, x^{3} + 59309632 \, x^{2} - 24558208 \, x + 4193392\right )} - 42536064 \, x + 7263168}{6322680 \, x^{8} - 37028184 \, x^{7} + 94872792 \, x^{6} - 138903408 \, x^{5} + 127105440 \, x^{4} - 74438112 \, x^{3} + 27246240 \, x^{2} - \sqrt{3}{\left (3650401 \, x^{8} - 21378232 \, x^{7} + 54774832 \, x^{6} - 80195920 \, x^{5} + 73384360 \, x^{4} - 42976864 \, x^{3} + 15730624 \, x^{2} - 3290176 \, x + 301072\right )} - 5698752 \, x + 521472}\right ) \]

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)