3.52 \(\int \frac{1-x}{(2+x) \sqrt{1-x^3}} \, dx\)

Optimal. Leaf size=27 \[ -\frac{2}{3} \tanh ^{-1}\left (\frac{(1-x)^2}{3 \sqrt{1-x^3}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.117911, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{2}{3} \tanh ^{-1}\left (\frac{(1-x)^2}{3 \sqrt{1-x^3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 101.795, size = 371, normalized size = 13.74 \[ - \frac{3 \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \left (\frac{\sqrt{3}}{3} + 1\right ) \left (- x + 1\right ) \operatorname{atanh}{\left (\frac{3^{\frac{3}{4}} \sqrt{1 - \frac{\left (x - 1 + \sqrt{3}\right )^{2}}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2}}{3 \sqrt{- 4 \sqrt{3} + 7 + \frac{\left (x - 1 + \sqrt{3}\right )^{2}}{\left (- x + 1 + \sqrt{3}\right )^{2}}}} \right )}}{\sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \left (\sqrt{3} + 3\right ) \sqrt{- x^{3} + 1}} + \frac{2 \sqrt [4]{3} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- x + 1\right ) F\left (\operatorname{asin}{\left (\frac{- x - \sqrt{3} + 1}{- x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{\sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \left (\sqrt{3} + 3\right ) \sqrt{- x^{3} + 1}} + \frac{12 \sqrt [4]{3} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (- x + 1\right ) \Pi \left (4 \sqrt{3} + 7; \operatorname{asin}{\left (\frac{x - 1 + \sqrt{3}}{- x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{\sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- 4 \sqrt{3} + 7} \left (- \sqrt{3} + 3\right ) \left (\sqrt{3} + 3\right ) \sqrt{- x^{3} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*sqrt((x**2 + x + 1)/(-x + 1 + sqrt(3))**2)*(sqrt(3)/3 + 1)*(-x + 1)*atanh(3**
(3/4)*sqrt(1 - (x - 1 + sqrt(3))**2/(-x + 1 + sqrt(3))**2)*sqrt(-sqrt(3) + 2)/(3
*sqrt(-4*sqrt(3) + 7 + (x - 1 + sqrt(3))**2/(-x + 1 + sqrt(3))**2)))/(sqrt((-x +
1)/(-x + 1 + sqrt(3))**2)*(sqrt(3) + 3)*sqrt(-x**3 + 1)) + 2*3**(1/4)*sqrt((x**
2 + x + 1)/(-x + 1 + sqrt(3))**2)*sqrt(sqrt(3) + 2)*(-x + 1)*elliptic_f(asin((-x
- sqrt(3) + 1)/(-x + 1 + sqrt(3))), -7 - 4*sqrt(3))/(sqrt((-x + 1)/(-x + 1 + sq
rt(3))**2)*(sqrt(3) + 3)*sqrt(-x**3 + 1)) + 12*3**(1/4)*sqrt((x**2 + x + 1)/(-x
+ 1 + sqrt(3))**2)*sqrt(-sqrt(3) + 2)*(-x + 1)*elliptic_pi(4*sqrt(3) + 7, asin((
x - 1 + sqrt(3))/(-x + 1 + sqrt(3))), -7 - 4*sqrt(3))/(sqrt((-x + 1)/(-x + 1 + s
qrt(3))**2)*sqrt(-4*sqrt(3) + 7)*(-sqrt(3) + 3)*(sqrt(3) + 3)*sqrt(-x**3 + 1))

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Mathematica [C]  time = 0.360765, size = 262, normalized size = 9.7 \[ -\frac{2 \sqrt{6} \sqrt{\frac{i (x-1)}{\sqrt{3}-3 i}} \left (2 \sqrt{3} \sqrt{-2 i x+\sqrt{3}-i} \sqrt{x^2+x+1} \Pi \left (\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right )+\sqrt{2 i x+\sqrt{3}+i} \left (i \sqrt{3} x+x+i \sqrt{3}-1\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right )\right )}{\left (\sqrt{3}+3 i\right ) \sqrt{-2 i x+\sqrt{3}-i} \sqrt{1-x^3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(1 - x)/((2 + 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]*(-1
+ I*Sqrt[3] + x + I*Sqrt[3]*x)*EllipticF[ArcSin[Sqrt[-I + Sqrt[3] - (2*I)*x]/(S
qrt[2]*3^(1/4))], (2*Sqrt[3])/(-3*I + Sqrt[3])] + 2*Sqrt[3]*Sqrt[-I + Sqrt[3] -
(2*I)*x]*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(3*I + Sqrt[3]), ArcSin[Sqrt[-
I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-3*I + Sqrt[3])]))/((3*I
+ Sqrt[3])*Sqrt[-I + Sqrt[3] - (2*I)*x]*Sqrt[1 - x^3])

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Maple [C]  time = 0.036, size = 240, normalized size = 8.9 \[{{\frac{2\,i}{3}}\sqrt{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}}-{\frac{2\,i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)/(2+x)/(-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))-2*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)/(3/2+1/2
*I*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)),(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. \[ -\int \frac{x - 1}{\sqrt{-x^{3} + 1}{\left (x + 2\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.335244, size = 63, normalized size = 2.33 \[ \frac{1}{3} \, \log \left (-\frac{x^{3} - 12 \, x^{2} - 6 \, \sqrt{-x^{3} + 1}{\left (x - 1\right )} - 6 \, x - 10}{x^{3} + 6 \, x^{2} + 12 \, x + 8}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*log(-(x^3 - 12*x^2 - 6*sqrt(-x^3 + 1)*(x - 1) - 6*x - 10)/(x^3 + 6*x^2 + 12*
x + 8))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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