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

Optimal. Leaf size=25 \[ -\frac{2}{3} \tan ^{-1}\left (\frac{(1-x)^2}{3 \sqrt{x^3-1}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.103321, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{2}{3} \tan ^{-1}\left (\frac{(1-x)^2}{3 \sqrt{x^3-1}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 90.8784, size = 369, normalized size = 14.76 \[ - \frac{3 \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \left (- \frac{\sqrt{3}}{3} + 1\right ) \left (- x + 1\right ) \operatorname{atan}{\left (\frac{3^{\frac{3}{4}} \sqrt{\sqrt{3} + 2} \sqrt{- \frac{\left (- x + 1 + \sqrt{3}\right )^{2}}{\left (x - 1 + \sqrt{3}\right )^{2}} + 1}}{3 \sqrt{\frac{\left (- x + 1 + \sqrt{3}\right )^{2}}{\left (x - 1 + \sqrt{3}\right )^{2}} + 4 \sqrt{3} + 7}} \right )}}{\sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \left (- \sqrt{3} + 3\right ) \sqrt{x^{3} - 1}} - \frac{2 \sqrt [4]{3} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (- x + 1\right ) F\left (\operatorname{asin}{\left (\frac{- x + 1 + \sqrt{3}}{- x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{\sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \left (- \sqrt{3} + 3\right ) \sqrt{x^{3} - 1}} - \frac{12 \sqrt [4]{3} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- x + 1\right ) \Pi \left (- 4 \sqrt{3} + 7; \operatorname{asin}{\left (\frac{- x + 1 + \sqrt{3}}{x - 1 + \sqrt{3}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{\sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \left (- \sqrt{3} + 3\right ) \left (\sqrt{3} + 3\right ) \sqrt{4 \sqrt{3} + 7} \sqrt{x^{3} - 1}} \]

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 - sqrt(3) + 1)**2)*(-sqrt(3)/3 + 1)*(-x + 1)*atan(3**
(3/4)*sqrt(sqrt(3) + 2)*sqrt(-(-x + 1 + sqrt(3))**2/(x - 1 + sqrt(3))**2 + 1)/(3
*sqrt((-x + 1 + sqrt(3))**2/(x - 1 + sqrt(3))**2 + 4*sqrt(3) + 7)))/(sqrt((x - 1
)/(-x - sqrt(3) + 1)**2)*(-sqrt(3) + 3)*sqrt(x**3 - 1)) - 2*3**(1/4)*sqrt((x**2
+ x + 1)/(-x - sqrt(3) + 1)**2)*sqrt(-sqrt(3) + 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(3) + 3)*sqrt(x**3 - 1)) - 12*3**(1/4)*sqrt((x**2 + x + 1)/(-x -
sqrt(3) + 1)**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 - sqrt(3)
+ 1)**2)*(-sqrt(3) + 3)*(sqrt(3) + 3)*sqrt(4*sqrt(3) + 7)*sqrt(x**3 - 1))

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Mathematica [C]  time = 0.349488, size = 260, normalized size = 10.4 \[ -\frac{2 \sqrt{6} \sqrt{\frac{i (x-1)}{\sqrt{3}-3 i}} \left (2 \sqrt{3} \sqrt{-2 i x+\sqrt{3}-i} \sqrt{x^2+x+1} \Pi \left (\frac{2 \sqrt{3}}{3 i+\sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right )+\sqrt{2 i x+\sqrt{3}+i} \left (i \sqrt{3} x+x+i \sqrt{3}-1\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right )\right )}{\left (\sqrt{3}+3 i\right ) \sqrt{-2 i x+\sqrt{3}-i} \sqrt{x^3-1}} \]

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 = 0.031, size = 240, normalized size = 9.6 \[ -2\,{\frac{-3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}-1}}\sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) }+2\,{\frac{-3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}-1}}\sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticPi} \left ( \sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}},i/6\sqrt{3}+1/2,\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)/(2+x)/(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))+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)*EllipticPi(((-1+x)/(-3/2-1/2*I*3^(1/2)
))^(1/2),1/6*I*3^(1/2)+1/2,((3/2+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 = 0.354341, size = 41, normalized size = 1.64 \[ -\frac{1}{3} \, \arctan \left (\frac{x^{3} - 12 \, x^{2} - 6 \, x - 10}{6 \, \sqrt{x^{3} - 1}{\left (x - 1\right )}}\right ) \]

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*arctan(1/6*(x^3 - 12*x^2 - 6*x - 10)/(sqrt(x^3 - 1)*(x - 1)))

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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)