3.21 \(\int \frac{2^{2/3}+2 x}{\left (2^{2/3}-x\right ) \sqrt{1-x^3}} \, dx\)

Optimal. Leaf size=40 \[ -\frac{2\ 2^{2/3} \tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} x\right )}{\sqrt{1-x^3}}\right )}{\sqrt{3}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.172656, antiderivative size = 40, 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 \[ -\frac{2\ 2^{2/3} \tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} x\right )}{\sqrt{1-x^3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 147.99, size = 479, normalized size = 11.98 \[ - \frac{6 \cdot 2^{\frac{2}{3}} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (- x + 1\right ) \operatorname{atan}{\left (\frac{3^{\frac{3}{4}} \sqrt{1 + \sqrt [3]{2}} \sqrt{1 - \frac{\left (x - 1 + \sqrt{3}\right )^{2}}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- 4 \sqrt{3} + 8}}{6 \sqrt{-1 + \sqrt [3]{2}} \sqrt{- 4 \sqrt{3} + 7 + \frac{\left (x - 1 + \sqrt{3}\right )^{2}}{\left (- x + 1 + \sqrt{3}\right )^{2}}}} \right )}}{\sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{-1 + \sqrt [3]{2}} \left (1 + \sqrt [3]{2}\right )^{\frac{3}{2}} \sqrt{- 4 \sqrt{3} + 8} \sqrt{- x^{3} + 1}} + \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- x + 1\right ) \left (2^{\frac{2}{3}} + 2 + 2 \sqrt{3}\right ) F\left (\operatorname{asin}{\left (\frac{- x - \sqrt{3} + 1}{- x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{3 \sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- x^{3} + 1} \left (- 2^{\frac{2}{3}} + 1 + \sqrt{3}\right )} - \frac{12 \cdot 2^{\frac{2}{3}} \sqrt [4]{3} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (- x + 1\right ) \Pi \left (\frac{\left (- 2^{\frac{2}{3}} + 1 + \sqrt{3}\right )^{2}}{\left (-1 + 2^{\frac{2}{3}} + \sqrt{3}\right )^{2}}; \operatorname{asin}{\left (\frac{x - 1 + \sqrt{3}}{- x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{\sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- 4 \sqrt{3} + 7} \sqrt{- x^{3} + 1} \left (- 2^{\frac{2}{3}} + 1 + \sqrt{3}\right ) \left (- \sqrt{3} - 2^{\frac{2}{3}} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6*2**(2/3)*sqrt((x**2 + x + 1)/(-x + 1 + sqrt(3))**2)*sqrt(-sqrt(3) + 2)*(-x +
1)*atan(3**(3/4)*sqrt(1 + 2**(1/3))*sqrt(1 - (x - 1 + sqrt(3))**2/(-x + 1 + sqrt
(3))**2)*sqrt(-4*sqrt(3) + 8)/(6*sqrt(-1 + 2**(1/3))*sqrt(-4*sqrt(3) + 7 + (x -
1 + sqrt(3))**2/(-x + 1 + sqrt(3))**2)))/(sqrt((-x + 1)/(-x + 1 + sqrt(3))**2)*s
qrt(-1 + 2**(1/3))*(1 + 2**(1/3))**(3/2)*sqrt(-4*sqrt(3) + 8)*sqrt(-x**3 + 1)) +
2*3**(3/4)*sqrt((x**2 + x + 1)/(-x + 1 + sqrt(3))**2)*sqrt(sqrt(3) + 2)*(-x + 1
)*(2**(2/3) + 2 + 2*sqrt(3))*elliptic_f(asin((-x - sqrt(3) + 1)/(-x + 1 + sqrt(3
))), -7 - 4*sqrt(3))/(3*sqrt((-x + 1)/(-x + 1 + sqrt(3))**2)*sqrt(-x**3 + 1)*(-2
**(2/3) + 1 + sqrt(3))) - 12*2**(2/3)*3**(1/4)*sqrt((x**2 + x + 1)/(-x + 1 + sqr
t(3))**2)*sqrt(-sqrt(3) + 2)*(-x + 1)*elliptic_pi((-2**(2/3) + 1 + sqrt(3))**2/(
-1 + 2**(2/3) + sqrt(3))**2, asin((x - 1 + sqrt(3))/(-x + 1 + sqrt(3))), -7 - 4*
sqrt(3))/(sqrt((-x + 1)/(-x + 1 + sqrt(3))**2)*sqrt(-4*sqrt(3) + 7)*sqrt(-x**3 +
1)*(-2**(2/3) + 1 + sqrt(3))*(-sqrt(3) - 2**(2/3) + 1))

_______________________________________________________________________________________

Mathematica [C]  time = 0.495861, size = 327, normalized size = 8.18 \[ -\frac{4 \sqrt [6]{2} \sqrt{-\frac{i (x-1)}{\sqrt{3}+3 i}} \left (6 i \sqrt{3} \sqrt{2 i x+\sqrt{3}+i} \sqrt{x^2+x+1} \Pi \left (\frac{2 \sqrt{3}}{i+2 i 2^{2/3}+\sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )+\sqrt{-2 i x+\sqrt{3}-i} \left (\left (-3 i \sqrt [3]{2}+4 \sqrt{3}+\sqrt [3]{2} \sqrt{3}\right ) x-\sqrt [3]{2} \sqrt{3}+2 \sqrt{3}-3 i \sqrt [3]{2}-6 i\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )\right )}{\sqrt{3} \left (1+2\ 2^{2/3}-i \sqrt{3}\right ) \sqrt{2 i x+\sqrt{3}+i} \sqrt{1-x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

_______________________________________________________________________________________

Maple [C]  time = 0.057, size = 253, normalized size = 6.3 \[{{\frac{4\,i}{3}}\sqrt{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}}+{\frac{2\,i{2}^{{\frac{2}{3}}}\sqrt{3}}{-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}-{2}^{{\frac{2}{3}}}}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{\frac{i\sqrt{3}}{-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}-{2}^{{\frac{2}{3}}}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.364757, size = 78, normalized size = 1.95 \[ \frac{1}{3} \, \sqrt{6} 2^{\frac{1}{6}} \arctan \left (\frac{\sqrt{6} 2^{\frac{5}{6}}{\left (x^{3} + 3 \cdot 2^{\frac{2}{3}} x^{2} - 6 \cdot 2^{\frac{1}{3}} x + 2\right )}}{12 \, \sqrt{-x^{3} + 1}{\left (2^{\frac{2}{3}} x - 2^{\frac{1}{3}}\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*sqrt(6)*2^(1/6)*arctan(1/12*sqrt(6)*2^(5/6)*(x^3 + 3*2^(2/3)*x^2 - 6*2^(1/3)
*x + 2)/(sqrt(-x^3 + 1)*(2^(2/3)*x - 2^(1/3))))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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