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

Optimal. Leaf size=39 \[ \frac{2\ 2^{2/3} \tanh ^{-1}\left (\frac{\sqrt{3} \left (\sqrt [3]{2} x+1\right )}{\sqrt{-x^3-1}}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.158851, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{2\ 2^{2/3} \tanh ^{-1}\left (\frac{\sqrt{3} \left (\sqrt [3]{2} x+1\right )}{\sqrt{-x^3-1}}\right )}{\sqrt{3}} \]

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

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Rubi in Sympy [A]  time = 153.832, size = 462, normalized size = 11.85 \[ \frac{3 \cdot 2^{\frac{2}{3}} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (x + 1\right ) \operatorname{atanh}{\left (\frac{3^{\frac{3}{4}} \sqrt{1 + \sqrt [3]{2}} \sqrt{1 - \frac{\left (x + 1 + \sqrt{3}\right )^{2}}{\left (- x - 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2}}{3 \sqrt{-1 + \sqrt [3]{2}} \sqrt{4 \sqrt{3} + 7 + \frac{\left (x + 1 + \sqrt{3}\right )^{2}}{\left (- x - 1 + \sqrt{3}\right )^{2}}}} \right )}}{\sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{-1 + \sqrt [3]{2}} \left (1 + \sqrt [3]{2}\right )^{\frac{3}{2}} \sqrt{- x^{3} - 1}} - \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (x + 1\right ) \left (- 2 \sqrt{3} + 2^{\frac{2}{3}} + 2\right ) F\left (\operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{3 \sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- x^{3} - 1} \left (- \sqrt{3} - 2^{\frac{2}{3}} + 1\right )} - \frac{12 \cdot 2^{\frac{2}{3}} \sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (x + 1\right ) \Pi \left (\frac{\left (-1 + 2^{\frac{2}{3}} + \sqrt{3}\right )^{2}}{\left (- 2^{\frac{2}{3}} + 1 + \sqrt{3}\right )^{2}}; \operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{- x - 1 + \sqrt{3}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{\sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{4 \sqrt{3} + 7} \sqrt{- x^{3} - 1} \left (- 2^{\frac{2}{3}} + 1 + \sqrt{3}\right ) \left (- \sqrt{3} - 2^{\frac{2}{3}} + 1\right )} \]

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]

3*2**(2/3)*sqrt((x**2 - x + 1)/(x - sqrt(3) + 1)**2)*(x + 1)*atanh(3**(3/4)*sqrt
(1 + 2**(1/3))*sqrt(1 - (x + 1 + sqrt(3))**2/(-x - 1 + sqrt(3))**2)*sqrt(sqrt(3)
+ 2)/(3*sqrt(-1 + 2**(1/3))*sqrt(4*sqrt(3) + 7 + (x + 1 + sqrt(3))**2/(-x - 1 +
sqrt(3))**2)))/(sqrt((-x - 1)/(x - sqrt(3) + 1)**2)*sqrt(-1 + 2**(1/3))*(1 + 2*
*(1/3))**(3/2)*sqrt(-x**3 - 1)) - 2*3**(3/4)*sqrt((x**2 - x + 1)/(x - sqrt(3) +
1)**2)*sqrt(-sqrt(3) + 2)*(x + 1)*(-2*sqrt(3) + 2**(2/3) + 2)*elliptic_f(asin((x
+ 1 + sqrt(3))/(x - sqrt(3) + 1)), -7 + 4*sqrt(3))/(3*sqrt((-x - 1)/(x - sqrt(3
) + 1)**2)*sqrt(-x**3 - 1)*(-sqrt(3) - 2**(2/3) + 1)) - 12*2**(2/3)*3**(1/4)*sqr
t((x**2 - x + 1)/(x - sqrt(3) + 1)**2)*sqrt(sqrt(3) + 2)*(x + 1)*elliptic_pi((-1
+ 2**(2/3) + sqrt(3))**2/(-2**(2/3) + 1 + sqrt(3))**2, asin((x + 1 + sqrt(3))/(
-x - 1 + sqrt(3))), -7 + 4*sqrt(3))/(sqrt((-x - 1)/(x - sqrt(3) + 1)**2)*sqrt(4*
sqrt(3) + 7)*sqrt(-x**3 - 1)*(-2**(2/3) + 1 + sqrt(3))*(-sqrt(3) - 2**(2/3) + 1)
)

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Mathematica [C]  time = 0.482406, size = 328, normalized size = 8.41 \[ -\frac{4 \sqrt [6]{2} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \left (\sqrt{2 i x+\sqrt{3}-i} \left (\left (-3 i \sqrt [3]{2}+4 \sqrt{3}+\sqrt [3]{2} \sqrt{3}\right ) x+\sqrt [3]{2} \sqrt{3}-2 \sqrt{3}+3 i \sqrt [3]{2}+6 i\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )-6 i \sqrt{3} \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^2-x+1} \Pi \left (\frac{2 \sqrt{3}}{i+2 i 2^{2/3}+\sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )\right )}{\sqrt{3} \left (1+2\ 2^{2/3}-i \sqrt{3}\right ) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{-x^3-1}} \]

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]*Sqr
t[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])

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Maple [C]  time = 0.049, size = 249, normalized size = 6.4 \[{{\frac{4\,i}{3}}\sqrt{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}}-{\frac{2\,i{2}^{{\frac{2}{3}}}\sqrt{3}}{{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}+{2}^{{\frac{2}{3}}}}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{\frac{i\sqrt{3}}{{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}+{2}^{{\frac{2}{3}}}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}} \]

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
))

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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)

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Fricas [A]  time = 0.36217, size = 173, normalized size = 4.44 \[ \frac{1}{6} \, \sqrt{6} 2^{\frac{1}{6}} \log \left (\frac{x^{6} - 18 \cdot 2^{\frac{1}{3}} x^{4} + 56 \, x^{3} + 2 \, \sqrt{6} 2^{\frac{1}{6}} \sqrt{-x^{3} - 1}{\left (18 \, x^{2} - 2^{\frac{2}{3}}{\left (x^{4} - 8 \, x\right )} + 2^{\frac{1}{3}}{\left (5 \, x^{3} + 2\right )}\right )} - 18 \cdot 2^{\frac{2}{3}}{\left (x^{5} - 2 \, x^{2}\right )} - 8}{x^{6} + 80 \, x^{3} + 6 \cdot 2^{\frac{2}{3}}{\left (x^{5} + 10 \, x^{2}\right )} + 6 \cdot 2^{\frac{1}{3}}{\left (5 \, x^{4} + 8 \, x\right )} + 16}\right ) \]

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]

1/6*sqrt(6)*2^(1/6)*log((x^6 - 18*2^(1/3)*x^4 + 56*x^3 + 2*sqrt(6)*2^(1/6)*sqrt(
-x^3 - 1)*(18*x^2 - 2^(2/3)*(x^4 - 8*x) + 2^(1/3)*(5*x^3 + 2)) - 18*2^(2/3)*(x^5
- 2*x^2) - 8)/(x^6 + 80*x^3 + 6*2^(2/3)*(x^5 + 10*x^2) + 6*2^(1/3)*(5*x^4 + 8*x
) + 16))

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

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

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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)