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

Optimal. Leaf size=38 \[ -\frac{2\ 2^{2/3} \tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} x\right )}{\sqrt{x^3-1}}\right )}{\sqrt{3}} \]

[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 = 0.152818, antiderivative size = 38, 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 \[ -\frac{2\ 2^{2/3} \tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} x\right )}{\sqrt{x^3-1}}\right )}{\sqrt{3}} \]

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 = 156.949, size = 452, normalized size = 11.89 \[ - \frac{3 \cdot 2^{\frac{2}{3}} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \left (- x + 1\right ) \operatorname{atanh}{\left (\frac{3^{\frac{3}{4}} \sqrt{1 + \sqrt [3]{2}} \sqrt{\sqrt{3} + 2} \sqrt{- \frac{\left (- x + 1 + \sqrt{3}\right )^{2}}{\left (x - 1 + \sqrt{3}\right )^{2}} + 1}}{3 \sqrt{-1 + \sqrt [3]{2}} \sqrt{\frac{\left (- x + 1 + \sqrt{3}\right )^{2}}{\left (x - 1 + \sqrt{3}\right )^{2}} + 4 \sqrt{3} + 7}} \right )}}{\sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{-1 + \sqrt [3]{2}} \left (1 + \sqrt [3]{2}\right )^{\frac{3}{2}} \sqrt{x^{3} - 1}} + \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (- x + 1\right ) \left (- 2 \sqrt{3} + 2^{\frac{2}{3}} + 2\right ) F\left (\operatorname{asin}{\left (\frac{- x + 1 + \sqrt{3}}{- x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{3 \sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{x^{3} - 1} \left (- \sqrt{3} - 2^{\frac{2}{3}} + 1\right )} + \frac{12 \cdot 2^{\frac{2}{3}} \sqrt [4]{3} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- x + 1\right ) \Pi \left (\frac{\left (-1 + 2^{\frac{2}{3}} + \sqrt{3}\right )^{2}}{\left (- 2^{\frac{2}{3}} + 1 + \sqrt{3}\right )^{2}}; \operatorname{asin}{\left (\frac{- x + 1 + \sqrt{3}}{x - 1 + \sqrt{3}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{\sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{4 \sqrt{3} + 7} \sqrt{x^{3} - 1} \left (- 2^{\frac{2}{3}} + 1 + \sqrt{3}\right ) \left (- \sqrt{3} - 2^{\frac{2}{3}} + 1\right )} \]

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)*s
qrt(1 + 2**(1/3))*sqrt(sqrt(3) + 2)*sqrt(-(-x + 1 + sqrt(3))**2/(x - 1 + sqrt(3)
)**2 + 1)/(3*sqrt(-1 + 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(-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(as
in((-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)*sqrt((x**2 + x + 1)/(-x - sqrt(3) + 1)**2)*sqrt(sqrt(3) + 2)*(-x + 1)*ellipti
c_pi((-1 + 2**(2/3) + sqrt(3))**2/(-2**(2/3) + 1 + sqrt(3))**2, asin((-x + 1 + s
qrt(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 = 0.524202, size = 325, normalized size = 8.55 \[ -\frac{4 \sqrt [6]{2} \sqrt{-\frac{i (x-1)}{\sqrt{3}+3 i}} \left (6 i \sqrt{3} \sqrt{2 i x+\sqrt{3}+i} \sqrt{x^2+x+1} \Pi \left (\frac{2 \sqrt{3}}{i+2 i 2^{2/3}+\sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )+\sqrt{-2 i x+\sqrt{3}-i} \left (\left (-3 i \sqrt [3]{2}+4 \sqrt{3}+\sqrt [3]{2} \sqrt{3}\right ) x-\sqrt [3]{2} \sqrt{3}+2 \sqrt{3}-3 i \sqrt [3]{2}-6 i\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )\right )}{\sqrt{3} \left (1+2\ 2^{2/3}-i \sqrt{3}\right ) \sqrt{2 i x+\sqrt{3}+i} \sqrt{x^3-1}} \]

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
]*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])

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Maple [C]  time = 0.041, size = 262, normalized size = 6.9 \[ -4\,{\frac{-3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}-1}}\sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) }-6\,{\frac{{2}^{2/3} \left ( -3/2-i/2\sqrt{3} \right ) }{\sqrt{{x}^{3}-1} \left ( -{2}^{2/3}+1 \right ) }\sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticPi} \left ( \sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}},{\frac{3/2+i/2\sqrt{3}}{-{2}^{2/3}+1}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*(-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))-6*2^(2/3)*(-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)/(-2^(2/3)+1)*EllipticPi(((-1+x
)/(-3/2-1/2*I*3^(1/2)))^(1/2),(3/2+1/2*I*3^(1/2))/(-2^(2/3)+1),((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. \[ -\int \frac{2 \, x + 2^{\frac{2}{3}}}{\sqrt{x^{3} - 1}{\left (x - 2^{\frac{2}{3}}\right )}}\,{d x} \]

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 = 0.358671, size = 173, normalized size = 4.55 \[ \frac{1}{6} \, \sqrt{6} 2^{\frac{1}{6}} \log \left (-\frac{x^{6} - 18 \cdot 2^{\frac{1}{3}} x^{4} - 56 \, x^{3} - 2 \, \sqrt{6} 2^{\frac{1}{6}} \sqrt{x^{3} - 1}{\left (18 \, x^{2} - 2^{\frac{2}{3}}{\left (x^{4} + 8 \, x\right )} - 2^{\frac{1}{3}}{\left (5 \, x^{3} - 2\right )}\right )} + 18 \cdot 2^{\frac{2}{3}}{\left (x^{5} + 2 \, x^{2}\right )} - 8}{x^{6} - 80 \, x^{3} - 6 \cdot 2^{\frac{2}{3}}{\left (x^{5} - 10 \, x^{2}\right )} + 6 \cdot 2^{\frac{1}{3}}{\left (5 \, x^{4} - 8 \, x\right )} + 16}\right ) \]

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. \[ - \int \frac{2^{\frac{2}{3}}}{x \sqrt{x^{3} - 1} - 2^{\frac{2}{3}} \sqrt{x^{3} - 1}}\, dx - \int \frac{2 x}{x \sqrt{x^{3} - 1} - 2^{\frac{2}{3}} \sqrt{x^{3} - 1}}\, dx \]

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)

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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)