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

Optimal. Leaf size=156 \[ \frac{2 \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}}-\frac{2\ 2^{2/3} \tanh ^{-1}\left (\frac{\sqrt{3} \left (\sqrt [3]{2} x+1\right )}{\sqrt{-x^3-1}}\right )}{3 \sqrt{3}} \]

[Out]

(-2*2^(2/3)*ArcTanh[(Sqrt[3]*(1 + 2^(1/3)*x))/Sqrt[-1 - x^3]])/(3*Sqrt[3]) + (2*
Sqrt[2 - Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 - Sqrt[3] + x)^2]*EllipticF[ArcS
in[(1 + Sqrt[3] + x)/(1 - Sqrt[3] + x)], -7 + 4*Sqrt[3]])/(3*3^(1/4)*Sqrt[-((1 +
x)/(1 - Sqrt[3] + x)^2)]*Sqrt[-1 - x^3])

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Rubi [A]  time = 0.366649, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}}-\frac{2\ 2^{2/3} \tanh ^{-1}\left (\frac{\sqrt{3} \left (\sqrt [3]{2} x+1\right )}{\sqrt{-x^3-1}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[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]])/(3*Sqrt[3]) + (2*
Sqrt[2 - Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 - Sqrt[3] + x)^2]*EllipticF[ArcS
in[(1 + Sqrt[3] + x)/(1 - Sqrt[3] + x)], -7 + 4*Sqrt[3]])/(3*3^(1/4)*Sqrt[-((1 +
x)/(1 - Sqrt[3] + x)^2)]*Sqrt[-1 - x^3])

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Rubi in Sympy [A]  time = 144.518, size = 454, normalized size = 2.91 \[ - \frac{2^{\frac{2}{3}} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (x + 1\right ) \operatorname{atanh}{\left (\frac{3^{\frac{3}{4}} \sqrt{1 + \sqrt [3]{2}} \sqrt{1 - \frac{\left (x + 1 + \sqrt{3}\right )^{2}}{\left (- x - 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2}}{3 \sqrt{-1 + \sqrt [3]{2}} \sqrt{4 \sqrt{3} + 7 + \frac{\left (x + 1 + \sqrt{3}\right )^{2}}{\left (- x - 1 + \sqrt{3}\right )^{2}}}} \right )}}{\sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{-1 + \sqrt [3]{2}} \left (1 + \sqrt [3]{2}\right )^{\frac{3}{2}} \sqrt{- x^{3} - 1}} + \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (- \sqrt{3} + 1\right ) \sqrt{- \sqrt{3} + 2} \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{3 \sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- x^{3} - 1} \left (- \sqrt{3} - 2^{\frac{2}{3}} + 1\right )} + \frac{4 \cdot 2^{\frac{2}{3}} \sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (x + 1\right ) \Pi \left (\frac{\left (-1 + 2^{\frac{2}{3}} + \sqrt{3}\right )^{2}}{\left (- 2^{\frac{2}{3}} + 1 + \sqrt{3}\right )^{2}}; \operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{- x - 1 + \sqrt{3}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{\sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{4 \sqrt{3} + 7} \sqrt{- x^{3} - 1} \left (- 2^{\frac{2}{3}} + 1 + \sqrt{3}\right ) \left (- \sqrt{3} - 2^{\frac{2}{3}} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.526008, size = 209, normalized size = 1.34 \[ \frac{2 \sqrt{\frac{x+1}{1+\sqrt [3]{-1}}} \left (-\frac{\left (\sqrt [3]{-1}-x\right ) \sqrt{\frac{\sqrt [3]{-1}-(-1)^{2/3} x}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}}+\frac{i 2^{2/3} \sqrt{x^2-x+1} \Pi \left (\frac{i \sqrt{3}}{\sqrt [3]{-1}+2^{2/3}};\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt [3]{-1}+2^{2/3}}\right )}{\sqrt{-x^3-1}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [A]  time = 0.029, size = 249, normalized size = 1.6 \[{-{\frac{2\,i}{3}}\sqrt{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}}+{\frac{{\frac{2\,i}{3}}{2}^{{\frac{2}{3}}}\sqrt{3}}{{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}+{2}^{{\frac{2}{3}}}}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{\frac{i\sqrt{3}}{{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}+{2}^{{\frac{2}{3}}}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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