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

Optimal. Leaf size=122 \[ -\frac{\log \left (\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}+\frac{\log \left (-\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+\frac{2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}+1\right )}{6\ 2^{2/3}} \]

[Out]

-(ArcTan[(1 - (2*2^(1/3)*x)/(1 - x^3)^(1/3))/Sqrt[3]]/(2^(2/3)*Sqrt[3])) + Log[1
+ (2^(2/3)*x^2)/(1 - x^3)^(2/3) - (2^(1/3)*x)/(1 - x^3)^(1/3)]/(6*2^(2/3)) - Lo
g[1 + (2^(1/3)*x)/(1 - x^3)^(1/3)]/(3*2^(2/3))

_______________________________________________________________________________________

Rubi [A]  time = 0.155538, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{\log \left (\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}+\frac{\log \left (-\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+\frac{2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}+1\right )}{6\ 2^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

-(ArcTan[(1 - (2*2^(1/3)*x)/(1 - x^3)^(1/3))/Sqrt[3]]/(2^(2/3)*Sqrt[3])) + Log[1
+ (2^(2/3)*x^2)/(1 - x^3)^(2/3) - (2^(1/3)*x)/(1 - x^3)^(1/3)]/(6*2^(2/3)) - Lo
g[1 + (2^(1/3)*x)/(1 - x^3)^(1/3)]/(3*2^(2/3))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 16.7135, size = 107, normalized size = 0.88 \[ - \frac{\sqrt [3]{2} \log{\left (\frac{\sqrt [3]{2} x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{6} + \frac{\sqrt [3]{2} \log{\left (\frac{2^{\frac{2}{3}} x^{2}}{\left (- x^{3} + 1\right )^{\frac{2}{3}}} - \frac{\sqrt [3]{2} x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{12} - \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \sqrt [3]{2} x}{3 \sqrt [3]{- x^{3} + 1}} + \frac{1}{3}\right ) \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2**(1/3)*log(2**(1/3)*x/(-x**3 + 1)**(1/3) + 1)/6 + 2**(1/3)*log(2**(2/3)*x**2/
(-x**3 + 1)**(2/3) - 2**(1/3)*x/(-x**3 + 1)**(1/3) + 1)/12 - 2**(1/3)*sqrt(3)*at
an(sqrt(3)*(-2*2**(1/3)*x/(3*(-x**3 + 1)**(1/3)) + 1/3))/6

_______________________________________________________________________________________

Mathematica [C]  time = 0.105468, size = 59, normalized size = 0.48 \[ \frac{x^2 \left (\frac{1-x^3}{x^3+1}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{2 x^3}{x^3+1}\right )}{2 \left (1-x^3\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

(x^2*((1 - x^3)/(1 + x^3))^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, (2*x^3)/(1 + x
^3)])/(2*(1 - x^3)^(2/3))

_______________________________________________________________________________________

Maple [F]  time = 0.058, size = 0, normalized size = 0. \[ \int{\frac{x}{{x}^{3}+1} \left ( -{x}^{3}+1 \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(-x^3+1)^(2/3)/(x^3+1),x)

[Out]

int(x/(-x^3+1)^(2/3)/(x^3+1),x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 1.44773, size = 348, normalized size = 2.85 \[ -\frac{1}{216} \cdot 4^{\frac{2}{3}} \sqrt{3}{\left (\sqrt{3} \left (-1\right )^{\frac{1}{3}} \log \left (\frac{3 \cdot 4^{\frac{2}{3}} \left (-1\right )^{\frac{2}{3}}{\left (5 \, x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} - 12 \cdot 4^{\frac{1}{3}}{\left (2 \, x^{5} - x^{2}\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 2 \, \left (-1\right )^{\frac{1}{3}}{\left (19 \, x^{6} - 16 \, x^{3} + 1\right )}}{x^{6} + 2 \, x^{3} + 1}\right ) - 2 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}} \log \left (-\frac{6 \cdot 4^{\frac{1}{3}} \left (-1\right )^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{2} - 3 \cdot 4^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x - 2 \, \left (-1\right )^{\frac{2}{3}}{\left (x^{3} + 1\right )}}{x^{3} + 1}\right ) + 6 \, \left (-1\right )^{\frac{1}{3}} \arctan \left (\frac{3 \cdot 4^{\frac{1}{3}} \sqrt{3} \left (-1\right )^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{2} + 3 \cdot 4^{\frac{2}{3}} \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x - \sqrt{3} \left (-1\right )^{\frac{2}{3}}{\left (x^{3} + 1\right )}}{3 \,{\left (3 \cdot 4^{\frac{1}{3}} \left (-1\right )^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{2} + \left (-1\right )^{\frac{2}{3}}{\left (x^{3} + 1\right )}\right )}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/216*4^(2/3)*sqrt(3)*(sqrt(3)*(-1)^(1/3)*log((3*4^(2/3)*(-1)^(2/3)*(5*x^4 - x)
*(-x^3 + 1)^(2/3) - 12*4^(1/3)*(2*x^5 - x^2)*(-x^3 + 1)^(1/3) - 2*(-1)^(1/3)*(19
*x^6 - 16*x^3 + 1))/(x^6 + 2*x^3 + 1)) - 2*sqrt(3)*(-1)^(1/3)*log(-(6*4^(1/3)*(-
1)^(1/3)*(-x^3 + 1)^(1/3)*x^2 - 3*4^(2/3)*(-x^3 + 1)^(2/3)*x - 2*(-1)^(2/3)*(x^3
+ 1))/(x^3 + 1)) + 6*(-1)^(1/3)*arctan(1/3*(3*4^(1/3)*sqrt(3)*(-1)^(1/3)*(-x^3
+ 1)^(1/3)*x^2 + 3*4^(2/3)*sqrt(3)*(-x^3 + 1)^(2/3)*x - sqrt(3)*(-1)^(2/3)*(x^3
+ 1))/(3*4^(1/3)*(-1)^(1/3)*(-x^3 + 1)^(1/3)*x^2 + (-1)^(2/3)*(x^3 + 1))))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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