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

Optimal. Leaf size=26 \[ -\frac{F_1\left (-\frac{2}{3};\frac{2}{3},1;\frac{1}{3};x^3,-x^3\right )}{2 x^2} \]

[Out]

-AppellF1[-2/3, 2/3, 1, 1/3, x^3, -x^3]/(2*x^2)

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Rubi [A]  time = 0.0630453, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{F_1\left (-\frac{2}{3};\frac{2}{3},1;\frac{1}{3};x^3,-x^3\right )}{2 x^2} \]

Antiderivative was successfully verified.

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

[Out]

-AppellF1[-2/3, 2/3, 1, 1/3, x^3, -x^3]/(2*x^2)

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Rubi in Sympy [A]  time = 6.27822, size = 20, normalized size = 0.77 \[ - \frac{\operatorname{appellf_{1}}{\left (- \frac{2}{3},\frac{2}{3},1,\frac{1}{3},x^{3},- x^{3} \right )}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-appellf1(-2/3, 2/3, 1, 1/3, x**3, -x**3)/(2*x**2)

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Mathematica [B]  time = 0.181723, size = 120, normalized size = 4.62 \[ \frac{\sqrt [3]{1-x^3} \left (\frac{4 x^3 F_1\left (\frac{1}{3};-\frac{1}{3},1;\frac{4}{3};x^3,-x^3\right )}{\left (x^3+1\right ) \left (x^3 \left (3 F_1\left (\frac{4}{3};-\frac{1}{3},2;\frac{7}{3};x^3,-x^3\right )+F_1\left (\frac{4}{3};\frac{2}{3},1;\frac{7}{3};x^3,-x^3\right )\right )-4 F_1\left (\frac{1}{3};-\frac{1}{3},1;\frac{4}{3};x^3,-x^3\right )\right )}-1\right )}{2 x^2} \]

Warning: Unable to verify antiderivative.

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

[Out]

((1 - x^3)^(1/3)*(-1 + (4*x^3*AppellF1[1/3, -1/3, 1, 4/3, x^3, -x^3])/((1 + x^3)
*(-4*AppellF1[1/3, -1/3, 1, 4/3, x^3, -x^3] + x^3*(3*AppellF1[4/3, -1/3, 2, 7/3,
x^3, -x^3] + AppellF1[4/3, 2/3, 1, 7/3, x^3, -x^3])))))/(2*x^2)

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Maple [F]  time = 0.08, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3} \left ({x}^{3}+1 \right ) } \left ( -{x}^{3}+1 \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 1.81875, size = 474, normalized size = 18.23 \[ \frac{4^{\frac{2}{3}} \sqrt{3}{\left (2 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}} x^{2} \log \left (-\frac{3 \cdot 4^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x^{2} + 3 \cdot 4^{\frac{1}{3}} \left (-1\right )^{\frac{1}{3}}{\left (x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + \left (-1\right )^{\frac{2}{3}}{\left (x^{6} + 2 \, x^{3} + 1\right )}}{x^{6} + 2 \, x^{3} + 1}\right ) - \sqrt{3} \left (-1\right )^{\frac{1}{3}} x^{2} \log \left (\frac{6 \cdot 4^{\frac{2}{3}} \left (-1\right )^{\frac{2}{3}}{\left (x^{8} - 4 \, x^{5} + x^{2}\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} + 3 \cdot 4^{\frac{1}{3}}{\left (x^{10} - 11 \, x^{7} + 11 \, x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - \left (-1\right )^{\frac{1}{3}}{\left (x^{12} - 32 \, x^{9} + 78 \, x^{6} - 32 \, x^{3} + 1\right )}}{x^{12} + 4 \, x^{9} + 6 \, x^{6} + 4 \, x^{3} + 1}\right ) + 6 \, \left (-1\right )^{\frac{1}{3}} x^{2} \arctan \left (\frac{6 \cdot 4^{\frac{2}{3}} \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x^{2} - 3 \cdot 4^{\frac{1}{3}} \sqrt{3} \left (-1\right )^{\frac{1}{3}}{\left (x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - \sqrt{3} \left (-1\right )^{\frac{2}{3}}{\left (x^{6} + 2 \, x^{3} + 1\right )}}{3 \,{\left (3 \cdot 4^{\frac{1}{3}} \left (-1\right )^{\frac{1}{3}}{\left (x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - \left (-1\right )^{\frac{2}{3}}{\left (x^{6} + 2 \, x^{3} + 1\right )}\right )}}\right ) - 18 \cdot 4^{\frac{1}{3}} \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right )}}{432 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/432*4^(2/3)*sqrt(3)*(2*sqrt(3)*(-1)^(1/3)*x^2*log(-(3*4^(2/3)*(-x^3 + 1)^(2/3)
*x^2 + 3*4^(1/3)*(-1)^(1/3)*(x^4 - x)*(-x^3 + 1)^(1/3) + (-1)^(2/3)*(x^6 + 2*x^3
+ 1))/(x^6 + 2*x^3 + 1)) - sqrt(3)*(-1)^(1/3)*x^2*log((6*4^(2/3)*(-1)^(2/3)*(x^
8 - 4*x^5 + x^2)*(-x^3 + 1)^(2/3) + 3*4^(1/3)*(x^10 - 11*x^7 + 11*x^4 - x)*(-x^3
+ 1)^(1/3) - (-1)^(1/3)*(x^12 - 32*x^9 + 78*x^6 - 32*x^3 + 1))/(x^12 + 4*x^9 +
6*x^6 + 4*x^3 + 1)) + 6*(-1)^(1/3)*x^2*arctan(1/3*(6*4^(2/3)*sqrt(3)*(-x^3 + 1)^
(2/3)*x^2 - 3*4^(1/3)*sqrt(3)*(-1)^(1/3)*(x^4 - x)*(-x^3 + 1)^(1/3) - sqrt(3)*(-
1)^(2/3)*(x^6 + 2*x^3 + 1))/(3*4^(1/3)*(-1)^(1/3)*(x^4 - x)*(-x^3 + 1)^(1/3) - (
-1)^(2/3)*(x^6 + 2*x^3 + 1))) - 18*4^(1/3)*sqrt(3)*(-x^3 + 1)^(1/3))/x^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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