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

Optimal. Leaf size=27 \[ -\frac{3 \left (1-x^2\right )}{2 \left (-(x+1) \left (1-x^2\right )\right )^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.0680533, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{3 (1-x) (x+1)}{2 \left (x^3+x^2-x-1\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 1.87336, size = 24, normalized size = 0.89 \[ - \frac{3 \left (- x + 1\right ) \left (x + 1\right )}{2 \left (x^{3} + x^{2} - x - 1\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0201148, size = 23, normalized size = 0.85 \[ \frac{3 (x-1) (x+1)}{2 \left ((x-1) (x+1)^2\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 20, normalized size = 0.7 \[{\frac{ \left ( -3+3\,x \right ) \left ( 1+x \right ) }{2} \left ( \left ( 1+x \right ) \left ({x}^{2}-1 \right ) \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/((1+x)*(x^2-1))^(2/3),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.262481, size = 27, normalized size = 1. \[ \frac{3 \,{\left (x^{3} + x^{2} - x - 1\right )}^{\frac{1}{3}}}{2 \,{\left (x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/2*(x^3 + x^2 - x - 1)^(1/3)/(x + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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