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

Optimal. Leaf size=20 \[ \frac{3 \sqrt [3]{x^2-1}}{2 (x+1)^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.0186093, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{3 \sqrt [3]{x^2-1}}{2 (x+1)^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 1.26123, size = 17, normalized size = 0.85 \[ \frac{3 \sqrt [3]{x^{2} - 1}}{2 \left (x + 1\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.018686, size = 23, normalized size = 1.15 \[ \frac{3 (x-1) \sqrt [3]{x+1}}{2 \left (x^2-1\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 18, normalized size = 0.9 \[{\frac{-3+3\,x}{2}\sqrt [3]{1+x} \left ({x}^{2}-1 \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.268034, size = 19, normalized size = 0.95 \[ \frac{3 \,{\left (x^{2} - 1\right )}^{\frac{1}{3}}}{2 \,{\left (x + 1\right )}^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/2*(x^2 - 1)^(1/3)/(x + 1)^(2/3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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