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