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 = 0.0298237, 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, Rules used = {2067, 2064, 37} \[ -\frac{3 (1-x) (x+1)}{2 \left (x^3+x^2-x-1\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 2067
Rule 2064
Rule 37
Rubi steps
\begin{align*} \int \frac{1}{\left ((1+x) \left (-1+x^2\right )\right )^{2/3}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\left (-\frac{16}{27}-\frac{4 x}{3}+x^3\right )^{2/3}} \, dx,x,\frac{1}{3}+x\right )\\ &=\frac{\left (32 \sqrt [3]{2} (-1-x)^{4/3} (-1+x)^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (-\frac{16}{9}-\frac{8 x}{3}\right )^{4/3} \left (-\frac{16}{9}+\frac{4 x}{3}\right )^{2/3}} \, dx,x,\frac{1}{3}+x\right )}{9 \left (-1-x+x^2+x^3\right )^{2/3}}\\ &=-\frac{3 (1-x) (1+x)}{2 \left (-1-x+x^2+x^3\right )^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.0098332, 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.
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Maple [A] time = 0.001, size = 20, normalized size = 0.7 \begin{align*}{\frac{ \left ( 3\,x-3 \right ) \left ( 1+x \right ) }{2} \left ( \left ( 1+x \right ) \left ({x}^{2}-1 \right ) \right ) ^{-{\frac{2}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left ({\left (x^{2} - 1\right )}{\left (x + 1\right )}\right )^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.39376, size = 53, normalized size = 1.96 \begin{align*} \frac{3 \,{\left (x^{3} + x^{2} - x - 1\right )}^{\frac{1}{3}}}{2 \,{\left (x + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (\left (x + 1\right ) \left (x^{2} - 1\right )\right )^{\frac{2}{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left ({\left (x^{2} - 1\right )}{\left (x + 1\right )}\right )^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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