Optimal. Leaf size=27 \[ -\frac {3 \left (1-x^2\right )}{2 \left (-\left ((x+1) \left (1-x^2\right )\right )\right )^{2/3}} \]
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Rubi [A] time = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 37
Rule 2064
Rule 2067
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.42, size = 20, normalized size = 0.74 \[ \frac {3 \, {\left (x^{3} + x^{2} - x - 1\right )}^{\frac {1}{3}}}{2 \, {\left (x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left ({\left (x^{2} - 1\right )} {\left (x + 1\right )}\right )^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 20, normalized size = 0.74 \[ \frac {3 \left (x -1\right ) \left (x +1\right )}{2 \left (\left (x +1\right ) \left (x^{2}-1\right )\right )^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left ({\left (x^{2} - 1\right )} {\left (x + 1\right )}\right )^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.42, size = 20, normalized size = 0.74 \[ \frac {3\,{\left (\left (x^2-1\right )\,\left (x+1\right )\right )}^{1/3}}{2\,\left (x+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\left (x + 1\right ) \left (x^{2} - 1\right )\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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