Optimal. Leaf size=20 \[ \frac{3 \sqrt [3]{x^2-1}}{2 (x+1)^{2/3}} \]
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Rubi [A] time = 0.0056524, 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, Rules used = {651} \[ \frac{3 \sqrt [3]{x^2-1}}{2 (x+1)^{2/3}} \]
Antiderivative was successfully verified.
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Rule 651
Rubi steps
\begin{align*} \int \frac{1}{(1+x)^{2/3} \left (-1+x^2\right )^{2/3}} \, dx &=\frac{3 \sqrt [3]{-1+x^2}}{2 (1+x)^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.030237, 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.
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Maple [A] time = 0.003, size = 18, normalized size = 0.9 \begin{align*}{\frac{3\,x-3}{2}\sqrt [3]{1+x} \left ({x}^{2}-1 \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 (x^{2} - 1\right )}^{\frac{2}{3}}{\left (x + 1\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 = 1.68169, size = 47, normalized size = 2.35 \begin{align*} \frac{3 \,{\left (x^{2} - 1\right )}^{\frac{1}{3}}}{2 \,{\left (x + 1\right )}^{\frac{2}{3}}} \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 + 1\right )\right )^{\frac{2}{3}} \left (x + 1\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 (x^{2} - 1\right )}^{\frac{2}{3}}{\left (x + 1\right )}^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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