Optimal. Leaf size=27 \[ -\frac {3 \left (1-x^2\right )}{2 \left (-\left ((1+x) \left (1-x^2\right )\right )\right )^{2/3}} \]
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Rubi [A]
time = 0.02, 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 = {2092, 2089, 37}
\begin {gather*} -\frac {3 (1-x) (x+1)}{2 \left (x^3+x^2-x-1\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 2089
Rule 2092
Rubi steps
\begin {align*} \int \frac {1}{\left ((1+x) \left (-1+x^2\right )\right )^{2/3}} \, dx &=\text {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 ) \text {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 = 0.02, size = 23, normalized size = 0.85 \begin {gather*} \frac {3 (-1+x) (1+x)}{2 \left ((-1+x) (1+x)^2\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 20, normalized size = 0.74
method | result | size |
gosper | \(\frac {3 \left (1+x \right ) \left (-1+x \right )}{2 \left (\left (1+x \right ) \left (x^{2}-1\right )\right )^{\frac {2}{3}}}\) | \(20\) |
risch | \(\frac {3 \left (1+x \right ) \left (-1+x \right )}{2 \left (\left (-1+x \right ) \left (1+x \right )^{2}\right )^{\frac {2}{3}}}\) | \(20\) |
trager | \(\frac {3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}}{2 \left (1+x \right )}\) | \(21\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 20, normalized size = 0.74 \begin {gather*} \frac {3 \, {\left (x^{3} + x^{2} - x - 1\right )}^{\frac {1}{3}}}{2 \, {\left (x + 1\right )}} \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (\left (x + 1\right ) \left (x^{2} - 1\right )\right )^{\frac {2}{3}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \frac {3\,{\left (\left (x^2-1\right )\,\left (x+1\right )\right )}^{1/3}}{2\,\left (x+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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