Integrand size = 15, antiderivative size = 25 \[ \int \frac {1}{\sqrt [3]{(-1+x)^4 (1+x)^2}} \, dx=-\frac {3 (-1+x) (1+x)}{2 \sqrt [3]{(-1+x)^4 (1+x)^2}} \]
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Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6851, 37} \[ \int \frac {1}{\sqrt [3]{(-1+x)^4 (1+x)^2}} \, dx=\frac {3 (1-x) (x+1)}{2 \sqrt [3]{(1-x)^4 (x+1)^2}} \]
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Rule 37
Rule 6851
Rubi steps \begin{align*} \text {integral}& = \frac {\left ((-1+x)^{4/3} (1+x)^{2/3}\right ) \int \frac {1}{(-1+x)^{4/3} (1+x)^{2/3}} \, dx}{\sqrt [3]{(-1+x)^4 (1+x)^2}} \\ & = \frac {3 (1-x) (1+x)}{2 \sqrt [3]{(1-x)^4 (1+x)^2}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt [3]{(-1+x)^4 (1+x)^2}} \, dx=-\frac {3 (-1+x) (1+x)}{2 \sqrt [3]{(-1+x)^4 (1+x)^2}} \]
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Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88
method | result | size |
gosper | \(-\frac {3 \left (-1+x \right ) \left (1+x \right )}{2 \left (\left (-1+x \right )^{4} \left (1+x \right )^{2}\right )^{\frac {1}{3}}}\) | \(22\) |
risch | \(-\frac {3 \left (-1+x \right ) \left (1+x \right )}{2 \left (\left (-1+x \right )^{4} \left (1+x \right )^{2}\right )^{\frac {1}{3}}}\) | \(22\) |
trager | \(-\frac {3 \left (x^{6}-2 x^{5}-x^{4}+4 x^{3}-x^{2}-2 x +1\right )^{\frac {2}{3}}}{2 \left (-1+x \right )^{3} \left (1+x \right )}\) | \(43\) |
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Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (21) = 42\).
Time = 0.23 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.88 \[ \int \frac {1}{\sqrt [3]{(-1+x)^4 (1+x)^2}} \, dx=-\frac {3 \, {\left (x^{6} - 2 \, x^{5} - x^{4} + 4 \, x^{3} - x^{2} - 2 \, x + 1\right )}^{\frac {2}{3}}}{2 \, {\left (x^{4} - 2 \, x^{3} + 2 \, x - 1\right )}} \]
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\[ \int \frac {1}{\sqrt [3]{(-1+x)^4 (1+x)^2}} \, dx=\int \frac {1}{\sqrt [3]{\left (x - 1\right )^{4} \left (x + 1\right )^{2}}}\, dx \]
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\[ \int \frac {1}{\sqrt [3]{(-1+x)^4 (1+x)^2}} \, dx=\int { \frac {1}{\left ({\left (x + 1\right )}^{2} {\left (x - 1\right )}^{4}\right )^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {1}{\sqrt [3]{(-1+x)^4 (1+x)^2}} \, dx=\int { \frac {1}{\left ({\left (x + 1\right )}^{2} {\left (x - 1\right )}^{4}\right )^{\frac {1}{3}}} \,d x } \]
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Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt [3]{(-1+x)^4 (1+x)^2}} \, dx=-\frac {3\,{\left ({\left (x-1\right )}^4\,{\left (x+1\right )}^2\right )}^{2/3}}{2\,{\left (x-1\right )}^3\,\left (x+1\right )} \]
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