Integrand size = 18, antiderivative size = 18 \[ \int \frac {3+x}{(-1+x)^2 \sqrt [3]{-1+x^2}} \, dx=-\frac {3 \left (-1+x^2\right )^{2/3}}{2 (-1+x)^2} \]
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Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {801} \[ \int \frac {3+x}{(-1+x)^2 \sqrt [3]{-1+x^2}} \, dx=-\frac {3 \left (x^2-1\right )^{2/3}}{2 (1-x)^2} \]
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Rule 801
Rubi steps \begin{align*} \text {integral}& = -\frac {3 \left (-1+x^2\right )^{2/3}}{2 (1-x)^2} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {3+x}{(-1+x)^2 \sqrt [3]{-1+x^2}} \, dx=-\frac {3 \left (-1+x^2\right )^{2/3}}{2 (-1+x)^2} \]
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Time = 0.91 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83
method | result | size |
trager | \(-\frac {3 \left (x^{2}-1\right )^{\frac {2}{3}}}{2 \left (x -1\right )^{2}}\) | \(15\) |
gosper | \(-\frac {3 \left (1+x \right )}{2 \left (x -1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}}\) | \(18\) |
risch | \(-\frac {3 \left (1+x \right )}{2 \left (x -1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}}\) | \(18\) |
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none
Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {3+x}{(-1+x)^2 \sqrt [3]{-1+x^2}} \, dx=-\frac {3 \, {\left (x^{2} - 1\right )}^{\frac {2}{3}}}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} \]
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\[ \int \frac {3+x}{(-1+x)^2 \sqrt [3]{-1+x^2}} \, dx=\int \frac {x + 3}{\sqrt [3]{\left (x - 1\right ) \left (x + 1\right )} \left (x - 1\right )^{2}}\, dx \]
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none
Time = 0.23 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67 \[ \int \frac {3+x}{(-1+x)^2 \sqrt [3]{-1+x^2}} \, dx=-\frac {3 \, {\left (x + 1\right )}^{\frac {2}{3}}}{2 \, {\left (x - 1\right )}^{\frac {4}{3}}} \]
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\[ \int \frac {3+x}{(-1+x)^2 \sqrt [3]{-1+x^2}} \, dx=\int { \frac {x + 3}{{\left (x^{2} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )}^{2}} \,d x } \]
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Time = 4.95 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {3+x}{(-1+x)^2 \sqrt [3]{-1+x^2}} \, dx=-\frac {3\,{\left (x^2-1\right )}^{2/3}}{2\,{\left (x-1\right )}^2} \]
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