Integrand size = 15, antiderivative size = 18 \[ \int \frac {1}{x^2 \sqrt [3]{-x+x^3}} \, dx=\frac {3 \left (-x+x^3\right )^{2/3}}{4 x^2} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2039} \[ \int \frac {1}{x^2 \sqrt [3]{-x+x^3}} \, dx=\frac {3 \left (x^3-x\right )^{2/3}}{4 x^2} \]
[In]
[Out]
Rule 2039
Rubi steps \begin{align*} \text {integral}& = \frac {3 \left (-x+x^3\right )^{2/3}}{4 x^2} \\ \end{align*}
Time = 0.55 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt [3]{-x+x^3}} \, dx=\frac {3 \left (x \left (-1+x^2\right )\right )^{2/3}}{4 x^2} \]
[In]
[Out]
Time = 0.77 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83
method | result | size |
trager | \(\frac {3 \left (x^{3}-x \right )^{\frac {2}{3}}}{4 x^{2}}\) | \(15\) |
pseudoelliptic | \(\frac {3 \left (x^{3}-x \right )^{\frac {2}{3}}}{4 x^{2}}\) | \(15\) |
risch | \(\frac {\frac {3 x^{2}}{4}-\frac {3}{4}}{x {\left (x \left (x^{2}-1\right )\right )}^{\frac {1}{3}}}\) | \(20\) |
gosper | \(\frac {3 \left (1+x \right ) \left (x -1\right )}{4 x \left (x^{3}-x \right )^{\frac {1}{3}}}\) | \(21\) |
meijerg | \(-\frac {3 {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {2}{3}}}{4 \operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{3}} x^{\frac {4}{3}}}\) | \(33\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^2 \sqrt [3]{-x+x^3}} \, dx=\frac {3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}}}{4 \, x^{2}} \]
[In]
[Out]
\[ \int \frac {1}{x^2 \sqrt [3]{-x+x^3}} \, dx=\int \frac {1}{x^{2} \sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )}}\, dx \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x^2 \sqrt [3]{-x+x^3}} \, dx=\frac {3 \, {\left (x^{3} - x\right )}}{4 \, {\left (x + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )}^{\frac {1}{3}} x^{\frac {7}{3}}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.61 \[ \int \frac {1}{x^2 \sqrt [3]{-x+x^3}} \, dx=\frac {3}{4} \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} \]
[In]
[Out]
Time = 4.95 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^2 \sqrt [3]{-x+x^3}} \, dx=\frac {3\,{\left (x^3-x\right )}^{2/3}}{4\,x^2} \]
[In]
[Out]