Integrand size = 11, antiderivative size = 14 \[ \int \frac {1}{-\sqrt [3]{x}+x} \, dx=\frac {3}{2} \log \left (1-x^{2/3}\right ) \]
[Out]
Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1607, 266} \[ \int \frac {1}{-\sqrt [3]{x}+x} \, dx=\frac {3}{2} \log \left (1-x^{2/3}\right ) \]
[In]
[Out]
Rule 266
Rule 1607
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (-1+x^{2/3}\right ) \sqrt [3]{x}} \, dx \\ & = \frac {3}{2} \log \left (1-x^{2/3}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.79 \[ \int \frac {1}{-\sqrt [3]{x}+x} \, dx=\frac {3}{2} \log \left (-1+\sqrt [3]{x}\right )+\frac {3}{2} \log \left (1+\sqrt [3]{x}\right ) \]
[In]
[Out]
Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.79
method | result | size |
meijerg | \(\frac {3 \ln \left (1-x^{\frac {2}{3}}\right )}{2}\) | \(11\) |
derivativedivides | \(\frac {3 \ln \left (x^{\frac {1}{3}}-1\right )}{2}+\frac {3 \ln \left (x^{\frac {1}{3}}+1\right )}{2}\) | \(18\) |
trager | \(\frac {\ln \left (3 x^{\frac {2}{3}}-3 x^{\frac {4}{3}}+x^{2}-1\right )}{2}\) | \(19\) |
default | \(\frac {\ln \left (-1+x \right )}{2}+\frac {\ln \left (1+x \right )}{2}-\frac {\ln \left (x^{\frac {2}{3}}+x^{\frac {1}{3}}+1\right )}{2}+\ln \left (x^{\frac {1}{3}}-1\right )+\ln \left (x^{\frac {1}{3}}+1\right )-\frac {\ln \left (x^{\frac {2}{3}}-x^{\frac {1}{3}}+1\right )}{2}\) | \(50\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.57 \[ \int \frac {1}{-\sqrt [3]{x}+x} \, dx=\frac {3}{2} \, \log \left (x^{\frac {2}{3}} - 1\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (10) = 20\).
Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.57 \[ \int \frac {1}{-\sqrt [3]{x}+x} \, dx=\frac {3 \log {\left (\sqrt [3]{x} - 1 \right )}}{2} + \frac {3 \log {\left (\sqrt [3]{x} + 1 \right )}}{2} \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.21 \[ \int \frac {1}{-\sqrt [3]{x}+x} \, dx=\frac {3}{2} \, \log \left (x^{\frac {1}{3}} + 1\right ) + \frac {3}{2} \, \log \left (x^{\frac {1}{3}} - 1\right ) \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.29 \[ \int \frac {1}{-\sqrt [3]{x}+x} \, dx=\frac {3}{2} \, \log \left (x^{\frac {1}{3}} + 1\right ) + \frac {3}{2} \, \log \left ({\left | x^{\frac {1}{3}} - 1 \right |}\right ) \]
[In]
[Out]
Time = 0.15 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.57 \[ \int \frac {1}{-\sqrt [3]{x}+x} \, dx=\frac {3\,\ln \left (x^{2/3}-1\right )}{2} \]
[In]
[Out]