Integrand size = 18, antiderivative size = 28 \[ \int \frac {-2+2 x+3 x^2}{-1+x^3} \, dx=\frac {4 \arctan \left (\frac {1+2 x}{\sqrt {3}}\right )}{\sqrt {3}}+\log \left (1-x^3\right ) \]
[Out]
Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1885, 1600, 632, 210, 266} \[ \int \frac {-2+2 x+3 x^2}{-1+x^3} \, dx=\frac {4 \arctan \left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}}+\log \left (1-x^3\right ) \]
[In]
[Out]
Rule 210
Rule 266
Rule 632
Rule 1600
Rule 1885
Rubi steps \begin{align*} \text {integral}& = 3 \int \frac {x^2}{-1+x^3} \, dx+\int \frac {-2+2 x}{-1+x^3} \, dx \\ & = \log \left (1-x^3\right )+\int \frac {1}{\frac {1}{2}+\frac {x}{2}+\frac {x^2}{2}} \, dx \\ & = \log \left (1-x^3\right )-2 \text {Subst}\left (\int \frac {1}{-\frac {3}{4}-x^2} \, dx,x,\frac {1}{2}+x\right ) \\ & = \frac {4 \arctan \left (\frac {1+2 x}{\sqrt {3}}\right )}{\sqrt {3}}+\log \left (1-x^3\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {-2+2 x+3 x^2}{-1+x^3} \, dx=\frac {4 \arctan \left (\frac {1+2 x}{\sqrt {3}}\right )}{\sqrt {3}}+\log \left (1-x^3\right ) \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04
method | result | size |
default | \(\ln \left (-1+x \right )+\ln \left (x^{2}+x +1\right )+\frac {4 \arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{3}\) | \(29\) |
risch | \(\ln \left (4 x^{2}+4 x +4\right )+\frac {4 \arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{3}+\ln \left (-1+x \right )\) | \(33\) |
meijerg | \(-\frac {2 x \left (\ln \left (1-\left (x^{3}\right )^{\frac {1}{3}}\right )-\frac {\ln \left (1+\left (x^{3}\right )^{\frac {1}{3}}+\left (x^{3}\right )^{\frac {2}{3}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{3}\right )^{\frac {1}{3}}}{2+\left (x^{3}\right )^{\frac {1}{3}}}\right )\right )}{3 \left (x^{3}\right )^{\frac {1}{3}}}+\ln \left (-x^{3}+1\right )+\frac {2 x^{2} \left (\ln \left (1-\left (x^{3}\right )^{\frac {1}{3}}\right )-\frac {\ln \left (1+\left (x^{3}\right )^{\frac {1}{3}}+\left (x^{3}\right )^{\frac {2}{3}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{3}\right )^{\frac {1}{3}}}{2+\left (x^{3}\right )^{\frac {1}{3}}}\right )\right )}{3 \left (x^{3}\right )^{\frac {2}{3}}}\) | \(133\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {-2+2 x+3 x^2}{-1+x^3} \, dx=\frac {4}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \log \left (x^{2} + x + 1\right ) + \log \left (x - 1\right ) \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.11 \[ \int \frac {-2+2 x+3 x^2}{-1+x^3} \, dx=\log {\left (x - 1 \right )} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {-2+2 x+3 x^2}{-1+x^3} \, dx=\frac {4}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \log \left (x^{2} + x + 1\right ) + \log \left (x - 1\right ) \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {-2+2 x+3 x^2}{-1+x^3} \, dx=\frac {4}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \log \left (x^{2} + x + 1\right ) + \log \left ({\left | x - 1 \right |}\right ) \]
[In]
[Out]
Time = 0.19 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.04 \[ \int \frac {-2+2 x+3 x^2}{-1+x^3} \, dx=\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )+\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )+\ln \left (x-1\right )-\frac {\sqrt {3}\,\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,2{}\mathrm {i}}{3}+\frac {\sqrt {3}\,\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,2{}\mathrm {i}}{3} \]
[In]
[Out]