Integrand size = 17, antiderivative size = 51 \[ \int \frac {2-3 x^2}{4+9 x^4} \, dx=-\frac {\log \left (2-2 \sqrt {3} x+3 x^2\right )}{4 \sqrt {3}}+\frac {\log \left (2+2 \sqrt {3} x+3 x^2\right )}{4 \sqrt {3}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1179, 642} \[ \int \frac {2-3 x^2}{4+9 x^4} \, dx=\frac {\log \left (3 x^2+2 \sqrt {3} x+2\right )}{4 \sqrt {3}}-\frac {\log \left (3 x^2-2 \sqrt {3} x+2\right )}{4 \sqrt {3}} \]
[In]
[Out]
Rule 642
Rule 1179
Rubi steps \begin{align*} \text {integral}& = -\frac {\int \frac {\frac {2}{\sqrt {3}}+2 x}{-\frac {2}{3}-\frac {2 x}{\sqrt {3}}-x^2} \, dx}{4 \sqrt {3}}-\frac {\int \frac {\frac {2}{\sqrt {3}}-2 x}{-\frac {2}{3}+\frac {2 x}{\sqrt {3}}-x^2} \, dx}{4 \sqrt {3}} \\ & = -\frac {\log \left (2-2 \sqrt {3} x+3 x^2\right )}{4 \sqrt {3}}+\frac {\log \left (2+2 \sqrt {3} x+3 x^2\right )}{4 \sqrt {3}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.86 \[ \int \frac {2-3 x^2}{4+9 x^4} \, dx=\frac {-\log \left (-2+2 \sqrt {3} x-3 x^2\right )+\log \left (2+2 \sqrt {3} x+3 x^2\right )}{4 \sqrt {3}} \]
[In]
[Out]
Time = 0.20 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.78
method | result | size |
risch | \(-\frac {\ln \left (2+3 x^{2}-2 x \sqrt {3}\right ) \sqrt {3}}{12}+\frac {\ln \left (2+3 x^{2}+2 x \sqrt {3}\right ) \sqrt {3}}{12}\) | \(40\) |
default | \(\frac {\sqrt {6}\, \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\frac {\sqrt {6}\, x \sqrt {2}}{3}+\frac {2}{3}}{x^{2}-\frac {\sqrt {6}\, x \sqrt {2}}{3}+\frac {2}{3}}\right )+2 \arctan \left (\frac {\sqrt {6}\, x \sqrt {2}}{2}+1\right )+2 \arctan \left (\frac {\sqrt {6}\, x \sqrt {2}}{2}-1\right )\right )}{48}-\frac {\sqrt {6}\, \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\frac {\sqrt {6}\, x \sqrt {2}}{3}+\frac {2}{3}}{x^{2}+\frac {\sqrt {6}\, x \sqrt {2}}{3}+\frac {2}{3}}\right )+2 \arctan \left (\frac {\sqrt {6}\, x \sqrt {2}}{2}+1\right )+2 \arctan \left (\frac {\sqrt {6}\, x \sqrt {2}}{2}-1\right )\right )}{48}\) | \(140\) |
meijerg | \(\frac {\sqrt {6}\, \left (-\frac {x \sqrt {2}\, \ln \left (1-\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}+\frac {3 \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}{2-\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \ln \left (1+\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}+\frac {3 \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}{2+\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {1}{4}}}\right )}{24}-\frac {\sqrt {6}\, \left (\frac {x^{3} \sqrt {2}\, \ln \left (1-\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}+\frac {3 \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {3}{4}}}+\frac {x^{3} \sqrt {2}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}{2-\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {3}{4}}}-\frac {x^{3} \sqrt {2}\, \ln \left (1+\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}+\frac {3 \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {3}{4}}}+\frac {x^{3} \sqrt {2}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}{2+\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {3}{4}}}\right )}{24}\) | \(284\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.82 \[ \int \frac {2-3 x^2}{4+9 x^4} \, dx=\frac {1}{12} \, \sqrt {3} \log \left (\frac {9 \, x^{4} + 24 \, x^{2} + 4 \, \sqrt {3} {\left (3 \, x^{3} + 2 \, x\right )} + 4}{9 \, x^{4} + 4}\right ) \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.96 \[ \int \frac {2-3 x^2}{4+9 x^4} \, dx=- \frac {\sqrt {3} \log {\left (x^{2} - \frac {2 \sqrt {3} x}{3} + \frac {2}{3} \right )}}{12} + \frac {\sqrt {3} \log {\left (x^{2} + \frac {2 \sqrt {3} x}{3} + \frac {2}{3} \right )}}{12} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.76 \[ \int \frac {2-3 x^2}{4+9 x^4} \, dx=\frac {1}{12} \, \sqrt {3} \log \left (3 \, x^{2} + 2 \, \sqrt {3} x + 2\right ) - \frac {1}{12} \, \sqrt {3} \log \left (3 \, x^{2} - 2 \, \sqrt {3} x + 2\right ) \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.78 \[ \int \frac {2-3 x^2}{4+9 x^4} \, dx=\frac {1}{12} \, \sqrt {3} \log \left (x^{2} + \sqrt {2} \left (\frac {4}{9}\right )^{\frac {1}{4}} x + \frac {2}{3}\right ) - \frac {1}{12} \, \sqrt {3} \log \left (x^{2} - \sqrt {2} \left (\frac {4}{9}\right )^{\frac {1}{4}} x + \frac {2}{3}\right ) \]
[In]
[Out]
Time = 13.56 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.41 \[ \int \frac {2-3 x^2}{4+9 x^4} \, dx=\frac {\sqrt {3}\,\mathrm {atanh}\left (\frac {2\,\sqrt {3}\,x}{3\,x^2+2}\right )}{6} \]
[In]
[Out]