Integrand size = 18, antiderivative size = 33 \[ \int \frac {1}{x \sqrt {-2+4 x+3 x^2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} (1-x)}{\sqrt {-2+4 x+3 x^2}}\right )}{\sqrt {2}} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {738, 210} \[ \int \frac {1}{x \sqrt {-2+4 x+3 x^2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} (1-x)}{\sqrt {3 x^2+4 x-2}}\right )}{\sqrt {2}} \]
[In]
[Out]
Rule 210
Rule 738
Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int \frac {1}{-8-x^2} \, dx,x,\frac {-4+4 x}{\sqrt {-2+4 x+3 x^2}}\right )\right ) \\ & = -\frac {\tan ^{-1}\left (\frac {\sqrt {2} (1-x)}{\sqrt {-2+4 x+3 x^2}}\right )}{\sqrt {2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {1}{x \sqrt {-2+4 x+3 x^2}} \, dx=-\sqrt {2} \arctan \left (\sqrt {\frac {3}{2}} x-\frac {\sqrt {-2+4 x+3 x^2}}{\sqrt {2}}\right ) \]
[In]
[Out]
Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88
method | result | size |
default | \(\frac {\sqrt {2}\, \arctan \left (\frac {\left (-4+4 x \right ) \sqrt {2}}{4 \sqrt {3 x^{2}+4 x -2}}\right )}{2}\) | \(29\) |
trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \sqrt {3 x^{2}+4 x -2}+2 x -2}{x}\right )}{2}\) | \(38\) |
[In]
[Out]
none
Time = 0.41 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x \sqrt {-2+4 x+3 x^2}} \, dx=\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x - 1\right )}}{\sqrt {3 \, x^{2} + 4 \, x - 2}}\right ) \]
[In]
[Out]
\[ \int \frac {1}{x \sqrt {-2+4 x+3 x^2}} \, dx=\int \frac {1}{x \sqrt {3 x^{2} + 4 x - 2}}\, dx \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {1}{x \sqrt {-2+4 x+3 x^2}} \, dx=\frac {1}{2} \, \sqrt {2} \arcsin \left (\frac {\sqrt {10} x}{5 \, {\left | x \right |}} - \frac {\sqrt {10}}{5 \, {\left | x \right |}}\right ) \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x \sqrt {-2+4 x+3 x^2}} \, dx=\sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 4 \, x - 2}\right )}\right ) \]
[In]
[Out]
Time = 0.35 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03 \[ \int \frac {1}{x \sqrt {-2+4 x+3 x^2}} \, dx=\frac {\sqrt {2}\,\ln \left (\frac {2\,x-2+\sqrt {2}\,\sqrt {3\,x^2+4\,x-2}\,1{}\mathrm {i}}{x}\right )\,1{}\mathrm {i}}{2} \]
[In]
[Out]