Integrand size = 16, antiderivative size = 32 \[ \int \frac {-4+3 x}{4-2 x+x^2} \, dx=\frac {\arctan \left (\frac {1-x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {3}{2} \log \left (4-2 x+x^2\right ) \]
[Out]
Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {648, 632, 210, 642} \[ \int \frac {-4+3 x}{4-2 x+x^2} \, dx=\frac {\arctan \left (\frac {1-x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {3}{2} \log \left (x^2-2 x+4\right ) \]
[In]
[Out]
Rule 210
Rule 632
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \frac {3}{2} \int \frac {-2+2 x}{4-2 x+x^2} \, dx-\int \frac {1}{4-2 x+x^2} \, dx \\ & = \frac {3}{2} \log \left (4-2 x+x^2\right )+2 \text {Subst}\left (\int \frac {1}{-12-x^2} \, dx,x,-2+2 x\right ) \\ & = \frac {\tan ^{-1}\left (\frac {1-x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {3}{2} \log \left (4-2 x+x^2\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {-4+3 x}{4-2 x+x^2} \, dx=-\frac {\arctan \left (\frac {-1+x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {3}{2} \log \left (4-2 x+x^2\right ) \]
[In]
[Out]
Time = 1.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84
method | result | size |
risch | \(\frac {3 \ln \left (x^{2}-2 x +4\right )}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (x -1\right ) \sqrt {3}}{3}\right )}{3}\) | \(27\) |
default | \(\frac {3 \ln \left (x^{2}-2 x +4\right )}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -2\right ) \sqrt {3}}{6}\right )}{3}\) | \(29\) |
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {-4+3 x}{4-2 x+x^2} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (x - 1\right )}\right ) + \frac {3}{2} \, \log \left (x^{2} - 2 \, x + 4\right ) \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {-4+3 x}{4-2 x+x^2} \, dx=\frac {3 \log {\left (x^{2} - 2 x + 4 \right )}}{2} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {\sqrt {3} x}{3} - \frac {\sqrt {3}}{3} \right )}}{3} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {-4+3 x}{4-2 x+x^2} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (x - 1\right )}\right ) + \frac {3}{2} \, \log \left (x^{2} - 2 \, x + 4\right ) \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {-4+3 x}{4-2 x+x^2} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (x - 1\right )}\right ) + \frac {3}{2} \, \log \left (x^{2} - 2 \, x + 4\right ) \]
[In]
[Out]
Time = 9.18 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {-4+3 x}{4-2 x+x^2} \, dx=\frac {3\,\ln \left (x^2-2\,x+4\right )}{2}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,x}{3}-\frac {\sqrt {3}}{3}\right )}{3} \]
[In]
[Out]