Integrand size = 16, antiderivative size = 11 \[ \int \frac {-2+x^2}{x \left (2+x^2\right )} \, dx=-\log (x)+\log \left (2+x^2\right ) \]
[Out]
Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {457, 78} \[ \int \frac {-2+x^2}{x \left (2+x^2\right )} \, dx=\log \left (x^2+2\right )-\log (x) \]
[In]
[Out]
Rule 78
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {-2+x}{x (2+x)} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (-\frac {1}{x}+\frac {2}{2+x}\right ) \, dx,x,x^2\right ) \\ & = -\log (x)+\log \left (2+x^2\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {-2+x^2}{x \left (2+x^2\right )} \, dx=-\log (x)+\log \left (2+x^2\right ) \]
[In]
[Out]
Time = 0.14 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09
method | result | size |
default | \(-\ln \left (x \right )+\ln \left (x^{2}+2\right )\) | \(12\) |
norman | \(-\ln \left (x \right )+\ln \left (x^{2}+2\right )\) | \(12\) |
risch | \(-\ln \left (x \right )+\ln \left (x^{2}+2\right )\) | \(12\) |
parallelrisch | \(-\ln \left (x \right )+\ln \left (x^{2}+2\right )\) | \(12\) |
meijerg | \(-\ln \left (x \right )+\frac {\ln \left (2\right )}{2}+\ln \left (1+\frac {x^{2}}{2}\right )\) | \(18\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {-2+x^2}{x \left (2+x^2\right )} \, dx=\log \left (x^{2} + 2\right ) - \log \left (x\right ) \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int \frac {-2+x^2}{x \left (2+x^2\right )} \, dx=- \log {\left (x \right )} + \log {\left (x^{2} + 2 \right )} \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.18 \[ \int \frac {-2+x^2}{x \left (2+x^2\right )} \, dx=\log \left (x^{2} + 2\right ) - \frac {1}{2} \, \log \left (x^{2}\right ) \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.18 \[ \int \frac {-2+x^2}{x \left (2+x^2\right )} \, dx=\log \left (x^{2} + 2\right ) - \frac {1}{2} \, \log \left (x^{2}\right ) \]
[In]
[Out]
Time = 0.22 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {-2+x^2}{x \left (2+x^2\right )} \, dx=\ln \left (x^2+2\right )-\ln \left (x\right ) \]
[In]
[Out]