Integrand size = 16, antiderivative size = 10 \[ \int \frac {1+x^4}{x \left (1+x^2\right )^2} \, dx=\frac {1}{1+x^2}+\log (x) \]
[Out]
Time = 0.01 (sec) , antiderivative size = 10, 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 = {1266, 908} \[ \int \frac {1+x^4}{x \left (1+x^2\right )^2} \, dx=\frac {1}{x^2+1}+\log (x) \]
[In]
[Out]
Rule 908
Rule 1266
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1+x^2}{x (1+x)^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{x}-\frac {2}{(1+x)^2}\right ) \, dx,x,x^2\right ) \\ & = \frac {1}{1+x^2}+\log (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^4}{x \left (1+x^2\right )^2} \, dx=\frac {1}{1+x^2}+\log (x) \]
[In]
[Out]
Time = 0.14 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10
method | result | size |
default | \(\frac {1}{x^{2}+1}+\ln \left (x \right )\) | \(11\) |
norman | \(\frac {1}{x^{2}+1}+\ln \left (x \right )\) | \(11\) |
risch | \(\frac {1}{x^{2}+1}+\ln \left (x \right )\) | \(11\) |
parallelrisch | \(\frac {x^{2} \ln \left (x \right )+1+\ln \left (x \right )}{x^{2}+1}\) | \(19\) |
meijerg | \(-\frac {x^{2}}{2 \left (x^{2}+1\right )}+\frac {1}{2}+\ln \left (x \right )-\frac {x^{2}}{2 x^{2}+2}\) | \(31\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.80 \[ \int \frac {1+x^4}{x \left (1+x^2\right )^2} \, dx=\frac {{\left (x^{2} + 1\right )} \log \left (x\right ) + 1}{x^{2} + 1} \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {1+x^4}{x \left (1+x^2\right )^2} \, dx=\log {\left (x \right )} + \frac {1}{x^{2} + 1} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.40 \[ \int \frac {1+x^4}{x \left (1+x^2\right )^2} \, dx=\frac {1}{x^{2} + 1} + \frac {1}{2} \, \log \left (x^{2}\right ) \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.40 \[ \int \frac {1+x^4}{x \left (1+x^2\right )^2} \, dx=\frac {1}{x^{2} + 1} + \frac {1}{2} \, \log \left (x^{2}\right ) \]
[In]
[Out]
Time = 0.17 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^4}{x \left (1+x^2\right )^2} \, dx=\ln \left (x\right )+\frac {1}{x^2+1} \]
[In]
[Out]