Integrand size = 11, antiderivative size = 14 \[ \int \frac {x^4}{-1+x^4} \, dx=x-\frac {\arctan (x)}{2}-\frac {\text {arctanh}(x)}{2} \]
[Out]
Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {327, 218, 212, 209} \[ \int \frac {x^4}{-1+x^4} \, dx=-\frac {\arctan (x)}{2}-\frac {\text {arctanh}(x)}{2}+x \]
[In]
[Out]
Rule 209
Rule 212
Rule 218
Rule 327
Rubi steps \begin{align*} \text {integral}& = x+\int \frac {1}{-1+x^4} \, dx \\ & = x-\frac {1}{2} \int \frac {1}{1-x^2} \, dx-\frac {1}{2} \int \frac {1}{1+x^2} \, dx \\ & = x-\frac {\arctan (x)}{2}-\frac {\text {arctanh}(x)}{2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.86 \[ \int \frac {x^4}{-1+x^4} \, dx=x-\frac {\arctan (x)}{2}+\frac {1}{4} \log (1-x)-\frac {1}{4} \log (1+x) \]
[In]
[Out]
Time = 0.18 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.36
method | result | size |
default | \(x +\frac {\ln \left (-1+x \right )}{4}-\frac {\ln \left (1+x \right )}{4}-\frac {\arctan \left (x \right )}{2}\) | \(19\) |
risch | \(x +\frac {\ln \left (-1+x \right )}{4}-\frac {\ln \left (1+x \right )}{4}-\frac {\arctan \left (x \right )}{2}\) | \(19\) |
parallelrisch | \(x +\frac {i \ln \left (x -i\right )}{4}-\frac {i \ln \left (x +i\right )}{4}-\frac {\ln \left (1+x \right )}{4}+\frac {\ln \left (-1+x \right )}{4}\) | \(31\) |
meijerg | \(-\frac {\left (-1\right )^{\frac {3}{4}} \left (4 \left (-1\right )^{\frac {1}{4}} x +\frac {x \left (-1\right )^{\frac {1}{4}} \left (\ln \left (1-\left (x^{4}\right )^{\frac {1}{4}}\right )-\ln \left (1+\left (x^{4}\right )^{\frac {1}{4}}\right )-2 \arctan \left (\left (x^{4}\right )^{\frac {1}{4}}\right )\right )}{\left (x^{4}\right )^{\frac {1}{4}}}\right )}{4}\) | \(52\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.29 \[ \int \frac {x^4}{-1+x^4} \, dx=x - \frac {1}{2} \, \arctan \left (x\right ) - \frac {1}{4} \, \log \left (x + 1\right ) + \frac {1}{4} \, \log \left (x - 1\right ) \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.36 \[ \int \frac {x^4}{-1+x^4} \, dx=x + \frac {\log {\left (x - 1 \right )}}{4} - \frac {\log {\left (x + 1 \right )}}{4} - \frac {\operatorname {atan}{\left (x \right )}}{2} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.29 \[ \int \frac {x^4}{-1+x^4} \, dx=x - \frac {1}{2} \, \arctan \left (x\right ) - \frac {1}{4} \, \log \left (x + 1\right ) + \frac {1}{4} \, \log \left (x - 1\right ) \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43 \[ \int \frac {x^4}{-1+x^4} \, dx=x - \frac {1}{2} \, \arctan \left (x\right ) - \frac {1}{4} \, \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{4} \, \log \left ({\left | x - 1 \right |}\right ) \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \frac {x^4}{-1+x^4} \, dx=x-\frac {\mathrm {atan}\left (x\right )}{2}-\frac {\mathrm {atanh}\left (x\right )}{2} \]
[In]
[Out]