Integrand size = 13, antiderivative size = 22 \[ \int \frac {1}{x \left (a^5+x^5\right )} \, dx=\frac {\log (x)}{a^5}-\frac {\log \left (a^5+x^5\right )}{5 a^5} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {272, 36, 29, 31} \[ \int \frac {1}{x \left (a^5+x^5\right )} \, dx=\frac {\log (x)}{a^5}-\frac {\log \left (a^5+x^5\right )}{5 a^5} \]
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \text {Subst}\left (\int \frac {1}{x \left (a^5+x\right )} \, dx,x,x^5\right ) \\ & = \frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,x^5\right )}{5 a^5}-\frac {\text {Subst}\left (\int \frac {1}{a^5+x} \, dx,x,x^5\right )}{5 a^5} \\ & = \frac {\log (x)}{a^5}-\frac {\log \left (a^5+x^5\right )}{5 a^5} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (a^5+x^5\right )} \, dx=\frac {\log (x)}{a^5}-\frac {\log \left (a^5+x^5\right )}{5 a^5} \]
[In]
[Out]
Time = 0.20 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95
method | result | size |
risch | \(\frac {\ln \left (x \right )}{a^{5}}-\frac {\ln \left (a^{5}+x^{5}\right )}{5 a^{5}}\) | \(21\) |
parallelrisch | \(\frac {5 \ln \left (x \right )-\ln \left (a +x \right )-\ln \left (a^{4}-a^{3} x +a^{2} x^{2}-a \,x^{3}+x^{4}\right )}{5 a^{5}}\) | \(46\) |
default | \(\frac {\ln \left (x \right )}{a^{5}}-\frac {\ln \left (a^{4}-a^{3} x +a^{2} x^{2}-a \,x^{3}+x^{4}\right )}{5 a^{5}}-\frac {\ln \left (a +x \right )}{5 a^{5}}\) | \(49\) |
norman | \(\frac {\ln \left (x \right )}{a^{5}}-\frac {\ln \left (a^{4}-a^{3} x +a^{2} x^{2}-a \,x^{3}+x^{4}\right )}{5 a^{5}}-\frac {\ln \left (a +x \right )}{5 a^{5}}\) | \(49\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x \left (a^5+x^5\right )} \, dx=-\frac {\log \left (a^{5} + x^{5}\right ) - 5 \, \log \left (x\right )}{5 \, a^{5}} \]
[In]
[Out]
Time = 0.12 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x \left (a^5+x^5\right )} \, dx=\frac {\log {\left (x \right )}}{a^{5}} - \frac {\log {\left (a^{5} + x^{5} \right )}}{5 a^{5}} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x \left (a^5+x^5\right )} \, dx=-\frac {\log \left (a^{5} + x^{5}\right )}{5 \, a^{5}} + \frac {\log \left (x^{5}\right )}{5 \, a^{5}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (a^5+x^5\right )} \, dx=-\frac {\log \left ({\left | a^{5} + x^{5} \right |}\right )}{5 \, a^{5}} + \frac {\log \left ({\left | x \right |}\right )}{a^{5}} \]
[In]
[Out]
Time = 0.34 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x \left (a^5+x^5\right )} \, dx=-\frac {\ln \left (a^5+x^5\right )-5\,\ln \left (x\right )}{5\,a^5} \]
[In]
[Out]