Integrand size = 13, antiderivative size = 22 \[ \int \frac {1}{x \left (a^3+x^3\right )} \, dx=\frac {\log (x)}{a^3}-\frac {\log \left (a^3+x^3\right )}{3 a^3} \]
[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} \[ \int \frac {1}{x \left (a^3+x^3\right )} \, dx=\frac {\log (x)}{a^3}-\frac {\log \left (a^3+x^3\right )}{3 a^3} \]
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {1}{x \left (a^3+x\right )} \, dx,x,x^3\right ) \\ & = \frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,x^3\right )}{3 a^3}-\frac {\text {Subst}\left (\int \frac {1}{a^3+x} \, dx,x,x^3\right )}{3 a^3} \\ & = \frac {\log (x)}{a^3}-\frac {\log \left (a^3+x^3\right )}{3 a^3} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (a^3+x^3\right )} \, dx=\frac {\log (x)}{a^3}-\frac {\log \left (a^3+x^3\right )}{3 a^3} \]
[In]
[Out]
Time = 0.20 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95
method | result | size |
risch | \(\frac {\ln \left (x \right )}{a^{3}}-\frac {\ln \left (a^{3}+x^{3}\right )}{3 a^{3}}\) | \(21\) |
parallelrisch | \(\frac {3 \ln \left (x \right )-\ln \left (a +x \right )-\ln \left (a^{2}-a x +x^{2}\right )}{3 a^{3}}\) | \(31\) |
default | \(\frac {\ln \left (x \right )}{a^{3}}-\frac {\ln \left (a^{2}-a x +x^{2}\right )}{3 a^{3}}-\frac {\ln \left (a +x \right )}{3 a^{3}}\) | \(34\) |
norman | \(\frac {\ln \left (x \right )}{a^{3}}-\frac {\ln \left (a^{2}-a x +x^{2}\right )}{3 a^{3}}-\frac {\ln \left (a +x \right )}{3 a^{3}}\) | \(34\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x \left (a^3+x^3\right )} \, dx=-\frac {\log \left (a^{3} + x^{3}\right ) - 3 \, \log \left (x\right )}{3 \, a^{3}} \]
[In]
[Out]
Time = 0.10 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x \left (a^3+x^3\right )} \, dx=\frac {\log {\left (x \right )}}{a^{3}} - \frac {\log {\left (a^{3} + x^{3} \right )}}{3 a^{3}} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x \left (a^3+x^3\right )} \, dx=-\frac {\log \left (a^{3} + x^{3}\right )}{3 \, a^{3}} + \frac {\log \left (x^{3}\right )}{3 \, a^{3}} \]
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (a^3+x^3\right )} \, dx=-\frac {\log \left ({\left | a^{3} + x^{3} \right |}\right )}{3 \, a^{3}} + \frac {\log \left ({\left | x \right |}\right )}{a^{3}} \]
[In]
[Out]
Time = 0.29 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x \left (a^3+x^3\right )} \, dx=-\frac {\ln \left (a^3+x^3\right )-3\,\ln \left (x\right )}{3\,a^3} \]
[In]
[Out]