Integrand size = 16, antiderivative size = 57 \[ \int \frac {1}{x \left (1-3 x^4+x^8\right )} \, dx=\log (x)-\frac {1}{40} \left (5+3 \sqrt {5}\right ) \log \left (3-\sqrt {5}-2 x^4\right )-\frac {1}{40} \left (5-3 \sqrt {5}\right ) \log \left (3+\sqrt {5}-2 x^4\right ) \]
[Out]
Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1371, 719, 29, 646, 31} \[ \int \frac {1}{x \left (1-3 x^4+x^8\right )} \, dx=-\frac {1}{40} \left (5+3 \sqrt {5}\right ) \log \left (-2 x^4-\sqrt {5}+3\right )-\frac {1}{40} \left (5-3 \sqrt {5}\right ) \log \left (-2 x^4+\sqrt {5}+3\right )+\log (x) \]
[In]
[Out]
Rule 29
Rule 31
Rule 646
Rule 719
Rule 1371
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {1}{x \left (1-3 x+x^2\right )} \, dx,x,x^4\right ) \\ & = \frac {1}{4} \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^4\right )+\frac {1}{4} \text {Subst}\left (\int \frac {3-x}{1-3 x+x^2} \, dx,x,x^4\right ) \\ & = \log (x)+\frac {1}{40} \left (-5+3 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{-\frac {3}{2}-\frac {\sqrt {5}}{2}+x} \, dx,x,x^4\right )-\frac {1}{40} \left (5+3 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{-\frac {3}{2}+\frac {\sqrt {5}}{2}+x} \, dx,x,x^4\right ) \\ & = \log (x)-\frac {1}{40} \left (5+3 \sqrt {5}\right ) \log \left (3-\sqrt {5}-2 x^4\right )-\frac {1}{40} \left (5-3 \sqrt {5}\right ) \log \left (3+\sqrt {5}-2 x^4\right ) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x \left (1-3 x^4+x^8\right )} \, dx=\log (x)+\frac {1}{40} \left (-5+3 \sqrt {5}\right ) \log \left (3+\sqrt {5}-2 x^4\right )+\frac {1}{40} \left (-5-3 \sqrt {5}\right ) \log \left (-3+\sqrt {5}+2 x^4\right ) \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.12
method | result | size |
default | \(\ln \left (x \right )-\frac {\ln \left (x^{4}-x^{2}-1\right )}{8}-\frac {3 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (2 x^{2}-1\right ) \sqrt {5}}{5}\right )}{20}-\frac {\ln \left (x^{4}+x^{2}-1\right )}{8}+\frac {3 \,\operatorname {arctanh}\left (\frac {\left (2 x^{2}+1\right ) \sqrt {5}}{5}\right ) \sqrt {5}}{20}\) | \(64\) |
risch | \(\ln \left (x \right )-\frac {\ln \left (3 x^{4}-\frac {9}{2}-\frac {3 \sqrt {5}}{2}\right )}{8}+\frac {3 \ln \left (3 x^{4}-\frac {9}{2}-\frac {3 \sqrt {5}}{2}\right ) \sqrt {5}}{40}-\frac {3 \ln \left (3 x^{4}+\frac {3 \sqrt {5}}{2}-\frac {9}{2}\right ) \sqrt {5}}{40}-\frac {\ln \left (3 x^{4}+\frac {3 \sqrt {5}}{2}-\frac {9}{2}\right )}{8}\) | \(70\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x \left (1-3 x^4+x^8\right )} \, dx=\frac {3}{40} \, \sqrt {5} \log \left (\frac {2 \, x^{8} - 6 \, x^{4} - \sqrt {5} {\left (2 \, x^{4} - 3\right )} + 7}{x^{8} - 3 \, x^{4} + 1}\right ) - \frac {1}{8} \, \log \left (x^{8} - 3 \, x^{4} + 1\right ) + \log \left (x\right ) \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x \left (1-3 x^4+x^8\right )} \, dx=\log {\left (x \right )} + \left (- \frac {1}{8} + \frac {3 \sqrt {5}}{40}\right ) \log {\left (x^{4} - \frac {3}{2} - \frac {\sqrt {5}}{2} \right )} + \left (- \frac {3 \sqrt {5}}{40} - \frac {1}{8}\right ) \log {\left (x^{4} - \frac {3}{2} + \frac {\sqrt {5}}{2} \right )} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x \left (1-3 x^4+x^8\right )} \, dx=\frac {3}{40} \, \sqrt {5} \log \left (\frac {2 \, x^{4} - \sqrt {5} - 3}{2 \, x^{4} + \sqrt {5} - 3}\right ) - \frac {1}{8} \, \log \left (x^{8} - 3 \, x^{4} + 1\right ) + \frac {1}{4} \, \log \left (x^{4}\right ) \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x \left (1-3 x^4+x^8\right )} \, dx=\frac {3}{40} \, \sqrt {5} \log \left (\frac {{\left | 2 \, x^{4} - \sqrt {5} - 3 \right |}}{{\left | 2 \, x^{4} + \sqrt {5} - 3 \right |}}\right ) + \frac {1}{4} \, \log \left (x^{4}\right ) - \frac {1}{8} \, \log \left ({\left | x^{8} - 3 \, x^{4} + 1 \right |}\right ) \]
[In]
[Out]
Time = 0.24 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x \left (1-3 x^4+x^8\right )} \, dx=\ln \left (x\right )+\ln \left (x^4-\frac {\sqrt {5}}{2}-\frac {3}{2}\right )\,\left (\frac {3\,\sqrt {5}}{40}-\frac {1}{8}\right )-\ln \left (x^4+\frac {\sqrt {5}}{2}-\frac {3}{2}\right )\,\left (\frac {3\,\sqrt {5}}{40}+\frac {1}{8}\right ) \]
[In]
[Out]