Integrand size = 16, antiderivative size = 66 \[ \int \frac {1}{x^5 \left (1-3 x^4+x^8\right )} \, dx=-\frac {1}{4 x^4}+3 \log (x)-\frac {1}{40} \left (15+7 \sqrt {5}\right ) \log \left (3-\sqrt {5}-2 x^4\right )-\frac {1}{40} \left (15-7 \sqrt {5}\right ) \log \left (3+\sqrt {5}-2 x^4\right ) \]
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Time = 0.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1371, 723, 814, 646, 31} \[ \int \frac {1}{x^5 \left (1-3 x^4+x^8\right )} \, dx=-\frac {1}{4 x^4}-\frac {1}{40} \left (15+7 \sqrt {5}\right ) \log \left (-2 x^4-\sqrt {5}+3\right )-\frac {1}{40} \left (15-7 \sqrt {5}\right ) \log \left (-2 x^4+\sqrt {5}+3\right )+3 \log (x) \]
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Rule 31
Rule 646
Rule 723
Rule 814
Rule 1371
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {1}{x^2 \left (1-3 x+x^2\right )} \, dx,x,x^4\right ) \\ & = -\frac {1}{4 x^4}+\frac {1}{4} \text {Subst}\left (\int \frac {3-x}{x \left (1-3 x+x^2\right )} \, dx,x,x^4\right ) \\ & = -\frac {1}{4 x^4}+\frac {1}{4} \text {Subst}\left (\int \left (\frac {3}{x}+\frac {8-3 x}{1-3 x+x^2}\right ) \, dx,x,x^4\right ) \\ & = -\frac {1}{4 x^4}+3 \log (x)+\frac {1}{4} \text {Subst}\left (\int \frac {8-3 x}{1-3 x+x^2} \, dx,x,x^4\right ) \\ & = -\frac {1}{4 x^4}+3 \log (x)+\frac {1}{40} \left (-15+7 \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 (15+7 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{-\frac {3}{2}+\frac {\sqrt {5}}{2}+x} \, dx,x,x^4\right ) \\ & = -\frac {1}{4 x^4}+3 \log (x)-\frac {1}{40} \left (15+7 \sqrt {5}\right ) \log \left (3-\sqrt {5}-2 x^4\right )-\frac {1}{40} \left (15-7 \sqrt {5}\right ) \log \left (3+\sqrt {5}-2 x^4\right ) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^5 \left (1-3 x^4+x^8\right )} \, dx=\frac {1}{40} \left (-\frac {10}{x^4}+120 \log (x)+\left (-15+7 \sqrt {5}\right ) \log \left (3+\sqrt {5}-2 x^4\right )-\left (15+7 \sqrt {5}\right ) \log \left (-3+\sqrt {5}+2 x^4\right )\right ) \]
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Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.08
method | result | size |
default | \(-\frac {1}{4 x^{4}}+3 \ln \left (x \right )-\frac {3 \ln \left (x^{4}-x^{2}-1\right )}{8}-\frac {7 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (2 x^{2}-1\right ) \sqrt {5}}{5}\right )}{20}-\frac {3 \ln \left (x^{4}+x^{2}-1\right )}{8}+\frac {7 \,\operatorname {arctanh}\left (\frac {\left (2 x^{2}+1\right ) \sqrt {5}}{5}\right ) \sqrt {5}}{20}\) | \(71\) |
risch | \(-\frac {1}{4 x^{4}}+3 \ln \left (x \right )-\frac {3 \ln \left (7 x^{4}-\frac {21}{2}-\frac {7 \sqrt {5}}{2}\right )}{8}+\frac {7 \ln \left (7 x^{4}-\frac {21}{2}-\frac {7 \sqrt {5}}{2}\right ) \sqrt {5}}{40}-\frac {3 \ln \left (7 x^{4}-\frac {21}{2}+\frac {7 \sqrt {5}}{2}\right )}{8}-\frac {7 \ln \left (7 x^{4}-\frac {21}{2}+\frac {7 \sqrt {5}}{2}\right ) \sqrt {5}}{40}\) | \(77\) |
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Time = 0.24 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.15 \[ \int \frac {1}{x^5 \left (1-3 x^4+x^8\right )} \, dx=\frac {7 \, \sqrt {5} x^{4} \log \left (\frac {2 \, x^{8} - 6 \, x^{4} - \sqrt {5} {\left (2 \, x^{4} - 3\right )} + 7}{x^{8} - 3 \, x^{4} + 1}\right ) - 15 \, x^{4} \log \left (x^{8} - 3 \, x^{4} + 1\right ) + 120 \, x^{4} \log \left (x\right ) - 10}{40 \, x^{4}} \]
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Time = 0.09 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^5 \left (1-3 x^4+x^8\right )} \, dx=3 \log {\left (x \right )} + \left (- \frac {3}{8} + \frac {7 \sqrt {5}}{40}\right ) \log {\left (x^{4} - \frac {3}{2} - \frac {\sqrt {5}}{2} \right )} + \left (- \frac {7 \sqrt {5}}{40} - \frac {3}{8}\right ) \log {\left (x^{4} - \frac {3}{2} + \frac {\sqrt {5}}{2} \right )} - \frac {1}{4 x^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^5 \left (1-3 x^4+x^8\right )} \, dx=\frac {7}{40} \, \sqrt {5} \log \left (\frac {2 \, x^{4} - \sqrt {5} - 3}{2 \, x^{4} + \sqrt {5} - 3}\right ) - \frac {1}{4 \, x^{4}} - \frac {3}{8} \, \log \left (x^{8} - 3 \, x^{4} + 1\right ) + \frac {3}{4} \, \log \left (x^{4}\right ) \]
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Time = 0.34 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^5 \left (1-3 x^4+x^8\right )} \, dx=\frac {7}{40} \, \sqrt {5} \log \left (\frac {{\left | 2 \, x^{4} - \sqrt {5} - 3 \right |}}{{\left | 2 \, x^{4} + \sqrt {5} - 3 \right |}}\right ) - \frac {3 \, x^{4} + 1}{4 \, x^{4}} + \frac {3}{4} \, \log \left (x^{4}\right ) - \frac {3}{8} \, \log \left ({\left | x^{8} - 3 \, x^{4} + 1 \right |}\right ) \]
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Time = 8.59 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^5 \left (1-3 x^4+x^8\right )} \, dx=3\,\ln \left (x\right )-\frac {1}{4\,x^4}+\ln \left (x^4-\frac {\sqrt {5}}{2}-\frac {3}{2}\right )\,\left (\frac {7\,\sqrt {5}}{40}-\frac {3}{8}\right )-\ln \left (x^4+\frac {\sqrt {5}}{2}-\frac {3}{2}\right )\,\left (\frac {7\,\sqrt {5}}{40}+\frac {3}{8}\right ) \]
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