Optimal. Leaf size=66 \[ -\frac{1}{4 x^4}-\frac{1}{40} \left (15+7 \sqrt{5}\right ) \log \left (-2 x^4-\sqrt{5}+3\right )-\frac{1}{40} \left (15-7 \sqrt{5}\right ) \log \left (-2 x^4+\sqrt{5}+3\right )+3 \log (x) \]
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Rubi [A] time = 0.0626971, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {1357, 709, 800, 632, 31} \[ -\frac{1}{4 x^4}-\frac{1}{40} \left (15+7 \sqrt{5}\right ) \log \left (-2 x^4-\sqrt{5}+3\right )-\frac{1}{40} \left (15-7 \sqrt{5}\right ) \log \left (-2 x^4+\sqrt{5}+3\right )+3 \log (x) \]
Antiderivative was successfully verified.
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Rule 1357
Rule 709
Rule 800
Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{x^5 \left (1-3 x^4+x^8\right )} \, dx &=\frac{1}{4} \operatorname{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} \operatorname{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} \operatorname{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} \operatorname{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 ) \operatorname{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 ) \operatorname{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*}
Mathematica [A] time = 0.034366, size = 61, normalized size = 0.92 \[ \frac{1}{40} \left (-\frac{10}{x^4}+\left (7 \sqrt{5}-15\right ) \log \left (-2 x^4+\sqrt{5}+3\right )-\left (15+7 \sqrt{5}\right ) \log \left (2 x^4+\sqrt{5}-3\right )+120 \log (x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 71, normalized size = 1.1 \begin{align*} -{\frac{3\,\ln \left ({x}^{4}-{x}^{2}-1 \right ) }{8}}-{\frac{7\,\sqrt{5}}{20}{\it Artanh} \left ({\frac{ \left ( 2\,{x}^{2}-1 \right ) \sqrt{5}}{5}} \right ) }-{\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}}{20}{\it Artanh} \left ({\frac{ \left ( 2\,{x}^{2}+1 \right ) \sqrt{5}}{5}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50719, size = 76, normalized size = 1.15 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70473, size = 193, normalized size = 2.92 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.195624, size = 66, normalized size = 1. \begin{align*} 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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14576, size = 89, normalized size = 1.35 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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