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 = 0.06, antiderivative size = 66, normalized size of antiderivative = 1.00, 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 31
Rule 632
Rule 709
Rule 800
Rule 1357
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*}
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Mathematica [A] time = 0.04, 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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fricas [A] time = 0.82, size = 76, normalized size = 1.15 \[ \frac {7 \, \sqrt {5} x^{4} \log \left (\frac {2 \, x^{8} - 6 \, x^{4} - \sqrt {5} {\left (2 \, x^{4} - 3\right )} + 7}{x^{8} - 3 \, x^{4} + 1}\right ) - 15 \, x^{4} \log \left (x^{8} - 3 \, x^{4} + 1\right ) + 120 \, x^{4} \log \relax (x) - 10}{40 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.51, size = 66, normalized size = 1.00 \[ \frac {7}{40} \, \sqrt {5} \log \left (\frac {{\left | 2 \, x^{4} - \sqrt {5} - 3 \right |}}{{\left | 2 \, x^{4} + \sqrt {5} - 3 \right |}}\right ) - \frac {3 \, x^{4} + 1}{4 \, x^{4}} + \frac {3}{4} \, \log \left (x^{4}\right ) - \frac {3}{8} \, \log \left ({\left | x^{8} - 3 \, x^{4} + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 71, normalized size = 1.08 \[ -\frac {7 \sqrt {5}\, \arctanh \left (\frac {\left (2 x^{2}-1\right ) \sqrt {5}}{5}\right )}{20}+\frac {7 \sqrt {5}\, \arctanh \left (\frac {\left (2 x^{2}+1\right ) \sqrt {5}}{5}\right )}{20}+3 \ln \relax (x )-\frac {3 \ln \left (x^{4}-x^{2}-1\right )}{8}-\frac {3 \ln \left (x^{4}+x^{2}-1\right )}{8}-\frac {1}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 56, normalized size = 0.85 \[ \frac {7}{40} \, \sqrt {5} \log \left (\frac {2 \, x^{4} - \sqrt {5} - 3}{2 \, x^{4} + \sqrt {5} - 3}\right ) - \frac {1}{4 \, x^{4}} - \frac {3}{8} \, \log \left (x^{8} - 3 \, x^{4} + 1\right ) + \frac {3}{4} \, \log \left (x^{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.35, size = 49, normalized size = 0.74 \[ 3\,\ln \relax (x)-\frac {1}{4\,x^4}+\ln \left (x^4-\frac {\sqrt {5}}{2}-\frac {3}{2}\right )\,\left (\frac {7\,\sqrt {5}}{40}-\frac {3}{8}\right )-\ln \left (x^4+\frac {\sqrt {5}}{2}-\frac {3}{2}\right )\,\left (\frac {7\,\sqrt {5}}{40}+\frac {3}{8}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 66, normalized size = 1.00 \[ 3 \log {\relax (x )} + \left (- \frac {3}{8} + \frac {7 \sqrt {5}}{40}\right ) \log {\left (x^{4} - \frac {3}{2} - \frac {\sqrt {5}}{2} \right )} + \left (- \frac {7 \sqrt {5}}{40} - \frac {3}{8}\right ) \log {\left (x^{4} - \frac {3}{2} + \frac {\sqrt {5}}{2} \right )} - \frac {1}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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