Optimal. Leaf size=48 \[ -\frac {1}{4 x^4}-\frac {\tan ^{-1}\left (\frac {2 x^4+1}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {1}{8} \log \left (x^8+x^4+1\right )-\log (x) \]
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Rubi [A] time = 0.05, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1357, 709, 800, 634, 618, 204, 628} \[ -\frac {1}{4 x^4}+\frac {1}{8} \log \left (x^8+x^4+1\right )-\frac {\tan ^{-1}\left (\frac {2 x^4+1}{\sqrt {3}}\right )}{4 \sqrt {3}}-\log (x) \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 709
Rule 800
Rule 1357
Rubi steps
\begin {align*} \int \frac {1}{x^5 \left (1+x^4+x^8\right )} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1+x+x^2\right )} \, dx,x,x^4\right )\\ &=-\frac {1}{4 x^4}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {-1-x}{x \left (1+x+x^2\right )} \, dx,x,x^4\right )\\ &=-\frac {1}{4 x^4}+\frac {1}{4} \operatorname {Subst}\left (\int \left (-\frac {1}{x}+\frac {x}{1+x+x^2}\right ) \, dx,x,x^4\right )\\ &=-\frac {1}{4 x^4}-\log (x)+\frac {1}{4} \operatorname {Subst}\left (\int \frac {x}{1+x+x^2} \, dx,x,x^4\right )\\ &=-\frac {1}{4 x^4}-\log (x)-\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,x^4\right )+\frac {1}{8} \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,x^4\right )\\ &=-\frac {1}{4 x^4}-\log (x)+\frac {1}{8} \log \left (1+x^4+x^8\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x^4\right )\\ &=-\frac {1}{4 x^4}-\frac {\tan ^{-1}\left (\frac {1+2 x^4}{\sqrt {3}}\right )}{4 \sqrt {3}}-\log (x)+\frac {1}{8} \log \left (1+x^4+x^8\right )\\ \end {align*}
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Mathematica [C] time = 0.10, size = 141, normalized size = 2.94 \[ \frac {1}{24} \left (-\frac {6}{x^4}+\sqrt {3} \left (\sqrt {3}+i\right ) \log \left (x^2-\frac {i \sqrt {3}}{2}-\frac {1}{2}\right )+\sqrt {3} \left (\sqrt {3}-i\right ) \log \left (x^2+\frac {1}{2} i \left (\sqrt {3}+i\right )\right )+3 \log \left (x^2-x+1\right )+3 \log \left (x^2+x+1\right )-24 \log (x)+2 \sqrt {3} \tan ^{-1}\left (\frac {2 x-1}{\sqrt {3}}\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 49, normalized size = 1.02 \[ -\frac {2 \, \sqrt {3} x^{4} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} + 1\right )}\right ) - 3 \, x^{4} \log \left (x^{8} + x^{4} + 1\right ) + 24 \, x^{4} \log \relax (x) + 6}{24 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 46, normalized size = 0.96 \[ -\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} + 1\right )}\right ) + \frac {x^{4} - 1}{4 \, x^{4}} + \frac {1}{8} \, \log \left (x^{8} + x^{4} + 1\right ) - \frac {1}{4} \, \log \left (x^{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 94, normalized size = 1.96 \[ -\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{12}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{2}-1\right ) \sqrt {3}}{3}\right )}{12}-\ln \relax (x )+\frac {\ln \left (x^{2}-x +1\right )}{8}+\frac {\ln \left (x^{2}+x +1\right )}{8}+\frac {\ln \left (x^{4}-x^{2}+1\right )}{8}-\frac {1}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.79, size = 41, normalized size = 0.85 \[ -\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} + 1\right )}\right ) - \frac {1}{4 \, x^{4}} + \frac {1}{8} \, \log \left (x^{8} + x^{4} + 1\right ) - \frac {1}{4} \, \log \left (x^{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 41, normalized size = 0.85 \[ \frac {\ln \left (x^8+x^4+1\right )}{8}-\ln \relax (x)-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,x^4}{3}+\frac {\sqrt {3}}{3}\right )}{12}-\frac {1}{4\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 48, normalized size = 1.00 \[ - \log {\relax (x )} + \frac {\log {\left (x^{8} + x^{4} + 1 \right )}}{8} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{4}}{3} + \frac {\sqrt {3}}{3} \right )}}{12} - \frac {1}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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