Optimal. Leaf size=39 \[ -\frac {\tan ^{-1}\left (\frac {2 x^4+1}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{8} \log \left (x^8+x^4+1\right )+\log (x) \]
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Rubi [A] time = 0.03, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1357, 705, 29, 634, 618, 204, 628} \[ -\frac {1}{8} \log \left (x^8+x^4+1\right )-\frac {\tan ^{-1}\left (\frac {2 x^4+1}{\sqrt {3}}\right )}{4 \sqrt {3}}+\log (x) \]
Antiderivative was successfully verified.
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Rule 29
Rule 204
Rule 618
Rule 628
Rule 634
Rule 705
Rule 1357
Rubi steps
\begin {align*} \int \frac {1}{x \left (1+x^4+x^8\right )} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{x \left (1+x+x^2\right )} \, dx,x,x^4\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^4\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {-1-x}{1+x+x^2} \, dx,x,x^4\right )\\ &=\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 )\\ &=\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 {\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 = 0.08, size = 138, normalized size = 3.54 \[ \frac {1}{24} \left (-\sqrt {3} \left (\sqrt {3}-i\right ) \log \left (x^2-\frac {i \sqrt {3}}{2}-\frac {1}{2}\right )-\sqrt {3} \left (\sqrt {3}+i\right ) \log \left (x^2+\frac {1}{2} i \left (\sqrt {3}+i\right )\right )-3 \log \left (x^2-x+1\right )-3 \log \left (x^2+x+1\right )+24 \log (x)+2 \sqrt {3} \tan ^{-1}\left (\frac {2 x-1}{\sqrt {3}}\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 32, normalized size = 0.82 \[ -\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} + 1\right )}\right ) - \frac {1}{8} \, \log \left (x^{8} + x^{4} + 1\right ) + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 36, normalized size = 0.92 \[ -\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} + 1\right )}\right ) - \frac {1}{8} \, \log \left (x^{8} + x^{4} + 1\right ) + \frac {1}{4} \, \log \left (x^{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 87, normalized size = 2.23 \[ -\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{12}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{2}-1\right ) \sqrt {3}}{3}\right )}{12}+\ln \relax (x )-\frac {\ln \left (x^{2}-x +1\right )}{8}-\frac {\ln \left (x^{2}+x +1\right )}{8}-\frac {\ln \left (x^{4}-x^{2}+1\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.05, size = 36, normalized size = 0.92 \[ -\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} + 1\right )}\right ) - \frac {1}{8} \, \log \left (x^{8} + x^{4} + 1\right ) + \frac {1}{4} \, \log \left (x^{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.30, size = 34, normalized size = 0.87 \[ \ln \relax (x)-\frac {\ln \left (x^8+x^4+1\right )}{8}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,x^4}{3}+\frac {\sqrt {3}}{3}\right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 41, normalized size = 1.05 \[ \log {\relax (x )} - \frac {\log {\left (x^{8} + x^{4} + 1 \right )}}{8} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{4}}{3} + \frac {\sqrt {3}}{3} \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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