Optimal. Leaf size=46 \[ \frac {x^4}{4}+\frac {\tan ^{-1}\left (\frac {1-2 x^4}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {1}{8} \log \left (x^8-x^4+1\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1357, 703, 634, 618, 204, 628} \[ \frac {x^4}{4}+\frac {1}{8} \log \left (x^8-x^4+1\right )+\frac {\tan ^{-1}\left (\frac {1-2 x^4}{\sqrt {3}}\right )}{4 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 703
Rule 1357
Rubi steps
\begin {align*} \int \frac {x^{11}}{1-x^4+x^8} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {x^2}{1-x+x^2} \, dx,x,x^4\right )\\ &=\frac {x^4}{4}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {-1+x}{1-x+x^2} \, dx,x,x^4\right )\\ &=\frac {x^4}{4}-\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 {x^4}{4}+\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 {x^4}{4}+\frac {\tan ^{-1}\left (\frac {1-2 x^4}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {1}{8} \log \left (1-x^4+x^8\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 46, normalized size = 1.00 \[ \frac {x^4}{4}-\frac {\tan ^{-1}\left (\frac {2 x^4-1}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {1}{8} \log \left (x^8-x^4+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 37, normalized size = 0.80 \[ \frac {1}{4} \, x^{4} - \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} - 1\right )}\right ) + \frac {1}{8} \, \log \left (x^{8} - x^{4} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 37, normalized size = 0.80 \[ \frac {1}{4} \, x^{4} - \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} - 1\right )}\right ) + \frac {1}{8} \, \log \left (x^{8} - x^{4} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 38, normalized size = 0.83 \[ \frac {x^{4}}{4}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{4}-1\right ) \sqrt {3}}{3}\right )}{12}+\frac {\ln \left (x^{8}-x^{4}+1\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.11, size = 37, normalized size = 0.80 \[ \frac {1}{4} \, x^{4} - \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} - 1\right )}\right ) + \frac {1}{8} \, \log \left (x^{8} - x^{4} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 39, normalized size = 0.85 \[ \frac {\ln \left (x^8-x^4+1\right )}{8}+\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}}{3}-\frac {2\,\sqrt {3}\,x^4}{3}\right )}{12}+\frac {x^4}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 42, normalized size = 0.91 \[ \frac {x^{4}}{4} + \frac {\log {\left (x^{8} - x^{4} + 1 \right )}}{8} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{4}}{3} - \frac {\sqrt {3}}{3} \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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