Optimal. Leaf size=37 \[ \frac {1}{8} \log \left (x^8+x^4+1\right )-\frac {\tan ^{-1}\left (\frac {2 x^4+1}{\sqrt {3}}\right )}{4 \sqrt {3}} \]
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Rubi [A] time = 0.03, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {1357, 634, 618, 204, 628} \[ \frac {1}{8} \log \left (x^8+x^4+1\right )-\frac {\tan ^{-1}\left (\frac {2 x^4+1}{\sqrt {3}}\right )}{4 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1357
Rubi steps
\begin {align*} \int \frac {x^7}{1+x^4+x^8} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {x}{1+x+x^2} \, dx,x,x^4\right )\\ &=-\left (\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,x^4\right )\right )+\frac {1}{8} \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,x^4\right )\\ &=\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}}+\frac {1}{8} \log \left (1+x^4+x^8\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 37, normalized size = 1.00 \[ \frac {1}{8} \log \left (x^8+x^4+1\right )-\frac {\tan ^{-1}\left (\frac {2 x^4+1}{\sqrt {3}}\right )}{4 \sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 30, normalized size = 0.81 \[ -\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} + 1\right )}\right ) + \frac {1}{8} \, \log \left (x^{8} + x^{4} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 30, normalized size = 0.81 \[ -\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} + 1\right )}\right ) + \frac {1}{8} \, \log \left (x^{8} + x^{4} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 31, normalized size = 0.84 \[ -\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{4}+1\right ) \sqrt {3}}{3}\right )}{12}+\frac {\ln \left (x^{8}+x^{4}+1\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.43, size = 30, normalized size = 0.81 \[ -\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} + 1\right )}\right ) + \frac {1}{8} \, \log \left (x^{8} + x^{4} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 32, normalized size = 0.86 \[ \frac {\ln \left (x^8+x^4+1\right )}{8}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,x^4}{3}+\frac {\sqrt {3}}{3}\right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 37, normalized size = 1.00 \[ \frac {\log {\left (x^{8} + x^{4} + 1 \right )}}{8} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{4}}{3} + \frac {\sqrt {3}}{3} \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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