Optimal. Leaf size=23 \[ \frac {\tan ^{-1}\left (\frac {2 x^4+1}{\sqrt {3}}\right )}{2 \sqrt {3}} \]
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Rubi [A] time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1352, 618, 204} \[ \frac {\tan ^{-1}\left (\frac {2 x^4+1}{\sqrt {3}}\right )}{2 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 1352
Rubi steps
\begin {align*} \int \frac {x^3}{1+x^4+x^8} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,x^4\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x^4\right )\right )\\ &=\frac {\tan ^{-1}\left (\frac {1+2 x^4}{\sqrt {3}}\right )}{2 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {2 x^4+1}{\sqrt {3}}\right )}{2 \sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 18, normalized size = 0.78 \[ \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} + 1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 18, normalized size = 0.78 \[ \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} + 1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 19, normalized size = 0.83 \[ \frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{4}+1\right ) \sqrt {3}}{3}\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.47, size = 18, normalized size = 0.78 \[ \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} + 1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.30, size = 17, normalized size = 0.74 \[ \frac {\sqrt {3}\,\mathrm {atan}\left (\sqrt {3}\,\left (\frac {2\,x^4}{3}+\frac {1}{3}\right )\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 26, normalized size = 1.13 \[ \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{4}}{3} + \frac {\sqrt {3}}{3} \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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