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 = 0.0212884, antiderivative size = 23, normalized size of antiderivative = 1., 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 1352
Rule 618
Rule 204
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*}
Mathematica [A] time = 0.0056358, size = 23, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{2 x^4+1}{\sqrt{3}}\right )}{2 \sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 19, normalized size = 0.8 \begin{align*}{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,{x}^{4}+1 \right ) \sqrt{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.51776, size = 24, normalized size = 1.04 \begin{align*} \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} + 1\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45901, size = 61, normalized size = 2.65 \begin{align*} \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} + 1\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.116641, size = 26, normalized size = 1.13 \begin{align*} \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{4}}{3} + \frac{\sqrt{3}}{3} \right )}}{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16556, size = 24, normalized size = 1.04 \begin{align*} \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} + 1\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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