Optimal. Leaf size=16 \[ \frac{x^3}{3}-\frac{1}{3} \tan ^{-1}\left (x^3\right ) \]
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Rubi [A] time = 0.0058482, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {275, 321, 203} \[ \frac{x^3}{3}-\frac{1}{3} \tan ^{-1}\left (x^3\right ) \]
Antiderivative was successfully verified.
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Rule 275
Rule 321
Rule 203
Rubi steps
\begin{align*} \int \frac{x^8}{1+x^6} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{1+x^2} \, dx,x,x^3\right )\\ &=\frac{x^3}{3}-\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,x^3\right )\\ &=\frac{x^3}{3}-\frac{1}{3} \tan ^{-1}\left (x^3\right )\\ \end{align*}
Mathematica [A] time = 0.0047174, size = 16, normalized size = 1. \[ \frac{x^3}{3}-\frac{1}{3} \tan ^{-1}\left (x^3\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 13, normalized size = 0.8 \begin{align*}{\frac{{x}^{3}}{3}}-{\frac{\arctan \left ({x}^{3} \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50229, size = 16, normalized size = 1. \begin{align*} \frac{1}{3} \, x^{3} - \frac{1}{3} \, \arctan \left (x^{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45512, size = 36, normalized size = 2.25 \begin{align*} \frac{1}{3} \, x^{3} - \frac{1}{3} \, \arctan \left (x^{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.095263, size = 10, normalized size = 0.62 \begin{align*} \frac{x^{3}}{3} - \frac{\operatorname{atan}{\left (x^{3} \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14401, size = 16, normalized size = 1. \begin{align*} \frac{1}{3} \, x^{3} - \frac{1}{3} \, \arctan \left (x^{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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