Optimal. Leaf size=16 \[ \frac {x^3}{3}-\frac {1}{3} \tan ^{-1}\left (x^3\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 203
Rule 275
Rule 321
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*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 1.00 \[ \frac {x^3}{3}-\frac {1}{3} \tan ^{-1}\left (x^3\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 12, normalized size = 0.75 \[ \frac {1}{3} \, x^{3} - \frac {1}{3} \, \arctan \left (x^{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 12, normalized size = 0.75 \[ \frac {1}{3} \, x^{3} - \frac {1}{3} \, \arctan \left (x^{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 13, normalized size = 0.81 \[ \frac {x^{3}}{3}-\frac {\arctan \left (x^{3}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.46, size = 12, normalized size = 0.75 \[ \frac {1}{3} \, x^{3} - \frac {1}{3} \, \arctan \left (x^{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.03, size = 12, normalized size = 0.75 \[ \frac {x^3}{3}-\frac {\mathrm {atan}\left (x^3\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 10, normalized size = 0.62 \[ \frac {x^{3}}{3} - \frac {\operatorname {atan}{\left (x^{3} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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