Optimal. Leaf size=16 \[ \frac {x^3}{3}-\frac {1}{3} \tan ^{-1}\left (x^3\right ) \]
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Rubi [A]
time = 0.00, 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 = {281, 327, 209}
\begin {gather*} \frac {x^3}{3}-\frac {\text {ArcTan}\left (x^3\right )}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 281
Rule 327
Rubi steps
\begin {align*} \int \frac {x^8}{1+x^6} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,x^3\right )\\ &=\frac {x^3}{3}-\frac {1}{3} \text {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 \begin {gather*} \frac {x^3}{3}-\frac {1}{3} \tan ^{-1}\left (x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 13, normalized size = 0.81
method | result | size |
default | \(\frac {x^{3}}{3}-\frac {\arctan \left (x^{3}\right )}{3}\) | \(13\) |
meijerg | \(\frac {x^{3}}{3}-\frac {\arctan \left (x^{3}\right )}{3}\) | \(13\) |
risch | \(\frac {x^{3}}{3}-\frac {\arctan \left (x^{3}\right )}{3}\) | \(13\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 12, normalized size = 0.75 \begin {gather*} \frac {1}{3} \, x^{3} - \frac {1}{3} \, \arctan \left (x^{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 12, normalized size = 0.75 \begin {gather*} \frac {1}{3} \, x^{3} - \frac {1}{3} \, \arctan \left (x^{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 10, normalized size = 0.62 \begin {gather*} \frac {x^{3}}{3} - \frac {\operatorname {atan}{\left (x^{3} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.35, size = 12, normalized size = 0.75 \begin {gather*} \frac {1}{3} \, x^{3} - \frac {1}{3} \, \arctan \left (x^{3}\right ) \end {gather*}
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 \begin {gather*} \frac {x^3}{3}-\frac {\mathrm {atan}\left (x^3\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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