Optimal. Leaf size=16 \[ -\frac {x^3}{3}+\frac {1}{3} \tanh ^{-1}\left (x^3\right ) \]
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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 = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {281, 327, 212}
\begin {gather*} \frac {1}{3} \tanh ^{-1}\left (x^3\right )-\frac {x^3}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
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} \tanh ^{-1}\left (x^3\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 30, normalized size = 1.88 \begin {gather*} -\frac {x^3}{3}-\frac {1}{6} \log \left (1-x^3\right )+\frac {1}{6} \log \left (1+x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 23, normalized size = 1.44
method | result | size |
meijerg | \(\frac {i \left (2 i x^{3}-2 i \arctanh \left (x^{3}\right )\right )}{6}\) | \(18\) |
default | \(-\frac {x^{3}}{3}-\frac {\ln \left (x^{3}-1\right )}{6}+\frac {\ln \left (x^{3}+1\right )}{6}\) | \(23\) |
risch | \(-\frac {x^{3}}{3}-\frac {\ln \left (x^{3}-1\right )}{6}+\frac {\ln \left (x^{3}+1\right )}{6}\) | \(23\) |
norman | \(-\frac {x^{3}}{3}-\frac {\ln \left (x -1\right )}{6}+\frac {\ln \left (x +1\right )}{6}+\frac {\ln \left (x^{2}-x +1\right )}{6}-\frac {\ln \left (x^{2}+x +1\right )}{6}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 22, normalized size = 1.38 \begin {gather*} -\frac {1}{3} \, x^{3} + \frac {1}{6} \, \log \left (x^{3} + 1\right ) - \frac {1}{6} \, \log \left (x^{3} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 22, normalized size = 1.38 \begin {gather*} -\frac {1}{3} \, x^{3} + \frac {1}{6} \, \log \left (x^{3} + 1\right ) - \frac {1}{6} \, \log \left (x^{3} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 20, normalized size = 1.25 \begin {gather*} - \frac {x^{3}}{3} - \frac {\log {\left (x^{3} - 1 \right )}}{6} + \frac {\log {\left (x^{3} + 1 \right )}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.03, size = 24, normalized size = 1.50 \begin {gather*} -\frac {1}{3} \, x^{3} + \frac {1}{6} \, \log \left ({\left | x^{3} + 1 \right |}\right ) - \frac {1}{6} \, \log \left ({\left | x^{3} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 12, normalized size = 0.75 \begin {gather*} \frac {\mathrm {atanh}\left (x^3\right )}{3}-\frac {x^3}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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