Optimal. Leaf size=13 \[ -x-\frac {x^3}{3}+\tanh ^{-1}(x) \]
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Rubi [A]
time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1598, 308, 212}
\begin {gather*} -\frac {x^3}{3}-x+\tanh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 308
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^5}{x-x^3} \, dx &=\int \frac {x^4}{1-x^2} \, dx\\ &=\int \left (-1-x^2+\frac {1}{1-x^2}\right ) \, dx\\ &=-x-\frac {x^3}{3}+\int \frac {1}{1-x^2} \, dx\\ &=-x-\frac {x^3}{3}+\tanh ^{-1}(x)\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(29\) vs. \(2(13)=26\).
time = 0.00, size = 29, normalized size = 2.23 \begin {gather*} -x-\frac {x^3}{3}-\frac {1}{2} \log (1-x)+\frac {1}{2} \log (1+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.35, size = 22, normalized size = 1.69
method | result | size |
meijerg | \(-\frac {i \left (-\frac {2 i x \left (5 x^{2}+15\right )}{15}+2 i \arctanh \left (x \right )\right )}{2}\) | \(21\) |
default | \(-\frac {x^{3}}{3}-x -\frac {\ln \left (x -1\right )}{2}+\frac {\ln \left (x +1\right )}{2}\) | \(22\) |
norman | \(-\frac {x^{3}}{3}-x -\frac {\ln \left (x -1\right )}{2}+\frac {\ln \left (x +1\right )}{2}\) | \(22\) |
risch | \(-\frac {x^{3}}{3}-x -\frac {\ln \left (x -1\right )}{2}+\frac {\ln \left (x +1\right )}{2}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 21, normalized size = 1.62 \begin {gather*} -\frac {1}{3} \, x^{3} - x + \frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.60, size = 21, normalized size = 1.62 \begin {gather*} -\frac {1}{3} \, x^{3} - x + \frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 19 vs.
\(2 (8) = 16\).
time = 0.03, size = 19, normalized size = 1.46 \begin {gather*} - \frac {x^{3}}{3} - x - \frac {\log {\left (x - 1 \right )}}{2} + \frac {\log {\left (x + 1 \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 23 vs.
\(2 (11) = 22\).
time = 1.08, size = 23, normalized size = 1.77 \begin {gather*} -\frac {1}{3} \, x^{3} - x + \frac {1}{2} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.98, size = 11, normalized size = 0.85 \begin {gather*} \mathrm {atanh}\left (x\right )-x-\frac {x^3}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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