Optimal. Leaf size=20 \[ -\frac {x^2}{2}-\frac {1}{2} \log \left (1-x^2\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 20, 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, 272, 45}
\begin {gather*} -\frac {x^2}{2}-\frac {1}{2} \log \left (1-x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^4}{x-x^3} \, dx &=\int \frac {x^3}{1-x^2} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {x}{1-x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (-1+\frac {1}{1-x}\right ) \, dx,x,x^2\right )\\ &=-\frac {x^2}{2}-\frac {1}{2} \log \left (1-x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 18, normalized size = 0.90 \begin {gather*} -\frac {x^2}{2}-\frac {1}{2} \log \left (-1+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.34, size = 19, normalized size = 0.95
method | result | size |
risch | \(-\frac {x^{2}}{2}-\frac {\ln \left (x^{2}-1\right )}{2}\) | \(15\) |
meijerg | \(-\frac {x^{2}}{2}-\frac {\ln \left (-x^{2}+1\right )}{2}\) | \(17\) |
default | \(-\frac {x^{2}}{2}-\frac {\ln \left (x -1\right )}{2}-\frac {\ln \left (x +1\right )}{2}\) | \(19\) |
norman | \(-\frac {x^{2}}{2}-\frac {\ln \left (x -1\right )}{2}-\frac {\ln \left (x +1\right )}{2}\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 18, normalized size = 0.90 \begin {gather*} -\frac {1}{2} \, x^{2} - \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 = 2.10, size = 14, normalized size = 0.70 \begin {gather*} -\frac {1}{2} \, x^{2} - \frac {1}{2} \, \log \left (x^{2} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 14, normalized size = 0.70 \begin {gather*} - \frac {x^{2}}{2} - \frac {\log {\left (x^{2} - 1 \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.02, size = 15, normalized size = 0.75 \begin {gather*} -\frac {1}{2} \, x^{2} - \frac {1}{2} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 14, normalized size = 0.70 \begin {gather*} -\frac {\ln \left (x^2-1\right )}{2}-\frac {x^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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