Optimal. Leaf size=29 \[ -\frac {1}{4 x^4}-\frac {1}{2 x^2}+\log (x)-\frac {1}{2} \log \left (1-x^2\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 29, 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, 46}
\begin {gather*} -\frac {1}{4 x^4}-\frac {1}{2 x^2}-\frac {1}{2} \log \left (1-x^2\right )+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 272
Rule 1598
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (x-x^3\right )} \, dx &=\int \frac {1}{x^5 \left (1-x^2\right )} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{(1-x) x^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{1-x}+\frac {1}{x^3}+\frac {1}{x^2}+\frac {1}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{4 x^4}-\frac {1}{2 x^2}+\log (x)-\frac {1}{2} \log \left (1-x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 29, normalized size = 1.00 \begin {gather*} -\frac {1}{4 x^4}-\frac {1}{2 x^2}+\log (x)-\frac {1}{2} \log \left (1-x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 26, normalized size = 0.90
method | result | size |
risch | \(\frac {-\frac {1}{4}-\frac {x^{2}}{2}}{x^{4}}+\ln \left (x \right )-\frac {\ln \left (x^{2}-1\right )}{2}\) | \(23\) |
default | \(-\frac {\ln \left (x +1\right )}{2}-\frac {\ln \left (x -1\right )}{2}-\frac {1}{4 x^{4}}-\frac {1}{2 x^{2}}+\ln \left (x \right )\) | \(26\) |
norman | \(\frac {-\frac {1}{4}-\frac {x^{2}}{2}}{x^{4}}-\frac {\ln \left (x +1\right )}{2}-\frac {\ln \left (x -1\right )}{2}+\ln \left (x \right )\) | \(27\) |
meijerg | \(-\frac {\ln \left (-x^{2}+1\right )}{2}+\ln \left (x \right )+\frac {i \pi }{2}-\frac {1}{4 x^{4}}-\frac {1}{2 x^{2}}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 27, normalized size = 0.93 \begin {gather*} -\frac {2 \, x^{2} + 1}{4 \, x^{4}} - \frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.70, size = 30, normalized size = 1.03 \begin {gather*} -\frac {2 \, x^{4} \log \left (x^{2} - 1\right ) - 4 \, x^{4} \log \left (x\right ) + 2 \, x^{2} + 1}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 22, normalized size = 0.76 \begin {gather*} \log {\left (x \right )} - \frac {\log {\left (x^{2} - 1 \right )}}{2} - \frac {2 x^{2} + 1}{4 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.55, size = 33, normalized size = 1.14 \begin {gather*} -\frac {3 \, x^{4} + 2 \, x^{2} + 1}{4 \, x^{4}} + \frac {1}{2} \, \log \left (x^{2}\right ) - \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 = 0.03, size = 23, normalized size = 0.79 \begin {gather*} \ln \left (x\right )-\frac {\ln \left (x^2-1\right )}{2}-\frac {\frac {x^2}{2}+\frac {1}{4}}{x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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