Optimal. Leaf size=18 \[ -\frac {1}{4}+\frac {\log \left (\left (-x+x^3\right )^2\right )}{x} \]
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Rubi [A]
time = 0.15, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1607, 6857,
464, 213, 2605, 212} \begin {gather*} \frac {\log \left (x^2 \left (1-x^2\right )^2\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 213
Rule 464
Rule 1607
Rule 2605
Rule 6857
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2+6 x^2+\left (1-x^2\right ) \log \left (x^2-2 x^4+x^6\right )}{x^2 \left (-1+x^2\right )} \, dx\\ &=\int \left (\frac {2 \left (-1+3 x^2\right )}{x^2 \left (-1+x^2\right )}-\frac {\log \left (x^2 \left (-1+x^2\right )^2\right )}{x^2}\right ) \, dx\\ &=2 \int \frac {-1+3 x^2}{x^2 \left (-1+x^2\right )} \, dx-\int \frac {\log \left (x^2 \left (-1+x^2\right )^2\right )}{x^2} \, dx\\ &=-\frac {2}{x}+\frac {\log \left (x^2 \left (1-x^2\right )^2\right )}{x}+4 \int \frac {1}{-1+x^2} \, dx-\int \frac {2-6 x^2}{x^2 \left (1-x^2\right )} \, dx\\ &=-4 \tanh ^{-1}(x)+\frac {\log \left (x^2 \left (1-x^2\right )^2\right )}{x}+4 \int \frac {1}{1-x^2} \, dx\\ &=\frac {\log \left (x^2 \left (1-x^2\right )^2\right )}{x}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 16, normalized size = 0.89 \begin {gather*} \frac {\log \left (x^2 \left (-1+x^2\right )^2\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 18, normalized size = 1.00
method | result | size |
default | \(\frac {\ln \left (x^{6}-2 x^{4}+x^{2}\right )}{x}\) | \(18\) |
norman | \(\frac {\ln \left (x^{6}-2 x^{4}+x^{2}\right )}{x}\) | \(18\) |
risch | \(\frac {\ln \left (x^{6}-2 x^{4}+x^{2}\right )}{x}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 44 vs.
\(2 (16) = 32\).
time = 0.33, size = 44, normalized size = 2.44 \begin {gather*} \frac {2 \, {\left ({\left (x + 1\right )} \log \left (x + 1\right ) - {\left (x - 1\right )} \log \left (x - 1\right ) + \log \left (x\right ) + 1\right )}}{x} - \frac {2}{x} - 2 \, \log \left (x + 1\right ) + 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 = 0.35, size = 17, normalized size = 0.94 \begin {gather*} \frac {\log \left (x^{6} - 2 \, x^{4} + x^{2}\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 14, normalized size = 0.78 \begin {gather*} \frac {\log {\left (x^{6} - 2 x^{4} + x^{2} \right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 17, normalized size = 0.94 \begin {gather*} \frac {\log \left (x^{6} - 2 \, x^{4} + x^{2}\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.98, size = 16, normalized size = 0.89 \begin {gather*} \frac {\ln \left (x^2\,{\left (x^2-1\right )}^2\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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