Optimal. Leaf size=39 \[ -\frac {1}{6} \log (1-2 x)-\frac {1}{6} \log (1-x)+\frac {1}{6} \log (x+1)+\frac {1}{6} \log (2 x+1) \]
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Rubi [A] time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1161, 616, 31} \begin {gather*} -\frac {1}{6} \log (1-2 x)-\frac {1}{6} \log (1-x)+\frac {1}{6} \log (x+1)+\frac {1}{6} \log (2 x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 616
Rule 1161
Rubi steps
\begin {align*} \int \frac {1-2 x^2}{1-5 x^2+4 x^4} \, dx &=-\left (\frac {1}{4} \int \frac {1}{-\frac {1}{2}-\frac {x}{2}+x^2} \, dx\right )-\frac {1}{4} \int \frac {1}{-\frac {1}{2}+\frac {x}{2}+x^2} \, dx\\ &=-\left (\frac {1}{6} \int \frac {1}{-1+x} \, dx\right )-\frac {1}{6} \int \frac {1}{-\frac {1}{2}+x} \, dx+\frac {1}{6} \int \frac {1}{\frac {1}{2}+x} \, dx+\frac {1}{6} \int \frac {1}{1+x} \, dx\\ &=-\frac {1}{6} \log (1-2 x)-\frac {1}{6} \log (1-x)+\frac {1}{6} \log (1+x)+\frac {1}{6} \log (1+2 x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 0.79 \begin {gather*} \frac {1}{6} \log \left (2 x^2+3 x+1\right )-\frac {1}{6} \log \left (2 x^2-3 x+1\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-2 x^2}{1-5 x^2+4 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.47, size = 27, normalized size = 0.69 \begin {gather*} \frac {1}{6} \, \log \left (2 \, x^{2} + 3 \, x + 1\right ) - \frac {1}{6} \, \log \left (2 \, x^{2} - 3 \, x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 33, normalized size = 0.85 \begin {gather*} \frac {1}{6} \, \log \left ({\left | 2 \, x + 1 \right |}\right ) - \frac {1}{6} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) + \frac {1}{6} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{6} \, \log \left ({\left | x - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 30, normalized size = 0.77 \begin {gather*} \frac {\ln \left (x +1\right )}{6}+\frac {\ln \left (2 x +1\right )}{6}-\frac {\ln \left (x -1\right )}{6}-\frac {\ln \left (2 x -1\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 29, normalized size = 0.74 \begin {gather*} \frac {1}{6} \, \log \left (2 \, x + 1\right ) - \frac {1}{6} \, \log \left (2 \, x - 1\right ) + \frac {1}{6} \, \log \left (x + 1\right ) - \frac {1}{6} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 15, normalized size = 0.38 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {3\,x}{2\,x^2+1}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 29, normalized size = 0.74 \begin {gather*} - \frac {\log {\left (x^{2} - \frac {3 x}{2} + \frac {1}{2} \right )}}{6} + \frac {\log {\left (x^{2} + \frac {3 x}{2} + \frac {1}{2} \right )}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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