Optimal. Leaf size=39 \[ -\frac{1}{2} \log (1-2 x)+\frac{1}{2} \log (1-x)-\frac{1}{2} \log (x+1)+\frac{1}{2} \log (2 x+1) \]
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Rubi [A] time = 0.0178852, antiderivative size = 39, normalized size of antiderivative = 1., 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} \[ -\frac{1}{2} \log (1-2 x)+\frac{1}{2} \log (1-x)-\frac{1}{2} \log (x+1)+\frac{1}{2} \log (2 x+1) \]
Antiderivative was successfully verified.
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Rule 1161
Rule 616
Rule 31
Rubi steps
\begin{align*} \int \frac{1+2 x^2}{1-5 x^2+4 x^4} \, dx &=\frac{1}{4} \int \frac{1}{\frac{1}{2}-\frac{3 x}{2}+x^2} \, dx+\frac{1}{4} \int \frac{1}{\frac{1}{2}+\frac{3 x}{2}+x^2} \, dx\\ &=\frac{1}{2} \int \frac{1}{-1+x} \, dx-\frac{1}{2} \int \frac{1}{-\frac{1}{2}+x} \, dx+\frac{1}{2} \int \frac{1}{\frac{1}{2}+x} \, dx-\frac{1}{2} \int \frac{1}{1+x} \, dx\\ &=-\frac{1}{2} \log (1-2 x)+\frac{1}{2} \log (1-x)-\frac{1}{2} \log (1+x)+\frac{1}{2} \log (1+2 x)\\ \end{align*}
Mathematica [A] time = 0.0058667, size = 29, normalized size = 0.74 \[ \frac{1}{2} \log \left (-2 x^2+x+1\right )-\frac{1}{2} \log \left (-2 x^2-x+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 30, normalized size = 0.8 \begin{align*} -{\frac{\ln \left ( 1+x \right ) }{2}}-{\frac{\ln \left ( 2\,x-1 \right ) }{2}}+{\frac{\ln \left ( -1+x \right ) }{2}}+{\frac{\ln \left ( 1+2\,x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.96804, size = 39, normalized size = 1. \begin{align*} \frac{1}{2} \, \log \left (2 \, x + 1\right ) - \frac{1}{2} \, \log \left (2 \, x - 1\right ) - \frac{1}{2} \, \log \left (x + 1\right ) + \frac{1}{2} \, \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31776, size = 68, normalized size = 1.74 \begin{align*} -\frac{1}{2} \, \log \left (2 \, x^{2} + x - 1\right ) + \frac{1}{2} \, \log \left (2 \, x^{2} - x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.098273, size = 26, normalized size = 0.67 \begin{align*} \frac{\log{\left (x^{2} - \frac{x}{2} - \frac{1}{2} \right )}}{2} - \frac{\log{\left (x^{2} + \frac{x}{2} - \frac{1}{2} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15209, size = 45, normalized size = 1.15 \begin{align*} \frac{1}{2} \, \log \left ({\left | 2 \, x + 1 \right |}\right ) - \frac{1}{2} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) - \frac{1}{2} \, \log \left ({\left | x + 1 \right |}\right ) + \frac{1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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