Optimal. Leaf size=49 \[ \frac{1}{4} \left (1-\sqrt{3}\right ) \log \left (2 x-\sqrt{3}+2\right )+\frac{1}{4} \left (1+\sqrt{3}\right ) \log \left (2 x+\sqrt{3}+2\right ) \]
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Rubi [A] time = 0.0257659, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {632, 31} \[ \frac{1}{4} \left (1-\sqrt{3}\right ) \log \left (2 x-\sqrt{3}+2\right )+\frac{1}{4} \left (1+\sqrt{3}\right ) \log \left (2 x+\sqrt{3}+2\right ) \]
Antiderivative was successfully verified.
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Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{-1+2 x}{1+8 x+4 x^2} \, dx &=-\left (\left (-1+\sqrt{3}\right ) \int \frac{1}{4-2 \sqrt{3}+4 x} \, dx\right )+\left (1+\sqrt{3}\right ) \int \frac{1}{4+2 \sqrt{3}+4 x} \, dx\\ &=\frac{1}{4} \left (1-\sqrt{3}\right ) \log \left (2-\sqrt{3}+2 x\right )+\frac{1}{4} \left (1+\sqrt{3}\right ) \log \left (2+\sqrt{3}+2 x\right )\\ \end{align*}
Mathematica [A] time = 0.0207772, size = 44, normalized size = 0.9 \[ \frac{1}{4} \left (\left (1+\sqrt{3}\right ) \log \left (2 x+\sqrt{3}+2\right )-\left (\sqrt{3}-1\right ) \log \left (-2 x+\sqrt{3}-2\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 31, normalized size = 0.6 \begin{align*}{\frac{\ln \left ( 4\,{x}^{2}+8\,x+1 \right ) }{4}}+{\frac{\sqrt{3}}{2}{\it Artanh} \left ({\frac{ \left ( 8\,x+8 \right ) \sqrt{3}}{12}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.62243, size = 55, normalized size = 1.12 \begin{align*} -\frac{1}{4} \, \sqrt{3} \log \left (\frac{2 \, x - \sqrt{3} + 2}{2 \, x + \sqrt{3} + 2}\right ) + \frac{1}{4} \, \log \left (4 \, x^{2} + 8 \, x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5568, size = 136, normalized size = 2.78 \begin{align*} \frac{1}{4} \, \sqrt{3} \log \left (\frac{4 \, x^{2} + 4 \, \sqrt{3}{\left (x + 1\right )} + 8 \, x + 7}{4 \, x^{2} + 8 \, x + 1}\right ) + \frac{1}{4} \, \log \left (4 \, x^{2} + 8 \, x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.253882, size = 42, normalized size = 0.86 \begin{align*} \left (\frac{1}{4} - \frac{\sqrt{3}}{4}\right ) \log{\left (x - \frac{\sqrt{3}}{2} + 1 \right )} + \left (\frac{1}{4} + \frac{\sqrt{3}}{4}\right ) \log{\left (x + \frac{\sqrt{3}}{2} + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30801, size = 62, normalized size = 1.27 \begin{align*} -\frac{1}{4} \, \sqrt{3} \log \left (\frac{{\left | 8 \, x - 4 \, \sqrt{3} + 8 \right |}}{{\left | 8 \, x + 4 \, \sqrt{3} + 8 \right |}}\right ) + \frac{1}{4} \, \log \left ({\left | 4 \, x^{2} + 8 \, x + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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