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 = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, 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 31
Rule 632
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*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 0.90 \[ \frac {1}{4} \left (\left (1+\sqrt {3}\right ) \log \left (2 x+\sqrt {3}+2\right )-\left (\sqrt {3}-1\right ) \log \left (-2 x+\sqrt {3}-2\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 51, normalized size = 1.04 \[ \frac {1}{4} \, \sqrt {3} \log \left (\frac {4 \, x^{2} + 4 \, \sqrt {3} {\left (x + 1\right )} + 8 \, x + 7}{4 \, x^{2} + 8 \, x + 1}\right ) + \frac {1}{4} \, \log \left (4 \, x^{2} + 8 \, x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 46, normalized size = 0.94 \[ -\frac {1}{4} \, \sqrt {3} \log \left (\frac {{\left | 8 \, x - 4 \, \sqrt {3} + 8 \right |}}{{\left | 8 \, x + 4 \, \sqrt {3} + 8 \right |}}\right ) + \frac {1}{4} \, \log \left ({\left | 4 \, x^{2} + 8 \, x + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 31, normalized size = 0.63 \[ \frac {\sqrt {3}\, \arctanh \left (\frac {\left (8 x +8\right ) \sqrt {3}}{12}\right )}{2}+\frac {\ln \left (4 x^{2}+8 x +1\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.14, size = 41, normalized size = 0.84 \[ -\frac {1}{4} \, \sqrt {3} \log \left (\frac {2 \, x - \sqrt {3} + 2}{2 \, x + \sqrt {3} + 2}\right ) + \frac {1}{4} \, \log \left (4 \, x^{2} + 8 \, x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 36, normalized size = 0.73 \[ \ln \left (x+\frac {\sqrt {3}}{2}+1\right )\,\left (\frac {\sqrt {3}}{4}+\frac {1}{4}\right )-\ln \left (x-\frac {\sqrt {3}}{2}+1\right )\,\left (\frac {\sqrt {3}}{4}-\frac {1}{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 42, normalized size = 0.86 \[ \left (\frac {1}{4} - \frac {\sqrt {3}}{4}\right ) \log {\left (x - \frac {\sqrt {3}}{2} + 1 \right )} + \left (\frac {1}{4} + \frac {\sqrt {3}}{4}\right ) \log {\left (x + \frac {\sqrt {3}}{2} + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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