Optimal. Leaf size=19 \[ -2 \tanh ^{-1}\left (\frac{x+1}{\sqrt{x^2+3 x+1}}\right ) \]
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Rubi [A] time = 0.0146195, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {838, 206} \[ -2 \tanh ^{-1}\left (\frac{x+1}{\sqrt{x^2+3 x+1}}\right ) \]
Antiderivative was successfully verified.
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Rule 838
Rule 206
Rubi steps
\begin{align*} \int \frac{1-x}{x \sqrt{1+3 x+x^2}} \, dx &=-\left (4 \operatorname{Subst}\left (\int \frac{1}{4-x^2} \, dx,x,\frac{2+2 x}{\sqrt{1+3 x+x^2}}\right )\right )\\ &=-2 \tanh ^{-1}\left (\frac{1+x}{\sqrt{1+3 x+x^2}}\right )\\ \end{align*}
Mathematica [B] time = 0.0063617, size = 49, normalized size = 2.58 \[ -\tanh ^{-1}\left (\frac{2 x+3}{2 \sqrt{x^2+3 x+1}}\right )-\tanh ^{-1}\left (\frac{3 x+2}{2 \sqrt{x^2+3 x+1}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.008, size = 38, normalized size = 2. \begin{align*} -\ln \left ({\frac{3}{2}}+x+\sqrt{{x}^{2}+3\,x+1} \right ) -{\it Artanh} \left ({\frac{2+3\,x}{2}{\frac{1}{\sqrt{{x}^{2}+3\,x+1}}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.97464, size = 65, normalized size = 3.42 \begin{align*} -\log \left (2 \, x + 2 \, \sqrt{x^{2} + 3 \, x + 1} + 3\right ) - \log \left (\frac{2 \, \sqrt{x^{2} + 3 \, x + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}} + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.21484, size = 123, normalized size = 6.47 \begin{align*} \log \left (4 \, x^{2} - \sqrt{x^{2} + 3 \, x + 1}{\left (4 \, x + 5\right )} + 11 \, x + 5\right ) - \log \left (-x + \sqrt{x^{2} + 3 \, x + 1} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{1}{x \sqrt{x^{2} + 3 x + 1}}\, dx - \int \frac{1}{\sqrt{x^{2} + 3 x + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19265, size = 76, normalized size = 4. \begin{align*} -\log \left ({\left | -x + \sqrt{x^{2} + 3 \, x + 1} + 1 \right |}\right ) + \log \left ({\left | -x + \sqrt{x^{2} + 3 \, x + 1} - 1 \right |}\right ) + \log \left ({\left | -2 \, x + 2 \, \sqrt{x^{2} + 3 \, x + 1} - 3 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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