Optimal. Leaf size=19 \[ -\frac{\tan ^{-1}\left (\frac{1-x}{\sqrt{2}}\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0100857, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {618, 204} \[ -\frac{\tan ^{-1}\left (\frac{1-x}{\sqrt{2}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{3-2 x+x^2} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{-8-x^2} \, dx,x,-2+2 x\right )\right )\\ &=-\frac{\tan ^{-1}\left (\frac{1-x}{\sqrt{2}}\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0050858, size = 16, normalized size = 0.84 \[ \frac{\tan ^{-1}\left (\frac{x-1}{\sqrt{2}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 17, normalized size = 0.9 \begin{align*}{\frac{\sqrt{2}}{2}\arctan \left ({\frac{ \left ( 2\,x-2 \right ) \sqrt{2}}{4}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41061, size = 19, normalized size = 1. \begin{align*} \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x - 1\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.9499, size = 55, normalized size = 2.89 \begin{align*} \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x - 1\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.098663, size = 22, normalized size = 1.16 \begin{align*} \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} - \frac{\sqrt{2}}{2} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06335, size = 19, normalized size = 1. \begin{align*} \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x - 1\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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