Optimal. Leaf size=19 \[ \tan ^{-1}\left (\frac{3-2 x}{\sqrt{x^2+2 x-4}}\right ) \]
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Rubi [A] time = 0.0118567, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {724, 204} \[ \tan ^{-1}\left (\frac{3-2 x}{\sqrt{x^2+2 x-4}}\right ) \]
Antiderivative was successfully verified.
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Rule 724
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{(1-x) \sqrt{-4+2 x+x^2}} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{-4-x^2} \, dx,x,\frac{6-4 x}{\sqrt{-4+2 x+x^2}}\right )\right )\\ &=\tan ^{-1}\left (\frac{3-2 x}{\sqrt{-4+2 x+x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0059554, size = 22, normalized size = 1.16 \[ \tan ^{-1}\left (\frac{6-4 x}{2 \sqrt{x^2+2 x-4}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 23, normalized size = 1.2 \begin{align*} -\arctan \left ({\frac{-6+4\,x}{2}{\frac{1}{\sqrt{ \left ( -1+x \right ) ^{2}-5+4\,x}}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49182, size = 36, normalized size = 1.89 \begin{align*} -\arcsin \left (\frac{2 \, \sqrt{5} x}{5 \,{\left | x - 1 \right |}} - \frac{3 \, \sqrt{5}}{5 \,{\left | x - 1 \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03564, size = 55, normalized size = 2.89 \begin{align*} -2 \, \arctan \left (-x + \sqrt{x^{2} + 2 \, x - 4} + 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} + 2 x - 4} - \sqrt{x^{2} + 2 x - 4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.38457, size = 24, normalized size = 1.26 \begin{align*} -2 \, \arctan \left (-x + \sqrt{x^{2} + 2 \, x - 4} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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