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 = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {724, 204} \begin {gather*} \tan ^{-1}\left (\frac {3-2 x}{\sqrt {x^2+2 x-4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 724
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*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 1.16 \begin {gather*} \tan ^{-1}\left (\frac {6-4 x}{2 \sqrt {x^2+2 x-4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 20, normalized size = 1.05 \begin {gather*} -2 \tan ^{-1}\left (\sqrt {x^2+2 x-4}-x+1\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 18, normalized size = 0.95 \begin {gather*} -2 \, \arctan \left (-x + \sqrt {x^{2} + 2 \, x - 4} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 18, normalized size = 0.95 \begin {gather*} -2 \, \arctan \left (-x + \sqrt {x^{2} + 2 \, x - 4} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 23, normalized size = 1.21 \begin {gather*} -\arctan \left (\frac {4 x -6}{2 \sqrt {4 x +\left (x -1\right )^{2}-5}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.99, size = 27, normalized size = 1.42 \begin {gather*} -\arcsin \left (\frac {2 \, \sqrt {5} x}{5 \, {\left | x - 1 \right |}} - \frac {3 \, \sqrt {5}}{5 \, {\left | x - 1 \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} -\int \frac {1}{\left (x-1\right )\,\sqrt {x^2+2\,x-4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {1}{x \sqrt {x^{2} + 2 x - 4} - \sqrt {x^{2} + 2 x - 4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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