Optimal. Leaf size=13 \[ \tan ^{-1}\left (\sqrt {x^2-4 x+3}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {688, 203} \begin {gather*} \tan ^{-1}\left (\sqrt {x^2-4 x+3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 688
Rubi steps
\begin {align*} \int \frac {1}{(-2+x) \sqrt {3-4 x+x^2}} \, dx &=4 \operatorname {Subst}\left (\int \frac {1}{4+4 x^2} \, dx,x,\sqrt {3-4 x+x^2}\right )\\ &=\tan ^{-1}\left (\sqrt {3-4 x+x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 12, normalized size = 0.92 \begin {gather*} \tan ^{-1}\left (\sqrt {(x-2)^2-1}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.16, size = 21, normalized size = 1.62 \begin {gather*} -2 \tan ^{-1}\left (\frac {\sqrt {x^2-4 x+3}}{x-3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 18, normalized size = 1.38 \begin {gather*} 2 \, \arctan \left (-x + \sqrt {x^{2} - 4 \, x + 3} + 2\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 = 1.38 \begin {gather*} 2 \, \arctan \left (-x + \sqrt {x^{2} - 4 \, x + 3} + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 13, normalized size = 1.00 \begin {gather*} -\arctan \left (\frac {1}{\sqrt {\left (x -2\right )^{2}-1}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.94, size = 9, normalized size = 0.69 \begin {gather*} -\arcsin \left (\frac {1}{{\left | x - 2 \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 = -0.08 \begin {gather*} \int \frac {1}{\left (x-2\right )\,\sqrt {x^2-4\,x+3}} \,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}{\sqrt {\left (x - 3\right ) \left (x - 1\right )} \left (x - 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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