Optimal. Leaf size=13 \[ \tan ^{-1}\left (\sqrt {3-4 x+x^2}\right ) \]
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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 = {702, 209}
\begin {gather*} \text {ArcTan}\left (\sqrt {x^2-4 x+3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 702
Rubi steps
\begin {align*} \int \frac {1}{(-2+x) \sqrt {3-4 x+x^2}} \, dx &=4 \text {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 = 0.08, size = 21, normalized size = 1.62 \begin {gather*} -2 \tan ^{-1}\left (\frac {\sqrt {3-4 x+x^2}}{-3+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.91, size = 13, normalized size = 1.00
method | result | size |
default | \(-\arctan \left (\frac {1}{\sqrt {\left (x -2\right )^{2}-1}}\right )\) | \(13\) |
trager | \(\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{2}-4 x +3}}{x -2}\right )\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, 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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Fricas [A]
time = 2.54, 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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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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Giac [A]
time = 1.37, 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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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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