Optimal. Leaf size=8 \[ -\sin ^{-1}(2-x) \]
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Rubi [A]
time = 0.00, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {633, 222}
\begin {gather*} -\text {ArcSin}(2-x) \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 633
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-3+4 x-x^2}} \, dx &=-\left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{4}}} \, dx,x,4-2 x\right )\right )\\ &=-\sin ^{-1}(2-x)\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(23\) vs. \(2(8)=16\).
time = 0.06, size = 23, normalized size = 2.88 \begin {gather*} -2 \tan ^{-1}\left (\frac {\sqrt {-3+4 x-x^2}}{-1+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 5, normalized size = 0.62
method | result | size |
default | \(\arcsin \left (-2+x \right )\) | \(5\) |
trager | \(\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{2}+1\right ) x +\sqrt {-x^{2}+4 x -3}+2 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.37, size = 8, normalized size = 1.00 \begin {gather*} -\arcsin \left (-x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 29 vs.
\(2 (4) = 8\).
time = 0.39, size = 29, normalized size = 3.62 \begin {gather*} -\arctan \left (\frac {\sqrt {-x^{2} + 4 \, x - 3} {\left (x - 2\right )}}{x^{2} - 4 \, x + 3}\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 {- x^{2} + 4 x - 3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 24 vs.
\(2 (4) = 8\).
time = 0.46, size = 24, normalized size = 3.00 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} + 4 \, x - 3} {\left (x - 2\right )} + \frac {1}{2} \, \arcsin \left (x - 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 4, normalized size = 0.50 \begin {gather*} \mathrm {asin}\left (x-2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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