Optimal. Leaf size=25 \[ \sqrt {-8+12 x-4 x^2}+\frac {1}{2} \sin ^{-1}(3-2 x) \]
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Rubi [A]
time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {654, 633, 222}
\begin {gather*} \frac {1}{2} \text {ArcSin}(3-2 x)+\sqrt {-4 x^2+12 x-8} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 633
Rule 654
Rubi steps
\begin {align*} \int \frac {5-4 x}{\sqrt {-8+12 x-4 x^2}} \, dx &=\sqrt {-8+12 x-4 x^2}-\int \frac {1}{\sqrt {-8+12 x-4 x^2}} \, dx\\ &=\sqrt {-8+12 x-4 x^2}+\frac {1}{8} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{16}}} \, dx,x,12-8 x\right )\\ &=\sqrt {-8+12 x-4 x^2}+\frac {1}{2} \sin ^{-1}(3-2 x)\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 38, normalized size = 1.52 \begin {gather*} 2 \sqrt {-2+3 x-x^2}+\tan ^{-1}\left (\frac {\sqrt {-2+3 x-x^2}}{-1+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.76, size = 24, normalized size = 0.96
method | result | size |
default | \(-\frac {\arcsin \left (-3+2 x \right )}{2}+2 \sqrt {-x^{2}+3 x -2}\) | \(24\) |
risch | \(-\frac {2 \left (x^{2}-3 x +2\right )}{\sqrt {-x^{2}+3 x -2}}-\frac {\arcsin \left (-3+2 x \right )}{2}\) | \(32\) |
trager | \(2 \sqrt {-x^{2}+3 x -2}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (2 x \RootOf \left (\textit {\_Z}^{2}+1\right )+2 \sqrt {-x^{2}+3 x -2}-3 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{2}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 23, normalized size = 0.92 \begin {gather*} 2 \, \sqrt {-x^{2} + 3 \, x - 2} - \frac {1}{2} \, \arcsin \left (2 \, x - 3\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. 47 vs.
\(2 (23) = 46\).
time = 3.06, size = 47, normalized size = 1.88 \begin {gather*} 2 \, \sqrt {-x^{2} + 3 \, x - 2} + \frac {1}{2} \, \arctan \left (\frac {\sqrt {-x^{2} + 3 \, x - 2} {\left (2 \, x - 3\right )}}{2 \, {\left (x^{2} - 3 \, x + 2\right )}}\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*} - \frac {\int \frac {4 x}{\sqrt {- x^{2} + 3 x - 2}}\, dx + \int \left (- \frac {5}{\sqrt {- x^{2} + 3 x - 2}}\right )\, dx}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.18, size = 23, normalized size = 0.92 \begin {gather*} 2 \, \sqrt {-x^{2} + 3 \, x - 2} - \frac {1}{2} \, \arcsin \left (2 \, x - 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.30, size = 46, normalized size = 1.84 \begin {gather*} \frac {5\,\mathrm {asin}\left (2\,x-3\right )}{2}+2\,\sqrt {-x^2+3\,x-2}+\ln \left (x\,1{}\mathrm {i}+\sqrt {-x^2+3\,x-2}-\frac {3}{2}{}\mathrm {i}\right )\,3{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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