Optimal. Leaf size=28 \[ \sqrt {x^2-4 x}+4 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2-4 x}}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {640, 620, 206} \begin {gather*} \sqrt {x^2-4 x}+4 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2-4 x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 640
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {-4 x+x^2}} \, dx &=\sqrt {-4 x+x^2}+2 \int \frac {1}{\sqrt {-4 x+x^2}} \, dx\\ &=\sqrt {-4 x+x^2}+4 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-4 x+x^2}}\right )\\ &=\sqrt {-4 x+x^2}+4 \tanh ^{-1}\left (\frac {x}{\sqrt {-4 x+x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 40, normalized size = 1.43 \begin {gather*} \frac {(x-4) x-4 \sqrt {-((x-4) x)} \sin ^{-1}\left (\sqrt {1-\frac {x}{4}}\right )}{\sqrt {(x-4) x}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.11, size = 32, normalized size = 1.14 \begin {gather*} \sqrt {x^2-4 x}+4 \tanh ^{-1}\left (\frac {\sqrt {x^2-4 x}}{x-4}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 27, normalized size = 0.96 \begin {gather*} \sqrt {x^{2} - 4 \, x} - 2 \, \log \left (-x + \sqrt {x^{2} - 4 \, x} + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 28, normalized size = 1.00 \begin {gather*} \sqrt {x^{2} - 4 \, x} - 2 \, \log \left ({\left | -x + \sqrt {x^{2} - 4 \, x} + 2 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 26, normalized size = 0.93 \begin {gather*} 2 \ln \left (x -2+\sqrt {x^{2}-4 x}\right )+\sqrt {x^{2}-4 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 29, normalized size = 1.04 \begin {gather*} \sqrt {x^{2} - 4 \, x} + 2 \, \log \left (2 \, x + 2 \, \sqrt {x^{2} - 4 \, x} - 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 23, normalized size = 0.82 \begin {gather*} 2\,\ln \left (x+\sqrt {x\,\left (x-4\right )}-2\right )+\sqrt {x^2-4\,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 {x}{\sqrt {x \left (x - 4\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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