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 = 0.0077534, antiderivative size = 28, normalized size of antiderivative = 1., 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} \[ \sqrt{x^2-4 x}+4 \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-4 x}}\right ) \]
Antiderivative was successfully verified.
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Rule 640
Rule 620
Rule 206
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*}
Mathematica [A] time = 0.0285069, size = 40, normalized size = 1.43 \[ \frac{(x-4) x-4 \sqrt{-(x-4) x} \sin ^{-1}\left (\sqrt{1-\frac{x}{4}}\right )}{\sqrt{(x-4) x}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.003, size = 26, normalized size = 0.9 \begin{align*} \sqrt{{x}^{2}-4\,x}+2\,\ln \left ( x-2+\sqrt{{x}^{2}-4\,x} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.929766, size = 39, normalized size = 1.39 \begin{align*} \sqrt{x^{2} - 4 \, x} + 2 \, \log \left (2 \, x + 2 \, \sqrt{x^{2} - 4 \, x} - 4\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96705, size = 69, normalized size = 2.46 \begin{align*} \sqrt{x^{2} - 4 \, x} - 2 \, \log \left (-x + \sqrt{x^{2} - 4 \, x} + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{x \left (x - 4\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06386, size = 38, normalized size = 1.36 \begin{align*} \sqrt{x^{2} - 4 \, x} - 2 \, \log \left ({\left | -x + \sqrt{x^{2} - 4 \, x} + 2 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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