Integrand size = 13, antiderivative size = 28 \[ \int \frac {x}{\sqrt {-4 x+x^2}} \, dx=\sqrt {-4 x+x^2}+4 \text {arctanh}\left (\frac {x}{\sqrt {-4 x+x^2}}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {654, 634, 212} \[ \int \frac {x}{\sqrt {-4 x+x^2}} \, dx=4 \text {arctanh}\left (\frac {x}{\sqrt {x^2-4 x}}\right )+\sqrt {x^2-4 x} \]
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Rule 212
Rule 634
Rule 654
Rubi steps \begin{align*} \text {integral}& = \sqrt {-4 x+x^2}+2 \int \frac {1}{\sqrt {-4 x+x^2}} \, dx \\ & = \sqrt {-4 x+x^2}+4 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-4 x+x^2}}\right ) \\ & = \sqrt {-4 x+x^2}+4 \text {arctanh}\left (\frac {x}{\sqrt {-4 x+x^2}}\right ) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int \frac {x}{\sqrt {-4 x+x^2}} \, dx=\frac {(-4+x) x-4 \sqrt {-4+x} \sqrt {x} \log \left (\sqrt {-4+x}-\sqrt {x}\right )}{\sqrt {(-4+x) x}} \]
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Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93
method | result | size |
default | \(\sqrt {x^{2}-4 x}+2 \ln \left (-2+x +\sqrt {x^{2}-4 x}\right )\) | \(26\) |
trager | \(\sqrt {x^{2}-4 x}-2 \ln \left (2-x +\sqrt {x^{2}-4 x}\right )\) | \(28\) |
risch | \(\frac {x \left (x -4\right )}{\sqrt {x \left (x -4\right )}}+2 \ln \left (-2+x +\sqrt {x^{2}-4 x}\right )\) | \(29\) |
pseudoelliptic | \(\sqrt {x \left (x -4\right )}+2 \ln \left (\frac {\sqrt {x \left (x -4\right )}+x}{x}\right )-2 \ln \left (\frac {\sqrt {x \left (x -4\right )}-x}{x}\right )\) | \(43\) |
meijerg | \(\frac {4 i \sqrt {-\operatorname {signum}\left (x -4\right )}\, \left (\frac {i \sqrt {\pi }\, \sqrt {x}\, \sqrt {-\frac {x}{4}+1}}{2}-i \sqrt {\pi }\, \arcsin \left (\frac {\sqrt {x}}{2}\right )\right )}{\sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x -4\right )}}\) | \(50\) |
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Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {x}{\sqrt {-4 x+x^2}} \, dx=\sqrt {x^{2} - 4 \, x} - 2 \, \log \left (-x + \sqrt {x^{2} - 4 \, x} + 2\right ) \]
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Time = 0.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {x}{\sqrt {-4 x+x^2}} \, dx=\sqrt {x^{2} - 4 x} + 2 \log {\left (2 x + 2 \sqrt {x^{2} - 4 x} - 4 \right )} \]
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Time = 0.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {x}{\sqrt {-4 x+x^2}} \, dx=\sqrt {x^{2} - 4 \, x} + 2 \, \log \left (2 \, x + 2 \, \sqrt {x^{2} - 4 \, x} - 4\right ) \]
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Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\sqrt {-4 x+x^2}} \, dx=\sqrt {x^{2} - 4 \, x} - 2 \, \log \left ({\left | -x + \sqrt {x^{2} - 4 \, x} + 2 \right |}\right ) \]
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Time = 0.15 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {x}{\sqrt {-4 x+x^2}} \, dx=2\,\ln \left (x+\sqrt {x\,\left (x-4\right )}-2\right )+\sqrt {x^2-4\,x} \]
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