Integrand size = 22, antiderivative size = 16 \[ \int \frac {x}{4-x^2+\sqrt {4-x^2}} \, dx=-\log \left (1+\sqrt {4-x^2}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2186, 31} \[ \int \frac {x}{4-x^2+\sqrt {4-x^2}} \, dx=-\log \left (\sqrt {4-x^2}+1\right ) \]
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Rule 31
Rule 2186
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{4+\sqrt {4-x}-x} \, dx,x,x^2\right ) \\ & = -\text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt {4-x^2}\right ) \\ & = -\log \left (1+\sqrt {4-x^2}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {x}{4-x^2+\sqrt {4-x^2}} \, dx=-\log \left (1+\sqrt {4-x^2}\right ) \]
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Time = 0.12 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06
method | result | size |
trager | \(-\ln \left (-1-\sqrt {-x^{2}+4}\right )\) | \(17\) |
default | \(-\frac {\ln \left (x^{2}-3\right )}{2}+\frac {\sqrt {-\left (-2+x \right )^{2}-4 x +8}-2 \arcsin \left (\frac {x}{2}\right )}{2 \left (2+\sqrt {3}\right ) \left (-2+\sqrt {3}\right )}+\frac {\sqrt {-\left (2+x \right )^{2}+4 x +8}+2 \arcsin \left (\frac {x}{2}\right )}{2 \left (2+\sqrt {3}\right ) \left (-2+\sqrt {3}\right )}-\frac {\sqrt {-\left (x -\sqrt {3}\right )^{2}-2 \sqrt {3}\, \left (x -\sqrt {3}\right )+1}-\sqrt {3}\, \arcsin \left (\frac {x}{2}\right )-\operatorname {arctanh}\left (\frac {2-2 \sqrt {3}\, \left (x -\sqrt {3}\right )}{2 \sqrt {-\left (x -\sqrt {3}\right )^{2}-2 \sqrt {3}\, \left (x -\sqrt {3}\right )+1}}\right )}{2 \left (2+\sqrt {3}\right ) \left (-2+\sqrt {3}\right )}-\frac {\sqrt {-\left (x +\sqrt {3}\right )^{2}+2 \sqrt {3}\, \left (x +\sqrt {3}\right )+1}+\sqrt {3}\, \arcsin \left (\frac {x}{2}\right )-\operatorname {arctanh}\left (\frac {2+2 \sqrt {3}\, \left (x +\sqrt {3}\right )}{2 \sqrt {-\left (x +\sqrt {3}\right )^{2}+2 \sqrt {3}\, \left (x +\sqrt {3}\right )+1}}\right )}{2 \left (2+\sqrt {3}\right ) \left (-2+\sqrt {3}\right )}\) | \(271\) |
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Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (14) = 28\).
Time = 0.23 (sec) , antiderivative size = 55, normalized size of antiderivative = 3.44 \[ \int \frac {x}{4-x^2+\sqrt {4-x^2}} \, dx=-\frac {1}{2} \, \log \left (x^{2} - 3\right ) + \frac {1}{2} \, \log \left (-\frac {x^{2} + 3 \, \sqrt {-x^{2} + 4} - 6}{x^{2}}\right ) - \frac {1}{2} \, \log \left (-\frac {x^{2} + \sqrt {-x^{2} + 4} - 2}{x^{2}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (12) = 24\).
Time = 1.34 (sec) , antiderivative size = 48, normalized size of antiderivative = 3.00 \[ \int \frac {x}{4-x^2+\sqrt {4-x^2}} \, dx=\frac {\log {\left (2 \sqrt {4 - x^{2}} \right )}}{2} - \frac {\log {\left (2 \sqrt {4 - x^{2}} + 2 \right )}}{2} - \frac {\log {\left (2 x^{2} - 2 \sqrt {4 - x^{2}} - 8 \right )}}{2} \]
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none
Time = 0.18 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {x}{4-x^2+\sqrt {4-x^2}} \, dx=-\log \left (\sqrt {-x^{2} + 4} + 1\right ) \]
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Time = 0.33 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {x}{4-x^2+\sqrt {4-x^2}} \, dx=-\log \left (\sqrt {-x^{2} + 4} + 1\right ) \]
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Time = 0.16 (sec) , antiderivative size = 87, normalized size of antiderivative = 5.44 \[ \int \frac {x}{4-x^2+\sqrt {4-x^2}} \, dx=-\frac {\ln \left (x-\sqrt {3}\right )}{2}-\frac {\ln \left (\frac {\sqrt {3}\,x\,1{}\mathrm {i}+\sqrt {4-x^2}\,1{}\mathrm {i}+4{}\mathrm {i}}{x+\sqrt {3}}\right )}{2}-\frac {\ln \left (x+\sqrt {3}\right )}{2}-\frac {\ln \left (\frac {-\sqrt {3}\,x\,1{}\mathrm {i}+\sqrt {4-x^2}\,1{}\mathrm {i}+4{}\mathrm {i}}{x-\sqrt {3}}\right )}{2} \]
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