Integrand size = 22, antiderivative size = 24 \[ \int \frac {x}{\left (3-x^2\right ) \sqrt {5-x^2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {5-x^2}}{\sqrt {2}}\right )}{\sqrt {2}} \]
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Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {455, 65, 213} \[ \int \frac {x}{\left (3-x^2\right ) \sqrt {5-x^2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {5-x^2}}{\sqrt {2}}\right )}{\sqrt {2}} \]
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Rule 65
Rule 213
Rule 455
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{(3-x) \sqrt {5-x}} \, dx,x,x^2\right ) \\ & = -\text {Subst}\left (\int \frac {1}{-2+x^2} \, dx,x,\sqrt {5-x^2}\right ) \\ & = \frac {\text {arctanh}\left (\frac {\sqrt {5-x^2}}{\sqrt {2}}\right )}{\sqrt {2}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\left (3-x^2\right ) \sqrt {5-x^2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {5-x^2}}{\sqrt {2}}\right )}{\sqrt {2}} \]
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Time = 0.34 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88
method | result | size |
pseudoelliptic | \(\frac {\operatorname {arctanh}\left (\frac {\sqrt {-x^{2}+5}\, \sqrt {2}}{2}\right ) \sqrt {2}}{2}\) | \(21\) |
trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}-7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )-4 \sqrt {-x^{2}+5}}{x^{2}-3}\right )}{4}\) | \(48\) |
default | \(\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (4+2 \sqrt {3}\, \left (x +\sqrt {3}\right )\right ) \sqrt {2}}{4 \sqrt {-\left (x +\sqrt {3}\right )^{2}+2 \sqrt {3}\, \left (x +\sqrt {3}\right )+2}}\right )}{4}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (4-2 \sqrt {3}\, \left (x -\sqrt {3}\right )\right ) \sqrt {2}}{4 \sqrt {-\left (x -\sqrt {3}\right )^{2}-2 \sqrt {3}\, \left (x -\sqrt {3}\right )+2}}\right )}{4}\) | \(100\) |
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Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).
Time = 0.24 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.00 \[ \int \frac {x}{\left (3-x^2\right ) \sqrt {5-x^2}} \, dx=\frac {1}{8} \, \sqrt {2} \log \left (\frac {x^{4} - 4 \, \sqrt {2} {\left (x^{2} - 7\right )} \sqrt {-x^{2} + 5} - 22 \, x^{2} + 89}{x^{4} - 6 \, x^{2} + 9}\right ) \]
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Time = 2.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54 \[ \int \frac {x}{\left (3-x^2\right ) \sqrt {5-x^2}} \, dx=- \frac {\sqrt {2} \left (\log {\left (\sqrt {5 - x^{2}} - \sqrt {2} \right )} - \log {\left (\sqrt {5 - x^{2}} + \sqrt {2} \right )}\right )}{4} \]
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Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (20) = 40\).
Time = 0.27 (sec) , antiderivative size = 112, normalized size of antiderivative = 4.67 \[ \int \frac {x}{\left (3-x^2\right ) \sqrt {5-x^2}} \, dx=\frac {1}{12} \, \sqrt {3} {\left (\sqrt {3} \sqrt {2} \log \left (\sqrt {3} + \frac {2 \, \sqrt {2} \sqrt {-x^{2} + 5}}{{\left | 2 \, x + 2 \, \sqrt {3} \right |}} + \frac {4}{{\left | 2 \, x + 2 \, \sqrt {3} \right |}}\right ) + \sqrt {3} \sqrt {2} \log \left (-\sqrt {3} + \frac {2 \, \sqrt {2} \sqrt {-x^{2} + 5}}{{\left | 2 \, x - 2 \, \sqrt {3} \right |}} + \frac {4}{{\left | 2 \, x - 2 \, \sqrt {3} \right |}}\right )\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (20) = 40\).
Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.75 \[ \int \frac {x}{\left (3-x^2\right ) \sqrt {5-x^2}} \, dx=\frac {1}{4} \, \sqrt {2} \log \left (\sqrt {2} + \sqrt {-x^{2} + 5}\right ) - \frac {1}{4} \, \sqrt {2} \log \left ({\left | -\sqrt {2} + \sqrt {-x^{2} + 5} \right |}\right ) \]
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Time = 0.83 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.25 \[ \int \frac {x}{\left (3-x^2\right ) \sqrt {5-x^2}} \, dx=\frac {\sqrt {2}\,\left (\ln \left (\frac {\frac {\sqrt {2}\,\left (\sqrt {3}\,x+5\right )\,1{}\mathrm {i}}{2}+\sqrt {5-x^2}\,1{}\mathrm {i}}{x+\sqrt {3}}\right )+\ln \left (\frac {\frac {\sqrt {2}\,\left (\sqrt {3}\,x-5\right )\,1{}\mathrm {i}}{2}-\sqrt {5-x^2}\,1{}\mathrm {i}}{x-\sqrt {3}}\right )\right )}{4} \]
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