Integrand size = 22, antiderivative size = 25 \[ \int \frac {x}{\sqrt {3-x^2} \left (5-x^2\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {3-x^2}}{\sqrt {2}}\right )}{\sqrt {2}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 25, 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, 209} \[ \int \frac {x}{\sqrt {3-x^2} \left (5-x^2\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {3-x^2}}{\sqrt {2}}\right )}{\sqrt {2}} \]
[In]
[Out]
Rule 65
Rule 209
Rule 455
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {3-x} (5-x)} \, dx,x,x^2\right ) \\ & = -\text {Subst}\left (\int \frac {1}{2+x^2} \, dx,x,\sqrt {3-x^2}\right ) \\ & = -\frac {\arctan \left (\frac {\sqrt {3-x^2}}{\sqrt {2}}\right )}{\sqrt {2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\sqrt {3-x^2} \left (5-x^2\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {3-x^2}}{\sqrt {2}}\right )}{\sqrt {2}} \]
[In]
[Out]
Time = 0.46 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84
method | result | size |
pseudoelliptic | \(-\frac {\arctan \left (\frac {\sqrt {-x^{2}+3}\, \sqrt {2}}{2}\right ) \sqrt {2}}{2}\) | \(21\) |
trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )-4 \sqrt {-x^{2}+3}}{x^{2}-5}\right )}{4}\) | \(48\) |
default | \(-\frac {\sqrt {2}\, \arctan \left (\frac {\left (-4+2 \sqrt {5}\, \left (x +\sqrt {5}\right )\right ) \sqrt {2}}{4 \sqrt {-\left (x +\sqrt {5}\right )^{2}+2 \sqrt {5}\, \left (x +\sqrt {5}\right )-2}}\right )}{4}-\frac {\sqrt {2}\, \arctan \left (\frac {\left (-4-2 \sqrt {5}\, \left (x -\sqrt {5}\right )\right ) \sqrt {2}}{4 \sqrt {-\left (x -\sqrt {5}\right )^{2}-2 \sqrt {5}\, \left (x -\sqrt {5}\right )-2}}\right )}{4}\) | \(100\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {x}{\sqrt {3-x^2} \left (5-x^2\right )} \, dx=-\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x^{2} - 1\right )} \sqrt {-x^{2} + 3}}{4 \, {\left (x^{2} - 3\right )}}\right ) \]
[In]
[Out]
Time = 2.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {x}{\sqrt {3-x^2} \left (5-x^2\right )} \, dx=- \frac {\sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {3 - x^{2}}}{2} \right )}}{2} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 101 vs. \(2 (20) = 40\).
Time = 0.29 (sec) , antiderivative size = 101, normalized size of antiderivative = 4.04 \[ \int \frac {x}{\sqrt {3-x^2} \left (5-x^2\right )} \, dx=-\frac {1}{20} \, \sqrt {5} {\left (\sqrt {5} \sqrt {2} \arcsin \left (\frac {2 \, \sqrt {5} \sqrt {3} x}{3 \, {\left | 2 \, x + 2 \, \sqrt {5} \right |}} + \frac {2 \, \sqrt {3}}{{\left | 2 \, x + 2 \, \sqrt {5} \right |}}\right ) - \sqrt {5} \sqrt {2} \arcsin \left (\frac {2 \, \sqrt {5} \sqrt {3} x}{3 \, {\left | 2 \, x - 2 \, \sqrt {5} \right |}} - \frac {2 \, \sqrt {3}}{{\left | 2 \, x - 2 \, \sqrt {5} \right |}}\right )\right )} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {x}{\sqrt {3-x^2} \left (5-x^2\right )} \, dx=-\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-x^{2} + 3}\right ) \]
[In]
[Out]
Time = 0.79 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.32 \[ \int \frac {x}{\sqrt {3-x^2} \left (5-x^2\right )} \, dx=-\frac {\sqrt {2}\,\ln \left (\frac {\frac {\sqrt {2}\,\left (\sqrt {5}\,x+3\right )}{2}+\sqrt {3-x^2}\,1{}\mathrm {i}}{x+\sqrt {5}}\right )\,1{}\mathrm {i}}{4}-\frac {\sqrt {2}\,\ln \left (\frac {\frac {\sqrt {2}\,\left (\sqrt {5}\,x-3\right )}{2}-\sqrt {3-x^2}\,1{}\mathrm {i}}{x-\sqrt {5}}\right )\,1{}\mathrm {i}}{4} \]
[In]
[Out]