Integrand size = 19, antiderivative size = 31 \[ \int \frac {1}{\sqrt {1-x^2} \left (4+x^2\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {5} x}{2 \sqrt {1-x^2}}\right )}{2 \sqrt {5}} \]
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Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {385, 209} \[ \int \frac {1}{\sqrt {1-x^2} \left (4+x^2\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {5} x}{2 \sqrt {1-x^2}}\right )}{2 \sqrt {5}} \]
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Rule 209
Rule 385
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{4+5 x^2} \, dx,x,\frac {x}{\sqrt {1-x^2}}\right ) \\ & = \frac {\arctan \left (\frac {\sqrt {5} x}{2 \sqrt {1-x^2}}\right )}{2 \sqrt {5}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\sqrt {1-x^2} \left (4+x^2\right )} \, dx=-\frac {\arctan \left (\frac {x \sqrt {5-5 x^2}}{2 \left (-1+x^2\right )}\right )}{2 \sqrt {5}} \]
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Time = 0.58 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77
method | result | size |
pseudoelliptic | \(-\frac {\sqrt {5}\, \arctan \left (\frac {2 \sqrt {5}\, \sqrt {-x^{2}+1}}{5 x}\right )}{10}\) | \(24\) |
default | \(-\frac {\sqrt {5}\, \arctan \left (\frac {\sqrt {-x^{2}+1}\, \sqrt {5}\, x}{2 x^{2}-2}\right )}{10}\) | \(29\) |
trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+5\right ) \ln \left (\frac {-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}+5\right ) x^{2}+20 x \sqrt {-x^{2}+1}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+5\right )}{x^{2}+4}\right )}{20}\) | \(50\) |
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none
Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\sqrt {1-x^2} \left (4+x^2\right )} \, dx=-\frac {1}{10} \, \sqrt {5} \arctan \left (\frac {2 \, \sqrt {5} \sqrt {-x^{2} + 1}}{5 \, x}\right ) \]
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\[ \int \frac {1}{\sqrt {1-x^2} \left (4+x^2\right )} \, dx=\int \frac {1}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 4\right )}\, dx \]
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\[ \int \frac {1}{\sqrt {1-x^2} \left (4+x^2\right )} \, dx=\int { \frac {1}{{\left (x^{2} + 4\right )} \sqrt {-x^{2} + 1}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (21) = 42\).
Time = 0.29 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65 \[ \int \frac {1}{\sqrt {1-x^2} \left (4+x^2\right )} \, dx=\frac {1}{20} \, \sqrt {5} {\left (\pi \mathrm {sgn}\left (x\right ) + 2 \, \arctan \left (-\frac {\sqrt {5} x {\left (\frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{5 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}\right )\right )} \]
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Time = 0.58 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.55 \[ \int \frac {1}{\sqrt {1-x^2} \left (4+x^2\right )} \, dx=\frac {\sqrt {5}\,\ln \left (\frac {\frac {\sqrt {5}\,\left (-1+x\,2{}\mathrm {i}\right )\,1{}\mathrm {i}}{5}-\sqrt {1-x^2}\,1{}\mathrm {i}}{x-2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{20}-\frac {\sqrt {5}\,\ln \left (\frac {\frac {\sqrt {5}\,\left (1+x\,2{}\mathrm {i}\right )\,1{}\mathrm {i}}{5}+\sqrt {1-x^2}\,1{}\mathrm {i}}{x+2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{20} \]
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