Integrand size = 23, antiderivative size = 10 \[ \int \frac {1}{\sqrt {2-4 x^2} \sqrt {1-x^2}} \, dx=\frac {\operatorname {EllipticF}(\arcsin (x),2)}{\sqrt {2}} \]
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Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {430} \[ \int \frac {1}{\sqrt {2-4 x^2} \sqrt {1-x^2}} \, dx=\frac {\operatorname {EllipticF}(\arcsin (x),2)}{\sqrt {2}} \]
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Rule 430
Rubi steps \begin{align*} \text {integral}& = \frac {F\left (\left .\sin ^{-1}(x)\right |2\right )}{\sqrt {2}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {2-4 x^2} \sqrt {1-x^2}} \, dx=\frac {\operatorname {EllipticF}(\arcsin (x),2)}{\sqrt {2}} \]
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Time = 2.62 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10
method | result | size |
default | \(\frac {F\left (x , \sqrt {2}\right ) \sqrt {2}}{2}\) | \(11\) |
elliptic | \(\frac {\sqrt {\left (2 x^{2}-1\right ) \left (x^{2}-1\right )}\, F\left (x , \sqrt {2}\right )}{\sqrt {4 x^{4}-6 x^{2}+2}}\) | \(36\) |
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none
Time = 0.08 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\sqrt {2-4 x^2} \sqrt {1-x^2}} \, dx=\frac {1}{2} \, \sqrt {2} F(\arcsin \left (x\right )\,|\,2) \]
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Time = 1.88 (sec) , antiderivative size = 39, normalized size of antiderivative = 3.90 \[ \int \frac {1}{\sqrt {2-4 x^2} \sqrt {1-x^2}} \, dx=\frac {\sqrt {2} \left (\begin {cases} \frac {\sqrt {2} F\left (\operatorname {asin}{\left (\sqrt {2} x \right )}\middle | \frac {1}{2}\right )}{2} & \text {for}\: x > - \frac {\sqrt {2}}{2} \wedge x < \frac {\sqrt {2}}{2} \end {cases}\right )}{2} \]
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\[ \int \frac {1}{\sqrt {2-4 x^2} \sqrt {1-x^2}} \, dx=\int { \frac {1}{\sqrt {-x^{2} + 1} \sqrt {-4 \, x^{2} + 2}} \,d x } \]
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\[ \int \frac {1}{\sqrt {2-4 x^2} \sqrt {1-x^2}} \, dx=\int { \frac {1}{\sqrt {-x^{2} + 1} \sqrt {-4 \, x^{2} + 2}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {2-4 x^2} \sqrt {1-x^2}} \, dx=\int \frac {1}{\sqrt {1-x^2}\,\sqrt {2-4\,x^2}} \,d x \]
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