Integrand size = 23, antiderivative size = 6 \[ \int \frac {1}{\sqrt {1-x^2} \sqrt {-1+2 x^2}} \, dx=-\operatorname {EllipticF}(\arccos (x),2) \]
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Time = 0.01 (sec) , antiderivative size = 6, 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 = {431} \[ \int \frac {1}{\sqrt {1-x^2} \sqrt {-1+2 x^2}} \, dx=-\operatorname {EllipticF}(\arccos (x),2) \]
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Rule 431
Rubi steps \begin{align*} \text {integral}& = -F\left (\left .\cos ^{-1}(x)\right |2\right ) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(27\) vs. \(2(6)=12\).
Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 4.50 \[ \int \frac {1}{\sqrt {1-x^2} \sqrt {-1+2 x^2}} \, dx=\frac {\sqrt {1-2 x^2} \operatorname {EllipticF}(\arcsin (x),2)}{\sqrt {-1+2 x^2}} \]
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Time = 2.48 (sec) , antiderivative size = 25, normalized size of antiderivative = 4.17
method | result | size |
default | \(\frac {F\left (x , \sqrt {2}\right ) \sqrt {-2 x^{2}+1}}{\sqrt {2 x^{2}-1}}\) | \(25\) |
elliptic | \(\frac {\sqrt {-\left (2 x^{2}-1\right ) \left (x^{2}-1\right )}\, \sqrt {-2 x^{2}+1}\, F\left (x , \sqrt {2}\right )}{\sqrt {2 x^{2}-1}\, \sqrt {-2 x^{4}+3 x^{2}-1}}\) | \(55\) |
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Result contains complex when optimal does not.
Time = 0.07 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {1-x^2} \sqrt {-1+2 x^2}} \, dx=-i \, F(\arcsin \left (x\right )\,|\,2) \]
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\[ \int \frac {1}{\sqrt {1-x^2} \sqrt {-1+2 x^2}} \, dx=\int \frac {1}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )} \sqrt {2 x^{2} - 1}}\, dx \]
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\[ \int \frac {1}{\sqrt {1-x^2} \sqrt {-1+2 x^2}} \, dx=\int { \frac {1}{\sqrt {2 \, x^{2} - 1} \sqrt {-x^{2} + 1}} \,d x } \]
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\[ \int \frac {1}{\sqrt {1-x^2} \sqrt {-1+2 x^2}} \, dx=\int { \frac {1}{\sqrt {2 \, x^{2} - 1} \sqrt {-x^{2} + 1}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {1-x^2} \sqrt {-1+2 x^2}} \, dx=\int \frac {1}{\sqrt {1-x^2}\,\sqrt {2\,x^2-1}} \,d x \]
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