Integrand size = 21, antiderivative size = 30 \[ \int \frac {1}{\sqrt {-1+x^2} \sqrt {2+4 x^2}} \, dx=\frac {\sqrt {1-x^2} \operatorname {EllipticF}(\arcsin (x),-2)}{\sqrt {2} \sqrt {-1+x^2}} \]
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Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {432, 430} \[ \int \frac {1}{\sqrt {-1+x^2} \sqrt {2+4 x^2}} \, dx=\frac {\sqrt {1-x^2} \operatorname {EllipticF}(\arcsin (x),-2)}{\sqrt {2} \sqrt {x^2-1}} \]
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Rule 430
Rule 432
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-x^2} \int \frac {1}{\sqrt {1-x^2} \sqrt {2+4 x^2}} \, dx}{\sqrt {-1+x^2}} \\ & = \frac {\sqrt {1-x^2} F\left (\left .\sin ^{-1}(x)\right |-2\right )}{\sqrt {2} \sqrt {-1+x^2}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {-1+x^2} \sqrt {2+4 x^2}} \, dx=\frac {\sqrt {1-x^2} \operatorname {EllipticF}(\arcsin (x),-2)}{\sqrt {2} \sqrt {-1+x^2}} \]
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Time = 2.51 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13
method | result | size |
default | \(-\frac {i F\left (i x \sqrt {2}, \frac {i \sqrt {2}}{2}\right ) \sqrt {-x^{2}+1}}{2 \sqrt {x^{2}-1}}\) | \(34\) |
elliptic | \(-\frac {i \sqrt {\left (x^{2}-1\right ) \left (2 x^{2}+1\right )}\, \sqrt {2}\, \sqrt {-x^{2}+1}\, F\left (i x \sqrt {2}, \frac {i \sqrt {2}}{2}\right )}{2 \sqrt {x^{2}-1}\, \sqrt {4 x^{4}-2 x^{2}-2}}\) | \(66\) |
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none
Time = 0.08 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.30 \[ \int \frac {1}{\sqrt {-1+x^2} \sqrt {2+4 x^2}} \, dx=-\frac {1}{2} \, \sqrt {-2} F(\arcsin \left (x\right )\,|\,-2) \]
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\[ \int \frac {1}{\sqrt {-1+x^2} \sqrt {2+4 x^2}} \, dx=\frac {\sqrt {2} \int \frac {1}{\sqrt {x^{2} - 1} \sqrt {2 x^{2} + 1}}\, dx}{2} \]
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\[ \int \frac {1}{\sqrt {-1+x^2} \sqrt {2+4 x^2}} \, dx=\int { \frac {1}{\sqrt {4 \, x^{2} + 2} \sqrt {x^{2} - 1}} \,d x } \]
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\[ \int \frac {1}{\sqrt {-1+x^2} \sqrt {2+4 x^2}} \, dx=\int { \frac {1}{\sqrt {4 \, x^{2} + 2} \sqrt {x^{2} - 1}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {-1+x^2} \sqrt {2+4 x^2}} \, dx=\int \frac {1}{\sqrt {x^2-1}\,\sqrt {4\,x^2+2}} \,d x \]
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