Integrand size = 23, antiderivative size = 42 \[ \int \frac {1}{\sqrt {2-2 x^2} \sqrt {-1-x^2}} \, dx=-\frac {\sqrt {1-\frac {1}{x^4}} x^2 \operatorname {EllipticF}\left (\csc ^{-1}(x),-1\right )}{\sqrt {2-2 x^2} \sqrt {-1-x^2}} \]
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Time = 0.01 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.55, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {259, 228} \[ \int \frac {1}{\sqrt {2-2 x^2} \sqrt {-1-x^2}} \, dx=\frac {\sqrt {x^2-1} \sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right ),\frac {1}{2}\right )}{2 \sqrt {-x^2-1} \sqrt {1-x^2}} \]
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Rule 228
Rule 259
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-2+2 x^4} \int \frac {1}{\sqrt {-2+2 x^4}} \, dx}{\sqrt {2-2 x^2} \sqrt {-1-x^2}} \\ & = \frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt {-1-x^2} \sqrt {1-x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.06 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\sqrt {2-2 x^2} \sqrt {-1-x^2}} \, dx=\frac {x \sqrt {1-x^4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},x^4\right )}{\sqrt {2-2 x^2} \sqrt {-1-x^2}} \]
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Time = 2.43 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.71
method | result | size |
default | \(\frac {i F\left (i x , i\right ) \sqrt {2}\, \sqrt {-x^{2}-1}}{2 \sqrt {x^{2}+1}}\) | \(30\) |
elliptic | \(-\frac {i \sqrt {x^{4}-1}\, \sqrt {x^{2}+1}\, F\left (i x , i\right )}{\sqrt {-x^{2}-1}\, \sqrt {2 x^{4}-2}}\) | \(43\) |
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none
Time = 0.07 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.21 \[ \int \frac {1}{\sqrt {2-2 x^2} \sqrt {-1-x^2}} \, dx=-\frac {1}{2} \, \sqrt {-2} F(\arcsin \left (x\right )\,|\,-1) \]
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Time = 11.06 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.74 \[ \int \frac {1}{\sqrt {2-2 x^2} \sqrt {-1-x^2}} \, dx=\frac {\sqrt {2} {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{2}, 1, 1 & \frac {3}{4}, \frac {3}{4}, \frac {5}{4} \\\frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4} & 0 \end {matrix} \middle | {\frac {1}{x^{4}}} \right )}}{16 \pi ^{\frac {3}{2}}} - \frac {\sqrt {2} {G_{6, 6}^{3, 5}\left (\begin {matrix} - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4} & 1 \\0, \frac {1}{2}, 0 & - \frac {1}{4}, \frac {1}{4}, \frac {1}{4} \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{x^{4}}} \right )}}{16 \pi ^{\frac {3}{2}}} \]
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\[ \int \frac {1}{\sqrt {2-2 x^2} \sqrt {-1-x^2}} \, dx=\int { \frac {1}{\sqrt {-x^{2} - 1} \sqrt {-2 \, x^{2} + 2}} \,d x } \]
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\[ \int \frac {1}{\sqrt {2-2 x^2} \sqrt {-1-x^2}} \, dx=\int { \frac {1}{\sqrt {-x^{2} - 1} \sqrt {-2 \, x^{2} + 2}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {2-2 x^2} \sqrt {-1-x^2}} \, dx=\int \frac {1}{\sqrt {-x^2-1}\,\sqrt {2-2\,x^2}} \,d x \]
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