Integrand size = 23, antiderivative size = 12 \[ \int \frac {1}{\sqrt {2-3 x^2} \sqrt {1-x^2}} \, dx=\frac {\operatorname {EllipticF}\left (\arcsin (x),\frac {3}{2}\right )}{\sqrt {2}} \]
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Time = 0.00 (sec) , antiderivative size = 12, 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} \[ \int \frac {1}{\sqrt {2-3 x^2} \sqrt {1-x^2}} \, dx=\frac {\operatorname {EllipticF}\left (\arcsin (x),\frac {3}{2}\right )}{\sqrt {2}} \]
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Rule 430
Rubi steps \begin{align*} \text {integral}& = \frac {F\left (\sin ^{-1}(x)|\frac {3}{2}\right )}{\sqrt {2}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {2-3 x^2} \sqrt {1-x^2}} \, dx=\frac {\operatorname {EllipticF}\left (\arcsin (x),\frac {3}{2}\right )}{\sqrt {2}} \]
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Time = 2.53 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08
method | result | size |
default | \(\frac {F\left (x , \frac {\sqrt {6}}{2}\right ) \sqrt {2}}{2}\) | \(13\) |
elliptic | \(\frac {\sqrt {\left (3 x^{2}-2\right ) \left (x^{2}-1\right )}\, \sqrt {-6 x^{2}+4}\, F\left (x , \frac {\sqrt {6}}{2}\right )}{2 \sqrt {-3 x^{2}+2}\, \sqrt {3 x^{4}-5 x^{2}+2}}\) | \(57\) |
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none
Time = 0.08 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\sqrt {2-3 x^2} \sqrt {1-x^2}} \, dx=\frac {1}{2} \, \sqrt {2} F(\arcsin \left (x\right )\,|\,\frac {3}{2}) \]
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Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (14) = 28\).
Time = 1.43 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.83 \[ \int \frac {1}{\sqrt {2-3 x^2} \sqrt {1-x^2}} \, dx=\begin {cases} \frac {\sqrt {3} F\left (\operatorname {asin}{\left (\frac {\sqrt {6} x}{2} \right )}\middle | \frac {2}{3}\right )}{3} & \text {for}\: x > - \frac {\sqrt {6}}{3} \wedge x < \frac {\sqrt {6}}{3} \end {cases} \]
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\[ \int \frac {1}{\sqrt {2-3 x^2} \sqrt {1-x^2}} \, dx=\int { \frac {1}{\sqrt {-x^{2} + 1} \sqrt {-3 \, x^{2} + 2}} \,d x } \]
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\[ \int \frac {1}{\sqrt {2-3 x^2} \sqrt {1-x^2}} \, dx=\int { \frac {1}{\sqrt {-x^{2} + 1} \sqrt {-3 \, x^{2} + 2}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {2-3 x^2} \sqrt {1-x^2}} \, dx=\int \frac {1}{\sqrt {1-x^2}\,\sqrt {2-3\,x^2}} \,d x \]
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