Integrand size = 23, antiderivative size = 12 \[ \int \frac {1}{\sqrt {1-x^2} \sqrt {2-x^2}} \, dx=\frac {\operatorname {EllipticF}\left (\arcsin (x),\frac {1}{2}\right )}{\sqrt {2}} \]
[Out]
Time = 0.01 (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} \[ \int \frac {1}{\sqrt {1-x^2} \sqrt {2-x^2}} \, dx=\frac {\operatorname {EllipticF}\left (\arcsin (x),\frac {1}{2}\right )}{\sqrt {2}} \]
[In]
[Out]
Rule 430
Rubi steps \begin{align*} \text {integral}& = \frac {F\left (\sin ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {1-x^2} \sqrt {2-x^2}} \, dx=\frac {\operatorname {EllipticF}\left (\arcsin (x),\frac {1}{2}\right )}{\sqrt {2}} \]
[In]
[Out]
Time = 2.49 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08
method | result | size |
default | \(\frac {F\left (x , \frac {\sqrt {2}}{2}\right ) \sqrt {2}}{2}\) | \(13\) |
elliptic | \(\frac {\sqrt {\left (x^{2}-1\right ) \left (x^{2}-2\right )}\, \sqrt {-2 x^{2}+4}\, F\left (x , \frac {\sqrt {2}}{2}\right )}{2 \sqrt {-x^{2}+2}\, \sqrt {x^{4}-3 x^{2}+2}}\) | \(53\) |
[In]
[Out]
none
Time = 0.07 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\sqrt {1-x^2} \sqrt {2-x^2}} \, dx=\frac {1}{2} \, \sqrt {2} F(\arcsin \left (x\right )\,|\,\frac {1}{2}) \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {1-x^2} \sqrt {2-x^2}} \, dx=\int \frac {1}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )} \sqrt {2 - x^{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {1-x^2} \sqrt {2-x^2}} \, dx=\int { \frac {1}{\sqrt {-x^{2} + 2} \sqrt {-x^{2} + 1}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {1-x^2} \sqrt {2-x^2}} \, dx=\int { \frac {1}{\sqrt {-x^{2} + 2} \sqrt {-x^{2} + 1}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\sqrt {1-x^2} \sqrt {2-x^2}} \, dx=\int \frac {1}{\sqrt {1-x^2}\,\sqrt {2-x^2}} \,d x \]
[In]
[Out]