Integrand size = 23, antiderivative size = 40 \[ \int \frac {1}{\sqrt {2-3 x^2} \sqrt {-1-x^2}} \, dx=\frac {\sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {-1-x^2}} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {432, 430} \[ \int \frac {1}{\sqrt {2-3 x^2} \sqrt {-1-x^2}} \, dx=\frac {\sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {-x^2-1}} \]
[In]
[Out]
Rule 430
Rule 432
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+x^2} \int \frac {1}{\sqrt {2-3 x^2} \sqrt {1+x^2}} \, dx}{\sqrt {-1-x^2}} \\ & = \frac {\sqrt {1+x^2} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {-1-x^2}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {2-3 x^2} \sqrt {-1-x^2}} \, dx=\frac {\sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {-1-x^2}} \]
[In]
[Out]
Time = 2.49 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.85
method | result | size |
default | \(\frac {i F\left (i x , \frac {i \sqrt {6}}{2}\right ) \sqrt {-x^{2}-1}\, \sqrt {2}}{2 \sqrt {x^{2}+1}}\) | \(34\) |
elliptic | \(-\frac {i \sqrt {\left (3 x^{2}-2\right ) \left (x^{2}+1\right )}\, \sqrt {x^{2}+1}\, \sqrt {-6 x^{2}+4}\, F\left (i x , \frac {i \sqrt {6}}{2}\right )}{2 \sqrt {-3 x^{2}+2}\, \sqrt {-x^{2}-1}\, \sqrt {3 x^{4}+x^{2}-2}}\) | \(76\) |
[In]
[Out]
none
Time = 0.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.58 \[ \int \frac {1}{\sqrt {2-3 x^2} \sqrt {-1-x^2}} \, dx=-\frac {1}{6} \, \sqrt {3} \sqrt {2} \sqrt {-2} F(\arcsin \left (\frac {1}{2} \, \sqrt {3} \sqrt {2} x\right )\,|\,-\frac {2}{3}) \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {2-3 x^2} \sqrt {-1-x^2}} \, dx=\int \frac {1}{\sqrt {2 - 3 x^{2}} \sqrt {- x^{2} - 1}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {2-3 x^2} \sqrt {-1-x^2}} \, dx=\int { \frac {1}{\sqrt {-x^{2} - 1} \sqrt {-3 \, x^{2} + 2}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {2-3 x^2} \sqrt {-1-x^2}} \, dx=\int { \frac {1}{\sqrt {-x^{2} - 1} \sqrt {-3 \, x^{2} + 2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\sqrt {2-3 x^2} \sqrt {-1-x^2}} \, dx=\int \frac {1}{\sqrt {-x^2-1}\,\sqrt {2-3\,x^2}} \,d x \]
[In]
[Out]