Integrand size = 11, antiderivative size = 12 \[ \int \frac {1}{\sqrt {16-x^4}} \, dx=\frac {1}{2} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{2}\right ),-1\right ) \]
[Out]
Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {227} \[ \int \frac {1}{\sqrt {16-x^4}} \, dx=\frac {1}{2} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{2}\right ),-1\right ) \]
[In]
[Out]
Rule 227
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} F\left (\left .\sin ^{-1}\left (\frac {x}{2}\right )\right |-1\right ) \\ \end{align*}
Time = 10.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {16-x^4}} \, dx=\frac {1}{2} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{2}\right ),-1\right ) \]
[In]
[Out]
Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 4.18 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.25
method | result | size |
meijerg | \(\frac {x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\frac {x^{4}}{16}\right )}{4}\) | \(15\) |
default | \(\frac {\sqrt {-x^{2}+4}\, \sqrt {x^{2}+4}\, F\left (\frac {x}{2}, i\right )}{2 \sqrt {-x^{4}+16}}\) | \(34\) |
elliptic | \(\frac {\sqrt {-x^{2}+4}\, \sqrt {x^{2}+4}\, F\left (\frac {x}{2}, i\right )}{2 \sqrt {-x^{4}+16}}\) | \(34\) |
[In]
[Out]
none
Time = 0.08 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\sqrt {16-x^4}} \, dx=\frac {1}{2} \, F(\arcsin \left (\frac {1}{2} \, x\right )\,|\,-1) \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (5) = 10\).
Time = 0.35 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.58 \[ \int \frac {1}{\sqrt {16-x^4}} \, dx=\frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {x^{4} e^{2 i \pi }}{16}} \right )}}{16 \Gamma \left (\frac {5}{4}\right )} \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {16-x^4}} \, dx=\int { \frac {1}{\sqrt {-x^{4} + 16}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {16-x^4}} \, dx=\int { \frac {1}{\sqrt {-x^{4} + 16}} \,d x } \]
[In]
[Out]
Time = 5.37 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\sqrt {16-x^4}} \, dx=\frac {x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{2};\ \frac {5}{4};\ \frac {x^4}{16}\right )}{4} \]
[In]
[Out]