Integrand size = 15, antiderivative size = 25 \[ \int \frac {x^4}{\left (1-x^4\right )^{3/2}} \, dx=\frac {x}{2 \sqrt {1-x^4}}-\frac {1}{2} \operatorname {EllipticF}(\arcsin (x),-1) \]
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Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {294, 227} \[ \int \frac {x^4}{\left (1-x^4\right )^{3/2}} \, dx=\frac {x}{2 \sqrt {1-x^4}}-\frac {1}{2} \operatorname {EllipticF}(\arcsin (x),-1) \]
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Rule 227
Rule 294
Rubi steps \begin{align*} \text {integral}& = \frac {x}{2 \sqrt {1-x^4}}-\frac {1}{2} \int \frac {1}{\sqrt {1-x^4}} \, dx \\ & = \frac {x}{2 \sqrt {1-x^4}}-\frac {1}{2} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 4.76 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {x^4}{\left (1-x^4\right )^{3/2}} \, dx=\frac {1}{2} x \left (\frac {1}{\sqrt {1-x^4}}-\operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},x^4\right )\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 4.42 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.60
method | result | size |
meijerg | \(\frac {x^{5} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {5}{4},\frac {3}{2};\frac {9}{4};x^{4}\right )}{5}\) | \(15\) |
default | \(\frac {x}{2 \sqrt {-x^{4}+1}}-\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, F\left (x , i\right )}{2 \sqrt {-x^{4}+1}}\) | \(45\) |
risch | \(\frac {x}{2 \sqrt {-x^{4}+1}}-\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, F\left (x , i\right )}{2 \sqrt {-x^{4}+1}}\) | \(45\) |
elliptic | \(\frac {x}{2 \sqrt {-x^{4}+1}}-\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, F\left (x , i\right )}{2 \sqrt {-x^{4}+1}}\) | \(45\) |
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none
Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {x^4}{\left (1-x^4\right )^{3/2}} \, dx=-\frac {{\left (x^{4} - 1\right )} F(\arcsin \left (x\right )\,|\,-1) + \sqrt {-x^{4} + 1} x}{2 \, {\left (x^{4} - 1\right )}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (15) = 30\).
Time = 0.40 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {x^4}{\left (1-x^4\right )^{3/2}} \, dx=\frac {x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {3}{2} \\ \frac {9}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} \]
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\[ \int \frac {x^4}{\left (1-x^4\right )^{3/2}} \, dx=\int { \frac {x^{4}}{{\left (-x^{4} + 1\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^4}{\left (1-x^4\right )^{3/2}} \, dx=\int { \frac {x^{4}}{{\left (-x^{4} + 1\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^4}{\left (1-x^4\right )^{3/2}} \, dx=\int \frac {x^4}{{\left (1-x^4\right )}^{3/2}} \,d x \]
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