Integrand size = 15, antiderivative size = 45 \[ \int \frac {1}{x^4 \left (1-x^4\right )^{3/2}} \, dx=\frac {1}{2 x^3 \sqrt {1-x^4}}-\frac {5 \sqrt {1-x^4}}{6 x^3}+\frac {5}{6} \operatorname {EllipticF}(\arcsin (x),-1) \]
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Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {296, 331, 227} \[ \int \frac {1}{x^4 \left (1-x^4\right )^{3/2}} \, dx=\frac {5}{6} \operatorname {EllipticF}(\arcsin (x),-1)-\frac {5 \sqrt {1-x^4}}{6 x^3}+\frac {1}{2 x^3 \sqrt {1-x^4}} \]
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Rule 227
Rule 296
Rule 331
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2 x^3 \sqrt {1-x^4}}+\frac {5}{2} \int \frac {1}{x^4 \sqrt {1-x^4}} \, dx \\ & = \frac {1}{2 x^3 \sqrt {1-x^4}}-\frac {5 \sqrt {1-x^4}}{6 x^3}+\frac {5}{6} \int \frac {1}{\sqrt {1-x^4}} \, dx \\ & = \frac {1}{2 x^3 \sqrt {1-x^4}}-\frac {5 \sqrt {1-x^4}}{6 x^3}+\frac {5}{6} 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 = 10.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.44 \[ \int \frac {1}{x^4 \left (1-x^4\right )^{3/2}} \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {3}{2},\frac {1}{4},x^4\right )}{3 x^3} \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 4.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.33
method | result | size |
meijerg | \(-\frac {{}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {3}{4},\frac {3}{2};\frac {1}{4};x^{4}\right )}{3 x^{3}}\) | \(15\) |
risch | \(\frac {5 x^{4}-2}{6 x^{3} \sqrt {-x^{4}+1}}+\frac {5 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, F\left (x , i\right )}{6 \sqrt {-x^{4}+1}}\) | \(54\) |
default | \(\frac {x}{2 \sqrt {-x^{4}+1}}-\frac {\sqrt {-x^{4}+1}}{3 x^{3}}+\frac {5 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, F\left (x , i\right )}{6 \sqrt {-x^{4}+1}}\) | \(59\) |
elliptic | \(\frac {x}{2 \sqrt {-x^{4}+1}}-\frac {\sqrt {-x^{4}+1}}{3 x^{3}}+\frac {5 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, F\left (x , i\right )}{6 \sqrt {-x^{4}+1}}\) | \(59\) |
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Time = 0.08 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x^4 \left (1-x^4\right )^{3/2}} \, dx=\frac {5 \, {\left (x^{7} - x^{3}\right )} F(\arcsin \left (x\right )\,|\,-1) - {\left (5 \, x^{4} - 2\right )} \sqrt {-x^{4} + 1}}{6 \, {\left (x^{7} - x^{3}\right )}} \]
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Time = 0.50 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^4 \left (1-x^4\right )^{3/2}} \, dx=\frac {\Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {3}{2} \\ \frac {1}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 x^{3} \Gamma \left (\frac {1}{4}\right )} \]
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\[ \int \frac {1}{x^4 \left (1-x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-x^{4} + 1\right )}^{\frac {3}{2}} x^{4}} \,d x } \]
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\[ \int \frac {1}{x^4 \left (1-x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-x^{4} + 1\right )}^{\frac {3}{2}} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^4 \left (1-x^4\right )^{3/2}} \, dx=\int \frac {1}{x^4\,{\left (1-x^4\right )}^{3/2}} \,d x \]
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