Integrand size = 15, antiderivative size = 63 \[ \int \frac {1}{x^8 \left (1-x^4\right )^{3/2}} \, dx=\frac {1}{2 x^7 \sqrt {1-x^4}}-\frac {9 \sqrt {1-x^4}}{14 x^7}-\frac {15 \sqrt {1-x^4}}{14 x^3}+\frac {15}{14} \operatorname {EllipticF}(\arcsin (x),-1) \]
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Time = 0.01 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {296, 331, 227} \[ \int \frac {1}{x^8 \left (1-x^4\right )^{3/2}} \, dx=\frac {15}{14} \operatorname {EllipticF}(\arcsin (x),-1)-\frac {9 \sqrt {1-x^4}}{14 x^7}+\frac {1}{2 x^7 \sqrt {1-x^4}}-\frac {15 \sqrt {1-x^4}}{14 x^3} \]
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Rule 227
Rule 296
Rule 331
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2 x^7 \sqrt {1-x^4}}+\frac {9}{2} \int \frac {1}{x^8 \sqrt {1-x^4}} \, dx \\ & = \frac {1}{2 x^7 \sqrt {1-x^4}}-\frac {9 \sqrt {1-x^4}}{14 x^7}+\frac {45}{14} \int \frac {1}{x^4 \sqrt {1-x^4}} \, dx \\ & = \frac {1}{2 x^7 \sqrt {1-x^4}}-\frac {9 \sqrt {1-x^4}}{14 x^7}-\frac {15 \sqrt {1-x^4}}{14 x^3}+\frac {15}{14} \int \frac {1}{\sqrt {1-x^4}} \, dx \\ & = \frac {1}{2 x^7 \sqrt {1-x^4}}-\frac {9 \sqrt {1-x^4}}{14 x^7}-\frac {15 \sqrt {1-x^4}}{14 x^3}+\frac {15}{14} 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 = 0.32 \[ \int \frac {1}{x^8 \left (1-x^4\right )^{3/2}} \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {7}{4},\frac {3}{2},-\frac {3}{4},x^4\right )}{7 x^7} \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 4.66 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.24
method | result | size |
meijerg | \(-\frac {{}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {7}{4},\frac {3}{2};-\frac {3}{4};x^{4}\right )}{7 x^{7}}\) | \(15\) |
risch | \(\frac {15 x^{8}-6 x^{4}-2}{14 x^{7} \sqrt {-x^{4}+1}}+\frac {15 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, F\left (x , i\right )}{14 \sqrt {-x^{4}+1}}\) | \(59\) |
default | \(-\frac {\sqrt {-x^{4}+1}}{7 x^{7}}-\frac {4 \sqrt {-x^{4}+1}}{7 x^{3}}+\frac {x}{2 \sqrt {-x^{4}+1}}+\frac {15 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, F\left (x , i\right )}{14 \sqrt {-x^{4}+1}}\) | \(73\) |
elliptic | \(-\frac {\sqrt {-x^{4}+1}}{7 x^{7}}-\frac {4 \sqrt {-x^{4}+1}}{7 x^{3}}+\frac {x}{2 \sqrt {-x^{4}+1}}+\frac {15 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, F\left (x , i\right )}{14 \sqrt {-x^{4}+1}}\) | \(73\) |
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Time = 0.09 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^8 \left (1-x^4\right )^{3/2}} \, dx=\frac {15 \, {\left (x^{11} - x^{7}\right )} F(\arcsin \left (x\right )\,|\,-1) - {\left (15 \, x^{8} - 6 \, x^{4} - 2\right )} \sqrt {-x^{4} + 1}}{14 \, {\left (x^{11} - x^{7}\right )}} \]
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Time = 0.61 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.59 \[ \int \frac {1}{x^8 \left (1-x^4\right )^{3/2}} \, dx=\frac {\Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, \frac {3}{2} \\ - \frac {3}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 x^{7} \Gamma \left (- \frac {3}{4}\right )} \]
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\[ \int \frac {1}{x^8 \left (1-x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-x^{4} + 1\right )}^{\frac {3}{2}} x^{8}} \,d x } \]
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\[ \int \frac {1}{x^8 \left (1-x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-x^{4} + 1\right )}^{\frac {3}{2}} x^{8}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^8 \left (1-x^4\right )^{3/2}} \, dx=\int \frac {1}{x^8\,{\left (1-x^4\right )}^{3/2}} \,d x \]
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