Integrand size = 15, antiderivative size = 71 \[ \int \frac {1}{x^6 \left (1-x^4\right )^{3/2}} \, dx=\frac {1}{2 x^5 \sqrt {1-x^4}}-\frac {7 \sqrt {1-x^4}}{10 x^5}-\frac {21 \sqrt {1-x^4}}{10 x}-\frac {21}{10} E(\arcsin (x)|-1)+\frac {21}{10} \operatorname {EllipticF}(\arcsin (x),-1) \]
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Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {296, 331, 313, 227, 1195, 435} \[ \int \frac {1}{x^6 \left (1-x^4\right )^{3/2}} \, dx=\frac {21}{10} \operatorname {EllipticF}(\arcsin (x),-1)-\frac {21}{10} E(\arcsin (x)|-1)-\frac {21 \sqrt {1-x^4}}{10 x}-\frac {7 \sqrt {1-x^4}}{10 x^5}+\frac {1}{2 x^5 \sqrt {1-x^4}} \]
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Rule 227
Rule 296
Rule 313
Rule 331
Rule 435
Rule 1195
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2 x^5 \sqrt {1-x^4}}+\frac {7}{2} \int \frac {1}{x^6 \sqrt {1-x^4}} \, dx \\ & = \frac {1}{2 x^5 \sqrt {1-x^4}}-\frac {7 \sqrt {1-x^4}}{10 x^5}+\frac {21}{10} \int \frac {1}{x^2 \sqrt {1-x^4}} \, dx \\ & = \frac {1}{2 x^5 \sqrt {1-x^4}}-\frac {7 \sqrt {1-x^4}}{10 x^5}-\frac {21 \sqrt {1-x^4}}{10 x}-\frac {21}{10} \int \frac {x^2}{\sqrt {1-x^4}} \, dx \\ & = \frac {1}{2 x^5 \sqrt {1-x^4}}-\frac {7 \sqrt {1-x^4}}{10 x^5}-\frac {21 \sqrt {1-x^4}}{10 x}+\frac {21}{10} \int \frac {1}{\sqrt {1-x^4}} \, dx-\frac {21}{10} \int \frac {1+x^2}{\sqrt {1-x^4}} \, dx \\ & = \frac {1}{2 x^5 \sqrt {1-x^4}}-\frac {7 \sqrt {1-x^4}}{10 x^5}-\frac {21 \sqrt {1-x^4}}{10 x}+\frac {21}{10} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {21}{10} \int \frac {\sqrt {1+x^2}}{\sqrt {1-x^2}} \, dx \\ & = \frac {1}{2 x^5 \sqrt {1-x^4}}-\frac {7 \sqrt {1-x^4}}{10 x^5}-\frac {21 \sqrt {1-x^4}}{10 x}-\frac {21}{10} E\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac {21}{10} 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 = 0.28 \[ \int \frac {1}{x^6 \left (1-x^4\right )^{3/2}} \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {3}{2},-\frac {1}{4},x^4\right )}{5 x^5} \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 4.46 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.21
method | result | size |
meijerg | \(-\frac {{}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {5}{4},\frac {3}{2};-\frac {1}{4};x^{4}\right )}{5 x^{5}}\) | \(15\) |
risch | \(\frac {21 x^{8}-14 x^{4}-2}{10 x^{5} \sqrt {-x^{4}+1}}+\frac {21 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (F\left (x , i\right )-E\left (x , i\right )\right )}{10 \sqrt {-x^{4}+1}}\) | \(66\) |
default | \(\frac {x^{3}}{2 \sqrt {-x^{4}+1}}-\frac {\sqrt {-x^{4}+1}}{5 x^{5}}-\frac {8 \sqrt {-x^{4}+1}}{5 x}+\frac {21 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (F\left (x , i\right )-E\left (x , i\right )\right )}{10 \sqrt {-x^{4}+1}}\) | \(82\) |
elliptic | \(\frac {x^{3}}{2 \sqrt {-x^{4}+1}}-\frac {\sqrt {-x^{4}+1}}{5 x^{5}}-\frac {8 \sqrt {-x^{4}+1}}{5 x}+\frac {21 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (F\left (x , i\right )-E\left (x , i\right )\right )}{10 \sqrt {-x^{4}+1}}\) | \(82\) |
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none
Time = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^6 \left (1-x^4\right )^{3/2}} \, dx=-\frac {21 \, {\left (x^{9} - x^{5}\right )} E(\arcsin \left (x\right )\,|\,-1) - 21 \, {\left (x^{9} - x^{5}\right )} F(\arcsin \left (x\right )\,|\,-1) + {\left (21 \, x^{8} - 14 \, x^{4} - 2\right )} \sqrt {-x^{4} + 1}}{10 \, {\left (x^{9} - x^{5}\right )}} \]
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Time = 0.55 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.52 \[ \int \frac {1}{x^6 \left (1-x^4\right )^{3/2}} \, dx=\frac {\Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {3}{2} \\ - \frac {1}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} \]
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\[ \int \frac {1}{x^6 \left (1-x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-x^{4} + 1\right )}^{\frac {3}{2}} x^{6}} \,d x } \]
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\[ \int \frac {1}{x^6 \left (1-x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-x^{4} + 1\right )}^{\frac {3}{2}} x^{6}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^6 \left (1-x^4\right )^{3/2}} \, dx=\int \frac {1}{x^6\,{\left (1-x^4\right )}^{3/2}} \,d x \]
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